CONFERENCE PROCEEDINGS International Student Conference “Science and Progress”

German-Russian Interdisciplinary Science Center

St. Petersburg – Peterhof November, 14-18 2011

Organizing committee Prof. Dr. S.F. Bureiko, Dean of Faculty Physics, SPSU Prof. Dr. A.M. Shikin, Coordinator of G-RISC, SPSU E.I. Spirin, Dean-assistant of Faculty of Physics, SPSU E.V. Serova, Head of Academic Mobility Department, SPSU Dr. A.G. Rybkin, G-RISC office, SPSU A.A. Popova, G-RISC office, SPSU

Program Committee Prof. Dr. E. Rühl, Coordinator of G-RISC, FU Berlin Prof. Dr. C. Laubschat, Faculty of Physics, TU Dresden Prof. Dr. A.M. Shikin, Coordinator of G-RISC, SPSU Prof. Dr. Yu.S. Tver’yanovich, Faculty of Chemistry, SPSU Prof. Dr. V.N. Troyan, Faculty of Physics, SPSU

Contacts Faculty of Physics Saint-Petersburg State University Ulyanovskaya ul. 3, Peterhof, St. Petersburg, Russia 198504 Tel. +7 (812) 428-46-56, Fax. +7 (812) 428-46-55 E-mail: [email protected] Website: www.phys.spbu.ru/grisc

3 Heads of sections A. Chemistry – Prof. Dr. Yu.S. Tver’yanovich, Faculty of Chemistry, SPSU

B. Geo- and Astrophysics – Prof. Dr. V.N. Troyan, Faculty of Physics, SPSU, Dr. V.G. Nagnibeda, Faculty of Mathematics and Mechanics, SPSU

C. Mathematics and Mechanics – Prof. Dr. V. Reitmann, Faculty of Mathematics and Mechanics, SPSU

D. Solid State Physics – Prof. Dr. A.P. Baraban, Faculty of Physics, SPSU

E. Applied Physics – Prof. Dr. A.S. Chirtsov, Faculty of Physics, SPSU

F. Optics and Spectroscopy – Prof. Dr. Yu.V. Chizhov, Prof. Dr. N.A. Timofeev, Faculty of Physics, SPSU

G. Theoretical, Mathematical – Prof. Dr. Yu.M. Pis’mak, and Computational Physics Faculty of Physics, SPSU

H. Biophysics – Prof. Dr. N.V. Tsvetkov, Faculty of Physics, SPSU

I. Resonance Phenomena – Prof. Dr. V.I. Chizhik, in Condenced Matter Faculty of Physics, SPSU

A. Chemistry Solid-contact ion-selective electrodes with ion-to-electron transducer layer composed of nanostructured materials

Ivanova Nataliya [email protected]

Scientific supervisor: Prof. Dr. Mikhelson K.N., Department of Physical Chemistry, Chemical Faculty, Saint-Petersburg State University

Introduction Ion-selective electrodes (ISEs): potentiometric sensors of ions, comprise rou- tinely used analytical tool for essay of various analytes of clinical, industrial, and environmental relevance. However ISEs of the conventional design, containing internal aqueous solution and internal reference electrode, don’t fit modern requirements. The so-called solid-contact ISEs (SC-ISEs) – without internal filling – would allow for easier miniaturization, for planar technology of manufacturing, and, eventually, for better quality combined with lower production cost. The fundamental problem with SC-ISEs refers to ion-to-electron conductivity transduction at a stable and reproducible potential difference at the interface between the ioni- cally conducting sensor layer (membrane) and electronically conducting substrate. This problem has been successfully solved for electrodes with glass and crystalline membranes. Between a glass membrane containing an alkali metal oxide and the lead they place a tin alloy doped with the respective alkali metal, thus utilizing the first-kind electrode concept. The second-kind electrode concept is implemented in SC-ISEs with crystalline membranes normally containing Ag2S: the inner side of the membrane is vacuum-sputtered with Ag, and then the lead is soldered to this silver layer. None of these approaches can be used for SC-ISEs with solvent polymeric membranes containing ionophores: neutral or charged species selectively binding ions, and in this way ensuring a selective potentiometric response of the sensor. Conducting polymers appear the most promising ion-to-electron transducers for this kind of sensors [1]. So far, however, the long-term stability and the piece-to-piece reproducibility of the potentials of ionophore-based SC-ISEs does not fit practical needs, and remains well below that of conventional ISEs. In this work we try graphenes, nanostructured polymeric composite of Cu(I), and also hexacyanoferrates as active components of ion-to-electron transducer layer in SC-ISEs. Materials and methods The electronically conducting substrate was always graphite encapsulated in a PVC body. The transducer layers were formed by drop-casting solutions of the respective materials on top of the substrate. As active components of ion-to-electron transducer layer we used nanostructured materials such as electroactive conjugated polymer: polymeric complex of Cu(I) with bichynolyl-containing polyamidoacid 6 (as 4.5 % solution in N-metylpyrrolidone – N-MP) – PAC-2, graphemes (as 2.5% solution in dimethylformamide - DMFA), both kindly provided by the Institute of Macromolecular compounds RAS, St.Petersburg, and fullerene – C60 (kindly provided by St.PSU). Optionally, the layers were doped with dispersed carbon black (CB) and/or RedOx pair: K3Fe(CN)6/ K4Fe(CN)6. For the transducer layer compositions see Table 1.

Table 1. Electrode Composition of the transducer layer 1 2,5 % solution of graphenes in DMFA 2 4,5 % solution of polymeric complex of copper Cu(I) PAC-2 in N-MP 3 No transducer layer (the so-called coated-wire electrode - CWE) 200 µl saturated solution of salts mixture K [Fe(CN) ]+ 4 3 6 +K4[Fe(CN)6]·3H2O+300 mg PVC+1,7 ml THF 5 Same as 4, but dispersed in an ultrasonic bath 6 Conventional ISEs with inner solution KCl 10-2 M based on valinomycin 300 mg PVC+1,7 ml THF+150µl saturated solution of salts mixture

7 K3[Fe(CN)6]+ K4[Fe(CN)6]·3H2O, prepared by 1 M aqueous solution KCl 300 mg PVC+150µl DOP+1,7 ml THF+150µl saturated solution of

8 salts mixture K3[Fe(CN)6]+ K4[Fe(CN)6]·3H2O, prepared by 1 M aqueous solution KCl Dry mixture of salts K [Fe(CN) ]+ K [Fe(CN) ]·3H O with suspension 9 3 6 4 6 2 of carbon black in the ratio 1:1

100 µl saturated solution of salts mixture K3[Fe(CN)6]+

10 K4[Fe(CN)6]·3H2O+ 100µl 2,5 % solution of graphenes in DMFA+300 mg PVC+1,7 ml THF

100 µl saturated solution of salts mixture K3[Fe(CN)6]+

11 K4[Fe(CN)6]·3H2O+ 100µl 2,5 % solution of graphenes in DMFA+600 mf mixture of PVC:carbon black=1:1+ 2 ml THF 300 mg PVC+1,7 ml THF+100µl saturated solution of salts mix- 12 ture K3[Fe(CN)6]+ K4[Fe(CN)6]·3H2O+200 mg fullerenes- C60

K+-selective membranes contained poly(vinylchloride) (PVC) and bis(2- ethylhexyl)phthalate (DOP) (1:3) doped with 0.03 M potassium tetrakis(p-Cl- phenyl)borate (KClTPB). The membranes were formed by drop-casting the membrane cocktail: the aforementioned substances dissolved in tetrahydrofuran (THF) on top of the transducer layer. Conventional ISE with internal aqueous 7 solution (0.01 M KCl) and Ag/AgCl internal reference electrode was used for the back-to-back comparison with SC-ISEs. Zero-current potentiometric measurements were accompanied by chronopoten- tiometry (ChP) and electrochemical impedance spectroscopy (EIS). The reference electrode was always saturated Ag/AgCl electrode, as counter electrode for ChP and EIS measurements we used glassy carbon rod. Results and discussion Electrodes 1 and 2 showed steady positive drift of the potentials, while electrodes 4, 5, 8, 7 and 12 showed negative drift. Only electrodes 3, 9, 10 and 11 showed relatively stable potential readings over time, although worse than conventional ISE 6, see Fig. 1.

Fig. 1. Drift of the potential in control 0.1 M KCl solution for SC-ISEs 3, 9, 10, 11 and conventional ISE 6. Electrode 10, although showing a relatively stable potential in the control solution 0.1 M KCl, slowly by slowly lost response in solutions below 10-3 M - see Fig. 2, while electrodes 1, 2, 3, 6, 7, 8 and 9 retained Nernstian response down to 10-5 M KCl – see Fig. 3.

Fig. 2. Behaviour of electrode 9 retaining Nernstain response down to 10-5 M KCl over 6 months of observation. 8 Fig. 3. Behavior of electrode 10 over time: gradual degradation of the response. The potential drift may be caused by slow Red-Ox reaction responsible for the ion-to-electron conductivity transduction. This transduction can be studied by impedance and chronopotentiometric measurements. For selected electrodes, appeared most promising we carried out EIS ansd ChP measurements. SC-ISEs 1, 2, and 12 showed only bulk impedance, while SC-ISE 11 showed depressed semicircle, most likely a superposition of a bulk and a slow charge-transfer process, Fig. 4.

Fig. 4. EIS curves for SC-ISEs 1, 2, 11, 12 – curves 1, 2, 3 and 4, respectively. Chronopotentiometric data support the latter conclusion: polarization then plotted vs. square root of time is almost linear for ISEs 1, 2 and significantly nonlinear for ISEs 11, 12 (Fig. 5) suggesting slow charge-transfer in the latter electrodes. Indeed, SC- ISEs 12, and especially 11 show relatively large charge-transfer resistance. Besides insufficient stability of the ion-to-electron transduction, the additional reason for the Fig. 5. Chronopotentiometric measurements: degradation in sensor response can polarization vs. square root of time. 9 be the existence of water layer between the membrane and the electrode substrate, which behaves unintentionally as an extremely non-buffered electrolyte reservoir [2]. Indeed, some SC-ISEs, in particularly electrode 9 exhibit large potential drifts when 0.1 M KCl is replaced with 0.1 M NaCl, suggesting the existence of thin water layer between the membrane and the transducer layer, see Fig. 6.

Fig. 6. Response of SC-K +-SEs upon replacement of 0.1 M KCl with 0.1 M NaCl and back.

Conclusions The insufficient stability of SC-ISEs under study is caused, most likely, both by relatively slow RedOx process at the transducer layer, and the existence of water layer beneath the membrane. Profs. A. Yakimanski and M. Goikhman, Institute of Macromolecular compounds RAS, St. Petersburg, are greatly acknowledged for providing with Cu(I) -polymeric nanocomposite.

References 1. Ivaska A. // Electroanalysis, 1991, 3, 247. 2. Fibbioli M., Morf W.E., Badertscher M., N.F. de Rooij, Pretsch E. // Electroanalysis, 2000, 12, 1286.

10 Using of semiconductor oxide films for detection of volatile organic compounds in gases

Lopatnikov Artem [email protected]

Scientific supervisor: Prof. Dr. Povarov V.G., Department of Analytical Chemistry, Faculty of Chemistry, Saint-Petersburg State University

Introduction To create a gas-sensitive films that are used in chemical sensors, the semiconductors based on transitive metal oxides are widely used [1-3]. The principle of detection is based on the phenomenon of heterogeneous cata- lytical reactions occurring on the surface of the solid and gas. The character of transformations of organic compounds depends on the nature of the substance and the catalyst used, as well as the conditions of catalytic reaction. At low temperatures only physical adsorption is observed, while temperature increases, the growing role is played by the forces of chemical interaction between the adsorbed molecules and surface atoms of solid-state or adsorbed oxygen, located on its surface [4]. The presence of volatile organic compounds leads to its adsorption on a surface of film with subsequent oxidation. The mechanism of oxidation is accompanied by formation of the number of ionic and ion-radical forms, which are temporarily increases the conductivity of film [5]. Thus, these changes can be captured and treated as an analytical signal.Fig. 1 shows a schematic diagram of this detection method Fig.1. Working principle and circuit scheme of gas-sensitive sensor analyzer. 1 - substrate, 2 - oxide film, 3 - silver contact, 4 - flow of gas-oxidant in a mixture with volatile organic compounds, 5 - variable resistance, 6 - signal detection, 7 - constant voltage source

During the current research work we used the films of tin dioxide modified with oxide of copper and pentoxide of vanadium. Tin dioxide film provides electrical conductivity over a wide temperature range, while copper and vanadium oxides are typical catalysts for oxidation processes. Results and Discussion The films of mixed metal oxides were deposited on the edge of a rod of alum silicate. Through contacts rods are connected with the electrical circuit and a vari- 11 able resistor, which is simultaneously connected to the ADC. The detector response received without additional amplification. During the experiment the detection system was heated at a required temperature range. The detector is located inside the main channel of the aluminum thermostat of serial gas chromatograph. For researching of the films activity the mixtures of test compounds with air were injected into the chromatograph evaporator. Heating of the film was carried out in steps. Fig. 2 shows analytical signal of ethanol vapors depending on the temperature. The peaks have the usual "chromato- graphic" shape, their width decreases with increasing temperature, and the height increases up to a temperature 375 oC.

Fig. 2. The peaks of ethanol (three for each temperature) on the film of (SnO2 + 3% CuO). The feed rate of air through the hollow column - 80 ml/min. The first peak at 350oC was registered against the background of random noise in the measuring circuit. To determine the sensitivity of the detector the dependence of the peak area on the amount of substance was investigated. It turned out that for all compounds under consideration the calibration dependences are generally nonlinear. However, in the range from 1 to 5 mkg of substance there is a linear plot. Fig 3. shows a calibration curve for n-hexane. The reproducibility of the analytical signal Fig. 3. The dependence of the peak areas on the was determined by repeated mass of n-hexane. injection of 1 ml mixture of n- 12 hexane and air at a concentration of 2 mg / ml in evaporator. It turned out, the mean relative deviation of the peak area based on 25 measurements was about 2%. Dependences obtained for tin dioxide films modified with vanadium pentaoxide are generally similar for ones obtained for films with the addition of copper dioxide. Differences in the behavior of the two films of different composition appear more completely in the comparison of analytical signals obtained simultaneously with respect to the same substance. For this purpose, we placed two sensors of different composition into a standard heating block of gas chromatograph. The films deposited on the edge of cylindrical rods made of alum silicate with two through- holes in parallel, located along the main axis of the cylinder. Two metal wires with silver edges were passed through the holes and slightly protruding above plane of the film-covered end of the rod, were served as the contacts. For each of the film we used the electrical measuring circuit, described previously. During use of two sensors on the gas chromatograph was equipped with a set packed column with a phase 5% SE-30 on inertone, which provided the first real chromatograms, one of which is shown in Fig. 4. Most of the compounds give responses of standard form, herewith ethoxyethan exhibits different behavior. Oxygen of ethoxyethan has the ability to capture electrons. But after sorption and a corresponding reduction in electrical conductivity subsequent oxidation begins to liberate electrons into the measuring circuit. As a result, the large positive deviation is replaced by Fig. 4. Typical view of analytical signal. a slight negative. A graphical dependence on each other’s signals from two detectors of different composition in relation to the same compound seems to be the most informative. Each compound is conformed to a curve of the special form, which make possible its identification. Fig. 5 shows the trajectories of the signals from a number of substances, which are obtained with a combination of films with different Fig. 5. A graphical dependence on each other’s composition. signals from two detectors of different composition in relation to the same compound. 13 This phenomenon can be used as a means of identification. Due to the fact, that the analytical signal, obtained this way, corresponds to only one aspect of the pro- ceeding on the surface of the membrane process - the concentration of free charge carriers in a given time, one can conclude that this method yields the identification of a limited number of compounds. Conclusions We can make the following conclusions: 1. The sensitivity of the detector is about 1 microgram. 2. There is no need for a signal amplifier. 3. The reason for the nonlinear dependence of the height of the response on the concentration is a saturation of the surface and blocking the active centers. 4. The linear dependence of the peak area of concentration allows for quantitative analysis of the separated peaks. 5. Individuality of oxidation process strongly affects the shape of a peak. 6. The constructed model of the detector can be installed on a gas chromatograph, using its standard components. 7. Installing the detector on a capillary column chromatogram will receive high-resolution analysis in the mode of temperature programming. 8. Simultaneous usage of several films with different oxidation catalysts can be used as a detector electronic tongue.

References 1. Ahlers S. et al. // Sensors and Actuators B 107 (2005) 587-599. 2. Zhang W.-M. et al. // Sensors and Actuators B 123 (2007) 454-460. 3. Kim K.-W. et al. //Sensors and Actuators B 123 (2007) 318-324. 4. Yamazoe N. // Sensors and Actuators B 108 (2005) 2-14. 5. Gouma P.I. // Rev. Adv. Mater. Sci. 5 (2003) 147-154.

14 Digital spectrographic analysis of human biological fluids for determination of microelements

Savinov Sergey [email protected]

Scientific supervisor: Prof. Drobyshev A.I., Department of Analytical Chemistry, Faculty of Chemistry, Saint Petersburg State University

Introduction Chemical elements being presented in the human body can be divided into two groups, namely, essential and toxic [1]. Toxic elements (xenobiotics), which penetrate into the body usually from the environment, even in low concentrations damage not only health but also human life [2]. Significant increase or decrease in the concentration of essential (vital) elements can also lead to disorder of normal body functioning [3]. In this regard, very important, promising and called-for trend in modern clinical diagnostics is a biomonitoring of chemical elements in human organs and tissues, as well as in biological fluids [4]. To implement this direction highly sensitive, multi-element, high-performance and available methods of analysis of bio-organic samples are required. The aim of this study was to develop a technique of direct (without sample treatment) atomic emission digital spectrographic analysis of human biological fluids (as an example, saliva) with spectrum excitation of dried residue of a sample from end of carbon electrode in a.c. arc for determining the concentration of microelements in them.

Materials and methods To create an a.c. arc generator IVS-28 was used. The decomposition of emission of arc plasma in a spectrum was performed by spectral device MFS-8. Control of the spectral plant, including the registration of the spectrum with a linear photodiode detector MAES-10, as well as photometry of spectral lines and analytical information processing carried out by computer software "Atom 3.2" (VMK "Optoelektronika", Novosibirsk). Carbon electrodes: upper, sharpened to a cone, and bottom, machined to a diameter of 3 mm with a hole on its end, were used. Drops of liquid samples and auxiliary solutions were applied to the bottom electrode by microsyringe MSh-10. For preparation of calibration solutions deion- ized water, metals, their oxides or salts, concentrated hydrochloric acid, nitric acid, sulfuric acid and hydrogen peroxide were used.

Results and discussion Development of technique of microelement determination in liquid samples Preparing of electrodes for an application on their end of liquid samples included cleaning of the end zones from possible contaminations presented in coal rods by firing of electrodes in a.c. arc during 20 seconds at a current strength 15 A. Then on the 15 end of the electrode with a hole a thin polymer film was created by applying and drying of 20 ml of solution of polystyrene in toluene (0.3%) to prevent a penetration of the sample depth into the surface layer of the electrode (during evaporation of drops of samples). As an easy ionized addition agent which is known [5] to inhibit the influence of sample matrix components on the plasma temperature, determining the efficiency of atomic excitation, we used sodium chloride applied as 10 ml drop of an aqueous solution (15 g/l) on the end of the electrode. 3,00

2,50

2,00 Al 309,3 nm

relative units 1,50

, Cu 327,5 nm 1,00 Mg 279,6 nm

Intensity Zn 213,9 nm 0,50

0,00 13 14 15 16 17 18 19 Current strenght, А Fig. 1. Dependence of the intensities of analytical lines on the current strength of a.c. arc.

0,90 0,80 0,70 0,60 0,50 Al 309,3 nm

relative units 0,40 Cu 327,5 nm , 0,30 Mg 279,6 nm RSD 0,20 Zn 213,9 nm 0,10 0,00 13 14 15 16 17 18 19 Current strenght, А Fig. 2. Dependence of relative standard deviations of the intensities of analytical lines on the current strength of a.c. arc.

An important point in the development of technique of biological fluids analysis for microelement determination is to optimize the conditions of spectrum excitation in order to obtain maximum value of the analytical signal and the least error of its measurement. With a view of it, experiments in which the spectra of the samples had been excited at different current strength of a.c. arc were made. On the grounds of the obtained dependences of the intensities of analytical lines 16 (Fig. 1) and their relative standard deviations (Fig. 2) on the current strength we chose 17 A as the optimal value. Verification of the developed technique To confirm the universality of the developed technique comparative measurements of the intensities of spectral lines when spectrum exciting of the source sample of saliva, as well as samples treated with concentrated nitric acid and concentrated hydrogen peroxide - the most commonly used reagents for the treatment of biological objects to destruct an organic matrix [6, 7] – were carried out. As can be seen in Fig. 3, the relative difference of intensities does not exceed 10%. Thus, developed technique allows to determine microelements in saliva samples without preliminary mineralization. 20 18 16 14 Nitric acid treatment 12

relative units 10 Hydrogen peroxide , treatment 8 Without treatment 6 IIntensity 4 2 0 Zn Mn Al P Ti Fig. 3. The intensities of analytical lines of elements by different ways of treatment of same sample. Determination of microelements in human saliva The developed technique has been used by us for determination of microelements in samples of saliva of Saint Petersburg residents. Based on obtained results we calculated average values of concentrations, as well as intervals of their variations by 58 analyzed samples. The following table shows our data, as well as from other papers. Although there is spread of values of mean contents of most elements obtained by different authors, however, they fall into intervals estimated by us. Element Current work [8] [9] [10] [11] Mg 10 (3,6-71) - 6,76 - - Si 2,1 (0,5-18) - 5,36 - - Zn 0,28 (0,01-1,3) 0,17 1,3 - 0,26 Fe 0,18 (0,01-1,7) - 0,44 - - Ti 0,086 (0,001-1,1) - 0,758 - - Cr 0,050 (0,010-0,12) 0,05 0,03 - - Mn 0,049 (0,009-0,67) - 0,042 0,025 0,003 Cu 0,010 (0,001-0,72) 0,07 0,05 0,005 0,02 Co 0,008 (0,001-0,036) - 0,003 - - Ag 0,001 (0,0001-0,077) - - - - Table 1. Average content of several elements (with intervals of their variations) in saliva, mg/l. 17 Conclusion Thus, we developed a technique of atomic emission digital spectrographic analysis of biological fluids with spectrum excitation of dried residue of a sample from the end of carbon electrode in the a.c. arc with a relative error of microelement determination 10 - 20%. This technique was used to analyze samples of saliva. Findings on the average contents of microelements and intervals of their variations are in good agreement with other researchers data.

References 1. Avtsyn A.P. et al. Microelementoses of man. Etiology, classification, organo- pathology (In Russian). - Moscow: Medicine, 1991, - 496 pp. 2. Chandramouli K. et al. // Archives of Disease in Childhood. V. 94, pp. 844–848 (2009). 3. Takser L. et al. // Environmental Research. V. 95, pp. 119-125 (2004). 4. Kakkar P., Jaffery F.N. // Environmental Toxicology and Pharmacology. V. 19, pp. 335–349 (2005). 5. Terek T., Mika J., Gegush E. Emission spectral analysis (In Russian). V.1 - Moscow: Mir, 1982, - 286 pp. 6. Chiappin S. et al. // Clinica Chimica Acta. V. 383, pp. 30–40 (2007). 7. Burguera J.L., Burguera M. // Spectrochimica Acta. Part B. V. 64, pp. 451-458 (2009). 8. Baranovskaya, I.A. (In Russian) // Kazan Medical Journal. V. 90, pp. 87-89 (2009). 9. Notova S.V., Ordzhonikidze G.Z., Nigmatullina Y.F. (In Russian) // Journal of Orenburg State University. V. 6, pp. 146-147 (2003). 10. Watanabe K. et al. // Journal of Trace elements in Medicine and Biology. V.23, pp. 93-99 (2009). 11. Wang D., Dua X., Zheng W. // Toxicology Letters. V. 176, pp. 40–47 (2008).

18 Synthesis of condensed imidazole derivatives with a Nodal nitrogen atom - pyrido[1,2-a]benzimidazoles

Sokolov Alexandr [email protected]

Scientific supervisor: Dr. Begunov R.S., Department of Organic and Biological Chemistry, Faculty of Biology and Ecology, Yaroslavl Demidov State University

Introduction At the present time there is a demand for a number of new chemical structures, the scope of which also emerged recently. Such young science as molecular biology, genetic engineering, medical genetics need to provide specialized knowledge-based reagents for research in their fields. Opportunities to implement these requirements relate to the development in recent years new methodologies for the synthesis of chemicals and the development of physico-chemical methods of analysis and mathematical (quantum mechanical) modeling of chemical processes. One group of compounds, which are high demand in contemporary applied and experimental sciences, are condensed imidazole derivatives containing a bridging nitrogen atom. For example, to such structures belong pyrido[1,2-a] benzimidazoles, which are bioisosteric analogues of nitrogenous bases of DNA and exhibit a high biological activity [1-2], and furthermore, due to the system of conjugated bonds have fluorescent properties. Compounds with imidazole fragment, despite the wide spread in nature, not cost-effective to obtain from natural raw materials. In the literature described many techniques for their synthesis [3-4], but the drawbacks, such as the use of expensive reagents, high temperatures for the process, the low selectivity and low yield of final products do not allow to use well-known method- ology for the creation of industrial production technologies of these products. Based on analysis of more than 240 literary sources, it was concluded that the most promising method of synthesis of substituted pyrido[1,2-a]benzimidazoles is reductive cyclization of pyridinium salts. This method allows to realizes the process selectively, with a high yield of target products. It is based on the increased activity of pyridine and, in particular, its derivatives - N-oxides and quaternary salts with a full positive charge on the endocyclic nitrogen atom to the aromatic nucleophilic substitution reactions by α-positions of the ring (Fig. 1). However, an obstacle to widespread use of this method of synthesis is insufficient knowledge about the laws of the process of reductive cyclization. Therefore were investigated the main factors determining the reaction and the possibility for further chemical modification of the resulting products.

19 Fig. 1. Scheme of reductive cyclization of pyridinium salts.

1 a) R=R1=H, b) R=R1=CH3, c) R=H, R1=CH3;

2 =CH, a) R2=H, R3=CF3, b) R2=H, R3=CN, c) R2=H, R3=COOH, d) R2=H, R3=CONH2, e) R2=R3=CN; f) R2=H, R3=NO2 g) X=N, R2=R3=H; 3,5 Х=CH; 4,6 Х=N. Results and Discussion In this paper we studied the influence of the nature of the reducing agent, solvent, temperature on the direction of intramolecular reductive amination of salts of 1-(2-nitro-(het) aryl) pyridinium on example of 7-trifluoromethylphenyl pyridinium chloride. For the experimental investigation of the influence of nature of reducing agent on the direction of the reaction, we used agents that work in alkaline or in acidic medium. In the application of Na2S (pH> 7) occur addition of OH ¯-ion to α-position of the pyridine ring of salt 3a, followed by opening of the pyridine ring and the formation of an aldehyde derivative 7 (Fig. 2).

Fig. 2. Scheme of reduction of pyridinium salts in alkaline medium.

In the future, as the reducing agents we used metal chlorides of variable

oxidation states (TiCl3, SnCl2), working in an acidic environment. Reduction of 3a with chloride titanium (III) shown in Fig.3 leads to the formation of the product, defined by NMR and mass spectrometry as 1-(2-amino-4- (trifluoromethyl) phenyl) pyridinium Fig. 3. Scheme of reduction of pyridini- chloride (8). um salts with chloride titanium (III). 20 Thus, in the course of the reaction occurs reduction of the nitro group to amino group, without the formation of condensed tricyclic structures. Intramolecular cyclization with the formation of product 3a -7-(trifluoromethyl) pyrido[1,2 –a]benzimidazole (5a) (Fig. 4) is realized with application of chloride of tin (II) in acidic aqueous-alcoholic medium (3% HCl).

Fig. 4. Scheme of reductive cyclization of pyridinium salts with chloride tin (II). Influence of solvent nature on the course of the reductive amination 3a was investigated on a number of alcohols, the results are presented in Table 1. As can be seen from the table that Table 1. The yield of the reductive amina- the best conditions for the synthesis tion of 1-(2-nitro-4-(trifluoromethyl) phenyl) of pyrido[1,2 –a]benzimidazoles pyridinium chloride (3a) [C = 0.16 mmol/l, (5) is an acidic aqueous-alcoholic t = 500C]. medium, and the nature of the alco- № Solvent Yield 5a, % hol has no significant effect on the 1 Propan-2-ol 81.5 yield of 5a. 2 Propan-2-ol(H O, HCl) 96.3 In addition to the medium, tem- 2 3 Ethanol (H2O, HCl) 91.7 perature is important factor in the 4 Methanol (H O, HCl) 96.0 reduction of nitro compounds and 2 5 Water (HCl) 16.3 nucleophilic substitution reactions. Found (Table 2) that the temperature rise of reduction of 1-(2-nitro-4- Table 2. The yield of products of the reductive (trifluoromethyl) phenyl) pyridini- amination of 1-(2-nitro-4-(trifluoromethyl) um chloride (3a) to 50 °C increases phenyl) pyridinium chloride the yield of product 5a, a further [C = 0.16 mmol / l propane-2-ol-H2O]. increase of temperature of the reac- Yield, % № Temperature, 0С tion mixture leads to decrease in the 5а 8 yield of target products, which may 1 0 82.9 8.9 explain the occurrence of alternative 2 10 84.8 6.1 chemical processes. 3 20 91.0 - Thus, the experimentally were 4 30 93.2 - selected the most suitable conditions 5 40 96.3 - for the process of intramolecular 6 50 97 - cyclization of salts 3: medium - 7 60 96.8 - propan-2-ol-water-HCl, t = 50 °C. Based on experimental data and 8 70 95.9 quantum-mechanical simulation was set key stage of reduction of the nitro group, which determines way of the process of reductive cyclization of pyridinium salts (Fig. 1) and was proposed the mechanism of intramolecular reductive cyclization. 21 This makes it possible to use the developed technique for synthesis a variety of pyrido[1,2- α]benzimidazoles (Fig. 1). Compounds with fused tricyclic core are interesting as intercalators of DNA. At the same time, most intercalary activity should have pyrido[1,2-a] benzimidazoles containing NH2-, C(O)NH2-groups, because they increase the affinity of the compounds to the nucleic acid molecules. So were synthesized amino derivatives of pyrido[1,2-a]benzimidazoles based on reaction of electrophilic substitution - nitration and reduction reactions (Fig. 5).

Fig. 5. Scheme of reductive cyclization of pyridinium salts and futher modification

- nitration and reduction. a) R=CF3; b) R=NH2 The reactions of electrophilic substitution in the 7-R-pyrido[1,2-a]benzimidazole proceed under relatively mild conditions (nitration - 3 h at 70° C), which allows to obtain the final products of high purity and good yields 81-94%. Reduction of compound 9 was carried out with chlorides of metals of variable oxidation state, with the best results were observed when using titanium chloride (III). 36-substituted pyrido[1,2-α]benzimidazoles, which were obtained in our research have been investigated as drugs, with mechanism of action is based on intercalation into DNA and inhibition of replication of target cells. Among them, the greatest biological activity have structures containing amino, amido, or both amino and amido groups. Such drugs have high intercalary activity, which exceeding the known DNA intercalators in a few times [5]. Conclusion a) Developed an effective method for the synthesis of tricyclic condensed compounds based on reductive cyclization of pyridinium salts. b) Investigated main factors determining direction of reductive cyclization and proposed its mechanism. c) Obtained variety of pyrido[1,2-α]benzimidazoles, which have high biological activity and can be used in genetics research and pharmacology as DNA intercalators. References 1. Rida S.M., El-Hawash S.A.M. Fahmy H.T.Y. // Arch. Pharm. Res.,V. 29, No. 10, рр. 826-833 (2006). 2. Dupuy M., Pinguet F., Chavignon O., Chezal J.-M., Teulade J.-C., Chapat J.-P., Blache Y. // Chem. Pharm. Bull.,V. 49, No. 9, рр. 1061-1065 (2001). 3. Tereshchenko A.D., Tolmachev A.A., Tverdokhlebov A.V. // Synthesis. No. 3, рp. 373-376 (2004). 4. O'Shaughnessy J., Aldabbagh F. // Synthesis. No. 7, pp. 1069-1076 (2005). 5. Ryzvanovich G.A., Begunov R.S., Rachinskaya O.A., Muravenko O.V., Sokolov A.A. // Chem.-pharm. Journal, V. 45, № 3, p. 13-15 (2011). 22

С. Mathematics and Mechanics Algebraic approximation of global attractors of discrete dynamical systems

Malykh Artem [email protected]

Scientific supervisor: Prof. Dr. Volker Reitmann, Department of Applied Cybernetics, Faculty of Mathematics and Mechanics, Saint-Petersburg State University

1. Introduction One of the typical goals related to dynamical systems is the approximation of global B-attractors. Many results in this area were shown by Foias and Temam [1, 2]. In their papers they have shown how to construct the approximation of an attractor with any given precision for a large class of continuous-time analytic dynamical systems. The approximation is done with the help of algebraic sets. In the first part of the present paper we repeat the definition of algebraic sets, describe the reasons for their use for the approximation of attractors and give a theorem for discrete-time dynamical systems similar to one which was shown by Foias and Temam [1, 2]. In the second part of the paper we will describe Whitney stratifications of algebraic sets and its connection with global B-attractors.

2. Basic tools of global attractors and algebraic sets Consider the dynamical system with discrete time: (ϕt , (M ,ρ )), { }()t∈T (1) ∈ ; t where (M, ρ) is a metric space, T {Z,Z+} for all the map φ :M→M is continuous and for all t, s∈T we have φt+s=φt◦φs. Let us recall the definition of a global B-attracting set and of a B-attractor [3]. t∈핋 Definition 1: A set XÌM is called global B-attracting for (1) if dist(φt(B), X) →0 for t→∞ and for all bounded sets BÌM. Here for all Y, ZÌM the function dist(Y, Z)=supu∈Y infv∈Z ρ(u,v) is the Hausdorff semi-distance. Definition 2: A set X is called global B-attractor for (1) if 1. X is a bounded and closed set; 2. φt(X)=X for all t∈T; 3. X is a B-attracting set for (1). In this paper we consider only dynamical systems where M=n. It is a rather often used approach to do the approximation of an attractor with the help of manifolds, however some attractors have not-integer Hausdorff dimension, i.e. have fractal structure. Therefore they cannot be approximated by manifolds with any given precision. That is why we use for approximation algebraic sets which are not necessary manifolds. Here is the definition:

24 Definition 3: Consider the following system of equations:

 Pu11( ,,…= un ) 0  Pu21( ,,…= un ) 0   …  Pumn( 1,,…= u) 0 ∈ n Where m, n N, P, P1,…,Pm are polynomials defined on  . Then the set of its solutions is called an algebraic set. Let us provide an example of algebraic set which is not a manifold. Consider the simple algebraic set:

V1={( uuuuuu 12 ,) |( 1 − 2)( 1 += 2) 0}

Fig. 1. As we can see in Fig. 1, in the one-dimensional neighborhood of the point (0, 0) this set is not homeomorphic to an open ball in , so it is not a manifold.

3. Algebraic approximation for a class of discrete-time dynamical systems Let us consider the system generated by the iteration

ut+1 =− Au tt − R( u), t ∈Z (2) where A:n n is a linear symmetric operator. Let the system (2) has a global B-attractor X. Suppose that R: n→n is an analytic mapping inside some open n → set DÌ : XÌD. Define the mapF (u) := -Aut – R(ut). Let F be an invertible map and F-1 be an analytic map inside D. Let D contain the point b such that F-1, R are given by convergent Taylor series in the open ball Br(b): XÌBr(b). Before stating our approximation theorem we give some auxiliary notation. Let f : n  be given by the convergent Taylor series in the neighborhood of the point b. Then l ∂∂k1 ... kn fb( ) → k1 kn flb,():u = ∑ (ub11−−)) ...( ubn n kk1,...,n = 0 (kk1 ... n )! n m is a finite part of series. For the map f :  → , f(u)=(f1(u),…,fm(u)) given by convergent Taylor series in the neighborhood of the point b we have similar notation: ff=…( ,, f ) lb,1lb,,mlb Now we can give the statement of the approximation theorem. 25 Theorem: Suppose we have a system of the form (2). Let λ1,…,λn be the eigenvalues of A and suppose that |λ1|≥ |λ2| ≥ … ≥ |λn| and |λn| < 1. Let m be the minimal n index such that |λm+1|<1 and Qm be the projector on the linear subspace of  generated by the eigenvectors corresponding to λm+1, …, λn. Define for any L, M, n N N, u  the sum L k k −1 −k JLM,, N( u)=−∑( 1) ( QAm) R Nb,,( F Mb( u)) ∈ ∈ k =1 ∈ ∈ Then for any ε > 0 there exist L, M, N N such that for any u0 X

Qum0− J LM ,, N( u 0) <ε and the set 0  LM,, N={uJ|0 LM,, N( u) −= Q m u } is an algebraic set. Proof: Here we will give only a sketch of the proof. First, we represent an -1 arbitrary point of the attractor u0 as an infinite sum depending on A, R, Qm, F and u0. Then we prove that a finite part of this sumK L(u0) can approximate u0 with any given precision and that the norm of the difference between u0 and KL(u0) can be estimated uniformly for any point of the attractor. Then we represent R and F-1 -1 by finite parts of their Taylor series around b (RN,b and F M,b, respectively). Now we substitute these representations into KLand get the sum JL,M,N. Then we show that JL,M,N(u0) can approximate KL(u0) with any given precision and that the norm of the difference between KL(u0) and JL,M,N(u0) can be estimated uniformly for any point of the attractor.

The function JL,M,N is a sum of polynomial functions. Therefore the set

0 n  LM,, N =∈{u |0 JLM,, N( u) − Qu m =} is an algebraic set. 4. Whitney stratification Besides the ability to approximate attractors with any given precision, algebraic sets have another useful property related to the analysis of a global B-attractor. Any algebraic set has a Whitney stratification. Informally speaking, a stratification is a presentation of a manifold as an union of disjoint manifolds. The Whitney stratification requires some additional properties of tangent spaces from these manifolds. The definition from [4] is following. n { n A stratification of a set SÌ is a partition of S into submanifolds Sj} of  such that the family of strata {Sj} is locally finite at each point of S. Let us denote n as Gn,m (0 ≤ m≤ n) the set of all linear m-dimensional subspaces of  . This set has the structure of a C smooth manifold with dim(Gn,m = m(n-m)). The manifold Gn,m is called Grassman∞ manifold. Definition 4. A stratification of a set S is called Whitney stratification if each pair of strata Si, Sj; i ≠ j satisfies the following condition: {}pq∞∞, {} If kk==11 kk

26 are sequences of points in Si and Sj , respectively, both converging to a point p of

Si, if the sequence of tangent spaces {}TS ∞ qk jk=1 converges to a subspace L in Gn,m, where m = dim(Sj), and if the sequence  ∞ {,}pqkkk =1 n of lines containing 0 and qk-pk converges to a line lÌ in G1,l, then lÌL. Let us consider an example of a Whitney stratification of a simple algebraic set. Example 1: Consider the algebraic set 22  2 Suuuuuuuu={(12 , ) ∈ | 11( − 2) ==0} {1 = 0} { 1 = 2 }

5 Let us write S in the form S =  ii=1 S , where 2 Su12={(0, ) ∈> | u 2 0} , 2 Suu22={(0, ) ∈< | 2 0} , 2 1/2 S3={( uu 12, ) ∈= | u 1 u 2 , u 2 > 0} , 2 1/2 S4={( uu 12, ) ∈ | u 1 =− ( u 2 ), u 2 > 0} , Fig. 2. S5 = {(0, 0)} .

It is easy to see that {Si}, i=1,…,5 is a Whitney stratification of S. One can find the detailed explanation in [5].

Suppose we have a Whitney stratification of a set S into strata {Si}, i = 1,...,N,

N N. And suppose that m = max(dim(Si)), i=1,…,N. Our main interest in the analysis of an attractor are strata with dimension lower than m (let us call them low-dimension∈ strata). Our suggestion is the following: if a low-dimension stratum appears near some region in the Whitney stratification of all approximating algebraic set, then in this region we have the fractal structure of the B-attractor.

5. Conclusion We described the concept of algebraic sets and the reason why they are useful in the analysis of global B-attractors. Also, a theorem with explicit form of approximating algebraic sets is given. In the last part we have briefly described the concept of stratification, its possible connection with approximating sets and the direction for further investigation.

References 1. Foias C. and Temam, R. // Phys. D. V. 32. P. 163–182. (1988). 2. Foias C., Temam R. // SIAM J. Math. Anal. V. 25. No 5, P. 1269–1302. (1994). 3. Boichenko V.A., Leonov G.A., Reitmann V., Dimension Theory for ODE. Wiesbaden: Vieweg-Teubner Verlag. (2005). 4. Whitney H. // Trans. Am. Math. Soc. V. 36. P. 63-89. (1934). 5. Leonov G.A., Malykh A.E., Reitmann V. // Proc. Conf. PHYSCON, 2009. 27 Taken’s time delay embedding theorem for dynamical systems on infinite-dimensional manifolds

Popov Sergey [email protected]

Scientific supervisor: Prof., Dr. V. Reitmann, Department of Applied Cybernetics, Faculty of Mathematics and Mechanics, Saint-Petersburg State University

1. Introduction In 1981 Takens proved a theorem that allows the phase space of dynamical systems evolving on smooth finite-dimensional manifolds to be reconstructed from an appropriate time series. Later this result was generalized by Robinson [1] to the case of dynamical systems on an arbitrary Hilbert space. In this paper dynamical systems defined on infinite-dimensional manifolds are considered. We provide a certain generalization of Robinson’s result for such dynamical systems. Moreover, unlike Robinson’s paper where the results were obtained by an embedding theorem for infinite-dimensional Hilbert spaces due to Hunt & Kaloshin [2], our result makes use of an embedding theorem for infinite- dimensional manifolds due to Okon [3]. We apply also the Taken's embedding theory to the investigation of the dynamical system arising from the microwave heating problem [4]. Some approximation of the fractal dimension of the invariant set is given. 2. Time series analysis Let M be a Cr - smooth n-dimensional manifold. Let’s consider a dynamical system on M generated by a diffeomorphism φ: M→ M. The motion corresponding to an initial point u∈M is denoted by φt(u), t≥0. In a typical experiment [5], the phase space of this dynamical system is un- known. We try to infer properties of the system by taking measurements. Since each state of the dynamical system is uniquely specified by a point u in the phase space, a measured quantity is a function from phase space to the real numbers h: M→. Let τ be the length of the interval between the measurements. Then we get the sequence of observations: iτ z00= hu( ), .., zi =ϕ h ( ( u )), i ∈ 1,.., N . Let d∈N be an arbitrary number. Then we get the vectors d . ξ=i:(zz i , i+11 , … , z id +−) ∈ , i = 0,1, … , N0 − d + 1 This sequence of vectors Nd0 −+1 {ξi }i=0 is called a time series. The question arises: how to reconstruct the dynamics of the initial system by this time series? To answer this question let’s construct an embedding function which is defined as follows: 11d − Φϕ,h ():u = (hu ( ), h (ϕϕ ( u )),... , h ( (u ))),. uM ∈ 28 The next theorem shows how to choose the number d in this function in order to get an embedding from M to d. Theorem 1 [5]: Let M be a compact Cr - smooth n-dimensional manifold. Let d∈N such that d ≥2n+1. Then the set (φ,h) of pairs for which the embedding function Фφ,h(u) is an embedding is open and dense in the space rr Diff( M )×≥ C ( M ,) for r 1 Here Diff r(M) is the linear space of Cr- diffeomorphisms on M and Cr(M,) is the linear space of Cr - smooth maps from M to . Note that the inequality d ≥2n+1 arises from Whitney's embedding theorem. This theorem states that the set of embeddings from the n-dimensional manifold M to d is an open and dence set in the space of Cr - smooth maps from M to d. Now let us consider Robinson’s theorem [1]. This theorem is a modification of Taken’s theorem to the case of an arbitrary Hilbert space. The formulation of this theorem uses the property of ‘prevalence’. This concept, which generalizes the notion of ‘almost every’ from finite-dimensional spaces to infinite-dimensional spaces, was introduced be Hunt, Sauer & Yorke [6]. Definition :1 A Borel subset S of a normed linear space V is prevalent if there is a finite-dimensional subspace E of V such that for each ν∈V, ν+e belongs to S for almost every e∈E (with respect to a Borel measure μ). The fact that S is prevalent means that if we start at any point in the ambient space V and explore along the finite-dimensional space of directions specified by E, then almost every point encountered will lie in S. Now let us consider the notion of the thickness exponent. If X is a subset of a Banach space V, then the thickness exponent of X in V, τ(X;V), is a measure of how well X can be approximated by linear subspaces of V.

More formally (see [1]), denote by εV(X,n) the minimum distance between X and any n - dimensional linear subspace of V. Then −log(n ) . τ=(XV;) lim . n→∞ log(εV (Xn , )) -1/τ This formula shows that if εV(X,n) ~n then τ is the thickness exponent of X . Now we can formulate Robinson’s theorem. Theorem 2 [1]: Let H be a Hilbert space and S be a compact set whose fractal ∈ dimension (see [7]) satisfies dimf(S)(2+τ(S,H))d, and suppose further that S is an invariant set for a Lipschitz map φ:H→H, such that:

1. The set Γ of points in S such that φ(x)=x satisfies dimf(Γ)<1/2, and 2. S contains no periodic orbits of φ of periods 2,...,k. Then a prevalent set of Lipschitz maps h:H→ make the embedding k Фφ,h:H→ one-to-one on S. Note that unlike Taken’s theorem where the Whitney embedding theorem was used, Robinsons’s theorem makes use of an embedding theorem for finite- dimensional sets due to Hunt & Kaloshin [2]. 29 3. Numerical results The fractal dimension plays an important role in the embedding theory. In this section some numerical results of estimating the fractal dimension are presented. Let us consider the one-dimensional two-phase microwave heating process (see [4]) which is defined in the following way:  1 ε()xw = ( w ) −σ (,) x θ w ,  tt µ x x x t  ( ) 2  A()θt =θ xx +σ( xw , θ) t ,   wtftwtft(0,)==12( ) , ( 1,) ( ) , (1)  θ0,tt =θ 0, 1, = 0,  ( ) ( )  θxx(0,tt) =θ 0, ( 1,) = 0,  wx( ,0)= 0, wt ( x ,0) = e00( x) , θ( x ,0) =θ ( x ) Here x∈(0,1), t∈[0,T), θ(x,t) is the temperature of the material, ε(x) is the electric permittivity, μ(x) is the magnetic permeability, σ(x,θ) is the electrical conductivity, A(θ) is the operator of entalphy which describes the two-phase nature of the process. We can consider an approximating problem for (1), hence we have finite- dimensional approximation of the given process. Let X be an approximation of the phase space of this process. In order to estimate the fractal dimension of X we can estimate the correlation dimension of X. The correlation dimension is defined as follows: lnC (ε ) dimcor ( X )= lim ε→+0 ln(ε ) where C(x) is the correlation integral defined by the following formula: 1 N C(ε) = lim Hε−|| xxij − || N →∞ 2 ∑ ( ) (2) N ij,1= where xi are N vectors from X, ||·|| is the distance on X and H(x) is the Heaviside function: 1, x ≥ 0 Hx( )=  0, x < 0.

Let θ0(t) be the solution of the approximating problem for (1) at a fixed point x0∈(0,1) and at the time moment t. Let τ>0 be a time delay. Then we can consider an embedding phase space of dimension m, which consists of the points m yj =θτθτ{ 00( j ), ( ( j + 1)), …θτ+ ,0 ( ( jm − 1))} , j = 1,2, … , n = Nm − + 1 . Here N is a sufficiently large natural number. Then we can calculate the correlation integral (2) by the following formula: 1 n ε = ε−mm − Cmn, ( ) 2 ∑ H(||) yyj k || n jk,1= This formula depends on the parameters m,n and ε.

30 Fig. 1. The estimation of the correlation dimension.

In the Fig.1 the connection between the estimates of the fractal dimension and the parameter m is shown. At small values of m the value dimcor increases. Starting from some m0 the increasing stops. The value dimcor corresponding to this m0 is the estimate of the correlation dimension of the phase space.

4. Generalization to the case of an infinite-dimensional manifolds Robinson’s theorem holds on an arbitrary Hilbert space. But many dynamical systems (for example the dynamical system arising from the Sine-Gordon equation), arising in real life, act on infinite-dimensional manifolds. We can try to generalize the result of Robinson to such infinite-dimensional manifolds. To do this we have to choose an appropriate embedding theorem like the theorems by Whitney and Hunt & Kaloshin. Such an appropriate theorem is proved by Okon [3] Using this theorem the following result [8] was obtained.

References 1. Robinson J.C. // Nonlinearity № 18 pp. 2135-2143 (2005). 2. Hunt B.R. and Kaloshin V.Yu. // Nonlinearity № 12, pp. 1263-1275 (1999). 3. Okon T. // Arch. Math. № 78 pp. 36-42 (2002). 4. Manoranjan R.V., Yin H.M. // Contin. and Discrete and Dynamical Systems, Serie A, № 15 pp. 1155-1168 (2006). 5. Takens F. // Lecture Notes in Mathematics № 898 (1981). 6. Hunt B.R., Sauer T., and Yorke J. // Bull. Amer. Math. Soc. № 27, pp. 217-238 (1993). 7. Boichenko V.A., Leonov G.A. and Reitmann V. Dimension Theory for Ordinary Differential Equation. - Teubner, 2005. 8. Popov S.A. // Preliminary version (2012).

31 A two-phase problem arising from a microwave heating process in nonhomogeneous material

Serkova Nadezhda [email protected]

Scientific supervisor: Prof. Dr. Reitmann V., Department of Applied Cybernetics, Faculty of Mathematics and Mechanics, Saint-Petersburg State University

Introduction In the present work process of local heating of material by microwave radiation is considered. The mathematical model of this process consists of Maxwell’s equations and the heat transfer equation. It also takes into account that the material can be in two phases – liquid or solid. We consider the one-dimensional arising from a microwave heating problem in special the case of nonhomogeneity with respect to Maxwell's equations. Studies on the effect of microwaves on the material are of great importance to the issues of industrial and medical applications. In particular, the effects of microwave radiation are used to treat cancer. The so called hyperthermia procedure is applied in combination with other types of treatment and increases their effectiveness [7]. In this case, the mathematical model of the material is considered as biological tissue, which is characterized by specific features, such as heterogeneity. In the first part of the paper we formulate the heating problem, define the concept of a weak solution and provide the theorem of the existence of a weak solution for described problem. In the second part we model the problem using a numerical method to investigate dependencies of temperature profile form on values of electric permittivity and magnetic permeability at different intervals with respect to Maxwell's equations.

One-dimensional heating problem Let us consider the initial-boundary problem that describes a microwave heating process in one-dimensional space:

The first equation is Maxwell’s equation; it is derived from two Maxwell’s equations in one-dimensional space. The second equation is the heat transfer equation, (3)-(5) are the boundary conditions, (6) are initial conditions. 32 Here θ(x,t) is the temperature of the material, ε(x) is the electric permittivity, μ(x) is the magnetic permeability, σ(x, θ(x,t)) is the conductivity. A(θ) is the nonlinear operator of enthalpy which shows the two-phase nature of the process and takes the form

where m denotes the melting temperature of the material. Since we consider the case of heterogeneity with respect to Maxwell’s equations, electromagnetic properties vary depending on the interval. So the electric permittivity and the magnetic permeability and take a piecewise constant form:

Here a is the boundary of the layers.

Existence of a weak solution In connection with problem (1)-(6) one can explore various issues, one being the existence of the solution. Let us give the definition of weak solution.

Definition. A couple of functions is a weak solution of the mentioned system if

is satisfied for any ∈ψ H1(0,T,H1(0,1)) and

is satisfied for any η∈H1(0,T,H1(0,1)). If we can take T=∞ such solution is called global weak solution. Here H1(0,T,H1(0,1)) is a special Sobolev space defined in [3]. The second identity contains the operator of the enthalpy A, so that a solution defined in this way is also called enthalpy solution. The question of existence of a weak solution is solved by the following theorem, which is the main result of the present paper.

33 Theorem. Let the following conditions hold for the system (1)-(6): 1) ε(x), μ(x) are piecewise constant function.

There are constants 0

There is a σ >0 such that 0 ≤ σ(x,θ) ≤ σ (1+θ) for all (x, θ)∈(0,1)×[0,∞) 2) 0 0 ∈ 3)There is an a0>0 such that Aθ≥ a for all θ , θ ≠ m 4)

Then the problem (1)-(6) has a global weak solution.

Numerical results Under certain additional restrictions one can show that the system (1)-(6) has a classical solution. We assume that these solutions are smooth enough and use an explicit difference scheme as a numerical method to model this system. Through the modeling we observe a changing of the temperature profile depending on different parameters. The temperature profile in case of homogeneity is presented in Fig. 1. One can

Fig. 1. Typical form of the temperature profile without heterogeneity: “hot spot” at the end of the interval.

see that at the end of the interval a so called “hot spot” appears. Solution norm goes to infinity in the neighborhood of some point. It means that temperature grows very fast at the end of the interval and the heat is localized in a bounded region of the space. It is clear that the uncontrolled rise of temperature is dangerous when the microwave radiation is used for medical purposes. In our experiments, we consider a changing of the type of the graph and localiza- tion of the “hot spots” with respect to this initial situation when the electric permit- 34 tivity and magnetic permeability at different intervals with respect to Maxwell’s equation and position of boundary of the layers were varied. Figs. 2-4 illustrate the results of our experiments. One can observe connections between variations of electromagnetic characteristics at the intervals with respect to Maxwell’s equation and heat distribution on

Fig. 2. Form of the temperature profile depending on parameters: growing of magnetic permeability at the second interval leads to the decreasing of maximum temperature of the “hot spot”.

Fig. 3. Form of the temperature profile depending on parameters: growing of electric permittivity at the second interval leads to the appearance of a second “hot spot” and the disappearance of the first one.

35 Fig. 4. Form of the temperature profile and maximum temperature depending on the location of the boundary of layers: location of the “second” “hot spot” is before the boundary of layers; maximum temperature on the second “hot spot” depends on its location. the whole interval. The maximum temperature of the first observed “hot spot” can be managed by varying the magnetic permeability. Changing of electric permittivity allows to obtain another “hot spot” instead of the first one. Its location and maximum temperature depend on position of the boundary of layers.

References 1. Landau L.D., Lifshitz E.I., Electrodynamics of continuous environments (In Russian) – Moscow: Nauka, 1992. 2. Yin H.M. // SIAM J. of Mathematical Analysis, 29, pp. 409-433 (2002). 3. Ladyzhenskaya O.A., Solonnikov V.A., Uraltseva N.N. Linear and quasilinear equations of second order and parabolic type(In Russian) – Moscow: Nauka, 1967. 4. Sun D., Manoranjan V.S., Yin H.M. // Discrete and Continuous Dynamical Systems, Supplement, pp. 956-964 (2007). 5. Serkova N.D. // Diploma project, SPbSU (2011). 6. Kalinin Y.N., Reitmann V., Yumaguzin N.Y. // Discrete and Continuous Dynamical Systems, Supplement, Vol.2 pp. 956-964 (2011). 7. Kumar S., Katiyar V.K. // Int. J. of Appl. Math and Mech. 3(3), pp. 1-17 (2007).

36 Lyapunov functions in upper Hausdorff dimension estimates of cocycle attractors

Slepukhin Alexander [email protected]

Scientific supervisor: Prof. Dr. Reitmann V., Department of Applied Cybernetics, Faculty of Mathematics and Mechanics, Saint-Petersburg State University

1. Introduction General upper estimates of the Hausdorff dimension of attractors of dynamical systems have been derived for the first time by A. Douady and J. Oesterlé in [1]. Later these results were generalized by other authors [2, 3]. For the first time Lyapunov functions have been introduced into the estimates of Hausdorff dimension by V.A. Boichenko and G.A. Leonov in [4]. The investigation of nonautonomous differential equations leads to the theory of cocycles and their attractors [5-8]. In a certain way one can consider random dynamical systems and the associated random attractors. Elements of the Douady-Oesterlé theory of upper Hausdorff dimension estimates for random attractors were developed by H. Crauel and F. Flandoli in [9]. In this paper we state two theorems about upper Hausdorff dimension estimates of cocycle attractors or invariant sets which include Lyapunov functions. These results can be looked as generalization of the estimates for attractors or invariant sets of autonomous systems [4, 10] to cocycle attractors. 2. Basic tools for cocycle theory Let (Θ,ρ ) be a compact complete metric space. A base flow ({σt} , (Θ,ρ )) is θ t∈ θ defined by a continuous mapping σ:×Θ→Θ, (t,θ)→ σt(θ) satisfying 0 1)σ=(⋅) id Θ 2)σts+ (⋅⋅⋅) =σ t() σ s() for all t, s∈ A cocycle over the base flow ({σt} , (Θ,ρ )) is defined by the pair t∈ θ

tn {ϕ( θ⋅,)}tR∈ ,( ,|| ⋅ ||) ,where θ ∈Θ 1) φt(θ, ·):n → n for all t∈, θ∈Θ 0 2) φ (θ, ·) = id n , for all θ∈Θ  3) φt+s(θ, ·) = φt(σs(θ), φs(θ, ·)) for all t, s∈, θ∈Θ In the sequel we shortly denote a cocycle

tn {ϕ( θ⋅,)}tR∈ ,( ,|| ⋅ ||) θ ∈Θ over the base flow ({σt} , (Θ,ρ )) by (φ, σ). The basics of the cocycle theory one t∈ θ ∈ n can find in [5]. If θ Θ→Z(θ)Ì is a map, we call Ẑ = {Z(θ)}θ∈Θ a nonautonomous set. The nonautonomous set Ẑ is said to be compact if all sets Z(θ)Ìn, θ∈Θ are compact, and invariant for the cocycle (φ,σ) if 37 ϕθtt( , ZZ( θ=)) ( σθ( )) for all t∈ and θ∈Θ.

The set Ẑ = {Z(θ)}θ∈Θ is said to be globally B-pullback attracting for the cocycle (φ,σ) if limdist (ϕσtt (()− ( θ ),BZ ), ( θ )) = 0 t→∞ for any θ∈Θ and any bounded B Ìn.

A nonautonomous set Â = {A(θ)}θ∈Θ is called global B-pullback attractor for the cocycle (φ,σ) if the set Âis compact, invariant and globally B-pulback attracting for the cocycle. 3. Upper Hausdorff dimension estimates for cocycles Let (M,ρ) be a metric space and ZÌM be an arbitrary subset of M. We denote the Hausdorff dimension of Z by dimHZ (c.f. [10]). n n Let L: → be a linear operator and let a1(L)≥...≥ an(L) denote [10] the singular values of L. Let d∈[0,n] be an arbitrary number. It can be represented as d=d0+s, where d0∈{0,1,...,n-1} and s∈(0,1]. Now we put s αL …α L α L, for d∈ 0, n ,  11( ) dd00( ) + ( ) ( ] ω=d (L):   1 , for d = 0 and we call ωd(L) the singular value function of L of order d. Suppose that (φ,σ) is a cocycle for which the maps φt(θ, ·):n → n are smooth enough for all t∈, and θ∈Θ. Let us make the following assumptions:

(A1) The nonautonomous set Ẑ = {Z(θ)}θ∈Θ is a compact invariant set for the cocycle (φ,σ). ∈ t n n t (A2) For each θ Θ and t>0 let ∂2φ (θ, ·):  →  be the differential of φ (θ, ·) with respect to the second argument, which has the following properties: a) For each ε>0 and t>0 the function tt t ||ϕθ( ,v) −ϕθ−∂ϕθ( , u) 2 ( , uvu)( −) || ηε (t, θ=) : sup vu, ∈θ Z( ) ||vu− || 0< ||vu − || ≤ε is bounded on Θ and converges to zero as ε→0 for each fixedt >0. t b) For each t>0 supsup ||∂θ2ϕ ( ,u ) ||op <∞ θ ∈Θu ∈Z () θ where ||L||op denotes the operator norm of L. Theorem 1. Suppose that the assumptions (A1) and (A2) are satisfied and the following conditions hold: 1) There exists a compact set K̃Ì n such that

U ZK(θ⊂)  . θ∈È (1) n 2)There exist a continuous function k:Θ×  →>0, a time τ>0 and a number ττ d∈(0,n] such that ku(σθϕθ ( ), ( , )) τ sup ω∂d (2ϕθ( ,u )) < 1. (,)θuK ∈Θ ×  ku(,θ )

Then dimHZ(θ)≤d for each θ∈Θ.

38 Theorem 1 is presented and proved in [11]. A shorter announcement of these result has been given in [12]. 4. Cocycles generated by differential equations Let us consider the nonautonomous ODE u = f( tu, ) (2) where f:×n → is a Ck -smooth (k≥1) vector field. With respect to the vector field (2) we introduce thehull of f given by H(f)={ f( ⋅+ tt,, ⋅) ∈} , where the closure is taken in the compact-open topology. One can show that H(f) is metrizable with a metric ρ. As a result we get the complete metric space (H(f),ρ) on which a base flow calledBebutov flow [6] is given by the shift map σt (f ) = ft ( ⋅+ ,) ⋅ for any f̃ ∈ H(f). We assume that H(f) is compact. A sufficient condition for this is the almost-periodicity of f(t,u) with respect to t. Suppose now that we have on Θ= H(f) the evaluation map given by n (,)θuu ∈Θ ×  → θ(0,). In particular we get for θ = f∈ H(f) fˆ ( tu, )= f(0, u) It follows that fˆ (σ=t ( f),, u) f( tu) for all t∈ and u∈n. Using this map we can associate to (2) the family of vector fields uf =ˆ( σθt ( ), u ), (3) where θ∈H(f) is arbitrary. The given system (2) is included into (3) as special case. Under the following additional assumptions on (2) one can show for system (3) the existence of a cocycle over the base flow ({σt} , (H(f),ρ)) (cf. [5]). t∈ (A3) The map (t,u)∈ ×n→f(t,u) is continuous and satisfies a local Lipschitz condition with respect to u. (A4) There exist locally integrable functions p,q:→ such that ||f ( tu , ) ||≤+ pt ( ) || u ||2 qt ( ) for all (t,u)∈ ×n. ∈ n For a point (θ0,u0) Θ× we denote by w(t,u0) the solution of the variational equation along the orbit of the cocycle through (θ0,u0) i.e., the equation  ˆ tt w=∂2 f( σθ( 0),, ϕθ( 00uw)) (4) ∈ n with the initial condition w(0,w0)  . Then we have for t≥0 t . t ∂ϕ2( θ 00,,u) w 0 = wtw( 0) i.e. ∂ϕ2( θ 00,uw) 0 is a solution of the variational equation (4).

Let λ1(θ,u)≥... ≥λn(θ,u) be the eigenvalues of the matrix 1 ˆˆT ∂θ+∂θ22fu( ,,) fu( ) 2  . Theorem 2. Suppose that there exist a continuous function V:Θ×n→ for which the generalized derivative 39 d tt Vuσθϕθ( ),,( 0 ) dt ( ) exists along the given trajectory. Suppose further that there are a number d∈(0,n] written as d=d0+s with d0∈ {0,1,..., n-1} and s∈(0,1] and a time τ>0 such that ϕθττ( , ZZ( θ=)) ( σθ( )) for all θ∈Θ, the condition (1) is satisfied and τ [,,λσθϕθttuu +…+λσθϕθtt, , + ∫ 10( ( ) ( )) d0 ( ( ) ( 0)) 0

tt d tt +λs d +10 σθϕθ( ),,( u) + V σθϕθ( ),,]0( u0) dt < 0 ( ) dt ( ) for all θ∈Θ and u0∈K̃. Then dimHZ(θ)≤d for all θ∈Θ. Theorem 2 is also presented in [11,12].

5. Upper Hausdorff dimension estimate for invariant set of nonautonomous Rössler System We consider the nonautonomous Rössler system [13] x=− y − z,, y = x z =− btz( ) + at( )( y − y2 ) (5) where the parameters are functions a,b:→>0 which we write as at( )=+ a0 a 1( t), bt( ) =+ b01 b( t)

Here a0 and b0 are positive constants; a1(·) and b1(·) are smooth functions satisfying the inequalities

atabtb1( ) ≤ε 01, ( ) ≤ε 0 (6) for all t∈ where ε∈(0,1) is a small parameter. Assume also, that there is an l>0 such that bt( ) ≤ε l (7) for all t∈ and the hull H(f) with f as right-hand side of (5) is compact in compact- open topology. A sufficient condition for this is the almost periodicity ofa and b. It follows that system (5) is a special type of system (2) for which the assumptions of Wakeman's theorem are satisfied. Thus (5) generates a cocycle

tn {ϕ( θ⋅,)} tR∈ ,( ,||⋅ ||) θ∈Hf( ) over the base flow σρt ,,H(f ) ({ }t∈ ( )) We assume that for this cocycle there exist a compact set Ẑ={Z(θ)}θ∈H(f), which satisfies (1) with a compactK ̃, and a time τ>0 such that ϕθττ( , ZZ( θ=)) ( σθ( )) for all θ∈H(f). Instead of (5) we consider the family of systems 2 x=− y − z, y = x , z =− bθθ ( tz ) + a ( t )( y − y ), where for brevity we have written ttˆ atθθ( )≡ aˆ ( σθ( )), bt( ) ≡ b( σθ( )) 40 Our aim is to estimate from above the Hausdorff dimension of Ẑ with the help of Theorem 2. To do so we have to check the inequality d (8) λ1,θθ(txyz,,,) +λ2, ( txyz ,,,) + s λ3, θ( txyz ,,,) + V θ( txyz ,,,)< 0 dt t for all t∈[0,τ], all (x,y,z)∈K̃, and all θ∈H(f) in which λ(k,θ)(t,x,y,z)≡λk(σ (θ), φt(θ,x,y,z)), k=1,2,3 are the eigenvalues of the symmetrized Jacobian matrix for the right-hand side of (8) ordered with respect to their size as λ1,θ≥λ2,θ≥λ3,θ and tt Vθ ( txyz,,,)≡ V( σθϕθ( ) ,( ,,,xyz)) is a Lyapunov-type function defined for (x,y,z)∈ K̃, θ∈H(f) and t∈[0,τ] by the relation t 1 Vσθ( ),, xz : =( 1 − s) ξ( z − bθ ( t) x) ( ) 2 , where ξ is a varying parameter.

We calculate the eigenvalues λk,θ and the derivative dVθ/dt and substitute them into (8). Direct computations with the use of (6), (7) and Theorem 2 finally give the estimate 2btθ ( ) 21( −ε)b0 dimH Z (θ≤−) 3 ≤−3 2 2 btθθ( )+ξ htxy( ,,;) (1+ ε)b0 +( ab 00 +2) + b 0 + 1 + ε⋅ C with some constant C which is calculated from the parameters of the system. It is clear that if we turn back to the autonomous Rössler system, i.e. tend ε→0, we will get the already known Hausdorff dimension estimate for a compact invariant set of the Rössler system (cf. [10]) 2b dim K ≤−3.0 H 22 b0+( ab 00 + 2) ++ b 0 1

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D. Solid State Physics Intercalation of Al as a method of formation of quasifreestanding graphene

Anna Popova, Alexander M. Shikin [email protected]

Scientific supervisor: Prof. Dr. Shikin A.M., Solid State Electronics Department, Faculty of Physics, Saint-Petersburg State University

1. Introduction It is well known that the unique electronic structure and physical-chemical properties of graphene will appreciably changed under the interaction of graphene with a substrate. This interaction leads to an energy shift of the electronic states in the valence band, to an appearance of the local energy gap and, in some cases, to a distortion of linear dispersion dependences of the π states of graphene [1-4] that can be followed by a loss of unique characteristic for graphene [5, 6]. The present work deals with investigation of a process of intercalation of Al underneath of a graphene, synthesized by cracking of propylene (C3H6) on top of monocrystalline substrate Ni(111), and analysis of changes of the electronic structure of graphene during the Al intercalation. On the one hand, one of the main aims of this work was a verification of the results of early published works [7, 8] about a possible energy shift of the Dirac point under the interaction of graphene with a Al (due to a possible charge transfer between Al and graphene). On the other hand, an intercalation of only noble metals [3] underneath of graphene was intensively studied before (i.e. metals with d-type of the valence band). It was showed that intercalation of metals with d electrons in valence band underneath of graphene leads to a hybridization of π states of graphene with d states of metal with the corresponding distortion of dispersion dependences of π states of graphene in the crossing region of these states and the formation of local energy gaps in the dispersion dependences of π states of graphene [3]. The valence band of Al does not include d electrons, so it was expected that intercalation of Al underneath of a graphene will lead to a blocking of strong covalent interaction of graphene with Ni substrate without any breaks in the dispersion dependences of π states in the valence band. Investigations of the electronic structure were carried out by angle-resolved photoelectron spectroscopy with the application of synchrotron radiation. Influence of different concentration of the deposited and intercalated Al atoms on the result of intercalation process of Al underneath of graphene/Ni(111) was studied.

2. Results and Discussion Intercalation of Al is very complicated process with definite stages of the formation of system. First of all, we should illustrate all these stages of the intercalated process up to stage of stable electronic structure typical for Al layer at the interface underneath of graphene (see Fig. 1). So, the stages (a) and (b) in Fig. 1 correspond 44 to a synthesis of graphene by cracking of propylene on top of monocrystalline substrate Ni(111). As a result a well-ordered graphene monolayer is formed [9]. The stage (c) in Fig. 1 is a deposition of Al on top of graphene with a subsequent annealing of the system at the temperature of 400°C during 5 minutes (stage (d) in Fig. 1). It was found that after annealing of the system the Al atoms intercalated underneath a graphene dissolve mainly in the Ni layer with formation of surface alloy enriched by Ni. In order to increase a relative concentration of Al in Ni-Al alloy an additional deposition of Al is necessary (stage (e) in Fig. 1) with subsequent annealing of the system at the same temperature ((f) in Fig. 1). Simultaneously with a growth of the relative concentration of Al in underlying surface Ni-Al alloy and a partial accumulation of Al takes place at the interface between graphene and the Ni-Al layer. At certain concentration of intercalated Al (after the next deposition of the Al on top of system (stage (g) in Fig. 1) and subsequent annealing of the system at the same temperature (stage (h) in Fig. 1)) a formation of continuous layer of Al at the interface under graphene takes place. In this case the electronic structure of graphene becomes similar to the electronic structure of quasifreestand-

Fig. 1. Schematic illustration of all stages of the experiment. (a) - monocrystalline substrate Ni(111); (b) –graphene synthesized by cracking of propylene on top of Ni(111); (c) – Al deposition on graphene; (d) – annealing of the system; (e) – extra deposition of Al; (f) – annealing of the system; (g) – third (extra) Al deposition; (h) – annealing of the system. ing graphene on top of Al. Changes of the valence band structure during the process of formation of the system are shown in Fig. 2. The photoemission spectra were measured in the normal emission geometry with angle resolution ~1o. The spectrum for graphene on Ni(111) (Fig. 2a and stage (b) in Fig. 1) is characterized by the binding energy of π states of graphene in the Г point of the surface Brillouin zone of about 10 eV. While for pyrolytic graphite binding energy of π states (with a weak interaction between 45 graphite layers) is about 8 eV (Fig. 2e) [10]. Peaks near the 0.2-0.5eV and 1.5 eV in the spectrum for the system graphene/Ni(111) correspond to 3d states of Ni. Annealing of the system with Al deposited on graphene (stage (d) in Fig. 1) at the temperature 400°C does not lead to any visible changes of energy position of π states of graphene in the valence band. Binding energy of π states is about 10 eV as it was before. But some tracks at the 8.55 eV of binding energy can be distinguished. This stage is characterized by a solution of the intercalated Al metal into Ni substrate or by a formation of the surface alloy with a stoichiometry near the Ni3Al. We didn’t observe any shift of π states of graphene that can be evidence of absence of blocking of the strong interaction of graphene with a substrate at this stage. At the same time 3d states of Ni is weaken and shift toward higher binding energies (BE’s becomes ~1.7 and 1.8 eV). According to early published works [11, 12] alloying of Al with d metal Ni leads to a shift of the Ni d band toward the higher binding energies in parallel with decreasing of the concentration of Ni in alloy. Therefore, these changes of energy position of Ni d peaks can be related to an alloying of Al with Ni (with predominant concentration of Ni) during the intercalation of Al at the first step. After the additional Al deposition (see stage (e) in Fig. 1) with subsequent annealing of the system (see stage (f) in Fig. 1) we observe considerable changes in the valence band spectra. Aluminum which is accumulated at the interface between graphene and substrate begins blocking strong covalent interaction of graphene with Ni d states in some areas under Fig. 2. Changes of the valence graphene monolayer. As a result an additional band photoemission spectra at peak of π states appears at the binding energy different stages of experiment (see ~8.55 eV. However the weak peak of π states Fig. 1). The spectra are measured at the binding energy ~10 eV also remains in in the normal emission geometry. the spectrum that leads to two-peak structure of (a) - graphene synthesized on top the π states. This is evidence that there are dif- of Ni(111); (b) - annealing of the ferent areas underneath a graphene monolayer system after the deposition of Al with weak and strong interaction of graphene (stage (d)in Fig. 1); (c) – anneal- with a substrate. In regions where intercalated ing of the system after the addi- Al atoms are located underneath of graphene tional Al deposition (stage (f) in monolayer a blocking of strong interaction of Fig. 1); (d) – annealing of the sys- graphene with a substrate takes place. In the tem after Al deposition (stage (h) places where Al atoms are located underneath a in Fig. 1); (e) – pyrolytic graph- graphene the binding energy of π states is about ite (for comparison). Photon en- ~8.55 eV and in the places where Al atoms are ergy is 60 eV. not located at the interface – the binding energy 46 of π states is about ~10 eV (as it is in the case of strong interaction of graphene with a Ni substrate).

Fig. 3. (a) – Series of photoelectron spectra, measured with angle resolution for graphene after the intercalation of Al (stage (h) in Fig. 1); (b) - corresponding dispersion dependences of π states of graphene measured in the ГК direction of the surface Brillouin zone. Dispersion dependences are represented in the form of dN/dE. Photon energy is 60 eV. Additional Al deposition on top of system (stage (g) in Fig. 1) and subsequent annealing of the system (stage (h) in Fig. 1) lead to further considerable changes in electronic structure of the valence states of the system. When the concentration of Al is enough for filling of the whole layer at the interface, only one peak of π states is observed in the valence band spectrum at the binding energy ~ 8.55 eV (Fig. 2e). Fig. 3 shows series of photoelectron spectra measured with angle resolution for this system and corresponding dispersion dependences of π states of graphene measured in the ГК direction of the Brillouin zone. Fig. 4 shows in details dispersion dependences of π states of graphene near the Fermi level in the region of the K point of the surface Brillouin zone. The top of lower cone of graphene is located at the energy of about 0.4 eV relative to the Fermi level (Fig. 4). The lower edge of the band of graphene π states in the region of the Г point is located at the energy of 8.55 eV (Fig. 3) that differs from case of graphene on top of Ni(111). The shift of the energy position of π states of graphene toward the Fermi level, in Fig. 4. Detailed dispersion depen- comparison with graphene/Ni(111) [3, 9], can dences of π states of graphene in testify to blocking of the strong interaction of the region of the K point of the sur- graphene with a substrate after intercalation of face Brillouin zone. 47 Al. Detailed analysis of the dispersion dependences of π states near the Fermi level in the region of the K point shows a some distortion of the dispersion dependences in the region of the crossing of π states with the substrate-derived states located at the energy of about ~1 eV due to covalent interaction with these states (Fig. 4).

Conclusions Intercalation of Al underneath graphene synthesized on Ni(111) leads to the “blocking” of strong interaction of graphene with a substrate and formation of electronic structure characteristic for a quasifreestanding graphene. Dispersion dependences of π states have a linear character in the region of the K point of the surface Brillouin zone and the Dirac point is located near the Fermi level. But a small energy gap is formed between occupied and unoccupied cones of π states near the K point. A charge transfer from Al to atoms of graphene after intercalation of Al is not observed in opposite to the prediction in work [7, 8]. At the initial stages of Al intercalation alloying of Al with underlying Ni layer takes place. It makes the process of intercalation of Al more complex in comparison with other metals.

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48 Calculation of Sound Speed in Artificial Opal

Andrey Uskov [email protected]

Scientific supervisor: Dr. Borisov B.F., Department of Solid State Physics, Faculty of Physics, Saint-Petersburg State University

Introduction Microcellular materials are widely used as catalysts, filters and adsorbents. They also could be used to investigate properties of materials in confined geometry. For example, it is possible to investigate properties of phase transitions in liquid metals embedded in porous glasses and artificial opals by measuring dependency of sound speed on temperature. There are at least three methods allowing to determine phase transitions. They are X-ray analysis, NMR, and acoustical methods. The first two methods have some serious disadvantages. Each measurement takes several minutes, thus dynamics of phase transitions could not be clarified. But the theory which could link elasticity of embedded material and sound speed is still absent [1]. Thus, it is still impossible to link quantity of liquid in pores with sound speed. The aim of this work is calculation of sound speed in artificial opal with empty pores and dependency of sound speed on pore filling factor.

Elasticity of opals The theory we have developed takes into account internal structure of artificial opals. They consist of closely packed silica spheres. Each sphere consists of smaller silica spheres of the second order. The spheres of the second order consist of the smaller spheres of the third order [1, 3]. Each silica sphere has several neighbor spheres. The summary force acting on this sphere depends on neighbor’s coordinates and Fig. 1. Internal structure of artificial opal [3]. elasticity of spheres. In general, the elasticity factors for different spheres could differ.

49 Fig. 2. Scheme of interaction between silica spheres. According to the second Newton’s law, motion of silica sphere could be de-

ma=∑cosαiii k∆ l scribed by the following equation: It could be shown that motion of sphere could be described by the following differential equation: ∂2uka2cos2α ∂2u = ii 2∑ 2 ∂tm∂x Where u is coordinate of the corresponding sphere along the wave vector. After introduction of the porosity of the material: 3 m=−(1 γρ) ma We can calculate elasticity of the whole material and link it with sound speed in material: k cos2 α c = ∑ ii aρm (1−γ)

And finally we evaluate elasticity factors of each contact between two silica spheres. Thus sound speed could be calculated as a function on compression factor of silica spheres, what is equal to diameter of silica spheres substituted by distance between to neighbor silica spheres and divided by two. 3 2 2π c c = 11 2R +δ (1−γρ) m 6 ln  δ 

Mercury porometry simulation Unfortunately, we have no possibility to measure compression factor directly, but we had mercury porometry data for samples under investigation. We simulated 50 porometry for ideal opals with different compression factors and compared with experimental results.

Fig. 3. Cross-sections of ideal opal without compression. In order to calculate porometry, we simulated cross-sections of ideal opals. Then, for each point the minimal distance to pore well was calculated. Actually, this distance has similar physical sense to pore size for current point. But the dependence of number of points on corresponding size doesn’t fit porometry data. This data should be normalized in the following way: we should take into account the fact that cross-section is two-dimensional and a single spherical pore makes quite complicated contribution to this dependence. Contributions for spherical pores are presented in the Fig. 4. Using this data, the dependence of volume of pores on it’s radius was calculated for different cross-sections. The average value between this dependencies has a physical sense of porometry. This data were compared to experimental results.

Fig. 4. Dependence of point number on corresponding size. 51 Comparison results are presented in the Fig. 6. It could be seen that experimental results fit theoretical ones for compression range about 5%. And the difference between theoretical and experimental results could be explained by two effects. The first one is dislocation of solica spheres in real sample. The second effect is deformation of silica spheres.

Fig. 5. Dependence of volume of pores on radius. (relative to silica sphere size).

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53 Modification of spin and electronic structure of graphene by intercalation of Bi

Evgeny Zhizhin [email protected]

Scientific supervisors: Prof. Dr. Vladimirov G.G., Prof. Dr. Shikin A.M., Department of Solid State Electronics, Faculty of Physics, St. Petersburg State University

Introduction Exploration of spin and electronic structure of various nanostructured systems have attracted increasing interest in recent years, due to wide development of Spintronics – a new branch of nanotechnology. Spintronics is based on exploiting the 'spin' of the electron [1, 2]. By the normal conditions, value of the spin-orbit splitting of electronic states of graphene is negligible [2], but the intercalation of Au atoms underneath graphene monolayer leads to the effects induced of the substrate-induced spin-orbit splitting of the π-states of graphene [3, 4]. However, the interaction of the π-states of graphene with the d - states of Au, in addition to the spin polarization of π states, accompanied by the formation of discontinuities in the dispersion dependences of the π- states [4]. The aim of our research was to investigate a possibility of the intercalation atoms of another metal with high atomic number Bi underneath graphene monolayer. Additionally, Bi is characterized by sp-type of the valence band structure, i.e. by the lack of the d-electrons in the valence band. We present in the current work the results of investigations of the features of the electronic structure of graphene with intercalated Bi underneath and the features of the spin structure of the formed system.

Experimental details The experiment was carried out in the Helmholtz-Zentrum (BESSY II) at the Russian-German beamline by photoelectron spectroscopy with angular and spin resolution. As a result the dispersion dependences were measured with spin resolution with using hemispherical energy analyzer SPECS "Phoibos 150". Graphene monolayer on the surface of Ni(111) was formed by cracking of propylene at heated surface of thin Ni(111) layer. The layer of Ni(111) with a thickness of about 100Ǻ was deposited on an atomically clean surface of W(110). Cleaning the surface of W(110) were produced by annealing in oxygen at a pressure of 5.10-7 mbar and the temperature 1200°С followed a short heating to temperatures of ∼2000°С in ultrahigh vacuum. Graphene was produced by a catalytic reaction of cracking of propylene (С3H6) at the surface of Ni(111) during for 5 min at a pressure of propylene 1.10-6 mbar and temperature of the sample 450°С. After that the sample was annealed in ultrahigh vacuum and at 400-450°С. Intercalation of Bi atoms underneath a graphene monolayer was produced by deposition of Bi on 54 the surface of a graphene monolayer with subsequent annealing of the system at a temperature of ∼300-350°С. The pressure in the research chamber during the experiment was on the level of 1-2.10-10 mbarr.

Experimental results and discussion Fig. 1 shows the series of the normal emission photoelectron spectra for the various stages of formation of the system with graphene synthesized on the surface of Ni(111) with subsequent intercalation of Bi underneath a graphene. It is seen that a graphene on Ni(111) is characterized by the binding energy of π- states 1 eV that testify to strong interaction of graphene with the Ni-substrate. Fig. 1. The stages of the formation After the deposition of Bi on a top of graphene of the MG/Bi/Ni(111) system. Series monolayer (Fig. 1c) and following annealing of the normal emission photoelec- of the system at a temperature of ∼300-350°С tron spectra: a) after deposition of (Fig. 1d) π-state of graphene is shifted toward ∼ 100 Å Ni on the surface of the the lower binding energy (~ 8,1 eV) that W(110), b) after the formation of indicates about a “blocking” of the strong graphene by cracking of propylene, interaction between graphene and substrate c) after deposition of ∼ 1 ML Bi on due to the intercalation of Bi. graphene, d) after intercalation of Figs. 2 and 3 show the dispersion depen- Bi underneath graphene. dences of π states for graphene interca-

Fig. 2. The dispersion dependence of Fig. 3. The dispersion dependence of the the valence band electronic states for valence band electron states for a gra- graphene on Ni(111) in the ГК direc- phene after intercalation of Bi atoms un- tion of the Brillouin zone synthesized derneath a graphene measured in the ГК by cracking of propylene. direction of the Brillouin zone. 55 lated of Bi underneath. From Fig. 2 we can see that the π-state of graphene for this system doesn’t reach the Fermi level, and dispersion dependence of the π states of graphene in the region of the K point of the Brillouin zone has parabolic character. Compared to the quasi-free graphene dispersion relations the π-state of graphene synthesized on the surface Ni(111) are significantly shifted towards higher binding energies approximately on 2 eV. The upper edge of the band of the π-states in the region of the К point of the Brillouin zone is located below the Fermi level. An intercalation atom of Bi underneath a graphene monolayer leads to a substantial change in the electronic structure of graphene as compared to the electronic structure of graphene on the surface of the Fig .4. The dispersion dependence of the π states Ni(111). of graphene after intercalation of Bi atoms mea- As a result of intercala- sured in the region of the K point of the Brillouin tion atoms of Bi a significant zone. energy shift of the π-states of graphene toward lower binding energies is observed that testify to a “blocking” of a strong interaction of graphene with the substrate. Due to the Bi intercalation the electronic structure of the valence band becomes similar that characteristic for a quasi-free graphene. Thereat, practically no any distortions in the dispersion dependences of the π- states of graphene and a formation of discontinuities in the dispersion dependences are observed. In Fig. 4 the detailed disper- Fig. 5. Photoelectron spectrum with the spin res- sion dependence of the π- states olution for π states of graphene measured in the of graphene in the region of the ГК direction of the Brillouin zone after interca- K point of the Brillouin zone lation of Bi. is shown. 56 The band edge of the π-states reaches practically the Fermi level. Only small energy gap is formed with a width of ~ 0,2 eV near the Fermi level. Fig. 5 shows the corresponding photoelectron spectrum with the spin resolution measured for π states of graphene in the direction ГК of the Brillouin zone after intercalation of Bi. It is seen that intercalation of atoms Bi underneath a graphene does not lead to any visible spin splitting of the π states of graphene. It means that the effect of the substrate-induced spin-orbit splitting of the π-states of graphene does not manifests itself after intercalation of Bi despite its high atomic number.

Conclusions As a result of the experiment one can make the following conclusions: 1. Intercalation of Bi underneath a graphene synthesized on Ni(111) leads to blocking of the strong interaction of graphene with substrate and formation of electronic structure similar that characteristic for quasi-free graphene with small energy gap near the Fermi level. 2. Availability of only high atomic number Z of intercalated atoms (Bi) is not sufficient for the effect of the induced spin-orbit splitting of the π-states of graphene.

Acknowledgment. The experiment was carried out done in the Helmholtz- Zentrum (BESSY II) at the Russian-German beamline. The work was supported in framework of G-RISC and Euler program.

References 1. Rashba E. I. // J. Supercond.— Vol. 15, no. 1 (2002). 2. Rybkin A.G. Electronic, energetic and spin structure of the thin layers of metals induces by spin-orbit interaction // St. Petersburg – 2010. 3. Varykhalov A. // PRL, 101, 157601 (2008). 4. Popova A.A. // FTT, vol. 53, no. 12 (2011).

E. Applied Physics Usage Pocket Comsol for the Numerical Nonstationary Nonlocal Plasma Modeling

Burkova Zoya [email protected]

Scientific supervisor: Dr. Chirtsov A.S., Department of General Physics, Faculty of Physics, Saint Petersburg State University

One of the most interesting kinds of plasma (ion-ion plasma) has been investigated for a long time [1]. The formation of ion-ion plasma is interesting to simulate due to a high range of relationships of different kinds of plasma characteristics can be received. It’s easy to use a numerical simulation to simulate wide range of systems, which is interesting for physics. Software package COMSOL is well adapted to solve such problems [2]. There are a lot of modules in this software. One of them is the plasma module. The foundation of the COMSOL Multiphysics Plasma Module is the Drift Diffusion interface which describes the transport of electrons in an electric field. The Drift Diffusion interface solves a pair of reaction/convection/diffusion equations, one for the electron density and the other for the electron energy density. The problem is that COMSOL can simulate only one task in one session. Because of that were used package Wolfram Mathematica to compare the results.

External circuit One of the simplest model has been investigated - DC glow discharges. Such discharge in the low pressure regime has long been used for gas lasers and fluorescent lamps. DC discharges are attractive to study because the solution is time independent. The DC discharge consists of two electrodes, one powered (the anode) and one grounded (the cathode). The electrons are emitted from the cathode surface and then accelerated by the strong electric field, where they acquire enough energy to initiate ionization. The positive column is coupled to an external circuit:

60 Domain equations The electron density and mean electron energy are computed by solving a pair of drift-diffusion equations for the electron density and mean electron energy. Convection of electrons due to fluid motion is neglected.

∂ ()nnee+∇⋅− ()µee⋅⋅ED−∇nRee = (1) ∂t   ∂ ()nnεε+∇⋅− ()µµµ⋅ ED−⋅∇nRεε +⋅E Γe = (2) ∂t   Γ =− µ ⋅ EDnn−⋅∇ ee()eee (3) and, ne denotes the electron density, Re is the electron rate expression, μe is the electron mobility which is either a scalar or tensor, E is the electric field, and De is the electron diffusivity, which is either a scalar or a tensor. The first term on the right side of Eq. 1 represents migration of electrons due to an electric field. The second term on the right side of Eq. 1 represents diffusion of electrons from regions of high electron density to low electron density.

Here, ne is the electron energy density, Re is the energy loss/gain due to inelas- tic collisions, μe is the electron energy mobility, E is the electric field, and De is the electron energy diffusivity. The subscript refers to electron energy. The third term on the left side of Eq. 2 represents heating of the electrons due to an external electric field. Note that this term can either be a heat source or a heat sink depending on whether the electrons are drifting in the same direction as the electric field or not. For a Maxwellian electron energy distribution function, the following relationships hold:  5 DTee==µµee,,εε  µµDT= ε e (4)  3

Where Te is the electron “temperature”. So, given the electron mobility. From which the electron diffusivity, energy mobility and energy diffusivity are computed.

Reaction Formula Type ε(eV) 1 e+Ar=>e+Ar Elastic 0 2 e+Ar=>e+Ars Excitation 11,5 3 e+Ars=>e+Ar Superelastic -11,5 4 e+Ar=>2e+Ar+ Ionization 15,8 5 e+Ars=>2e+Ar+ Ionization 4,2 6 Ars+Ars=>e+Ar+ Ar+ Penning ionization - 7 Ars+Ar=>Ar+Ar Metastable quenching - Plasma chemistry Argon is one of the simplest mechanisms to implement at low pressures. The electronically excited states can be lumped into a single species which results in a

61 chemical mechanism consisting of only 3 species and 7 volumetric reactions and 2 surface reactions: Volumetric reactions Surface reactions Reaction Formula Sticking coefficient 1 Ars=>Ar 1 2 Ar+=>Ar 1

Where: Ar-argon in ground state, Ars – argon in metastable state, Ar+- ionized argon, e - electron. Results The electron density occurs along the axial length of the column. The electron density peaks in the region between the cathode fall and positive column. This region is sometimes referred to as Faraday dark space. The electron density also decreases rapidly in the radial direction. This is caused by diffusive loss of electrons to the outer walls of the column where they accumulate a surface charge. The build

Fig. 1. Surface plot of electron density of argon with metastable level.

Fig. 2. Surface plot of electron density of argon without metastable level. 62 up of negative charge leads to a positive potential in the center of the column with respect to the walls. The electron density of argon with metastable level in Fig. 1 and without in Fig. 2 are plotted along the axial length of the column. The solution shows that the measurement of Argon without metastable level is by a factor of ten lower then Argon with metastable level.

Fig. 3. Difference between data of Ars and Ar (2D).

Fig. 4. Difference between data of Ars and Ar (3D).

The software Wolfram Mathematica has been used to compare the results. Plot data saved in a text file inCOMSOL exported in Wolfram Mathematica. Difference between the data of Ars and Ar is plotted in Figs. 3 and 4. The results received in this work a good coordinated with theory and allows to continue the investigation of the plasma behavior using program package COM- SOL and Mathematica.

References 1. Кудрявцев А.А. // Письма в ЖТФ, 1996, т.22, вып.17, с. 11-14. 2. http:// www.comsol.com 63 Factorization of charge formfactors for clusterized light nuclei in reactions e+16O and e+12C

Danilenko Valeria [email protected]

Scientific supervisor: Prof. Dr. Gridnev K.A., Department of Nuclear Physics, Faculty of Physics, Saint-Petersburg State University

Introduction A Bose–Einstein condensate is a state of matter of a dilute gas of weakly interacting bosons confined in an external potential and cooled to temperatures very near absolute zero (0 K or −273.15°C). Under such conditions, a large fraction of the bosons occupy the lowest quantum state of the external potential, at which point quantum effects become apparent on a macroscopic scale. There are some cases in which one can use a model of binding alpha-particles [1] in order to describe the structure of the nuclei. It can also useful for identifying the properties of the elastic scattering of the electrons. We use a model of binding alpha particles in out approach – the nucleus consists of interacting alpha-particles and nuclear binding energy is a sum of alpha-particles pair interactions. We consider the coordinates of alpha-particles to be frozen and calculate the positions of them under the following requirements: 1. The configuration of alpha-particles corresponds to the minimum of the potential energy. 2. The nucleus must have a nearly spherical form [2]. The process of formation of alpha-particles in nucleus is dynamical. They form a crystal lattice in the nucleus and the interaction is counted per bond. The appropriate interaction potential is the form of the one parameter Yukawa potential: e−γr Vr()=−V , nucl 0 r where V0 is fitted through the experimental data and γ is inverse value of the Compton wavelength of the neutral π-meson [2]. There is a conjecture that nuclei of the most widespread elements in the universe might be viewed as a tight packing of alpha-particles. The binding energies within such a framework show a good agreement with experimental values. The basis of the model is the assumption that the binding energy can be expressed as a sum of energies coming from interactions between alpha-particles and their self-energies. So the nuclear binding energy can be written as (1)

EAB =+0 ()6Nnαα+ C, where A0 determinates the energy of interaction, Nα is the number of alpha-particles in the nucleus, nα is the number of bonds between alpha-particles and C is the Coulomb energy of the nucleus calculated as 64 3e2 C =−ZZ(),1 A−13/ 5r0 where A is the mass number and r0=1.25 fm. The term with the number of bonds stands for the interactions between alpha-particles [1]. Using of the model: 1. In general we allow nuclei to consist not only of alpha-particles but also of neutrons not bound into alpha-particles, so that the model can be used for predicting binding energies of nuclei with arbitrary mass number. Generalization is done under the following requirements: The dependence of binding energy on the number of neutron pairs at first behaves linearly, but then acquires a slope, and there are no particular shell effects – the separation energy of two neutrons is monotonically decreasing. This could be explained: it is energetically favorable to place neutrons inside different parts of alpha-nucleus, assuming that their bonding to alpha-particles contributes more than bonding between neutron pairs themselves. This gives initial liner behavior until the neutron pairs due to inflating the nucleus starting to distort the alpha-bonds. This generalizes (1) as follows:

EABn=+02()6Nnαα++βNC,

which brings in a linear term βN2n, but the number of bonds between alpha- particles has to be recalculated. 2. If we consider a tetrahedral structure, under very general assumption about the alpha-particles potential one can calculate the location of alpha-particles within such construction. Depending on their size, such constructions have a different number of bonds per particle which brings to a picture similar to the shell model of nuclei. If we consider the experimental value of a separation energy of an alpha-particle as a a function of the number of alpha-particles we will se that apart from peaks stemming from magical numbers, there are a few nuclei with relatively high alpha-particles binding energy. These “alpha- magical” numbers are 3, 7, 13…Similar magic numbers were obtained through the analysis of clusters in molecular physics [1]. So we call the nuclei clusterized if we use the model of binding alpha-particles for describing them. In our research we considered folded formfactors for the clusterized light nuclei. At certain conditions the scattering amplitude can be expressed through the formfactor. In case of electrons it is a charge formfactor. This formfactor in the case of cluster structure can be factorized as two multipliers: one for distribution of clusters in nuclei and the second for the cluster formfactor itself. The cross-section is calculated using the following: 2 2 dσ  zz()e  1 22 dσ  = 12 Fq() = Fq()  2    dΩΩ2mv 4 ϑ  d    sin res 2 65 The factorization of the formfactor:

Fq()=∗Fααη We consider 2 possible types of alpha-particles density distribution, namely surface and volume distribution. In the first case the formfactor will be proportional to the spherical Bessel function of order zero j0 in the second case it will be the spherical Bessel function of order one j1 [3] sin(kr) η1 ==jk()r α 0 kr 1 cos(kr) η2 ==jk()r sin(kr) − α 1 ()kr 2 kr As the formfactor for clusters itself we used: 2 Frα =−exp( α ) The calculations were made for 2 reactions on different energies: Reaction Energy, Mev e+160 87,2 e+12C 21,34 e+12C 22,5 e+12C 24,5 e+12C 27,3 e+160 75

The obtained results of the angle distribution of the cross-sections are presented in Fig. 1. 3,5E -14

3E -14

2,5E -14 87,2 Mev 2E -14 21,34 Mev

1,5E -14 22,5 Mev 24,5 Mev 1E -14 26,3 Mev Cross-section, mBrn/ster 5E -15 75 Mev 0 0 20 40 60 80 100 Angle, grad

Fig. 1. Angle distributions of the cross-sections in the system of center of mass.

The calculations show that the alpha-particle density distribution in the nuclei in question is a surface distribution. The results were obtained with the formfactor 1.

66 Conclusion We have used the factorized formfactors for clusterized light nuclei. Factorized formfactors give us a possibility to scan nuclear surface and some intrinsic layers of a nucleus. The obtained result shows that the alpha-particle density distribution in the considered nuclei is a surface distribution.

References 1. Gridnev K.A., Torilov S.Yu., Gridnev D.K., Kartawenko V.G., Greiner W. // Int. J. Mod. Phys. 14 (2005), 635. 2. Torilov S.Yu., Gridnev K.A., Greiner W. // Int. J. Mod. Phys. 16 (2007), 1757. 3. Arscen J.B., Weber H.J. Mathematical methods for physicists. - Academic Press, 2005.

67 Study of interaction forces between constant magnet and high-temperature superconductor

Marek Veronika [email protected]

Scientific supervisor: Dr. Chirtsov A.S., Department of General Physics - I, Faculty of Physics, Saint-Petersburg State University

Introduction The high-temperature superconductivity is interesting in terms of both theory and experiment. Presently a great number of articles and overviews dedicated to this phenomenon exist (i.e. [1, 2]). But a complete theory has not been already created although the high-temperature superconductivity was discovered 25 years ago. However this topic well demonstrates laws of electrodynamics that is why it is traditionally studied in classical physics courses. A study is usually accompanied by an experiment’s show but an implementation of this experiment is joined with a number of problems (e.g. sizes of a superconductive ceramics and of a magnet are small). Therefore an interactive demonstrations set and a video clip (composed of a real experiment and an animation) about the mechanism of levitation of a constant magnet over a high-temperature superconductor were created [3, 4] (Fig. 1).

(a) (b) (c) Fig. 1. a – one of interactive models: magnet (current ring) + superconductor (set of induced round currents); b – clip’s screenshot: a real experiment; c – clip’s screenshot: a three-dimensional animated model of the system.

Experiment During the video clip production, an existence of the space interval where the magnet is in neutral equilibrium instead of the expected stable one was found. To study this phenomenon a setup which allows the measurement of magnet’s altitude depending on the applied external force (electrically regulated) was thought up. Systematical measurements of two types were implemented. In the first one the applied force has been monotonically increased up to its maximum and then likewise decreased to the value defined by own magnet’s weight (m = 200 mg). 68 In the second experiment the nonmonotonically varying force was used. Thereby obtained plots (Fig. 2) show that hysteresis effects don’t exist in case of large distances between the magnet and the superconductive ceramics. In the case of twice smaller distance one can observe a strong hysteresis effect and an existence

a b Fig. 2: Appearance of hysteresis effects in case of monotonically (a) and nonmonotonically (b) change of the applied force. of several magnet equilibrium positions. Moreover after the removal of the external force the magnet doesn’t return to its initial position and its levitation altitude noticeably decreases.

Results and Discussion To explain the observed phenomenon two possible models were considered. The first one – the “abrupt collapse model” – supposes that when current flowing on the superconductor exceeds the critical value icr, the destruction of the superconductive state in the corresponding areas of ceramics occurs. The model of critical Bean state [5] supposes only a limitation of the current by the critical value. It was assumed that, when the induced currents exceed the critical value, a circle of disturbed superconductivity appears. Inside this circle the induced currents turn either to zero or to the critical value (depending on the model used), that was considered by adding the new compensative currents. This partial destruction of superconductivity, which conserves after the magnet’s removal from the ceramics, was considered as the cause of the magnet’s non-return to the initial state. By means of these assumptions, the mentioned models and the image method, an analytical description of the phenomena was worked out. This description allows in principle the calculation of the critical current’s density from the experimentally obtained plots, but actually it doesn’t take into account several features of the system as the following: the finiteness of the superconductive ceramics, the nonsimple- connectedness of its geometrical form and a breach of boundary conditions for the magnet induction B under the additive currents. Thus we have the additive currents which consider geometrical features of the problem and ones which assure the boundary conditions. Thereby the Laplace problem (which evidently can be solved only numerically) for a vector potential A with mixed-boundary conditions was considered. Therefore the relaxation method 69 was applied, that with a glance of cylindrical symmetry of the problem leads to the following connection between vector potential’s values in different points: Ar(),,φ z ≈ 1 ≈−Ar()δφrz,, +−Ar()δφrz,, +−2Ar()δφrz,, +−Ar((),,φδzz++Ar(),,φδzz 6   In initial approximation A was supposed to be zero everywhere except the layer near the part of ceramics with the additional effective currents. For this and overlying layers the current has to satisfy Neiman’s boundary conditions:

∂Aϕ 4π − ==B i' ∂z r c ϕ Because of a slow convergence of the method in the area of superconductivity’s destruction, the “cloud” computing was used. The thereby found vector potential allows the calculation of the additional currents in those parts of ceramics which preserve the superconductivity. Fig. 3 represents an example of the total current calculation results. The distant plot area corresponds to the central hole in the superconductive ceramics, the close one - to the area out of ceramics. The central part for the first model (Fig. 3a) contains “dents” appeared because of the currents’ exceeding the critical value (that leads to a broadening of the destruction superconductivity area). For the Bean model (Fig. 3b) this is expressed in appearance of the “plateau” instead of “dents”. Note that the influence of new superconductivity destruction zones appearing during calculation of vector potential (and therefore currents’ distribution) is also taken into account. The lifting force acting on the magnet is defined by the current distribution and can be calculated as a sum of interactions between the magnet and the total round currents iƩ(R) on the ceramics’ surface: ∞ mR23H Fi= Σ ()R dR ∫ 2252/ 0 (HR+ ) a b

Fig. 3. Results of the calculation of the total current on the superconductor’s surface for the “abrupt collapse model” (a) and the Bean model (b). 70 The area of disturbed superconductivity reaches its maximum size in a point of maximum approaching the magnet to the ceramics which means the maximum of the applied force. Then under decreasing of the applied force and under the repulsion of induced surface currents, the magnet moves away from the superconductor but reaches lesser altitude. That happens because of the superconductivity’s destruction which turns to zero a part of the induced currents in the area adjoining to the central hole in the magnet. Thus there are two causes seeking for compensating each other: an increasing of the induced currents’ density and an appearance of the conductivity’s destruction. In the real experiment it’s a levitation magnet’s altitude which assures the equilibrium between the gravity force and the external applied force. In the numerical experiment, carried out to verify an applicability of the considered models, the role of this adjustable parameter is played by the critical current value. In case of its right choice, the calculated values of magnetic repulsion forces must be equal. Conclusions The computational modelling of the experiment showed, the “abrupt collapse model” is more applicable for description of the lifting force hysteresis than the Bean’s one. By means of above described method the critical current value (in conventional units) was calculated in the network of the first model. Further im- provements of the model could be made by more precise consideration of a magnet field configuration or by creation of the hybrid model on the base of described ones. Evidently it’s necessary to create much precise and perfect device for the experiment’s implementation. It should be added that above described effects was observed only on the one of high-superconductive ceramics. On the others one can observe the trivial altitude dependence on the applied force: under small deviations from magnet’s equilibrium

a b

Fig. 4. Results of ceramics surface studies. a – the sample, demonstrating hysteresis effects, b – the trivial sample. position weak hesitations appear, and no hysteresis effects are found out. That’s why the comparative analysis of two superconductive samples has was carried out on optic microscope. The results are represented in the Fig. 4. As one can notice, a size of microgrits composing the “hysteresis” sample is much greater than one of the trivial ceramics. 71 References 1. Delft D., Kes P. The discovery of superconductivity // Physics Today. September 2010, pp. 38-42. 2. Гинзбург В. Л., Ландау Л.Д. // ЖЭТФ, 1950, Т. 20, С. 1064. 3. Марек В.П., Чирцов А.С. Использование возможностей мультимедиа и компьютерного моделирования для организации самостоятельной работы студентов и их подготовки к работам физ. практикумов // В сб. «Материалы Х Межд. Конф. «Физика в системе современного образования» (ФССО-09)», 31 мая- 4 июня 2009 г., СПб, 2009, Т. 2, С. 193-195. 4. Марек В.П. Использование возможностей мультимедиа и компьютерного моделирования для создания иллюстраций по курсу общей физики // В сб. «Тезисы докладов молодежной научной конференции «Физика и прогресс», 18-20 ноября 2009 г, Санкт-Петербург, Россия, С. 137. 5. Bean C.P. // Rev. Mod. Phys. 1964. V. 36. N 1. P. 31-36.

72 Usage of stereoscopic 3D-visualization technologies

Marek Veronika [email protected]

Scientific supervisor: Dr. Chirtsov A.S., Department of General Physics - I, Faculty of Physics, Saint-Petersburg State University

Introduction The three-dimensional (3D) computer models are greatly used in up-to-date educational process, especially in scientific areas, by reason of their proximity to real experiments. However these 3D objects are rather complicated to be appre- hended by means of its plane 2D projections. This problem was partially solved due to the possibility of rotating the model relative to the plane of the monitor. But today another solution is proposed by 3D stereoscopic images and virtual reality technologies which are becoming more and more popular at present [1]. Until recently principal products in this area were dedicated to the creation of complex expensive virtual reality systems. Currently a great number of simple rather cheap methods of stereoscopic computer images making and also a lot of software for its creation and demonstration exist. All of this actually allows using these new technologies in education, but evidently only in those cases when it gives irrefut- able advantages in comparison with traditional forms of studying. Main methods of stereoscopic images creation are known since the XIX century: for this it is necessary to project two pictures recorded by two spatially separated observers on the retinas of both eyes. This procedure can be fulfilled in different ways. One of the simplest and best-known is the anaglyphic method. It’s based on the chromatic selection of images for each eye and uses glasses with color filters for the images’ viewing. Essential disadvantages of this method are low quality of the color rendering and easy fatigability of the onlooker. Therefore the anaglyphic method isn’t suitable for the studying process but however it can be used for a rapid assessment of the scene “voluminosity”. More suitable for educational purposes is the polarization method. Here images for stereomates are created by light beams with different polarizations; the division of images is made with a help of polarization glasses.

Equipment A setup for computer stereo visualizations was worked out (Fig. 1). It includes two multimedia projectors ACER PD520 (based on DLP technology) on movable stands, linear polarizers with perpendicular polarization planes and a screen. An adjustment of projectors’ position was made by means of the calibrating images. Note also that the screen has to save the polarization in the scattered light. Special screens with metallic sputtering have this property but they rather expensive, that’s

73 a b

Fig. 1. A principal scheme (a) and a photo (b) of the developed equipment for the stereovisu- alisations. why for this first test setup the screen was made of means at hand with a mat aluminic foil on the surface. For viewing stereo images a polarized glasses with a perpendicular polarization for each eye was utilized. For stereo images and clips playback Stereoscopic Player [2] was used. Results Let’s consider several three-dimensional objects obtained by a computer modelling which are actually complex for the two-dimensional demonstration. They are for example a toroidal coil with a low number of turns or a problem about a magnetic field in a system “grounded sphere + electric charge” which is solved by the image method. In the first case, it’s better to observe the coil’s field at minimum from two different angles, in the second one, several elements outside a screen’s plane have to be eliminated for reducing image’s complexity. The usage of 3D technologies allows the solution of mentioned problems. It’s a collection of interactive 3D-models created in the special programs - “construction sets”, which was a base for first experiments on the new technology [3]. The models the most difficult for visualisation have been chosen, for instance “Beams motion in slightly misaligned a b

Fig. 2. Examples of the stereo images (anaglyphic method) based on results of the computer modelling: a – “Beams motion in slightly misaligned resonator”, b – “Compound objective”. resonator”, “Magnetic field of a constant magnet in the presence of superconductor”, “Absolute optical system”, “Compound objective” etc (Fig. 2). Another tested method for stereo demonstrations creation is implemented by means of Autosdesk 3ds Max software [4].

74 a b

Fig. 3. The vector model of one-electron atom in external magnetic field (fine and hyperfine-structure splitting are taken into account): a – video clip’s frame,b – anaglyphic variant.

Thereby a number of clips for the course “Quantum theory of atomic and molecular spectra” [5] have been created (Fig. 3). These clips visualize the vector model of an atom and different approximations for atomic and molecular static states computation. According to the simplest one, an electron moves in the elliptical quasi-classical a orbit round of the nuclear and the orbit is oriented orthogonally to the electron’s angular momentum l. Then the spin-orbit coupling, the nuclear spin and the external magnetic field (leads to Zeeman effect) are added. The next step in stereo technologies usage is the three-dimensional projection of the four-di- b mensional Minkowski space-time. An educational video clip on this topic was worked out (by means of Autodesk 3ds Max). It demonstrates the light cone and also one- and two-sheet 4D-hyperboloids whose�� lateral surface is set by the following rela- tivistically invariant expressions:

22 2 ct −=r 0 c ct22−=rA22± where A is a constant.

Fig. 4. Different variants of light cone interpretation: a – traditionally used 2D-projection; b, c – anaglific 3D-projections; d – a frame from stereo clip’s about Minkowski space-time structure. d

75 The upper part of two-sheet 4D-hyperboloid corresponds to the space of 4D- velocities. The sections of mentioned quadric surfaces by planes ct = constant in the 4D-space are 3D-spheres. The 3D projection of the 4D-dimensional Minkowski space-time is a set of parallel 3D planes ct = const situated perpendicular to the time axis. Such presenta-

a b Fig. 5. Video clip’s frames: a – light cone plotting; b – 4D-velocity space. tion is evidently better than a traditional image of the light cone (Figs. 4, 5). The carried out video clip was recorded on virtual cameras imitating right and left eyes of an observer that allowed create a stereo clip.

References 1. Андреев C.В., Денисов Е.Ю., Кириллов Н.Е. // Программные продукты и системы, № 3, 2007, с. 37-40. 2. Stereoscopic Player [Электронный ресурс] / 3dtv.at website [сайт]. [2005 - 2011]. URL: http://www.3dtv.at/Index_en.aspx. 3. Колинько К.П., Никольский Д.Ю., Чирцов А.С. Многофункциональный компьютерный учебник по фундаментальному курсу физики. Разделы: “Движение частиц в силовых полях”, “Релятивистская динамика”, “Геометрическая оптика” // В сборнике трудов IV Международной конференции “Физика в системе современного образования”, Волгоград, 15-19 сентября 1997 г. 4. Autodesk 3ds Max Products. [Электронный ресурс] / Autodesk: [сайт]. [2010]. URL: http://usa.autodesk.com/3ds-max/. 5. Чирцов А.С. Конспект лекций по курсу «Атомные спектры» (ч. 1). – СПб: «Соло», 2007.

76 Evaluation of the influence of readout cables in the CBM Silicon Tracking System Prokofyev Nikita [email protected]

Scientific supervisor: Dr. Kondratyev V.P., Department of Nuclear Physics, Faculty of Physics, Saint-Petersburg State University

Introduction The Compressed Baryonic Matter (CBM) experiment is designed to explore the QCD phase diagram in the region of high net-baryon densities. The Silicon Tracking System (STS) is the central detector to perform charged-particle tracking and high-resolution momentum measurement. It consists of eight planar stations of about 3.5 m2 total active area. Each station combined from ladders - units of 10 sectors that are planned to be assembled in the lab and then mounted into the experimental setup. A sector may consist of a individual double-sided silicon micro- strip sensor or of two or three conjoined sensors. The thickness of every sensor is 300 μm. The signal is being read out via low-mass cables of up to 50 cm length in order to keep the active area of the detector free of electronics. The cables consist of micro-line-structured aluminum layers on polyimide carrier foils. The cables have a multi-layer structure: two layers of aluminum wires, shielding, spacers and carriers with a total thickness of 348 μm. At the first look, the thicknesses of cables and sensors are of comparable values, and it is difficult to say a priori, how significant the cable presence in the active area is. Even if cable material is more transparent for particles than sensor material - silicon. However, in a tracking station several layers of cables will be stacked and increase the material budget. To ensure that cables are transparent enough and do not cause significant noise, simulations with realistic STS models are being performed as described below.

Description of the simulation models The first STS station is shown in Fig. 1 (top view). Each ten sectors by vertical combined to ladder. Other stations contain more sensors, and different types of ladders, but first station is the simplest to describe simulation models. Two models were used as "starting points" to compare simulation and reconstruction results. Both are standard STS geometries, containing only sensors. First is the model with STS geometry sts_v11a.geo (v11a). Based on this geometry is the Realistic Cable Model (RCM), that contains cables, described as 9-layer boxes with overall thickness of 348 μm. This model produced from the exact drawings. Detailed description of cables, including exact drawings and radiation lengths is available here [1]. Second model to refer results is the sts_v11b.geo (v11b). Based on this geometry are the Silicon-Equivalent Model (SEM), and several models for reconstruction performance studies (not taking into account the cables): with sensor thickness 400 μm each, 500, 600 and 800 μm. Significant difference between the RCM and

77 the SEM is that SEM have no special GEANT volumes for cables. Instead of this, thicknesses of sensors were increased at the corresponding value.

Fig. 1. First STS station, view in the direction of the beam. This picture looks same for all models described. Dimensions: 50 x 30 cm. The dashed circle corresponds to 25º geometrical acceptance, the solid ellipse to a horizontally extended acceptance in order to provide efficient reconstruction of low momentum electrons.

Fig. 2. Standard STS geometry v11a, first Fig. 3 Standard STS geometry v11b, station side view, beam along the Z axis. first station side view, beam along Units of both scales are in cm. the Z axis. Units of both scales are in cm.

One cable has same thickness to radiation length ratio as 116 μm of silicon. As we can see in Fig. 4, first line of sensors has initial thickness (300 μm), next line has thickness increased on 232 μm (overall, 532 μm = sensor and 2 cables), following lines: 764, 996 and 1228 μm In the terms of thickness divided by radiation length, it is 0.32%, 0.57%, 0.82%, 1.05% and 1.31%. The thickness map for the first STS station is shown in Fig. 5. Differences between sts_v11a.geo and sts_v11b.geo are significant from the geometrical point of view. Nevertheless, the reconstruction results are nearly the same, that is why we can compare RCM and SEM and do not take into account the fact that these models have different positions of sensors.

Realistic Cable Model In principle, the realistic model has some advantages. First of all, the drawings have realistic look: every geometrical volume have prototype (see Fig. 6). Secondly, 78 description of model allows easily change such parameters as cable size, thickness and media. Also, we may suppose that such description looks realistic for GEANT and expect trusted results. On the other hand, in that model we have over 9000 GEANT volumes instead of about 1000. This slows down simulations significantly. Another problem is Kalman Filter (KF). KF is used at the reconstruction stage to

Fig. 6. Realistic Cable Model, first station side view. Units of both scales are in cm. decrease noise. Current algorithm of KF does not take into account any media except sensors, which, moreover, should be similar in thickness. This fact led us to concept of Silicon-Equivalent Model and also we realized the necessity of simulations with models that contains sensors of similar thickness (more thick than standard). Such models have been created and studied, results are presented below. Main parameters to evaluate cable influence is reconstruction efficiency at

1 GeV, power of the K0s and Λ0 peaks and signal to noise ratio for these peaks. Obviously, cable presence in the active volume of the detector leads to signal suppression and noise increase. Graphically it is shown in Fig. 7. Other reconstruction characteristics are compared in Table 1. Power of the peak is measured in relative units due to the fact that all reconstructions have equal number of Monte-Carlo events and calculated as integral under the peak with eliminated background.

v11a RCM v11b SEM Peak power (K ) 0s 1.22 0.95 1.22 1.08 ± 0.03 S/N (K ) 0s 0.56 0.16 0.57 0.12 ± 4% Peak power (Λ ) 0 1.54 1.22 1.54 1.07 ± 0.03 S/N (Λ ) 0 1.38 0.51 1.37 0.32 ± 4%

Table 1. Reconstruction parameters for models with and without cables.

79 Silicon-Equivalent Model The main idea of the SEM is to use only silicon sensors. Such model contains about 1000 GEANT volumes. Unfortunately, SEM is still working improperly with Kalman Filter. But it seems much easier to adjust KF to work with sensors of different thicknesses than with passive cable volumes. Nowadays, reconstruction results with the SEM is nearly the same as with the RCM, that shown in Table 1. Fig. 8 shows signal suppression and background increase in the SEM comparing to the standard value.

Fig. 7. Suppression of the Λ0 peak by Fig. 8. Suppression of the Λ0 peak in the cable presence (Realistic Cable Model). Silicon-Equivalent Model. Red filled Red filled spectrum is the result without spectrum corresponds to the standard cables (v11a). result (v11b).

Simulations with different sensor thickness After the SEM, next logical step is to perform simulations with sensors of equal thickness, but thicker than standard. Main feature in that case is the correct

Fig. 9. The Λ0 peak and increasing background for the different sensor thickness: 300 μm (blue line), 400 μm (magenta dashed), 500 μm (green dashed), 600 μm (black) and 800 μm (red line).

80 work of Kalman Filter. Several thickness values were studied: standard 300, and thicker 400, 500, 600 and 800 μm. Results for 600 μm are especially important, because one of the possibilities to the future CBM experiment is to use back-to-back single-sided micro-strip silicon sensors with thickness of 300 μm each. Increasing the sensor thickness leads to increase of the background, graphically it is shown in Fig. 9. Another reconstruction results is presented in Table 2. As we can see, the ghost probability grows at the factor of 10 from 300 to 800 μm. The _0 peak is not changing by absolute value but background increases and power of peak decreases. The S/N rate has significant reduction at 600 μm. Also we should keep in mind that this is simulations without any cable material.

Sensor thickness 300 μm 400 μm 500 μm 600 μm 800 μm

Ghost probability 0.013 0.022 0.035 0.055 0.116 Rec. efficiency at 1 0.96 0.95 0.95 0.93 0.93 GeV (±0.5%)

Ppeak(Λ0) (±0.03) 1.54 1.58 1.49 1.20 0.79

S/N (Λ0) (±4%) 1.38 0.92 0.62 0.34 0.16

Table 2. Reconstruction parameters for different sensor thickness.

Summary Based on the data obtained are the following conclusions. The Realistic Cable Model and the Silicon-Equivalent Model provide roughly the same results. The S/N ratio for Λ0 is better in the RCM, but the SEM may be improved by adjusting Kalman Filter. Nevertheless, both the RCM and the SEM provides passable results. The usefulness of back-to-back single-sided micro-strip sensors for reaction Au+Au at 25 GeV is being doubted by the reconstruction results with “thick” sensors.

References 1. Heuser J. M. et al. // CBM Progress Report 2010, p. 14 (2011).

81 Application of graph theory to modeling of the complex hydraulic systems

Strizhenko Olga [email protected]

Scientific supervisor: Prof. Dr. Slavyanov S.U., Department of Computational Physics, Faculty of Physics, Saint-Petersburg State University

Introduction This work covers the actual problem of design and maintenance of complex hydraulic systems (or pipeline network), which play very important role in the life of mankind. People use given systems for oil and gas transportation, water and heat supply in the cities, irrigation etc. Hydraulic system is defined as a set of different facilities (pump stations, pressure regulators, shutters etc.) and connecting them pipelines, closed or open channels involved in the transportation of compressible and incompressible fluids (water, oil, gas etc.) [1]. Hydraulic systems are a well known example of a complex and large scale distributed parameter system. By this reason the modeling approaches, numerical methods and optimization of operating modes of fluid transport networks are of permanent interest for researchers and engineers who create more and more perfect simulators (OLGA [2], PipeSim [3], Stoner etc). But all these simulators have common concepts of graph representation of hydraulic network, which appeared several decades ago at the inception of the theory of hydraulic circuits. In this work author described common concepts of graph representation of pipeline network, supplement topology matrices of graph model to formulate Kirchhoff’s laws for flow distribution problem solving, structure of the hydraulic simulator prototype «Pipeline Network» and extension of the prototype for the tasks of leak detection as well.

Graph Representation of Pipeline Network For the beginning we need to understand what is pipeline network. Pipeline network is defined as a set of diffrerent facilities (pump stations, pressure regulators, shutters and etc.) and connecting them pipelines, closed or open channels involved in the transportation of compressible and incompressible fluids (water, oil, gas and etc.). It should be noted that hydraulic network contains points of branch and connection of pipes, and additional elements. For example if we discuss leak detection problem we should consider detectors as aditional element of pipeline network which is located in specific point in the pipeline or can move along pipeline over time that depends on particular type of detector used in the network to gather data (temperature, pressure, rate) about current state of system or a single pipeline as well. Using data from detectors we can identify leakage. 82 Let’s consider the graph representation of the basic elements of pipeline network (Fig. 1 (a)). Edges represent straight sections of pipelines, which have concrete diameter, length, hydraulic resistance and another properties. Nodes are elements that have a dual functionality: 1) Node describes changes of geometry and properties of the hydraulic network like turn, branch, connection of different types of pipeline and hold local resistances, corresponding to given changes. 2) Node is a holder of facilities like pumps, pressure regulators and so on, which affect a flow rate, pressure and resistance in the hydraulic system. Detectors can be represented as external for graph nodes with known values of pressure, flow rate or temperature. Data from these points can be interpreted as boundary conditions for the flow distribution problem or as control values for alarm generation in the leak detection system (Fig. 1 (b)).

(a) (b) Fig. 1. (a) –Graph representation of hydraulic network; (b) – Leak detection system.

Formulation of hydraulic Kirchhoff’s laws Modeling of flow distribution in the hydraulic system (finding of pressure in the all nodes and rates in the all edges in the network) is used to design pipeline network optimally or predict a behavior of pipeline network after changes in the topology of network, after adding new facility to consideration and so on. This modeling is based on two Kirchhoff’s hydraulic laws which can be easily described by incidence and loop matrices of pipeline graph model: The first Kirchhoff’s law, describing local mass conservation law of fluid flow for each node can be easily represented by using incidence matrix: Ax = Q (1) where A – incidence matrix of graph, x – flow rate vector,Q – vector of additional flow from outside the system. The second Kirchhoff’s law, which can be interpreted as energy conservation law in the loop of the network, is represented by using loop matrix: By = 0 (2) where B – loop matrix of graph, y – drop pressure vector. 83 Law of the state flow in the pipeline is: yH+=SXx (3) where H – push vector, S – matrix of pipeline hydraulic resistance, X – diagonal β-1 matrix with elements ǀxiiǀ , β – some coefficient. Generally speaking parameters S, H, Q in the equations are not constants and depend on current flow distribution in the system: SS==(,xp), HH(,xp),QQ= (,xp) (4) Finally we get a complete system of non-linear equations: Ax = Q  (5) By = 0  yH+=SXx Solving this system we can find flow distribution in the pipeline network.

The development of leak detection model The first stage in these research activities is the model development of leak problem in the oil pipeline. The model will be developed by taking into consideration that a leak is treated as an outlet segment, in a part of pipeline network. The outlet segment has certain diameter and very short length. In the model, the diameter of the outlet segment will represent the size of leak, where the diameter could be adjusted to represent the leak opening. The length of outlet segment is taken short; short enough to make the pressure drop effect along the short segment is negligible, compared with the pressure drop along the main line [4].

Leak detection techniques Pipeline failure can be caused by corrosion and wear, intentional damage, unin- tentional damage or operation outside the design boundaries. When leak occurs at any point in a pipeline, there is a sudden change in pressure, flow, etc. characteristics [5]. Analyzing data from sensors about current state of pipeline, we can detect a leakage. In this work only two leak detection techniques were used: Volume Balance Method (VBM) and Pressure Point Analysis Method (PPA) (Fig. 1 (b)). First method is based on the principle of conservation of mass. For a pipeline the flow entering and leaving the pipe can be measured. The mass of the fluid can be estimated from the dimensions of the pipe and by measuring process variables like volumetric flow rate, pressure and temperature. When the mass of the fluid exiting from the pipe section is less than estimated mass, a leak is determined [6]. The principle behind the operation of single point analysis is that the pressure in a pipeline will decline as result of a leak. Further, certain statistical properties can be computed to determine if a pressure is declining in a significant manner. PPA requires that all events other than leaks that may cause a pressure to decline, such as operational changes to the pipeline, must be identified so that the leak detection can be inhibited until such time as the pipeline returns to a steady operation. The technique operates as follows. A buffer of he most recent measured values of 84 pressure is kept for analysis. The data is divided into two periods and the mean and variance of two samples are computed.

nnoldnew ()nnoldn+−ew 2 µµoldn− ew t0 = (6) nn+ 2 2 oldnew ()nnoldn−+1 σ ew ( nnewo−1)σ ld where μ – is the sample mean of the data, σ2 – is the sample variance of the data, n – number of points in the sample. The result of equation (6) is an observed value of a random variable, which has a Student’s t-distribution with n1 +n2 -2 degrees of freedom. Using the value computed in (6) and number of freedom the significance level of the test can be found from a table of Student’s distribution. This is taken as probability of a leak being present [7]

Results and discussions On the course of research the simulator prototype with following functionalities has been developed (Fig. 2):

Fig. 2. Graphical User Interface.

Graphical editor of hydraulic model • Flow distribution • Leakage detection (Fig. 3)

85 (a) (b)

Fig. 3. (a) –Pressure log over time, red line – time of leak occurrence; (b) – Probability of leak over time (PPA output).

References 1. Merenkov A., Hasilev V. Theory of hydraulic circuits.- Moscow: Nauka, 1985 – 276 p (in Russian). 2. Bendiksen K.H., Malnes D., Moe R., Nuland S. The Dynamic Two-Fluid Model OLGA: Theory and Application. SPE Production Engineering, May 1991, pp. 171-180. 3. José M. Chaves-González, Miguel A. Vega-Rodríguez, Juan A. Gómez-Pulido, Juan M. Sánchez-Pérez. PipeSim: Pipeline-Scheduling Simulator // 8th International Symposium on Computers in Education (SIIE'2006), pp. 109-116. León, Spain, October 2006. 4. Pudjo Sukarno, Kuntjoro Adji Sidarto, Amoranto Trisnobudi, Delint Ira Setyoadi, Nancy Rohani & Darmadi. // J. Eng. Sci. Vol. 39 B, No. 1, 2007. 5. Olunloyo V.O. S., Ajofoyinbo A. M. A model for real time leakage detection in pipelines: A case of an integrated GPS receiver // Proceedings of the 3rd International Conference on Applied Mathematics, Simulation, Modeling, 2009. 6. Stuart L. Scott, Maria A. Barrufet. Worldwide Assessment of Industry Leak Detection Capabilities for Single & Multiphase Pipelines, OTRC Library Number: 8/03A120, 2003. 7. Whaley R.S., Nicholas R.E., Van Reet J.D. Tutorial on software based leak detection techniques, Pipeline Simulation Interest Group, 1992.

F. Optics and Spectroscopy A modern implementation of Rozhdestvenski interferometer

Agishev1 N.A., Medvedeva2 T.A., Ryabchikov1 E.L. [email protected] 1 Faculty of Physics, Saint-Petersburg State University, Russia. 2 State educational institution lyceum №419 Petrodvorets, Saint- Petersburg, Russia.

Scientific supervisor: Ass. Prof. Anisimov Yu.I., Department of General Physics-1, Faculty of Physics, Saint-Petersburg State University

Introduction Historically, Rozhdestvenski interferometer was used to study the anomalous dispersion in the material. Researches of anomalous dispersion, performed at the beginning of last century, gave a significant contribution to the development of quantum mechanics and helped to build a reliable model of the atomic nucleus. Various data can be derived from the interference pattern with the absorption lines such as: the concentration of atoms in the vapor substance, the oscillator strengths of atoms. Moreover for reliable measurements it is needed high quality of the image. In his works Rozhdestvenski was faced with several challenges: the photographic method of recording the spectrum required bright light source, which led to uneven heating of the mirrors of the interferometer and its misalignment. In the present work a number of modern methods for reception of the better image of an interferential picture are offered: use of sources of intensive radiation in a narrow spectral range (light-emitting diodes), and application of systems of registration of a picture on the basis of CCD-matrixes of the high resolution is considered. Rozhdestvenski interferometer coupled with spectrograph is used in researching discharge tubes produced by EDD technology [1]. They may be promising in laser researches, for example for creating new active media. The purpose of this study was to modernize the "classical" experimental assembly with a Rozhdestvenski interferometer, coupled with a spectrograph, for use in further experiments by the definition of non-stationary concentrations of vapors of chemical substances in the discharge tubes produced by EDD technology.

Results and Discussion Modernization of the experimental assembly has touched sources of light and photodetectors. Scheme of the installation is presented in Fig. 1. LED emitted (S1) beam of light gets to an interferometer through collimator L1. Rozhdestvenski interferometer consisted of semitransparent mirrors A1, A2 and complete reflective mirrors B1, B2. Optical cell with vapours of investigated substance was placed in one of interferometer arms. The parallel-sided plate k2 was situated in other interferometer arm. Plate k was used to adjust the interferometer. At the output of 88 the interferometer the light beam passed the lens L2 and entered the spectrograph. Interference fringes decomposed in spectrum was registered by photodetector CCD.

Fig. 1. Scheme of the installation. By changing photographic plates to CCD-matrixes of modern cameras processing of the image of hooks had been essentially simplified, but spectral range had been limited. Work is underway on the use of CCDs in the UV range. The resonance transitions of most studied substances contained in this range. Powerful (10–20 A) spectral lamps of high and average pressure and the high- voltage discharge in a porcelain capillary which was earlier used as a sources of optical radiation in Rozhdestvenski interferometer has been replaced by modern light-emitting diodes. Their advantages are obvious. Above all this is ease of use in continues and pulsed modes and the almost complete absence of electrical interference. The spectral characteristics of LEDs were studied by use of specially designed automatical monochromator coupled with a PC. The results of these measurements are presented in Fig. 2. The spectral range of used LEDs covers the range from near UV to near IR range. Research into the LEDs’ response speed showed the possibility of creation of experimental installation with a time resolution of 0.2–0.5 microseconds. This is 10 times less then registration time of known "classic" schemes based on Rozhdestvenski interferometer. 89 The used Rozhdestvenski interferometer has the great historical heritage. This optical device has half a century ago received a silver medal at an exhibition in Brussels. The mechanical and optical quality of the device is that it took only minor adjustments to use it nowadays.

1,0

ts 0,8 uni ve ve ti 0,6 la re y, it 0,4 ens t In 0,2

0,0 400500 600700 λ, Fig. 2. Spectral characteristics of light-emitting diodes. To demonstrate the working capability of the experimental assembly Rozhdestvenski hooks on the sodium doublet were obtained. This experimental demonstration became possible by assistance of staff of the Departments of Optics and General Physics I.

Fig. 3. Hooks in the sodium doublet. A thickness of the plate k = 2.75 mm.

Fig. 4. Hooks in the sodium doublet. A thickness of the plate k = 9.12 mm.

Fig. 5. Hooks in the sodium doublet. A thickness of the plate k = 16.40 mm.

90 Specially designed sodium cell filled with helium under the pressure of 76 Torr was used as a vapor source. The cell was heated up by electric oven to a temperature of 450 K. Using the dependence of the saturated sodium vapor’s pressure on temperature, one can estimate the concentration of sodium vapors. For a given temperature the vapor concentration was 5.3±0.4·1012 cm-3. A white LED working in continuous mode was used as light source. The LED’s voltage at 100 mA current was 3 V. Interferogram registration was made by digital camera (Olympus Sp-350) with the 8 Mpx CCD. The obtained images are shown in Figs. 3-5.

Conclusions There was obtained a value of (5.5±0.7)·1012 cm-3 of atomic concentrations of sodium vapors on basis of the interference pattern. Value calculated from dependence of the saturated sodium vapor’s pressure on temperature was (5.3±0.4)·1012 cm-3. The both values had the same order. The ultimate goal of our researches based on Rozhdestvenski interferometer coupled with spectrograph is creation of experimental assembly operated in pulsed mode with high time resolution (0.5 microseconds). This will allow to measure non-stationary concentration of vapors of chemical substances in the discharge tubes produced by EDD technology by Rozhdestvenski hooks.

References 1. Anisimov Yu.I., Mashek A.Ch., Metel’skii K.E., Ryabchikov E.L. // Optics and Spectroscopy, 2009, Vol. 107, No. 3, pp. 368–370.

91 Investigation of the two-photon induced fluorescence in Rb vapor excited by Ti:Sapphire femtosecond laser pulses

Bondarchik Julia [email protected]

Scientific supervisor: Prof. Dr. Pastor A.A., Department of Optics, Faculty of Physics, Saint-Petersburg State University

Introduction Nowadays the tendency is that the 21st century will likely be known as the century of the photon. Optics is one of the key technologies in the 21st century because today photonic technologies replace traditional ones. It is connected with the advantages of optics, such as speed, lack of interactions, massive parallelism and small energy consumption. On the other hand it has some cons, for instance an all-optical transistor does not exist yet. The solution will appear with using nonlinear optical materials [1]. Therefore, it is important to develop this chapter of optics devoted to nonlinear processes. With reference to the earlier research [2], we would like to investigate coherent two-photon interaction of femtosecond laser radiation and resonant medium in rubidium vapor. The basic idea is to excite rubidium atoms from ground 5S state to state 5D through an intermediate 5P state by means of laser pulses with a wavelength 790 nm, generated by the Ti:Sapphire coherent femtosecond laser. By variation of pulses time delay and changing the input wavelengths it might be possible to observe different temporal dynamic. In this article special attention will be paid to the experiment’s background, specifically theoretical overview and preparation for experimental setup.

General concepts and definitions As far as we are going to investigate two-photon absorption it is necessary to give a definition for this process, namely: two-photon absorption (TPA) is the simultaneous absorption of two photons in order to excite a molecule from the ground state to a higher energy electronic state [3]. For TPA identical as well as different frequen- cies can be used, in our experiment two identical laser pulses were used. Hence, the sum of the energies of the two photons Fig. 1. Rubidium energy level scheme. has to be equal to the energy difference 92 between the involved ground (5S) and upper (5D) states of the molecule. According to the theory, excited rubidium atoms emit lines with wavelength 5 µm (5D-6P) and 420 nm (6P-5S). Rubidium energy level scheme is presented in Fig. 1. Two-photon absorption occurs only in nonlinear optical molecules and even then only at high light intensity. Thus, only with the invention of lasers (Light Amplification by Stimulated Emission of Radiation) it became possible to observe nonlinear processes. The essence of the laser is the gain – the factor by which an input beam is amplified by a medium. A laser can be classified by different types of gain medium (gas, liquid, solid or plasma) and each of them is used for different purposes. Experimental setup In our experiment for the excitation of the pulses Ti:Sapphire femtosecond laser “Pulsar 10” designed by Amplitude Technologies company was used. The Pulsar laser system is a compact femtosecond laser source providing more than 10 mJ pulse energy at 10 Hz repetition rate. The pulse duration shorter than 45 fs leads to a high peak power. The system is a Ti:Sapphire laser based on the so-called “Chirped Pulse Amplification” (CPA) scheme. The system amplifies pulses from a Ti: Sapphire oscillator and consists of a stretcher, a regenerative amplifier, as well as multiphase amplifiers with respective pump lasers and a compressor. For our research the cell containing rubidium was made from fused silica and thereby allows the heating of the vapor in the range of 300 – 500 К. Visible fluorescence at a wavelength of 420 nm was obtained and fixed by computer-controlled spectrograph Ocean Optics SD 2000. The schematic diagram of the experiment is presented in Fig. 2.

Fig. 2. Schematic diagram of the experiment. As a light source Ti:Sapphire femtosecond laser was used; then, cell containing rubidium vapor under investigation was located between focusing optics. Consequently, output signal from substance under study was registered by computer-controlled spectrograph and indicated occurring nonlinear process, which can be explained as two-photon transition between ground 5S and upper 5D states in rubidium vapor. Theoretical model Nonlinear optics is the utilization of the fact that the when the electric field of the light wave is sufficiently high, the induced polarization in a medium is not linearly proportional to the electric field, but depends on its higher power as well. The application of an electric fieldE to a dielectric material causes the constituent atoms and molecules to become polarized. The medium responds to the fieldE by developing a polarization P which represents the net induced dipole moment per

93 unit volume. In a linear dielectric medium the induced polarization P is proportional to the electric fieldE at that point, and they are related by

PE=ε0χ , where χ is the electric susceptibility. However, the linearity breakdown at high fields and theP vs. E behavior deviates from the linear relationship. As soon as P becomes a function of E, we can expand it in terms of increasing powers of the electric field E. It is customary to represent the induced polarization as: 2 3 PE=+εχ01 εχ02EE+ εχ03 , where χ1, χ2, and χ3 are the linear, second-order and third-order susceptibilities. The coefficients decrease rapidly for higher terms and are not shown in the above equation. The importance of the second and third terms, i.e. nonlinear effects, depend on the field strengthE . Non-linear effects begin to become observable when fields are very large, that invariably require lasers. In our research special attention goes to the nonlinear third-order susceptibility, since it consists of two parts: real and imaginary. Real part of third-order susceptibility is responsible for third-harmonic generation, stimulated Raman scattering and etc, while imaginary part describes two-photon absorption [4]. Thus, in case under review it is necessary to consider third-order nonlinear optical processes, which is a phenomenon due to an induced material polarization that is proportional to the third power of the electric field. In this section we present a theoretical model to fit the present experiment. Firstly, it is important to review semi classical approximation for nonlinear effects. According to quantum mechanics, electron movement inside the atom we can describe with wave function which constitutes from Schrödinger’s equation: ∂Ψ iH =Ψˆ ; ∂t and besides using perturbation theory we can assume ˆˆ ˆ H=+ H0 Ht'( ) , where Ĥ0 is defined by interaction between the electron and the nucleus, whereas Ĥ' – by interaction of the electron with the external field. In case that the interaction energy of dipole with the field in the dipole approximation is defined by multiplication of dipole moment μ by field intensity we consider:  Hˆ '() t= −µˆ Et () . The basic reason why we use dipole approximation is that at constant external field reacts only dipole, while quadrupole responds to field gradient. As far as in the visible and ultraviolet range the wavelength of the light is much bigger than the characteristic dimensions of the atom, we can assume that at each period of time the atom is inside an almost constant field.

For Ĥ =Ĥ̂0 solution of Schrödinger equation is described by hydrogen atom: ˆ Hi0 =εi i. 94 So any function can be presents as: ∞ ΨΨ==∑ aii where aii i=1

And besides for Ψ=Ψ(t) → ai=ai(t). Since the normalization of wave function, any change its in time constitutes as rotation. As any operator can be expressed as ˆˆ A==∑ Aij i j, Aij iAj ij solution of Schrödinger’s equation will look like: i t Ψ(t ) =Ψ ( t ) ⋅ exp − Hˆ ( t ) dt 0  ∫ . t0 i For Hˆˆ= H Ψ( t ) =Ψ ( t ) ⋅ exp − Httˆ ( − ) . 0 0  00 We should take into consideration that by definition the evolution operator of the system is ˆ Ψ()t = Utt (,00 ) ⋅Ψ ( t ) . Foregoing result was obtained in Schrödinger’s conception. The next step is to represent wave function in Heisenberg’s conception, where this function does not depend on time: ˆ−−11 ˆˆ Ψui =U ⋅Ψ() t = UU ⋅ ⋅Ψ () t00 = Ψ () t . Since the mean values of the physical quantities should be equal for different representations of wave function, it is possible to assume that ˆˆ ˆ AA=Ψii Ψ i =Ψ uu A Ψ u , and moreover ˆˆˆˆ−1 Aui() t=⋅ U At () ⋅ U . Consequently we obtain that the time evolution operator describes the transition from one representation to another [5].

Conclusion Thus, observed visible fluorescence at 420 nm should be explained as a result of two-photon excitation of rubidium atoms from the ground state 5S to the excited 5D state through the intermediate quasiresonant 5P state. The process mentioned above can be associated with two various options of two-photon transition: through state 5P1/2 or state 5P3/2. Achieved results will be used in our further experiments devoted to coherent excitation of the atomic transition in rubidium by femtosecond pulses of laser radiation at the wavelength of 790 nm.

References 1. Vartiainen E. Class lecture, Topic: Applied Optics. Faculty of Technology, Lappeenranta University of Technology, Lappeenranta, Finland, 2011.

95 2. Ariunbold G.O., Sautenkov V.A., Scully M.O. // Journal of the Optical Society of America – Optical Physics, Vol. 28, No. 3, pp. 462-467, Mar. 2011. 3. Two-photon absorption. http://en.wikipedia.org/wiki/Two-photon_absorption, Oct. 25, 2011 [Nov. 23, 2011]. 4. Yariev A. Third-Order Optical Nonlinearities – Stimulated Raman and Brilloun Scattering in Quantum Electronics, 3rd ed. New York: John Wiley & Sons, 1989, pp. 453- 458. 5. Pastor A. Class lecture, Topic: Nonlinear optics. Faculty of Physics, Saint Petersburg State University, Saint Petersburg, Russia, 2010. 6. Krausz F. Photonics I. The theory of light & its advanced applications. http:// www.attoworld.de/Documents/pdf/lectures/ photonicsI/Vorlesung_photonics1.pdf, 2004 [Dec. 7, 2011].

96 The research of optical spectra of oil fraction in IR-area

Chernova Ekaterina [email protected]

Scientific supervisor: Prof. Dr. Nemetz V.M., Department of Optics, Faculty of Physics, Saint-Petersburg State University

Introduction Nowadays spectroscopy becomes more popular in oil production due to the short time spending for measurements. (1) In this field of science spectra interpret- ing is one of the important factors in an adequacy of analysis. The main goal of researching the information value of oil fractions’ optical spectra is a generation of a prior method for the optimization oil-products’ compounding process and for the creation physicochemical parameters prediction model. The application of these results can be important for the invention methods for the quantitative analysis of oil and oil-fractions. Traditional spectroscopic approaches have limited possibilities due to the basing on the idea of extraction narrow frequency region with one or two characteristic components from the complicated spectrum. Whereas different component fragments overlap in oil-spectra, attempts to find solution by such method were unsuccessful. On the other hand, the method of principal components (2) allows identifying fraction samples, in spite of lack possibility to separate absorption bands. It is one of the methods of multidimensional statistic analysis. Its characteristics and application to oil products will be discussed in this paper.

Results and Discussion Mathematical spectra processing. Applying a mathematical tool requires formalization of spectral information. In common case the obtained data notation is of the 2×N-dimensional matrix form (Fig. 1).

Fig.1. The form of obtained data.

In this matrix equation reference goes here λi are spectrum points, in which measurement were made and Ii are intensities in these points. There are two cases: as intensity can be absolute or normalized quantity of analytical signal. The obtained 2×N-dimensional matrix can be replaced by a N-dimensional vector. This variant is qualified if the principle of a functioning automated measuring system is taken into account. This principle is such that measuring result is a set of analytic signal steps in certain, as a rule, equidistant points. In the case of analytical spectroscopy optical characteristics of systems (intensity, transmission density) 97 are counted in equidistant spectrum points. Moreover, for the different samples the same spectrum points are chosen. Thus, all sample spectra can be presented as a set of intensity values, while wavelength values, which are the same for all samples and do not have specific information, can be neglected. As a result the spectrum is transformed to the vector: (Iλ1, Iλ 2 , Iλ3....IλN ) Here λ1, λ2, λ3….. λN are certain wavelengths, N – spatial dimension. In the case of spectroscopic approach N has a value of several thousand. Vector representation means that the spectrum is presented as a vector in multidimensional space. The magnitude of the vector’s projection on N axis is proportional to the analytical signal intensities at the corresponding wavelengths. Therefore, the spectrum as an object image (sample, fraction) in the N-dimensional space can be represent as a vector, its terminus giving corresponding point in N-dimensional space. If there are M objects (samples, fractions) then they are represented in this space as the points and their quantity is M. The simplest characteristic of distinction in a points' location is Euclidean distance between them. As a result of multiple spectrum measurements and of transforming these spectra in vectors every object- sample can be represented as a certain distribution in the N-dimensional space. The method of principal components There are many methods of reducing dimensions; however, in practice the method of principal components is the most widespread. According to this technique, the set of points-patterns occupies certain area in N-dimensional space. Coordinate’s dependencies in this space leads to the fact that these areas have a certain shape stretched along certain axis, the direction of which can be different from basis vectors (Fig. 2).

Fig. 2. The illustration of principal components selection in two-dimensional space.

In the case when XY are basis axes the correlation between the points’s coordinates is being observed well (X increase – also Y increase, X decrease – also Y decrease). As a result, the area of all samples has evidently stretched along the 98 certain axis shape. If this axis U is considered as a new basis axis and the second axis V is chosen as perpendicular to the first, then the fact of new axes linear independence is obvious. In addition to the scatter between points is maximal along new axis U in a new coordinate system. Therefore, this direction is the most informative according to the given configuration. The meaning of finding of the new axis is to retrieve information maximum from the system. This fully illustrates the conception of MPC – the searching new axes in an initial space. These directions should be characterized by well-defined ranking of information quantity called principal axes. The fist axis has an information maximum, the second one has a maximum after the first and the third has a maximum after the first and the second. After building the new system for further consideration suitable amount of axes can be chosen. Consideration of the correlation of amount of information and laboriousness further calculation is very important. There are many mechanisms for counting information quantity, which correspond to the principal components. Among well-known algorithms there are SIMCA, NIPALS and searching of covariance matrix’s eigenvector. Oil and oil’s spectra A Crude oil (petrolium) consists of compounds boiling at the different temperatures that can be divided into variety of different generic fractions by distillation. This term ”fraction” has been used in the meaning of composition. However, the petroleum can only be arbitrarily defined in terms of boiling point and carbon number (3) (Fig. 3). ”The molecular boundaries of petroleum cover a wide range of boiling points and carbon numbers of hydrocarbon compounds and other compounds containing nitrogen, oxygen, and sulfur, as well as metallic (porphyrin) constituents” (3).

Fig. 3. Boiling point-carbon number profile for petroleum.

Research fractions were obtained by thermal petroleum distillation of different oil’s samples by apparatus “ARN-2”. Each of oil’s samples was in amount of 4 liters. Therefore, there were 32 oil fractions; each of them was the result of 99 10-degree rectification from the boiling temperature to 390 °C. Obtained liquids were keeping in a cooling cabinet. It was decided to use the optical absorptive spectral method. Absorption spectra were obtained under normal conditions: normal pressure and temperature

Fig.4.. Absorption spectrum of the fraction from 62 to 70 °C. by Fourier-spectrometer “Bruker” in IR-area (Measurement range: 550 – 5000 cm-1 (2-18 µm)). According to literature there were no scientific papers related to spectroscopic researching of such products. One of the spectra is presented in Fig. 4. The method’s of principal components application. The goal of this work is to research convergence of real compound’s spectrum and spectrum synthesized from its rectification spectra. Computation of synthesized spectrum was based on the formula: , Ds (λk ) = (∑ Di (λk ) × Ci ) /( ∑Ci ) i i Here Ds(λk) – synthesized optical density at the wavelength λk. Di(λk) – optical density i-fraction at the wavelength λk. Design formula for principal components of synthesized spectrum:

P1s = (∑ P1i × Ci ) /( ∑Ci ) i i P2 s = (∑ P2i × Ci ) /( ∑Ci ) i i ….

PN s = (∑∑PN i × Ci ) /( Ci ) ii 100 Here P1s…PNs – principal components synthesized spectrum. Amount of components is N. P1i…PNi – principal components of spectrum i-fraction (N components). Only fractions which consist in compound’s mixture are summarized. Results of calculation synthesized spectrum and initial spectrum imagery are presented in Fig. 5.

Fig. 5. Plane of first and second principal components.

In Fig. 5 there is experimental point scatter corresponding to fraction with lower boiling temperature 100ºС (100а, 100в, 100с).Indicative points of se180sint and se180real are located at center of area formed by image-points of initial fractions' spectra. It shows that se180 compound is a mixture of corresponding fractions, which means that it is possible to obtain prior information about compound spectrum in case of availability of composing fractions' spectra. Moreover, this result shows that intermolecular interaction is not affecting spectral properties of compound much. Conclusion On the basis of the work done it has been confirmed that the method of principal components can be used for petroleum fraction researches. This means that it is possible to obtain a prior set of physicochemical parameters of petroleum compounds. References 1. Vasilyev, A. V., Grinenko, E. V. Shchukin, A. O., Fedulin T. 2007. Infrared spectroscopy of organic and natural compounds. SPb. SPbGLTA 2. Konushenko, I. O. 2008. New methods identification of liquid mixtures. Diploma thesis. Saint-Petersburg State University, Laboratory of spectrum analysis. 3. James G. Speight. 2002. Handbook of Petroleum Product Analysis. John Wiley & Sons, Inc., publication

101 Luminescence spectra of YVO4 and Y2O3 nanopowders

Kolesnikov Ilya [email protected]

Scientific supervisor: Dr. Kurochkin A.V., Department of General Physic I, Faculty of Physics, Saint-Petersburg State University

Introduction In recent years, rare-earth-doped nanocrystalline phosphors have attracted great interest. Unique luminescence properties, especially characteristic narrow f–f bands typical for the Ln3+ ions make them promising materials for multicolor displays and lightning, biological labeling, lasers and optical amplifiers and optical sensors. The main attention is paid to YVO4 and Y2O3 as the most prospective hosts. The aim of the present work is investigation of the luminescent properties of different nanocrystalline powders doped with europium ions and determination the optimal composition and synthesis parameters. Also in this paper the kinetics 3+ 5 of luminescence was studied and the lifetime of the Eu level D0 of europium ions was calculated. Results and Discussion Excitation spectra Luminescence properties of nanocrystalline phosphors depend on various factors such as composition of the host, the synthesis method and parameters, the size and shape of particle, and the concentration of the rare-earth ions.

4800

4600

4400

4200 intensity, a.u. 4000

3800

3600 200 220 240 260 280 300 320 340 360 380 wavelength, nm

3+ o Fig. 1. Excitation spectrum of YVO4:Eu 16 mol.% 1000 C.

The measured excitation spectrum of the luminescence band at 619 nm for 3+ YVO4:Eu 16 at.% powder is shown in Fig. 1. The observed broad band with a maximum at about 280 nm can be attributed to the charge transfer from the oxygen ligands to the central vanadium atom inside the 3- VO4 ion. On the other hand, this band can be also assigned to the charge transfer 102 (CT) transition between Eu3+ and O2-, an electron transfers from O2- (2p6) orbital to the empty orbital of 4f6 for Eu3+ [1, 2]. In such a way it can be concluded that 3- the broad band around 280 nm is assigned to the overlap of VO4 absorption and charge transfer transition between Eu3+ and O2-. Thus, the pump energy is absorbed by the lattice host and then excitation is transferred to the ions of europium. Emission spectra In order to study the 1800000 0-2 o 5 7 YVO4:Eu 8% 800C effect of lattice host, the 1600000 DJ – FJ’ o Y2O3:Eu 8% 800C powders of YVO4 and Y2O3 1400000 3+

doped with Eu (8 mol. %) u. 1200000 a. were synthesized at 800 1000000 y, oC by the Pechini method. it 800000

ns te 2-6

in 600000 The emission spectra of 0-1 400000 the samples are shown in 1-1 0-0 0-4 200000 0-3 Fig. 2. The luminescence 0 3+ intensity of Eu in yttrium -200000 vanadate was found to be 500 550 600 650700 wavelength, nm about 6 times higher than 3+ in yttrium oxide. Therefore, Fig.2. Emission spectra of YVO4:Eu 8 mol.% and 3+ further studies were carried Y2O3:Eu 8 mol.% under 325 nm excitation. out with yttrium vanadate host. The emission spectrum , a.u. 3+ of YVO4:Eu powder is 5 characterized by the D0 7 – F2,4 electric dipole transitions of Eu3+ resulted in luminescence bands at 619 and 699 nm. Other contri- normalized intensity butions of weaker intensity 5 7 are the D0 – F1,3 magnetic 0 dipole transitions at 594 and 590 600 610 620 630 640 652 nm. The higher intensity wavelength, nm 3+ of the electric dipole transi- Fig. 3. Detailed emission spectrum of YVO4: Eu tions can be explained by 16 mol.% 700 oC. low symmetry of Eu3+ local site in the YVO4 host lattices (D2d, without inversion center). The weak intensity bands at 538 and 609 nm are corresponded to transitions 5 5 7 7 from D1 and D2 levels to F1 and F6, respectively. 5 7 5 7 The intensity ratio of transitions D0 – F2 and D0 – F1 can be used for analysis of the local surrounding of the luminescent center and its symmetry. This ratio is called the asymmetry coefficient k. Fig.3 shows the detailed spectrum of YVO4: Eu3+ 16 mol.% 700 oC in spectral region from 590 to 640 nm.

103 The asymmetry coefficients were calculated for the series of samples with different sintering temperatures. The values of these coefficients are listed in Table 1 . 5 7 5 7 3+ Table 1. The intensity ratio of D0 – F2 to D0 – F1 for YVO4:Eu 16 mol.% phosphors. Intensity Asymmetry Temperature, °C 5 7 5 7 D0 – F1 D0 – F2 coefficient,k 700 8,26·106 1,28·108 15,50 850 4,39·107 5,93·108 13,51 900 1,29·107 1,96·108 15,19 950 2,4·107 3,48·108 14,50 1000 2,58·107 3,57·108 13,84

The observed dependence of the asymmetry coefficient demonstrates a tendency of the Eu3+ ions local symmetry increase with synthesis tem- , a.u. perature growth. Fig. 4 shows dependence

intensity of the luminescence intensity at 619 nm on the concentration 3+ 0 of Eu ions (2, 4, 8, 12, 16 3+ mol.%) for YVO4:Eu pow- -2 0 2 4 6 8 10 12 14 16 18 ders synthesized at 900 °C. concentration, % The luminescence intensity 3+ o Fig. 4. Emission spectra of YVO4: Eu 900 C on 619 nm grows with increasing number under 325 nm excitation. of Eu3+ ions up to 8 mol.%. Further increase of Eu3+ concentration leads to a reduction in the intensity. This decrease is due to the concentration quenching. Consequently the optimum a.u. y, concentration of Eu3+ ions can be determined as 8 mol.% in intensit yttrium vanadate host. Fig. 5 shows the effect of the synthesis temperature on 0 luminescence inten-sity of 700 750 800 850 900 950 1000 the band at 619 nm (transition 5 7 temperature, C D0 – F2) for nanopowders 3+ 3+ Fig. 5. Emission spectra of YVO4:Eu 16 mol.% YVO4:Eu 16 mol.%. It was on 619 nm under 325 nm excitation. found that the sample syn- 104 thesized at 1000 °С demonstrates the highest luminescence intensity and can be considered as the most efficient phosphor.

Kinetics of luminescence 5 In order to determine life-time of the D0 level the luminescence kinetics has been investigated. The experimental data were approximated by “biexponential fitting”:

. 5 The observed non-exponential decay of D0 luminescence is most probably connected with two different sites of the Eu3+ ions (at the surface - surface defects of the nanoparticles and Eu3+ ions inside the particle/grain). So, the effective 3+ lifetime of the studied YVO4:Eu nanopowders can be determined from the following equation:

3+ The efficient life-times of nanopowders YVO4:Eu synthesized at different temperatures are listed in Table 2.

5 7 3+ Table 2. The life-times of transition D0 – F2 for YVO4:Eu 16 mol.% phosphors.

o Two-exponential fitting Temperature, С τav , ms τ1, ms τ2, ms 700 0,18 0,54 0,59 850 0,16 0,54 0,54 900 0,19 0,62 0,54 950 0,18 0,50 0,46 1000 0,22 0,51 0,43

Conclusions 3+ 3+ a) The emission spectra of YVO4:Eu and Y2O3:Eu were investigated. b) Dependence of luminescence on the synthesis temperature and the concentration of rare-earth ions were studied. 3+ o c) The asymmetry coefficients were calculated. YVO4:Eu 1000 C has the lowest coefficient. 5 3+ d) The kinetics of the D0 level of YVO4:Eu was investigated. The fluorescence life-time presents a maximal value for the samples annealed at 700 °C. References 1. Zhou Y.H. et al. // Optical Materials 27, pp. 1426–1432 (2005). 2. Devaraju M.K. et al. // J. Crystal Growth, Vol. 311, pp. 580–584 (2009). 3. Yuhua Wang et al. // Materials Research Bulletin 41, pp. 2147–2153 (2006). 4. Georgescu S. et al. // Romanian Reports in Physics, V. 60, No. 4, pp. 947–955 (2008). 5. Hreniak D. et al. // Journal of Luminescence 131, pp. 473–476 (2011). 105 Resonance grating based on InGaAs/GaAs quantum well

Kozhaev Mikhail, Kapitonov Yury [email protected]

Scientific supervisor: PhD Petrov V.V., Department of Photonics, Faculty of Physics, Saint-Petersburg State University

Introduction Significant problem of modern physics is creation of an optical computer that will make calculations in purely optical way without dissipation of energy in the element [1]. One of the most promising candidates for the working medium of logic elements of optical computer are InGaAs/GaAs quantum wells (QW). Resonant reflection spectroscopy associated with the birth of 2D-excitons is used for studying optical properties of QW [2]. Light reflection from the «excitonic mirror» is coherent in contrast to the photoluminescence. However, the resonant reflection from the QW, located inside the specimen, is difficult to detect against the background of non-resonant reflection from the sample surface. The only way to obtain interpretable QW reflection spectrum is to have zero reflection from the sample surface by using the Brewster geometry for the incident light: the angle of incidence equals to the Brewster angle, and polarization is in the plane of incidence (p-polarization). To overcome this fundamental limitation we created resonant grating by spatial modulation of the QW properties. In this case diffraction peaks were observed. Travelling direction of this signal differs from the reflection direction. In this case influence of non-resonant reflection from the sample surface is eliminated for all incidence angles and light polarizations. Spatial modulation of QW was made using an ion beam irradiation of the GaAs substrate prior to the MBE growth of the heterostructure.

Results and Discussion Lithography The first step of sample creation was made by focused ion beam (FIB). Array of lines with length 400 μm was created on epi-ready GaAs substrate by focused beam of Ga+ ions with an energy of 30 keV. The steps between the lines was 9 μm, and the linear dose was 0.1 and 2 nA*s/cm. Thus, in addition to milling, which in this case will not be significant on the substrate surface will occur implantation, defect formation and deposition of hydrocarbons (Fig. 1).

Molecular beam epitaxy

Then using molecular beam epitaxy (MBE) method three In0.02Ga0.98As/GaAs QW with different thicknesses (and hence with different exciton resonance spectral position) were grown on the substrate (Fig. 2). Fig. 3 shows different sample places: with and without lithography.

106 Ga+ hydrocarbons Ga+ e-

10 μm Implantation Deposition of and defect hydrocarbons

Fig. 1. Left picture - image of lithography made by scanning electron microscopy (SEM). In addition to etching (whose influence in this case because of the low dose only slightly) in lithography place occurs implantation and defect and hydrocarbons deposition.

150 nm

4.0 nm Cap GaAs 150 nm QW 3 In(2%)GaAs 3.0 nm Intermediate layer GaAs 150 nm QW 2 In(2%)GaAs 2.0 nm Intermediate layer GaAs QW 1 270 nm In(2%)GaAs

Bu er GaAs

0.1 nA*s/cm 2 nA*s/cm Substrate with lithography GaAs

Fig. 2. Left figure shows sample layers and the depth of QW. Right figure shows influence of different lithography doses on grown layers and QW.

10 μm

Fig. 3. Surface maps were obtained using atomic force microscopy from two different sample places: without lithography (left), and with 2 na*s/cm doses (right). 107 Optical properties of the sample The sample was studied at a temperature 9 K using a Ti:sapphire femtosecond laser with a wide (about 20 meV) emission spectrum using Brewster geometry (Fig. 4). Brewster geometry allows getting interpretable reflection spectra because

Cryostat with Sample the sample s-pol Ti:Sa fs-laser (9-300 К)

mirror 1 diaphragm

mirror 3

p-pol

Monochromator CCD mirror 2 half mirror lens Fig. 4. Left figure shows the incidence light polarization effect to the reflection from the sample surface in Brewster geometry. On the right shows the optical setup scheme. there is a polarization in which the nonresonant reflection from the sample surface is absent. Thus the reflection spectroscopy is limited by the angle and polarization of the incident light. But if QW properties are periodically laterally modulated then diffraction signal propagation direction differs from reflection signal propagation direction. Thereby incident light angle and polarization deviation don’t lead to loss of the information about QW. Fig. 5 shows reflection and diffraction spectra in Brewster geometry for

Fig. 5. Reflection and diffraction spectra comparison for p-and s-polarized incident light. Sample temperature is 9 K. Brewster geometry is used. s- and p- polarization. In contrast to s-pol diffraction spectra, the s-pol reflection spectrum can’t be interpreted because of the useful signal interference with the nonresonant reflection from sample surface. 108 Studied sample had three QW which are located at different heights from the substrate. Reflection peaks intensities from these QW are approximately the same order (full line in Fig. 6). However, for diffraction spectra picture is quite different. For the sample place in which QW were modulated by 2 nA*s/cm before growth, diffraction peaks intensities differ by an order (dotted line in Fig. 6) although distances from QW 1 to substrate and from QW 2 to substrate varies slightly more

Fig. 6. Reflection and diffraction spectra for places with preliminary lithography linear dose 0.1 and 2 nA*s/cm. Sample temperature is 9 K. Brewster geometry is used. Oscillations likely are related to the interference of the light which scattered from the sample surfaces. Spectra are normalized to the first QW peak height. than twice. For the sample place in which QW were modulated by 0.1 nA*s/cm before growth, diffraction peaks from QW 2 and QW 3 are mostly lost in noise (dashed line in Fig. 6). This observation allows to “turn on” only some of the QW peaks in diffraction spectrum. Conclusion The diffraction grating has little effect on the quantum wells quality. However the diffraction signal propagation direction differs from the nonresonant light propagation direction reflected from the surface. Creation of multiple QW structure on the substrate with lithography leads to difference in the reflection intensities from QW, depending on their geometrical position. Thus, the resonant diffraction grating is a new tool for a more detailed study of the physical processes occurring in the quantum wells, including non-linear optical properties. In addition, such facilities may have practical applications, such as an optical gate - one of the key photonic logic elements, and other devices. References 1. Gerlovin et I.Ya. al. // Nanotechnology, 11(4), 383-386 (2000). 2. Poltavtsev S.V. et al. // Physica Status Solidi (C) Current Topics in Solid State Physics, 6(2), 483-487 ,(2009). 109 Application of 2D-correlation spectroscopy method for interpretation of spectra and enhancing the spectral resolution

Maximova Ekaterina, Lev Derzhavets [email protected], [email protected]

Scientific supervisors: Dr. Levin S.B., Department of Mathematical Physics, Faculty of Physics, Saint-Petersburg State University; Dr. Bulanin K.M., Department of Molecular Spectroscopy, Faculty of Physics, Saint-Petersburg State University

Introduction The idea of 2D-correlation spectroscopy appeared not too long ago , so the number of publications in this field is not significant yet. 2D-correlation spectra consist of two orthogonal components - synchronal and asynchronal spectra. Both of them carry significant and independent information about the behavior of the studied system. The aim of current work is to make acquaintance with the 2D-correlation method for the interpretation of spectra and enhancing the spectral resolution. Also, we considered the continuous approximation of spectral amplitude two-dimensional surface defined in the frequency-time surface: I,()ννt=Ω()Pt()

n 2 ν −a ν − nu where j ( j ) Ων()=A∑ j e j=1

nt Bj and Pt()= ∑ j 2 2 j=1 (tt− 0 ) + η j

This kind of approximation is convenient for analytical construction of 2D- correlation spectra. For approximation of the peaks, we used the Nelder-Mead method (simplex algorithm). Furthermore, such a constructions will give opportunity to interpret more effectively complex processes being under study [1]. For approximation of the peaks, we used the Nelder-Mead method (simplex algorithm). Correlations obtained by using the discrete set of experimental data, proposed by Noda [2, 3] are the additional criterion for the quality of approximation.

110 2D-correlation spectroscopy method

Nelder-Mead method For the peaks approximation, we used the Nelder-Mead method (simplex algorithm). The Nelder-Mead algorithm is designed to solve the classical unconstrained optimization problem of minimizing a given nonlinear function f : Rn →R. The method • uses only function values at some points in Rn, and • does not try to form an approximate gradient at any of these points. Hence it belongs to the general class of direct search methods . The Nelder-Mead method is simplex-based. A simplex S in Rn is defined as the n 2 convex hull of n+1 vertices x0,…, xn  R . For example, a simplex in R is a tri- angle, and a simplex in R3 is a tetrahedron.

A simplex-based direct search method begins with a set of n+1 points x0,…, n xn  R that are considered as the vertices of a working simplex S , and the corresponding set of function values at the vertices fj =f(xj) , for j=0,…, n . The initial working simplex S has to be nondegenerate, i.e., the points x0,…, xn must not lie in the same hyperplane. The method then performs a sequence of transformations of the working simplex S, aimed at decreasing the function values at its vertices. At each step, the transformation is determined by computing one or more test points, together 111 with their function values, and by comparison of these function values with those at the vertices. This process is terminated when the working simplex S becomes sufficiently small in some sense, or when the function values fj are close enough in some sense (provided f is continuous). The Nelder-Mead algorithm typically requires only one or two function evaluations at each step, while many other direct search methods use n or even more function evaluations.

Results and Discussion As approximating functions was taken the following basis:

2 2 −−νν1 −−νν2 ()0 b1 ()0 b2 y,()ν t=e +e 1 2 2 2 2 2 (tt− 0 ) + ηη1 (tt− 0 ) + 2 The offered approximation allows us to interpret the complicated processes more effectively from the point of view of chemical reactions occurring in the system Quality ν 1 and ν 2 were taken value of positions of the maxima of the peaks 0 0 in real 2107.332 cm-1 and 1130.564 cm-1, respectively. Approximating with respect to the frequency was carried out [4]. 1,2 Search for factors b1,2,t0 , ƞ1,2 produced by least squares method, i.e. was taken following sum of squares [4]:

m 2 − S=mj∑()yt( ) y j j=1

Where y(tj) – values of analytical expression in the experimental points and yj –experimental values at corresponding points. With such an analytic function is not difficult to find correlations :

a b

Fig. 1. a) synchronous spectrum, b) asynchronous spectrum.

112 References 1. Isao Noda, Yukihiro Ozaki. Two-dimensional Correlation Spectroscopy – Applications in Vibrational and Optical Spectroscopy.- England, 2004. 2. Tonkov M.V., Filippov N.N. // Chem. Fizika, 10(7), 922 (1991). 3. Noda I. // Appl. Spectrosc., 47, 1329 (1993). 4. Lev Derzhavets. Application of two-dimensional correlation spectroscopy to im- prove the resolution of spectral bands. BSc. Thesis, St.-Petersburg University, 2010.

113 Observation of the fine structure for rovibronic spectral lines in visible part of emission spectra of D2

Umrikhin I.S., Zhukov A.S. [email protected], [email protected]

Scientific supervisor: Prof., Dr. Lavrov B.P., Department of Optics, Faculty of Physics, St. Petersburg State University

For the first time the fine (triplet) structure of rovibronic lines has been ob- 3 3 - served in the visible part of the D2 spectrum (mainly within the Λg → c Πu band systems). General trends are illustrated by the example of the R-branch of the 3 - 3 - i Πg ,v’=1 →c Πu ,v”=1 band.

The studies of the D2 spectrum have been started just after the discovery of the molecule. However, our knowledge of optical spectrum of molecular deuterium is still insufficient in spite of tremendous efforts by spectroscopists over the previous century [1]. Up to now most of spectral lines have not yet been assigned [2], and wavenumbers of rovibronic transitions in visible part of the D2 spectrum were obtained without resolving the fine structure of triplet-triplet spectral lines, in spite of its observation by Fourier [3] and laser [4] spectroscopy in infrared. The goal of present work was to study an opportunity of resolving the fine structure in visible by means of spectroscopic technique developed in [1, 5].

The experimental setup was described in [5]. Pure D2 plasma of constricted glow discharge with cold cathode and water cooled walls under j = 0.4 A/cm2, T = 640±50 K was used as a light source. The flux of radiation through a hole in an anode was focused on the entrance slit of the 2.65 m Ebert-Fastie spectrograph equipped with additional camera lens and computer-controlled CMOS matrix detector. In our conditions the resolving power was mainly limited by Doppler -1 broadening (linewidth ΔνD ≈ 0.15 cm ) and overlap of adjacent lines.

Wavenumbers of rovibronic transitions of the D2 molecule were previously obtained by photographic recording of spectra [2]. Our way of determining the wavenumbers is based on linear response of CMOS matrix detector on the spectral irradiance and digital intensity recording. Both things provide an extremely important advantage of our technique over traditional photographic recording with microphotometric comparator reading. It not only makes it easier to measure the relative spectral line intensities but also makes it possible to investigate the shape of the individual line profiles and, in the case of overlap of the contours of adjacent lines (so-called blending), to carry out numerically the deconvolution operation (inverse to the convolution operation) and thus to measure the intensity and wavelength of even blended lines. As is well known, it is this blending that makes it very hard to analyze dense multiline spectra of the D2 molecule [2].

114 For small regions of the spectrum (≈0.5 nm wide) the observed spectral intensity distribution is approximated by superposition of a certain number of Voigt profiles:     ∞ 2  exp(−t )  Fxjj()= A 2 dt , (1) ∫  2  −∞    ∆υD xx− j  + − t  2     2∆υL  2∆υL   where ΔνD and ΔνL are Doppler and Lorentzian linewidths equal for all the profiles within a spectral region under the study. Aj is the intensity and xj – center of the profile. One region corresponds to one third of the matrix 550 − 600 pixels wide. Thus for each region we obtain 550-600 experimental values of the observed obs intensity Ji from each photodetector of the CMOS matrix. To find the optimal values for adjustable parameters (line centers, relative intensities and one common values ΔνD and ΔνL for all profiles) we need to solve the system of 550-600 nonlinear equations: J obs − J synt (x ) = 0 , (2) i i K synt where J (xi ) = ∑ Fj (xi ) j=1 is “synthesized” intensity value with Fj in the form (1) and K – number of approximated Voigt profiles, xi – coordinate of the photodetector on the CMOS matrix. We approximate the observed intensity distribution by the minimal number of profiles K, which give us random spread of the deviations (2). For different spectral regions we use K=10÷50 profiles for one region, thus we obtain from 20 to 100 parameters for 550-600 equations in the system (2). Therefore system of equations (2) is overdetermined and, hence, is inconsistent because the experimental data always involve measurement errors. The straightforward general solution for solving such problems is the least- squares method. In our case, it consists in minimizing the sum of mean-square deviations (2): M 2 obs synt 2 (3) r = ∑(Ji − J (xi )) i=1 obs where M indicates a number of experimental intensity values Ji in the spectral region under the study. To find optimal values of the profile parameters minimizing expression (3) we used special computer program based on Levenberg-Marquardt's algorithm. If the experimental errors are random and distributed according to a normal (Gaussian) law, the solution obtained by least-squares method corresponds to the maximum likelihood principle. The obtained values for K, Aj, xj, j=1…K,

ΔνD, ΔνL are optimal for the observed intensity distribution. Thus it is possible to obtain the optimal values of the intensity, center coordinate and linewidth for each resolved spectral line in the spectral region under the study. In the case of long-focus spectrometers the dependence of the wavelength on the coordinate along direction of dispersion is close to linear in the vicinity of the center 115 of the focal plane. It can be represented as a power series expansion over of the small parameter x/F (The x-coordinate actually represents small displacement from the center of the matrix detector, F is the focal length of the spectrometer mirror), which in our case does not exceed 2x10-3 [5]. On the other hand, the wavelength dependence of the refractive index of air n(λ) is also close to linear inside a small enough part of the spectrum. Thus, when recording narrow spectral intervals, the product λvac(x) = λ(x) n(λ(x)) has the form of a power series of low degree. This circumstance makes it possible to calibrate the spectrometer directly in vacuum wavelengths λvac = 1/ν, thereby avoiding the technically troublesome problem of accurate measuring the refractive index of air under the various conditions under which measurements are made. We used for calibration experimental vacuum wavelength values from [2] as standard reference data. Those data show small random spread around smooth curve representing dependence of the wavelengths on positions of corresponding lines in the focal plane of the spectrometer. Moreover those random errors are in good accordance with normal Gaussian distribution function. Thus it is possible to obtain precision for new wavenumber values better than that of the reference data due to smoothing. The calibration curve of the spectrometer was obtained by the polynomial least-squares fitting of the data. Our measurements showed that, using a linear hypothesis is inadequate and a third-degree polynomial is exces- sive, while an approximation by a second-degree polynomial provides calibration accuracy better than 2x10-3 nm. 3 - The results are illustrated in Fig. 1 by the example of the R-branch of the i Πg 3 - −1 ,v’=1 → c Πu ,v”=1 band. Observed wavenumber values (in cm ) and Iv/Ir for all spectral lines of the branch are presented in Table 1. One can see that separation of

60 J, counts R4

40 R3 R6 30 R5 20 R7

0 1702017022 1702417026 -1

Fig. 1. Part of the observed D2 spectrum containing 5 spectral lines of the R-branch 3 - 3 - of the i Πg ,v’=1 →c Πu ,v”=1 band. Fine structure components (visible doublet) are marked by solid bold vertical lines. Experimental intensity J in relative units is shown by open circles. Solid line represents the intensity distribution calculated as a sum of optimal Voigt profiles. Dotted lines represent wavenumbers of observed spectral lines, solid vertical lines – wavenumbers of lines reported in [2]. 116 observed doublets is close to 0.2 cm−1. This is in coincidence with results obtained by means of laser [4] and FTIR [3] spectroscopy in infrared part of the spectrum. Joint analysis of splitting in such visible "doublets" (about 0.2 cm-1) and relative intensities of two main components of visible "doublets" (about 2.0) show that they represent partly resolved fine structure of lines determined by triplet splitting of 3 3 - lower rovibronic levels of various Λg→ c Πu electronic transitions. The observed ratio of intensities of the doublet components is close to 2.0 predicted by Burger-Dorgello-Ornstein sum rule when one assumes that the multiplet splitting in upper rovibronic states may be neglected while in the lower rovibronic 3 states c Πu,v=1,N” two fine structure sublevels (J”=N”-1 and J”=N”+1) are close to each other and located noticeably lower than that with J”=N”. These assumptions are in agreement with IR tunable laser observations (EJ"=2 ≈ EJ"=0

Data report- Present work ed in [2]

N′′ νv νr Δνvr Iv/Ir ν 1 16996,58(3) 16996,36(4) 0,22(5) 1,93(5) 16996,58 2 17010,39(3) 17010,21(3) 0,18(4) 1,88(2) 17010,36 3 17019,74(3) 17019,57(4) 0,17(5) 1,99(4) 17019,71 4 17025,13(3) 17024,96(3) 0,17(4) 1,94(3) 17025,11 5 17026,99(3) 17026,82(4) 0,17(5) 2,05(7) 17026,97 6 17025,70(3) 17025,53(4) 0,17(5) 1,84(4) 17025,71 7 17021,57(4) 17021,40(4) 0,18(6) 1,82(11) 17021,60 8 17014,91(4) 17014,70(4) 0,21(6) 1,90(60) 17014,87 9 17005,86(4) 17005,70(5) 0,16(7) 1,80(20) 17005,79 10 16994,57(4) 16994,40(4) 0,16(6) 1,89(14) 16994,53

Present work was financially supported in part by the Russian Foundation for Basic Research, Grant No. 10-03-00571-a.

117 References 1. Lavrov B.P., Umrikhin I.S. // J. Phys. B. 2008. v. 41. 105103. (25pp); 2. Freund R.S., Schiavone J.A., Crosswhite H.M. // J. Phys. Chem. Ref. Data. V.14. No 1. P. 235, 1985. 3. Dabrowski I and Herzberg G. // Acta Phys. Hung. 1984. v.55, n.1-4, pp. 219- 228. 4. Davies P.B., Guest M.A. and Johnson S.A. // J. Chem. Phys. 1988. v. 88, n.5, pp. 2884-2890. 5. Lavrov B.P., Mikhailov A.S., Umrikhin I.S. // J. Opt. Technol., v. 78, I. 3, p. 180, 2011. 6. Lavrov B.P., Umrikhin I.S. // e-Print arXiv:1112.2277v1 [physics.chem-ph] 10 Dec 2011. −1 Table 1. Wavenumber values (in cm ) and relative intensities Iv/Ir for violet and 3 - 3 - red components of the R-branch lines for (1 − 1) band of the i Πg → c Πu electronic transition. The νv and νr are wavenumbers of violet and red components of the doublets; Δνvr = νv - νr .

118

G. Theoretical, Mathematical and Computational Physics Conservation laws and energy-momentum tensor in Lorentz-Fock space

Angsachon Tosaporn [email protected]

Scientific supervisor: Prof. Dr. Manida S.N., Department of High Energy and Elementary Particles Physics, Faculty of Physics, Saint-Petersburg State University

Introduction

In this paper we consider the conservation laws for classical particles in AdS4. At first we parameterize a geodesic line and construct the conserved quantities with analog of five dimensional Minkowski space M(2,3). Consequently we change

AdS4 space to AdS-Beltrami space and write out conserved quantities in Beltrami coordinates. Furthermore we take a limit for small velocity (xi c) and we get the conserved quantities in Lorentz-Fock space. And finally the energy-momentum tensor for dust material is constructed. ≪

Consererved quantities in embedding anti de-Sitter space(AdS4) We define anti de-Sitter space as four dimensional hyperboloid A (2,3) 2 A B 2 AdS4 = {X M , X = ABX X = R } (1) embedded in five dimensional Minkowski space. Induced metric for this space can be expressed ϵ η 2 A B ds = ( ABdX dX ) AdS (2) where AB = diag{1,1,-1,-1,-1} and indices A,B are running values -1,0,1,2,3. Now we consider an action forη the classical∥ massive particle, which can be expressedη in five dimensional coordinates S = -mc [(V2)1/2+a(X2-R2)]d (3) where X( ) is a parameterized timelike curve with a constraint X2 = R2. VA( ) = dXA/d are the velocity the∫ particle. We knowλ, that an action (3) is invariant under transformation:λ λ λ B XA XA+ ABX , (4) where AB is and antisymmetric infinitesimal parameter. From Noether theorem we can find the conserved quantities→ alongω a timelike geodesic line m ω K=( XVXV−)(5) AB 2 ABBA RV We can write this conserved quantities in the form of two timelike vectors and m(ηξηξ− ) K = ABBA (6) AB 222 ξ η ηξηξ−⋅() where = (0,c, i) , = (R,0, i). Finally we can find a mass-shell equation ξ ξ η η AB 2 (7) KKmAB = 2 120 Conserved quantities in Beltrami coordinates In this part we will write our conserved quantities in Beltrami coordinates. Beltrami coordinates are determined by the relation

x = RX /X-1 i i) (8)

Let H, Pi, Ki, Ji be conservedμ μ quantities in Beltrami coordinates and these quantities are expressed in the formulas = (λc,λξ +η m HK==0(− 1)  (9) xxx2()− λ  2()xx×  2 1−−iii + c2 R 2 Rc 22

mx()ii− λ x Kii== RK0  (10) xxx2()− λ  2()xx×  2 1−−iii + c2 R 2 Rc 22

εijkxx j k Ji = Rcm  (11) xxx2()− λ  2()xx×  2 1−−+iii c2 R 2 Rc 22

mxi Pii== cK(− 1) Rcm  (12) xxx2()− λ  2()xx×  2 1−−iii + c2 R 2 Rc22

Conservation laws for nonrelativistic cosmological particles We consider the conservation laws for particles which are moving with the small velocity (ẋi c) in the space with constant curvature R. The conserved quantities and mass-shell equation will be written as ≪ m H = (λxx − )2 (13) 1− ii R2

mxx(i− λ i) Ki = (14) (λxx − )2 R 1− ii R2 mx i (15) π i≡limc→∞ cPi= ()λxx − 2 1− ii R2 222 (16) H−Ki =m The space with this mass-shell equation is called Lorentz-Fock space. If we put

,R ׀xi׀ ,T+t, R/T c0 and consider the small vicinity point in this space t T = then we get λ m ≡ H = ≪ (17)≪ 2 xi 1− 2 c0

121

mx π = i (18) i 2 xi 1− 2 c0

π m( x− tx ) K +=iii (19) i c 2 0 xi 1− 2 c0 Therefore we see that this “cosmological” dynamics under the nonrelativistic limit does not differ from the standard relativistic dynamics with speed of light c0.

The energy-momentum tensor in Lorentz-Fock space In this part we consider the construction of energy-momentum tensor for dust particles in Lorentz-Fock space. We are starting from the 4-velocity in this space which is expressed in the formula 1 vi u µ = ( 1 , ) (20) (vtx− )2 c 1− ii R2 From this velocity we can down the index of 4-velocity with the help of a metric of Lorentz-Fock space in this form R222− x2 Rx R2 2 2i2iij ds = g dx dx =+−R24 dt23 dtdxiδ ij 22 dx dx (21) μ ν ct ct ct μν Therefore the 4-velocity in the lower index is written out in the formula 2 22 , 2 R R−+ xiii xvt R xi− vti u0 = 44 ui = 33 (22) ct (vtx− )2 ct (vtx− )2 1− ii 1− ii R2 R2 The energy-momentum tensor for dust particles is defined in this equation

Tuuµν= ρ µ ν (23) Finally, this tensor can be expressed in each component R4 ρ (())R2+− x vt x 2 T==ρ uu iii (24) 00 0 0 ct88 ()vt− x 2 1− ii R2

4 2 R ρ (())()R+−− x vt x x vt T==ρ uu kkkii (25) 00iict77 ()vt− x 2 1− ii R2

4 R ρ ()()x−− vt x vt T==ρ uu iijj (26) ij i j ct66 ()vt− x 2 1− ii R2

122 References 1. Cacciatori A., Gorini V., Kamenshchik A.Yu. Special Relativity in the 21-st century. // Ann. Phys. 17, p. 728-768 (2008), arxiv:hep-th/0807.3009. 2. Manida S.N. Fock-Lorentz transformations and time-varying speed of light, arxiv:gr-qc/9905046.

123 Renormalization-group and ε- expansion: representation of anomalous dimensions as nonsingular integrals

Batalov Lev [email protected]

Scientific supervisor: Prof. Dr. Adzhemyan L.Ts., Department of Statistical Physic, Faculty of Physics, Saint-Petersburg State University

Introduction The analytic calculation of critical exponents in critical behavior models with renormalization-group method (RG) and ε- expansion is very difficult in high order of perturbation theory [1], because it needs to compute singular in ε integrals (ε= 4 - d, where d is dimension of space). In this paper we find such implementation of RG-method with ε-expansion, in which the problem is reduced to computing of finite integrals and we can to automate it. It’s importantly to choose the most usable scheme of renormalization for solution of this problem. The operation of renormalization for diagram Г is written: ΓR = ΓR (1) It names R-operation of Bogolubov-Parsuk often and removes divergences in subgraphs and after remained surface divergence. First it needs to find the divergent part of diagram. The scheme of null momentum (NM) it most usable for calculations. It is reduced to subtraction from function F(k) n initial terms of a Taylor series: 1 n k m 1 F( k )− | =da (1 −∂ a )nn+1 F ( ak ) (2) ∑ k=0 ∫ a m=0 mn!!0 For R-operation for graph χ we get: 1 1 nnii+1 Rχ= daiiai(1 −∂ a )χ ({ a } ) (3) ∏ ∫ i i ni ! 0 In equation (3) The product consists of all divergent subgraphs (including whole diagram χ) with canonical dimension ni ≥ 0, a is the parameter of extension inside i –subgraph of momentums flowing into this subgraph. The NM-scheme is not easy, but it become such after some modification. However, in new scheme we are able to write interesting for us values in usable for computation form (3).

Renormalization-group functions We shall consider the solution of this problem for example φ4 - theory. The space dimension is d = 4-ε. The renormalization action for this theory is [2]:

1 22 22 S=−( mZpZ + +δϕ m) + ... 2 12 (4) 124 Surface UV-divergences for ε = 0 there are in diagrams of 1-irreducible functions Γ2 = <φφ> and Г4 = <φφφφ> having quadratic and logarithmic divergence respectively. 2 Counterterms δZi = Zi-1 ; i = 1,2,3 and δm remove these divergences in each order of perturbation theory at the coupling constant g. Calculation of counterterms in (4) is changed R-operation for diagrams of basic theory. The scheme of renormalization is determined the K-operation, which selects divergent subgraphs of 1-irreducible diagrams. We write this operation in NM-scheme, and we will change the normalization point after. K-operation for NM-scheme is:

KΓ≡Γ4040 K,()| Kfpf{ } ≡p=0 (5) 2 KΓ=2( K 0 + pK 222 ) Γ , Kf ({ p} ) ≡∂ (2 f )|p=0 (6) p We separated the initial partial sum of series in momentum. The quantity of terms in it is determined the dimension of diagram. Renormalization constants are convenient in notation of normalized functions: K Γ Γ=−ΓΓ=−K , 03 (7) 2 224gµ ε

In notation Γ renormalization constants are:

Zi=−Γ=1 KR im |=µ , i 1 , 2 , 3 (8) We select the renormalization scheme, which corresponds previous conditions (8), but it is defined in normalization point m=μ. This scheme is named NP-scheme. Schemes NM and NP differ only by the finite renormalization of parameters g, m2 and constants Zi. It is conditioned through the relation between residues at higher poles of renormalization constants and in first pole. This relation provides UV-finity of renormalization functions. Values of critical exponents in NM and NP-scheme are equal.

We consider renormalization constants Z1,Z2. these constants defineβ - function and anomalous dimensions of field. Proposed method of computation approach for both renormalization schemes NM and NP. However in NP-scheme the result is formulated the simplest way. Renormalization constants in NM-scheme and NP-scheme depend only from dimensionless charge g and space dimension. This way, the form of renormalization- group equations (if the renormalization mass μ is arbitrary) and relation between RG-functions and renormalization constants in these schemes are the same. The system of RG-functions includes the β - function: ε  β=−g  +γg (9) 2  and γ- functions. These functions relate with renormalization constants Zi by the equation: γi=∂βgln Zi (10)

125 In particular case,

γg=∂βgln Z g (11) From (9)-(11) we get: gε 1 β =− (12) 2 1+∂gg ln Zg

∂ gε ggln Zi (13) γ i =− 2 1+∂gg ln Zg The RG-functions defined singular inε renormalization constants haven’t poles in ε [3]. The removing of these poles for renormalized models is guaranteed by the renormalization theory. The removing of poles in the calculation shows us, that it is true. It is inter- estingly, that in our case the removing of poles has analytic character without calculation of diagrams. It takes the problem of approximation for the high poles in numerical calculations away.

The result is: 2 f2 γ 2 = (14) 1+ f2

2 f3 γ 3 (15) 1+ f3

2 where f=− Rm( ∂Γ2 ) | , i = 2, 3 i m im=µ Conclusion Equations (14-15) are the main result of this work. We realized the automatic program for computation of critical exponents with them. We used the GRC program from the GRACE packet to construct diagrams. We added to it a program for finding relevant subgraphs. The corresponding integrand was generated for each diagram needed to calculate values (15) using representation for the R-operation (3). We calculated integrals using the Monte-Carlo method. We applyed the spherical d-dimensional coordinate system and the Feynman representation to compute these integrals.

The anomalous dimensions γ2 and γ3 were calculated in the four-loop approximation (it was needed to compute 204 diagrams). Based on them, we found β- function (12) and defined the coordinate of the fixed point u* up to ε4 inclusively, and the * critical exponent η=γ2|u=u was computed after.

References 1. Kleinert H., Schulte-Frohlinde V. Critical Properties of φ4-Theories. - Word Sci. Publ., River Edge, NJ, 2001. 2. Васильев А.Н. Квантовополевая ренормгруппа в теории критического поведения и стохастической динамике. – СПб.: изд-во ПИЯФ, 1998. 3. de Alcantara Bonfim O.F., Kirkham J.E., McKane A.J. // J. Phys A, 13:7 L247–L251 (1980).

126 Surface states in semi-infinite superlattice with rough boundary

Bylev Alexander [email protected]

Scientific supervisor: Prof. Dr. Kuchma A.E., Department of Statistical Physics, Faculty of Physics, Saint-Petersburg State University

Abstract Surface states in semi-infinite superlattices with rough boundary are studied using simple one-band approximation for nondegenerate bands of a crystal lattice. Superlattice is modeled by δ-like potential wells, surface potential well has different depth than other potential wells and roughness is modeled by dependence of surface potential power on coordinates in surface plane. It is shown that wave function of surface state will decrease in spreading direction as a result of scattering on roughness. Expression for coefficient of damping of surface state in longitudinal direction is derived; contributions caused by scattering along the surface and by transformation of surface wave to volume wave were distinguished; numerical calculations of this coefficient are done. Introduction Boundary between solid body and vacuum or other media could be a source of special states of electrons, called surface states. Such states are localized near the boundary of the body. Possibility of existence of such states was firstly studied in [1]. In our work we studied surface states in semi-infinite superlattices. Superlattices are referred to call solid body structures with additional periodic potential with bigger period than period of a crystal lattice. Parameters of superlattice potential could be widely changed, due to it is possible to controllable change band structure of a spectrum. Superlattices were discovered early in the 20th century and were theoretically studied in [2]. Now superlattices are widely used in electronics and optics. In general case surface is not a sharp change from nonpertubed periodic potential to the outer space. It’s also necessary to take into account, that surface might be covered by adsorbent layer. This kind of roughness will cause scattering of a surface wave, which is an electron surface state, on the surface roughness. This scattering will also cause transformation of a surface wave to volume waves. It’s interesting to estimate damping of a surface state caused by this scattering.

Model of considered system In one band approximation [3] for nondegenerate electron bands in crystal Schrodinger equation for envelope wave function is: 2  −+∆Ψ()rV()rrΨΨ()= Er() 2m 127 In order to simplify the problem we consider the following potential consisting of δ-wells (Fig. 1). Roughness of surface is described by dependence of power of surface potential on coordinates in surface plane [4].

Fig. 1. Here ρ→ = (x,y) z- axis in direction perpendicular to the surface of superlattice, → a- period of superlattice, ν0 - superlattice potential, u0+ u1( ρ ) - surface potential → and u1(ρ ) describes roughness of surface layer, u̅ - outer space potential So we are solving the following equation: 22∂ ∞  −∆− +θ−− + ρδ − δ− Ψρ ==Ψρ ρ 2 u( z ) ( u01 u ())() z v0∑ ( z na ) (,) z E (,) z 2m ∂z n=1 We suppose that roughness is random function with zero average value. We also accept that roughness is statistically homogeneous with correlation function:   2   u1(ρ)u1(ρ′) = ∆U W (ρ − ρ′)

Damping of surface state According to our model we suggest that wave function of surface state is      Ψ(ρ, z) = eik0ρ Ψ(q , z) + a(k )eikρ Ψ(q, z) 0 ∑ , k ≠k0

   ik0ρ ikρ where e Ψ(q0 , z) - surface wave, e Ψ(q, z) - surface waves or Bloch waves (these functions are solutions of Schredinger equation for z>0) when  2mE  qk2 +=22qk+=2 εε= 0 0  2  and ε < u . On the surface z=0 the derivative of wave function must satisfy the following conditions (functions Ψ(q,z) is chosen so that Ψ(q,0)=1)    ′ ′ − Ψ (q0 ,+0) + Ψ (q0 ,−0) − u0 − ∑u1(k0 − k )a(k ) = 0  k ≠k    0    ′ ′ ′ ′ (− Ψ (q,+0) + Ψ (q,−0) − u0 )a(k ) − u1(k − k0 ) − ∑u1(k − k )a(k ) = 0 ′  k ≠k 0 128 We solve these equations by perturbation theory in the 2nd order. Then we get    q uk()()−− kuk k 0 λ(q ) − cos qa −η ( q ) + u = 10 1 0 ( 0 0) 00∑ sin qa0 k q (λ(q ) − cos qa) −η ( q ) + u0 sin qa where v λ(q ) =± L L22 − 1, L = cos qa −0 sin qa , η ( q ) = u − q 2q This equation determines wave vector of surface state of electrons. Rhs of this equation is complex quantity because of complexity of λ(q) for Bloch waves and imaginary contributions from zeros of denominator. That’s why wave number of surface state of electron k0 becomes a complex quantity too. Imaginary part of wave number k describes the damping of surface wave propagating along rough 0 boundary. Phase velosity of surface wave is changing too. In our analysis we use the following correlation function   (k −k )2 2 − 0     ∆U 2 2π 2 pc u1(k0 − k )u1(k − k0 ) = 2 2 e (2l) pc where pc is inverse correlation length of roughness. → ̅ Assuming that damping is small, we put q0̅ and k0 of plane surface case into rhs of previous equation. In this case rhs of this equation will simply give us correction to surface potential u and we get the following expression for imaginary 0 part of this correction

22 2 ε−qk + ε 2 − 0 2 ∆U dq q 2 ε−qk Im ∆u =− 1−Lqe2 ( ) 2 pc I 0 − 0022∫ 2 ppcc−∞ 2Fq () sin qa 

kk22+ − i 0 2 2 ∆U qe2 pc  kk −π ⋅ ii0 ∑ 22I0  i ppccFq′()i  where

q F( q )≡( λ (q ) − cos qa) −η ( q ) + u0 sin qa In this equation integral gives us contribution caused by transformation of surface wave to volume Bloch waves, and the sum gives us contribution caused by scattering to the states propagating along the surface of superlattice. As a result ̅ we get the correction to k0

q0 ∆u0 (q0 , k0 ) δ k0 = − k0 F ′(q0 )

129 In case of u̅ =0 and u >ν estimations of contributions of scattering along the 0 0 surface and of transformation to volume wave are 2 2 2 kk+ − 0  ∆U 2  kk  Im δπk =− ()uaeI2 pc 0 ()0 s 2 0 0  2  pc  pc 

2 2 2 3 k +k −va0 − 0  ∆U ()va0 e 2 p2  kk  Im δk =− e c I 0 ()0 Bloch 2 2 2 0  2  p −va0 p c ()ua00− va + ()va0 e  c 

As we can see, contribution of transformation of surface waves to volume waves −ν a is strongly suppressed by e 0 factor. And both contributions become comparable when ν0 goes to u0 . Contributions to parameter of damping caused by scattering as functions of inverse correlation length of roughness are illustrated by Figs. 2, 3.

0.20

0.15

0.10

0.05

2 4 6 8 10 Fig. 2.

0.00020

0.00015

0.00010

0.00005

2 4 6 8 10 Fig. 3.

130 Fig. 2 gives contribution of scattering to the states propagating along the surface and Fig. 3 gives contribution of transformation to volume waves. In case of small and large inverse correlation length (in comparison with k0 ) damping goes to zero that is physically proved true. Small inverse correlation length practically corresponds to the plane surface and in case of large inverse correlation length averaging of roughness occurs.

References 1. Tamm I.E. // Sow. Phys, 1932, v.1, p. 733. 2. Keldysh L.V. // FTT, 1962, v.4, p. 2265. 3. Peter Y.Yu, Manuel Cordona. Fundamentals of semiconductors. - Springer, 2010. 4. Kuchma A.E., Kovalevsky D.V., Voronin N.V. // Vestnik S.-Peterburgskogo Universiteta, Ser. 4. 2008. Number 4. pp. 3-15.

131 Modeling of thermal-hydraulic processes in complex domains by conservative immersed boundary method

Chepilko Stepan [email protected]

Scientific supervisor: Yudov Y.V., Alexandrov Research Institute of Technology

Introduction Today main approach within Computational Fluid Dynamics (CFD) for numerical flow simulations in domains with complex boundaries is so-called boundary- fitted methods on unstructured grids. Due to its universality they are implemented in most widely used CFD-codes (CFX, Star-CD). In these methods grid is generated basing on boundary representation inside computational domain. But in recent decade there is a growing interest in developing of alternative immersed boundary methods, where computational domain is embedded in Cartesian grid, what significantly simplifies spatial discretization of equations in most cells and reduces volumes of data for storage. The aim of present work is implementation of conservative Cartesian cut-cell immersed boundary method based on developing of work [1] for modeling of two-dimensional viscous incompressible laminar flows of Newtonian fluids in Boussinesq approximation. Stated goal was realized in F77 program with following key features: spatial discretization by finite volume second-order accurate method on collocated Cartesian grid, temporal discretization by two-step fractional step procedure, direct numerical solver for pressure Poisson equation, piecewise linear modelling of complex boundaries, formation of computational control volumes of rectangular fully-fluid cells and trapezoidal partly-fluid ones near the boundary with cell-merging technique for small cut cells and, finally, accurate discretization of governing equations at cut cells by means of second-order accurate interpolation stencils, slightly modified as offered in [1]. Accuracy and fidelity of the programmed solver have been validated by simulating a number of benchmark laminar flows including natural convection in closed cavities. Ability to simulate flows with arbitrary complex boundaries has been demonstrated. Numerical scheme The governing equations are Navier-Stokes system for incompressible Newtonian viscous fluid with scalar transport equation written in integral form for each control volume CV with control surface CS ∫ vndS = 0 CV   ∂=vp−∇dV −+vv()ndSnηϕ()∇+vdSs()dV t ∫∫∫ ∫ ∫ CV CV CS CS CV

132   ∂=φφdV −+()vn dS Γ ()nd∇ φ S , t ∫∫∫ CV CS CS where ν is velocity vector, n – external normal, p – pressure, φ - scalar quantity, s(φ) – source term, η - kinematical viscosity, Γ – diffusive coefficient. Two-step fractional step method [1] is used for advancing solution in time. In this time-stepping scheme, the solution is advanced from time level “n” to “n+1” through an intermediate convection-diffusion step, where the momentum equations without the pressure gradients terms are first advanced in time using fully explicit scheme. This step is followed by pressure-correction step, when pressure Poisson equation has to be solved and divergence-free velocity field at time level “n+1” is obtained by pressure correction. Finally, scalar field at time level “n+1” is computed explicitly. The spatial discretization is performed on a 2-D Cartesian mesh using a cell-centered collocated arrangement of primitive variables p, u, v, φ with introduction of face-center velocities U, V, which are used for surface flux computation, for strict implementation of mass conservation at pressure-correction stage and for elimination of unphysical pressure field problem. Inclusion of immersed boundaries is realized as follows. The immersed boundary is first represented by a series of piece-wise linear segments. Based on this representation one determines Cartesian cells that are cut by the boundary. Cut-cells which center lies in the fluid are reshaped by discarding the part of these cells that lies in the solid. Pieces of cut cells whose center lies in the solid are absorbed by neighboring fluid cells for removing numerical instability of the solver. This results in the formation of trapezoidal control volumes for cells near the boundary and of rectangular ones for internal fluid cells (Fig. 1). The key aspect of immersed boundary solver is an accurate discretization of governing equations at cut cells, which includes evaluation of mass, convective, diffusive fluxes and pressure gradients on the cell-faces of trapezoidal cells from neighboring cell-center values with second-order accuracy. For this aim surface integrals for fluxes are computed by mid-point rule, which requires accurate evaluation of integrand at the face-center. For regular internal cells the integrand is calculated with second- Fig. 1. Variables arrangement and order accuracy using a linear profile between control volumes formation. nodes. But for cut faces spatial interpolation stencils are used (slightly modified as offered in [1]), which generally represent value at node P as weighted average of values at neighboring nodes nb.

Results and discussions Accuracy and fidelity of the programmed solver was validated by simulating a number of benchmark problems with laminar recirculating flows in closed en- 133 closures (cavities) with complex geometry. Here we collect results of computation both of hydrodynamic flows and of natural convection ones. Each flow problem is fully characterized by a set of specific dimensionless numbers - Reynold’s (Re), Prandtl’s (Pr), Rayleigh’s (Ra), which are combinations of characteristic dimension parameters. The first problem is laminar flow in cavity with inclined side walls and moving lid. Streamlines and velocity profiles, compared with [2], are shown in Figs. 2, 3.

Fig. 2. Llid-driven cavity with Fig. 3. Lid-driven cavity with inclined side inclined side walls, streamlines walls, velocity centerline profiles u(y) (left) and for 30° (top) and 45° (bottom) v(x) (right) for 30°at Re=1000. at Re=1000.

Fig. 4. Local heat transfer coefficients for Ra=48000, Pr=0.706, ε=0.652 (top) and Ra=49300, Pr=0.706, ε=-0.623 (bottom). Top curve is for internal wall circle, bottom – for external.

The second problem is natural convection between eсcentric circles with eccentricity ε and hot internal and cold external boundaries. Angle distribution of 134 Fig. 5. Streamlines for positive (a) and negative (b) eccentricity. Parameters are the same as in Fig. 4. local heat transfer coefficients along hot and cold walls and its agreement with experimental data [2] and other numerical studies [3] (DINUS-code) are shown in Fig. 4. Small difference for ε>0 for top part of internal circle might be caused by local flow instability. Streamlines for both cases are shown in Fig. 5. The last investigated problem is flow in square cavity with hot and cold side walls and adiabatic top and bottom ones. For demonstration of abilities of the solver there were studied two possibilities: computational domain embedding into ordinary non-rotated grid and into 45°-rotated grid, which is equivalent to simultaneous rotation of cavity and gravitational force. Streamlines and centerline velocity profiles, compared with [4], are depicted in Figs. 6, 7.

Fig. 7. 45°-rotated square cavity, veloc- Fig. 6. 45°-rotated square cavity, ity centerline profiles profiles u(y) (left) streamlines for Ra=106, Pr=0.71. and v(x) (right) for Ra=106, Pr=0.71.

Detailed numerical validation includes comparison of maximum centerline velocities (umax, vmax), their locations (y,x) and Nusselt (Nu) number (full heat flux over heated wall) with benchmark simulations on fine grids [4] and is summarized in Table 1 for various Rayleigh numbers 104-106. Full comparison is performed

135 for non-rotated cavity (α=0°) and partial – for 45°-rotated one (α=45°). One could observe small difference <0,8% for non-rotated case and <1,1% for rotated one.

Table 1. Comparative study with benchmark solutions for Ra=104-106. Simulation u max y v max x Nu Ra=104 128x128, α=0 16.182 0.824 19.626 0.121 2.244 Benchmark solution [12] 16.176 0.826 19.624 0.12 2.245 Ra=105 128x128, α=0 34.88 0.856 68.53 0.0663 4.519 128x128, α=45○ 34.53 - 68.43 - - Benchmark solution [12] 34.74 0.855 68.64 0.066 4.522 Ra=106 128x128, α=0 64.77 0.848 219.22 0.0351 8.83 128x128, α=45○ 64.13 - 217.48 - - Benchmark solution [12] 64.83 0.850 220.5 0.039 8.827

Conclusions There was realized conservative 2-dimensional cut-cell immersed boundary method for laminar incompressible flow modelling. There were investigated main features of proposed approach in spite of many simplifications, such as explicit convective-diffusion scheme, direct pressure solver, Dirihlet flow boundary conditions, etc. Universality of approach was checked for various complex geometries. The program was tested for numerous benchmark flow problems including natural convection in complex cavities.

References 1. Ye T., Mittal R., Udaykumar H.S., Shyy W. // Journal of Computational Physics, V. 156, pp. 209-240 (1999). 2. Ray S., Date A. // Numerical Heat Transfer. Part B, Vol. 38, pp. 93-131 (2000). 3. Yudov Yu.V. // Mathematical models and computer simulations, Vol. 3, 2, pp. 185-195 (2011). 4. Darbandi M., Schneider G.E. // Numerical Heat Transfer. Part A, Vol. 35, pp. 695-715 (1999).

136 Development of functional integration techniques for drift-diffusion processes on Riemannian manifolds

Chepilko Stepan [email protected]

Scientific supervisor: Dr. Dmitrieva L.A., Faculty of Physics, Saint- Petersburg State University

Introduction Today complete self-consistent rigorous mathematical foundation of functional integration (FI) is absent. Moreover, there exists a deep gap between theoretical physicists, who successfully use so-called path integrals as a formal tool, and mathematicians, who obtain different exact FI results in various fields. The main problem in FI foundation, to our knowledge, is a strong interplay between stochastic processes, evolutionary equations and Riemannian geometry. So the aims of present work are follows: to consider most general problem of drift-diffusion Ito process and to describe its relation with variable coefficients parabolic type PDE and Riemannian geometry; to present rigorous functional integral representation for Cauchy problem of associated PDE based on measure, generated by PDE fundamental solution [1]; to present path integral construction based on multiplicative semigroup representation in which local asymptotic either of exact fundamental solution or equivalent approximating kernel operator with the same generator could be used [2, 3]; to connect both pictures together and to point out an ambiguities in formal limiting expressions and local path integral discretizations [4]. Functional integration Let us consider time-homogeneous Ito stochastic differential equation (SDE) in space X = Rm ii ia dq ()tb=+()qdtea ()qdWt(), where Wa(t) is m-dimensional Wiener process with property dWadWb~δabdt as dt→0.

Solution of this equation is stochastic Ito process qx (t), which for 0≤τ < t ≤T satisfies following integral equation t t qtii()=+xbi ((qs))ds +=eqia((sd)) Ws(),(qxτ) ∈ X x ∫ ∫ a x (1) τ τ Here the last integral is stochastic integral over Wiener process (in Ito sense). Since dimensions of Wiener and Ito processes are the same Ito process qi(t) is diffusion i i process with drift coefficient b(q) and diffusion matrix ea(q). Moreover, every Ito process is markovian process, so its transition probability function generates evolutionary family in Banach space B(X) of functions on X

Ut()−=ττfx() Ef[(qt( ))] =−fy()Pt(,xd,)y x ∫ (2) X 137 where P(t-τ,x,dy) is transition probability function of process qi(t). In time- ho- mogeneous case evolutionary operator family forms simply operator semigroup in terms of variable s=t-τ

U (s1) U (s2 ) = U (s1 + s2 ) with generator D defined as Us()− I Df ()x = lim fx() s→0 s From Ito formulae one could find that generator is 1 i j i D = 2 ea (q)ea (q)∂i j + b (q)∂i It could be shown, that function u(s,x)=U(s)f(x), s=t-τ is solution of following initial Cauchy problem for parabolic type PDE

∂=sus(,xD)(us,)xu,(0,)xf= ()x called backward Kolmogorov equation. Important fact is that transition probability density p(s,x,y) of Ito process is a fundamental solution of Kolmogorov (backward and forward) equations

∂ s p(s, x, y) = Dx p(s, x, y) ∂ s p(s, x, y) = Dy * p(s, x, y)    p(0, x, y) = δ (x − y)  p(0, x, y) = δ (x − y) Stochastic process q(t’), τ ≤ t’ ≤ t is defined by a system of finite-dimensional distributions µ (A ×...× A ) = P(q(t )∈ A , k =1,...,n) t1...tn 1 n k k where Ak is subset of σ-algebra Σ of subsets of space X. This system introduces finite-additive measureμ on algebra of cylindrical sets Q QM()=∈{(qt):((qt),...,qt())}MM, ∈Σn tt1 ... n nn1 nn by relation µµ[(QM)] = ()M tt11... nnnt...tn According to Kolmogorov’s measure extension theorem measure μ admits σ-additive extension on minimal σ-algebra of all cylindrical sets Q in space of functions [τ,t]→X. Let us consider as simple example a diffusion equation in Rm 1 2 1 2 i ∂tu = 2 σ ∆ Rn u = 2 σ ∂ ∂iu which has following well-known fundamental solution 2 m (x− y) − − p(t, x, y) = (2πtσ 2 ) 2 e 2tσ 2

So probability that Brownian particle, started at point x(t0)=x0=x will occupy cylindrical set {a1≤ x(t1)≤b1,..., an≤ x(tn)≤bn} is n ()xx− 2 n m b1 bn − ii−1 − ∑ 2()tt− σ2 µπ=−((2 tt))σ2 2 ... edi=1 ii−1 x ...ddx ∏ ii−1 ∫ ∫ 1 n i=1 a1 an Extension of this measure to space of continuous but nowhere differentiable (τ,t) functions Sx (X) is called Wiener measure μW. Since one has a measure on func- (τ,t) tional space Sx (X) of trajectories x(t) one could define an ordinary Lebesque integral for functionals F[x(t)] I = F[x]dµ ∫ W (τ ,t ) Sx ( X ) 138 But this expression is unconstructive, since explicit expression for μW is still undefined. So one could consider “path” integral n ()xx− 2 n m − ii−1 − ∑ 2 2 2 i=1 2(ttii− −1 )σ It*l=−im ((2πσiitF−1 )) nn(,xx1 ... )e dx1...dxn n→∞ ∏ ∫ i=1 X n where Brownian trajectory x(t) is approximated by piecewise constant curve n x (t)=xk=x(tk), tk ≤ t ≤ tk+1 1 and Fn (x1,...,xn)=F(xn(t)). It turns out, that for Wiener measure I=I* for some set of functionals F. Measure on functional space of trajectories of Ito process, generated by fundamental solution of corresponding backward Kolmogorov equation 1 i j i ∂ su(s, x) = ( 2 ea (q)ea (q)∂i j + b (q)∂i )u(s, x) we will call Ito measure. The following statement is very important. If q1, q2 are i two Ito processes with drift coefficientsb 1,b2 and the same diffusion matrix ea

dqkk()tb=+()qdkkte((qd), Wt( )), k = 12, then corresponding Ito measures μ1, μ2 are absolutely continuous with density dµ  tt 2 [(qq⋅=)] exp(αα((sd)), Ws())|− 1 |(qs())||2 ds 11 ∫∫2 1  dµ1  ττ -1 where α=e (q)(b2(q)- b1(q)). Let us now introduce geometric objects. Ito process qi(t) induces structure of Riemannian manifold (M,g) in Rm. Let us m ij i j i equipe R with inverse metric g (x)=ea (x)ea (x) and Levi-Chevita connection Γ jk Then one could rewrite generator in covariant form 11ij i i D=22 ea() qe a () q ∂+ ij b () q ∂=i ∆ M +f ∂ i where f i (x) is covariant drift vector (bi (x) is not a vector due to Ito formulae) fbi=+ i1 g lj Γ i 2 lj 1 1 and ΔM – invariant Laplacian on M − i j i j k 2 2 i j ∆M = g ∂i j − g Γi j ∂k = g ∂i (g g ∂ j ) Solution of initial Cauchy problem for drift-diffusion on M at interval [τ,t] 1 i ∂ su(s, x) = ( 2 ∆M + f ∂i )u(s, x) (3)  (3) u(0, x) = f (x) y could be expressed by Lebesque functional integral over condition Ito measure μI us(,xd)(= yf yd)(µ y q) ∫ ∫ I (,τ t ) X SXxy, () (τ,t) where space Sx,y (X) consists of trajectories of Ito process (1), started at point x and finished aty. Using measure equivalence theorem one could express solution y u(s,x) as FI over condition Riemannian Wiener measure μW t t y  i ja ij  us(,xd)(=−yf yd)(µ qg)exp fedW 1 gffdt ' ∫ ∫ W  ∫ ij a 2 ∫ ij  (,τ t )  τ τ  X SXxy, () y where μW is generated by fundamental solution of diffusion equa0tion on M 139 1 ∂tu(t, x) = 2 ∆M u(t, x)

(τ,t) and space Sx,y (X) consists of solutions of following Ito SDE (Brownian motion on M) tt qi () t=− x1 glj ((')) q  t Γi ((')) q t dt ' + ei ((')) q  t dWa (') t ∫∫2 lj a ττ Path integration Using Chapman-Kolmogorov equation for transition probability density p(s,x,y) (or, equivalently semigroup property of fundamental solution) and taking limit of n→∞, one obtains “path” integral multiplicative representation of fundamental solution  s   s  ps(,xy,)= lim,p xy, 11...p ,,yyn dy ...dyn n→∞ ∫     n  n +11  n +  X Existence of the limit is a consequence of following general Chernoff theorem for operator semigroups. Let S(t) be an operator semigroup with generator D and n property ||S(t)||=1+O(t), t→0. Then for every f from B(X) for n→∞, ∑ i=1ti→t, St( )⋅⋅ ... St ( ) f → etD f 1 n For S(t) one could take Ito evolutionary family U(s) (2) with exact transition probability density p(s,x,y). So from the above theorem one obtains coincidence of Lebesque FI representation of fundamental solution and its “path” integral limit. Exactly this relation (I=I*) we have already cited for Wiener integral with explicitly known fundamental solution. The key issue of path integral approach is an ability to replace exact p(s,x,y) for finite times by its local asymptotic for s→0 and “close” in some sense points x,y. Chernoff's theorem allow one not only to use local asymptotic of exact transition probability density in multiplicative semigroup representation, but to use in some sense equivalent operator family S(s) (usually of kernel operators), called approximating, instead of exact one U(s). Condition of equivalency of multiplicative representations by approximating operator family are follows: ||S(s)||=1+O(s), s→0 and S(s) − I 1 i lim = D = 2 ∆M + f ∂i s→0 s One way to find1 exact local asymptotic is to try following WKB-ansatz for fundamental solution [3] for ρ2(x,y)

However, continuous formal expression (6) must be interpreted only as a limit of multiplicative representation. For our kernel example (5) one has to interpret (6), where L is given in work [3], only as limit (4) with p replaced by p. There are several variations of formal “Lagrangians” (6) that are suggested in literature. Very often formal “Lagrangians” are written in discretized from (see [4]), but correctness and fixation of ambiguities in these expressions in general case on non-constant metric g is still an open question.

Conclusions Problems in rigorous FI foundations were reviewed. The following existed, but not clearly formulated in literature items and results were put together: general problem of relation between drift-diffusion process, Riemannian geometry and evolutionary equations of parabolic type was investigated. Basing of this interplay solution of backward Kolmogorov equation was represented as Lebesque functional integral over Ito measure for corresponding stochastic process. Difference in functional and path integral definitions was mentioned including problem of its correspondence. Path integral multiplicative representation was build using local asymptotic of fundamental solution; problem of variety and ambiguity in formal continuous Lagrangians for path integrals was discussed.

References 1. Daleckiy Ju. // Mathematical Analysis (Russian), pp. 83–124 (1966). 2. Smolyanov O., Weizsacker H., Wittich O. // arXiv:math/0409155v3 (2008). 3. Alimov A.L., Buslaev V.S. // Vestnik LGU (Russian), 1, pp. 1-14 (1972). 4. Kleinert H. Path integrals // World Scientific (2009).

~~141 Multifractal generalization of the detrending moving average approach to time series analysis~~

~~Ganin Denis [email protected]~~

~~Scientific supervisor: Dr. Kuperin Y.A., Department of Nuclear Physics, Faculty of Physics, Saint-Petersburg State University~~

Introduction The analysis of financial time series has been the focus of intense research by the physics community in the last years [1, 2]. An important aspect concerns concepts as scaling and the scale invariance of return fluctuations [3]. The aim was to characterize the statistical properties of the series with the hope that a better understanding of the underlying stochastic dynamics could provide useful information to create new models which are able to reproduce experimental facts. It has been recently noticed that time series of returns in stock markets are of multifractal (multiscaling) character. Among other approaches the detrending moving average (DMA) algorithm [4] is a widely used technique to quantify the long-term correlations of non-stationary time series. In the present paper we develop a multifractal generalization of DMA that is multifractal detrending moving average (MFDMA) algorithms for the analysis of one-dimensional multifractal measures and multifractal time series. The performance ability of the elaborated MFDMA methods has been investigated using synthetic multifractal measures, namely binomial measure with the parameter p=0.25, and fractal Brownian motion with different values of the parameter H. We have found that the estimated multifractal scaling exponent τ(q) and the singularity spectrum f(α) are in good agreement with the theoretical values. It has been shown that the MFDMA algorithm also out-performs the multifractal detrended fluctuation analysis (MFDFA) proposed in [5]. The proposed MFDMA method was applied to analyzing the time series of the DJIA Index and its multifractal nature has been fairly detected.

Results and Discussion For the beginning we have tested the method by applying it to the binomial measure, for which there is a analytical representation for the form of the singularity spectrum f(α) and for the set of Renyi dimensions [6]. Compare our results with the theoretical ones. The graph of the singularity spectrum f(α) (Fig. 1 (c)) has the form, which was predicted theoretically (see, for instance [6]). Also, we can calculate the spectrum using the program FracLab. For clarity, depict both depending in one graph (Fig. 1 (c)). It is obvious that the graphs are practically identical; it indicates good accuracy of the proposed here method. In addition, if we construct a spectrum of the generalized Renyi dimensions (D(q)) (Fig. 1 (d)), we find that the data obtained are completely consistent with theoretical predictions. 142 a b

~~c d~~

Fig. 1. a) Binomial measure (parameter p=0.25); b) Multifractal scaling exponent τ(q) for the binomial measure (p=0.25); c) Graphs of the singularity spectrum f(α) for binomial measure (p=0.25), constructed by the method MFDMA (green line) and using FracLab (blue line); d) Spectrum of Renyi dimensions for the binomial measure (p=0.25) [6].

We also apply a new method to a monofractal Brownian motion [5]. For the calculations we take initial time series with Hurst exponent H=0.5, containing 4096 points. From the graph of the spectrum Renyi dimensions (Fig. 2 (d)), one can see that D(q)=1, as expected, since the Renyi dimension must be equal to dimension of measure, that is unity. The graph of the multifractal scaling exponent τ(q) is a linear function (Fig. 2 (b)), which is consistent with the theory, because the time series in question is monofractal. Applying the method MFDMA, we find that the singularity spectrum f(α) is almost completely degenerated to a point (Fig. 2 (c)), as it should be for the monofractal Brownian motion. It is known that FracLab program builds the same form of the singularity spectrum f(α) for a multifractal series and for a monofractal series, that is obviously wrong. For clarity, we represent both results in the same graph (Fig. 2 (c)). It is obvious that method MFDMA has the advantage in comparison the methods used in FracLab program, because the method MFDMA gives a true form of the singularity spectrum f(α).

~~143 a b~~

~~c d~~

Fig. 2. a) Series of Brownian motion (Hurst exponent H=0.5); b) Multifractal scaling exponent τ(q) for the Brownian motion (H=0.5); c) Graphs of the singularity spectrum f(α) for the Brownian motion (H=0.5), constructed by the method MFDMA (blue line) and using FracLab (green line); d) Spectrum of Renyi dimensions for the Brownian motion (H=0.5). After that we have applied the our method to real data, such as the values of the Dow Jones Index [3]. In this case, we use the logarithmic increments of time series rather than the Dow Jones Index itself. We can determine the spectrum of Renyi dimensions D(q) which is practically constant and equal to unity (Fig. 3 (d)). We see in the graph (Fig. 3 (c)) that the singularity spectrum f(α) isn’t degenerated to a point, as it was for a monofractal Brownian motion. It indicates that economic time series of values of the Dow Jones Index have multifractal nature [3]. Now, compare the form of the singularity spectrum f(α) calculated for a fractal Brownian motion and for the Dow Jones Index. The graph (Fig.4) clearly shows that the peak of spectrum of the Dow Jones Index is shifted relative to peak of the spectrum of fractal Brownian motion, and the scope of the spectrum Dow Jones index is much broader. This result suggests that the time series of economic indexes have multifractal nature, as expected.

Conclusion The implemented method MFDMA gives better results compared with calculations in Fraclab program. Algorithm MFDMA detects multifractality of time series much better. This method may be considered as one of the most accurate in identifying the main multifractal characteristics of nonstationary time series. 144 a b

~~c d~~

Fig. 3. a) Series of Dow Jones indexes (it contains 3000 points); b) Multifractal scaling exponent τ(q) for Dow Jones; c) Graphs of the singularity spectrum f(α) for Dow Jones, constructed by the method MFDMA (blue line) and using FracLab (green line); d) Spectrum of Renyi dimensions for Dow Jones.

~~Fig. 4. The singularity spectrum f(α), obtained for a fractal Brownian motion (blue line) and for a values of the Dow Jones Index (green line), (enlarged scale).~~

References 1. Bouchaud J.P., Potters M. Theory of Financial Risk. - Cambridge University Press, Cambridge, 2000. 2. Mantegna R.N., Stanley H.E. An introduction to Econophysics. - Cambridge University Press, Cambridge, 1999. 3. Mantegna R.N., Stanley H.E. // Nature, 376, 46–49 (1995). 4. Alssio E., Carbone A., Castelli G., Frappietro V. // Eur. Phys. J. B, 27, 197 (2002). 5. Kantelhardt J.W. at al // Physica A, 316, 87-114 (2002). 6. Feder A. Fractals. - Moscow: Mir, 1991, -254 p. 145 Propagation of photons and massive vector mesons between a parity breaking medium and vacuum

~~Kolevatov Sergey [email protected]~~

~~Scientific supervisor: Prof. Dr. Andrianov A.A., Department of High Energy and Elementary Particles Physics, Faculty of Physics, Saint-Petersburg State University~~

Introduction The problem of crossing the boundary between the vacuum and a parity breaking medium, which may occur in the presence of so-called axion fields, is important for modern physics because axions are realistic candidates for the role of dark matter. This assumption is attractive by the fact that these particles were result of the hypothesis which explained the strong CP problem in QCD, and only after some time it was shown that they, in principle, could be dark matter. However, the experiments showed that on scales comparable to the size of the Universe axion fields are not observed. Yet, not applicable to the specific effects at large distances, does not preclude the existence of these fields on the scale of stars and even galaxies. One may think of an axion background accumulated by very dense stars like neutron ones or even of bosonic axion stars [1]. Another interesting area for observation of parity breaking is the heavy ion physics [2]. In the occurrences of axion-like background in astrophysics or heavy ion physics the existence of a boundary between the parity-odd medium and the vacuum is quite essential. For star condensed axions there is evidently a boundary where axion background disappears and photons distorted by it escape to vacuum.

~~Results and Discussion We start from the Lagrange density which describes the propagation of a vector field in the presence of a pseudoscalar axion-like background,~~

~~(1)~~

where Aµ and acl stand for the vector and background pseudoscalar fields respectively, is the dual field strength, while B is the auxiliary Stuckelberg scalar field with real . The positive dimensionless coupling g > 0 and the (large) mass parameter M»m do specify the intensity and the scale of the pseudoscalar-vector interaction. Notice that we have included the Proca mass term for the vector field because, as it is discussed in [2], the latter is required to account for the strong 146 interaction effects in heavy ion collisions supported by massive vector mesons (ρ, ω, ...) in addition to photons. Moreover, as thoroughly debated in [3], the mass term for the vector field appears to be generally necessary to render the dynamics self-consistent in the presence of a Chern-Simons lagrangian and is generally induced by radiative corrections from the fermionic matter lagrangian. The auxiliary Stuckelberg lagrangian, which further violates gauge invariance beyond the mass term for the vector field, has been introduced to provide – just owing to the renowned Stuckelberg trick – the simultaneous occurrences of power counting renormalizability and perturbative unitarity for a general interacting theory. Moreover, its presence allows for a smooth massless limit of the quantized vector field. We shall consider a slowly varying classical pseudoscalar background of the kind, (2) where θ(...) is the Heaviside step distribution, in which a fixed constant four vector ζ with dimension of a mass has been introduced, in a way to violate Lorentz and CPT invariances in the Minkowski half space ζ ⋅x < 0. In what follows we shall suppose that ζ2 ≠0. If we now insert the specific form (2) of the the pseudoscalar background in the pseudoscalar-vector coupling lagrangian we can write the equivalent Lagrange density,

~~(3) in which the gauge invariance is badly broken by all the terms but the one, i.e. the Maxwell’s radiation lagrangian. Then the field equations read,~~

~~(4)~~

This system gives us different solutions in different half-spaces. The first is well- known Proca-Stuckelberg vector field, the second - Maxwell-Chern-Simons vector field, which have been extensively discussed and applied in [3] for the massive case and in [4] for the massless case. This solutions face one another at the hyperplane ζ ⋅x = 0. Hence locality of the quantized wave fields does require equality on the surface separating the classical pseudoscalar background from the vacuum: namely,

~~(5)~~

~~We discuss the case of a spatial Chern-Simons vector ζµ = (0, -ζx, 0, 0) so that~~

~~δ(ζ⋅x) = -δ(x)/ζx . If we now set such objects:~~

~~147 and look at the boundary conditions, we have the result that there are two different Fock vacua: namely,~~

~~The operator equalities can be written,~~

~~(6)~~

~~(7)~~

~~Using this relations one can find,~~

~~Moreover we get,~~

The latter quantity, can thereof be interpreted as the relative probability amplitude that a birefringent particle of mass m, frequency ω and wave vector (k1A, k2, k3) and chiral polarization vector

is transmitted from the left face to the right face through the hyperplane x1 = 0 to become a Proca-Stuckelberg particle with equal mass m, frequency ω and wave vector (k1A, k2, k3) but polarization vector

As an effect of this transmission, the first component of wave vector of a birefringent massive particle changes, while the longitudinal massive quanta do not change it’s wave vector. The similar result was obtained in the classic solutions of the Euler-Lagrange equations. Moreover, classical solutions give us the coefficient of reflection. It means that we know which part of coming into the vacuum particles is reflected and which is passed through: − k 1A k10 (8) K refl = (8) k 1A+ k10 where k1A is the first component of wave vector, corresponding for the polarization A

~~148~~

~~(9)~~

Easy to see that the longitudinal polarization does not feel the boundary and completely passes into the next area without reflection. Finally, using our classical solutions and the conventional Strum-Liouville theory we can determine the Green function in the momentum space, All calculations can be found in [5].

Conclusion The main results were obtained for the space-like Chern-Simons vector, such symmetry breaking is possible in the fireball, around neutron stars or in the axion stars. Relations, describing the passage through and reflection of incoming and outgoing particles from the area of any polarization were obtained. In particular, it was shown that the longitudinal polarization does not feel the boundary and completely passes into the next area without reflection. However, we need methods for testing the described phenomena. Since we cannot detect outgoing particles near the bound, we have to know what is going on after they are released outside. The influence of a boundary between parity-odd medium and vacuum on the decay width of photons and vector mesons represents a very interesting problem which deserves to be a subject of further investigation.

~~149 Acknowledgements. The work was partially supported by the non-profit foundation “Dynasty”. I am grateful to A.A. Andrianov and R. Soldati for productive and valuable colaboration.~~

References 1. Mielke E.W. and Perez J.A. // Phys. Lett. B 671 (2009) 174. 2. Andrianov A., Andrianov V.A., Espriu D. and Planells X. // Abnormal dilepton yield from local parity breaking in heavy-ion collisions, arXiv:1010.4688. 3. Alfaro J., Andrianov A., Cambiaso M., Giacconi P. and Soldati R. // Int. J. Mod. Phys. A 25 (2010) 3271 [arXiv:0904.3557]. 4. Andrianov A.A., Giacconi P. and Soldati R. // Journal of High Energy Physics 02 (2002) 030 [hep-th/0110279]. 5. Andrianov A.A., Kolevatov S.S. and Soldati R. // Journal of High Energy Physics, 11 (2011) 007 [arXiv:1109.3440].

~~150 Analytical solution of two-dimensional Scarf II model by means of SUSY~~

~~Krupitskaya Ekaterina [email protected]~~

~~Scientific supervisor: Prof. Dr. Ioffe M.V., Department of High Energy and Elementary Particles Physics, Faculty of Physics, Saint-Petersburg State University~~

Introduction The importance of each new exactly solvable model in one-dimensional quantum mechanics is well known. The approach of supersymmetric quantum mechanics (SUSY QM) and, in particular, shape invariance [1] has been fully exploited for construction and investigation of such models by generating a partnership between pairs of dynamical systems which allows us to establish the solvability of one in terms of other by means of intertwining relations with supercharges of first order in derivatives. Within the search for larger class of problems which can be solved by supersymmetrical methods, extensions of SUSY QM have been elaborated with different realizations if the intertwining operators (supercharges). The next step was the suggestion that for two-dimensional models one can use ordinary intertwining relations but with supercharge of second order in derivatives [2, 3]. The two main methods of SUSY-separation of variables were formulated. One of the methods leads to partial (quasi-exact) solvability of the model [3, 4], and another one - to complete(exact) solvability of two-dimensional generalizations of Morse [5] and Pӧschl-Teller [6] models. In our paper one more two-dimensional model - with potential, which is natu- rally associated with solvable one-dimensional hyperbolical version of Scarf model (Scarf II) [7] - can be solved analytically as well by means of supersymmetrical separation. This potential was obtained recently among new two-dimensional models with shape invariance property [8, 10]. Just this property will allow to solve the problems with the whole hierarchy of generalized Scarf II potentials. At first, the two-dimensional generalized Scarf II model was completely solved for the specific parameter value a = -1: both energy values and corresponding wave functions of all bound states were built analytically. And then the procedure was generalized to the models with arbitrary negative integer values of parameter a.

Investigation We start from the supersymmetrical intertwining relation HQ()12++==QH();QH−−()12HQ() (1) for two partners two-dimensional Hamiltonians of Schrödinger form  H(ii )=−∆ (2) + V ( ) ( x ); i = 1,2; x = ( xx ); ∆(2) ≡∂ 2 +∂ 2 (2) 1, 2 1 2 151 with mutually conjugated supercharges Q± of second order in derivatives. We have considered two-dimensional generalization of Scarf II potential:  11 V(1),(2) ( x )=− 2 λ2 aa ( 1)( − ) − cosh22 (λλxx ) cosh ( ) +− (3) (2kxkkxk1 sinh( 2λ+ 12) ) (2 1 sinh( 2λ+ 22) ) −+ 22, (4cosh( 2λλxx12)) (4cosh( 2 )) And the second order supercharges are:

~~+−† QQ=() =∂∂+ 4+− 2atanh ( λ x+ ) ∂+ − 2 acoth ( λ x− ) ∂+ +~~

~~2 (2k1 sinh(2λ+ xk 12 ) ) a tanh()λ x+− coth () λ+ x 2 − (4cosh (2λx1 )) (4)~~

~~(2k1 sinh( 2λ+ xk 22) ) − 2 (4cosh( 2λx2 ))~~

It is evident that potentials are not amenable to standard separation of variables. The first step of the our approach was to choose such values of parameters, that one of the Hamiltonians H(2) does allow standard separation of variables. Then, we have a chance to find the spectrum and wave functions of the partner Hamiltonian H(1) which does not allow standard separation. Indeed, one can choose the parameter a = −1 to cancel the terms prohibited from separation. For simplicity, we have also fixed the parameter λ = 1/2. Thus

~~(1)  (2) 11 H() x= −∆ − ( 22− )()()+Ux12 + Ux cosh (xx++ / 2) sinh ( / 2) (22)  ( ) (5) H( x)= −∆ + Ux( 12) + Ux( ) where one-dimensional potential U is defined as:~~

−2kx12sinh()+ k Ux()= . 4cosh 2 ()x (6) The next step was to solve the model which does allow separation of variables. That means, we need to find the solution of one- dimensional model with potential U. For the general case of U, this is impossible to perform analytically. But for specific form (6) for U(x), the solution is known explicitly [9]. After some transformations we have obtained the eigenfunctions and eigenvalues of H(2): (2) (2) 2 2 EEnm,,= mn =∈ n + ∈ m = −(An − ) − ( Am − ); (2)±±(2) (7) ψ=±ψ=ηηnmmnnm, ,(xx 1) ( 2) ±ηη mn( xx 12) ( ). where − A η=n (x ) (cosh( x )) exp ( − Barctan ( sinh ( x ))) × (−iB −− A 1/2, + iB −− A 1/2) ×Pn (isinh ( x )) 152 (α,β) In the formulas above, A and B are positive parameters (A,B > 0), and Pn are the n−th power Jacobi polynomials of their argument. The positive parameters A,

~~B can be expressed in terms of coupling constants k1 < 0, k2 : 2 2 12/ Ak=−12//−+1212((kk14)(+−k1 2 +1))~~

~~1 2 2 12/ (8) Bk=+((2 14)(+−k1 k2 +1)) 22~~

~~The condition n, m~~

(ii) The levels that are absent in the spectrum of H(2), if there are zero-modes of Q ̶ among the wave functions of H(1); (iii) The levels that are absent in the spectrum of H(2), if some wave function of H become non-normalizable after Q ̶ act on it. After the investigation these possibilities we obtain, that there are no levels of the second and the third type. And the first type levels could correspond only to antisymmetric function Ψ(2) .Thus, the discrete spectrum of H(1) is nondegenerate and consists of bound state levels with such energies with corresponding wave functions: (1) (1) 2 2 Enm,,= E nm =−( An − ) − ( Am − ), (12) + ( )− (10) ψnm,,(xQ) =ψnm ( x) To solve the problem with arbitrary parameter we had to extend our results. (1) To do this it is necessary to construct the hierarchy of Hamiltonians H (ak) with (1) (1) ak = −k, k = 1,2,..., with the previous one H ≡ H (a1). The hierarchy is based on the alternate application shape-invariance and intertwining relations. One can see that Hamiltonians are actually shape-invariant: ()11() Ha()kk= Ha()+1 (11) Therefore, the infinite chain (hierarchy) of Hamiltonians can be built: ()2 ()1 ()2 ()1 Ha()1 ÷=Ha()1 Ha()2 ÷=Ha()2 ... ()1 ()2 ()1 ÷=Ha()NN−1 Ha( ))(÷=HaN ) ..., 153 (1),(2) where the sign ÷ means that the corresponding Hamiltonians H (ak) are inter- ± twined bysupercharges Q (ak). After analysis, which we have carried out, one can conclude that eigenfunctions and eigenvalues of Hamiltonian H(1) are: ()12() 22 EEnm,,==nm −−()An−−()Am, (12) ()21() + ()2 + + ψψ()aann==()−11Qa( n− ))(ψ aQnn−11)(= aQ− )(an−2 )... + ()2 ...Qa()1 ψ ()a1 , where |n-m|≥k. + The latter condition follows from the existence of zero modes of Q (ak).

Conclusions It was demonstrated that the two-dimensional quantum model, which can be called as two-dimensional Scarf II model is exactly solvable for arbitrary value ak = −k. All hamiltonians of this hierarchy of Hamiltonians has nondegenerate spectrum. The values of energies are given by (7) for |n-m|≥k, and the corresponding wave functions are given by (12). Together with generalized Morse model [11] and generalized Pӧschl-Teller model [12] the complete analytical solution of this two-dimensional model, demonstrates that supersymmetrical approach is a powerful method to solve the problems which are not amenable to conventional separation of variables. The extended version of this work - M.V. Ioffe, E.V. Krupitskaya, D.N. Nishnianidze “Analytical solution of two-dimensional Scarf II model by means of SUSY methods” - is accepted for publication in “Annals of Physics”. Acknowledgements. The work was partially supported by the grant RFFI 09-01- 00145-a. I am also indebted to the non-profit foundation “Dynasty” for financial support. I am grateful to M.V. Ioffe and D.N. Nishnianidze for productive and valuable colaboration. References 1. Junker G. Supersymmetric Methods in Quantum and Statistical Physics, Springer, Berlin, 1996. 2. Andrianov A.A., Ioffe M.V., Nishnianidze D.N. // Phys.Lett., A201 (1995) 103; Andrianov A.A., Ioffe M.V., Nishnianidze D.N. // Theor. and Math.Phys., 104 (1995) 1129. 3. Ioffe M.V. // J.Phys.A37 (2004) 10363. 4. Cannata F., Ioffe M.V., Nishnianidze D.N. // J.Phys.:Math.Gen., A35 (2002) 1389. 5. Andrianov A.A., Cannata F. // J. Phys. A 37 (2004) 10297. 6. Andrianov A.A., Borisov N.V., Ioffe M.V., Eides M.I. // Phys. Lett. A 109 (1985) 143; Andrianov A.A., Borisov N.V., Eides M.I., Ioffe M.V. // Theor. Math. Phys. 61 (1984) 965. 7. AndrianovA.A., Borisov N.V., Ioffe M.V. // Phys. Lett. B 181 (1986)141. 8. Andrianov A.A.,Ioffe M.V. // Phys. Lett. B 205 (1988) 507. 154 9. Aoyama H., Sato M., Tanaka T. // Phys. Lett. B 503 (2001) 423; Aoyama H., Sato M., Tanaka T. // Nucl. Phys. B 619 (2001) 105. 10. Ioffe M.V., Nishnianidze D.N. // Phys. Rev. A 76 (2007) 052114. 11. Ioffe M.V., Nishnianidze D.N., Valinevich P.A. // J. Phys. A 43 (2010) 485303.

~~155 Effects of turbulent mixing on critical behaviour: Renormalization group analysis of the ATP model Malyshev Aleksei [email protected]~~

~~Scientific supervisor: Prof. Dr. Antonov N.V., High Energy and Elementary Particles Physics, Faculty of Physics, Saint-Petersburg State University~~

Critical behaviour of a system, subjected to strongly anisotropic turbulent mixing, is studied by means of the field theoretic renormalization group (RG). Specifically, relaxational stochastic dynamics of a non-conserved multicomponent order parameter of the Ashkin-Teller-Potts (ATP) model, coupled to a random velocity field with prescribed statistics, is considered. The velocity is taken Gaussian, d−1+ξ white in time, with correlation function of the form ∝δ(t-t′)/|k⊥| , where k⊥ is the component of the wave vector, perpendicular to the distinguished direction ("direction of the flow"). It is shown that, depending on the values of parameters that define self-interaction of the order parameter and the relation between the exponent ξ and the space dimension d, the system exhibits various types of large-scale behaviour. In addition to known asymptotic regimes, existence of a new, non-equilibrium and strongly anisotropic, type of critical behaviour (universality class) is established, and the corresponding critical dimensions are calculated to the leading order of the double expansion in ξ and ε = 6 − d (one-loop approximation). Numerous systems of very different physical nature reveal interesting singular behaviour in the vicinity of their critical points (second order phase transitions). Their correlation functions exhibit self-similar (scaling) behaviour with universal critical dimensions. Most typical equilibrium phase transitions belong to the 4 universality class of the On-symmetric φ model of an n-component scalar order parameter φ. Universal characteristics of the critical behaviour in this case depend only on n and the space dimension d and can be calculated within systematic perturbation schemes. Another important example is provided by the Ashkin-Teller-Potts class of models. Such models have numerous physical applications: magnetic materials and solids with nontrivial symmetry, Edwards-Anderson spin-glass models within the replica formalism, and so on. In general, the ATP models describe systems which locally have n states, but the energy of any given configuration depends on whether pairs of neighboring sites are in the same state or not. The case n = 2 corresponds to nematic-to-isotropic transitions in the liquid crystals, while the formal limit n = 0 corresponds to the percolation problem. The behaviour of a real system near its critical point is extremely sensitive to external disturbances, gravity, geometry of the experimental setup, presence of impurities and so on. Some disturbances can change the type of the phase transition (second-order to first-order one, and vice versa) and even produce completely new types of critical behaviour with rich and rather exotic properties. In the presence 156 of a distinguished direction, scaling behaviour can become strongly anisotropic. In this paper we study effects of turbulent mixing on the dynamical critical behaviour of the systems, described by the generalized ATP model, paying special attention to anisotropy of the flow.

Relaxational dynamics of a non-conserved n-component order parameter φa(x) with x≡{t, x} is described by a stochastic differential equation δϕH ( ) (1) ∂taϕ(x) =−λ0 +ηa(x), δϕa (x) where ∂t=∂/∂t, λ0 is the (constant) kinetic coefficient, andη a(x) is a Gaussian random noise with zero mean and the pair correlation function ()d ηa(x) η b ( x′) = δab Dxxηη( − ′′′′), Dxx( −=) 2()λδ0 tt − δ (xx −) ,(2) d being the dimension of the x space. Near the critical point, the Hamiltonian H(φ) of the ATP model is taken in the form

1 2 τ 00g Hd(ϕ)=−∂+x ϕa ( x) ϕa ( x) ϕϕa( xx) a ( ) +Rabc ϕϕϕ a( xxx) b( ) c ( ) , (3) ∫ 2 2 3! 2 where ∂i = ∂/∂xi is the spatial derivative, ∂ = ∂i∂i is the Laplacian, τ0∝(T−Tc) measures deviation of the temperature (or its analog) from the critical value and g0 is the coupling constant. Summations over repeated indices are always implied

~~(a, b, c =1, ..., n and i =1, ..., d); after the functional differentiation in (2) one has to replace φ(x)→ φ(x).~~

In the original ATP model Rabc is the irreducible invariant third-rank symmetric tensor of the symmetry group of the hypertetrahedron in n dimensions. The Rabc components are conveniently expressed in terms of the set of n+1vectors eα which define vertices of the hypertetrahedron [1] R= eeeααα, (4) abc ∑α a b c where eα satisfy nnn++11 α α α α β αβ ∑∑ea =0, eea b =+( n1)δδab , ∑ ee a a =+−( n1) 1. αα===111 a (5)

~~Using equations (5) all the contractions with the tensor Rabc can be calculated. For example, the contractions of two and three tensors have the forms~~

~~Rabc R abe= R1δ ce, R aec R chb R bfa= RR2 ehf , (6) 22 RnnRnn1=+−=+−( 1) ( 1,) 2( 1) ( 2.)~~

Coupling with the velocity field v={vi(t,x)}, which describes the turbulent mixing, is introduced by the replacement ∂t →∇t = ∂t+vi∂i in (1), where ∇t is the Lagrangian (Galilean covariant) derivative. The velocity ensemble is defined as follows. Letn be a unit constant vector that determines some distinguished direction ("direction of the flow"). Then any vector can be decomposed into the components perpendicular and parallel to the flow, for example,

~~x=x⊥+nx|| with x⊥⋅n=0. The velocity field will be taken in the formv =nv(t,x⊥). For v(t,x⊥) we assume a Gaussian distribution with zero mean and the pair correlation function:~~

~~dk ⊥ −d +−1 ξ vt( ,,x⊥) vt( ′′ x ⊥) =−δ ( t t ′) D0 d −1 exp(ikxx⊥⊥⊥⊥⋅−( ′))k . (7) ∫ (2π )~~

157 Here D0 is a constant amplitude factor, ξ is an arbitrary exponent (with the most realistic Kolmogorov value ξ=4/3) and k⊥=|k⊥|. In order to ensure multiplicative renormalizability of the model, it is necessary 2 2 2 to split the Laplacian in (3) into the parallel and perpendicular parts ∂ →∂|| +f0∂⊥

by introducing a new parameter f0 >0. In the anisotropic case, these two terms will be renormalized in a different way. According to the general theorem [2], the stochastic problem (1)−(7) is equivalent to the field theoretic model of the extended set of fieldsΦ= {ϕa′,ϕa,v} with action functional 1/4 ϕϕaa′′Dη 22 λ000gf S(Φ=) +ϕ−∇+λ∂+∂−τa′′ t 0 ⊥ f0 || 0 ϕ−a RSabcϕ a ϕ b ϕ c + v (v ), (8) 22{ ( )} 1/4 where we segregated the factor f0 from the charge g0. The first few terms represent the De Dominicis-Janssen action functional for the stochastic problem (1)−(3) at fixedv ; it involves the auxiliary scalar response fieldϕ a′. All the required integrations over x≡{t, x} and summations over the vector indices are implied. The last term in (8) corresponds to the Gaussian averaging over v with the correlator (7) and has the form

ξ = 0, so that the UV divergences in the correlation functions manifest themselves as poles in ε = 6−d, ξ and their linear combinations. The careful analysis, augmented by symmetry considerations, shows that all the counterterms needed to cancel the UV divergences in our model are present in the action (8). Here important role is played by the Galilean symmetry and the invariance with respect to the symmetry group of the hypertetrahedron. Thus our model appears multiplicatively renormalizable and we conclude that the renormalized action can be written in the form ϕϕ′′D SZ()Φ =aaη +ϕ′  −Z ∇ +λ Z ∂22 + Zf ∂ − Z τ ϕ − R 1 2 at2 ( 3⊥ 4 || 5 ) a (9) λµgfε/2 1/4 −Z RSϕϕϕ′ + ()v 6 2 abc a b c v Here λ, τ, f, g and w are renormalized analogs of the bare parameters (with the subscripts "0'') and μ is the reference mass scale. The one-loop calculation of the renormalization constants Zi is easily performed. In the minimal subtraction scheme they contain only simple poles in ε and ξ and have the forms: uR uR uR w 22uR uR ZZ= =−1,1,1111 Z =− Z =− − ,1,1. Z =− 12 Z =− (10) 12 233ε3 ε 4 εξ5 ε6 ε 158 Here we have passed to more convenient coupling constants u→g/128π3 and 3 w→w/24 π . The parameters R1 and R2 are related to the dimension n of the order parameter by the expression (6). Although we are especially interested in the cases n = 2 and 0, for completeness the coefficients R1 and R2 in what follows are assumed to be arbitrary. Expression (9) can be obtained by the multiplicative renormalization of the fieldsφ a→ φaZφ, φa'→ φa'Zφ' and the parameters: εξ/2 (11) λλττ00000=====Zλτ,,,, ZffZggZwwZfgwµ µ

(no renormalization of the velocity field is needed: Zv=1). The renormalization constants in Eqs. (9) and (11) are related as follows: 2 Z1234==== ZZλϕ′′′′,,,, Z ZZ ϕϕ Z ZZZ ϕλϕ Z ZZZZ ϕλf ϕ (12) 1/2 1/4 2 Z5=== ZZZZϕλτϕ′, Z6 ZZZ λu f ZZ ϕϕ′, ZZZwf λ 1.

~~The last equality is a corollary of the fact that Sv(v) is not renormalized.~~

The RG equation for the renormalized Green functions GR=〈Φ…Φ〉R in our model has the following form  {DNNGeRG ++=ϕϕϕϕγγµ′′} R ( , ,) 0, where Nϕ and Nϕ′ are the numbers of corresponding fields entering into GR, e denotes the set of the renormalized counterparts for the bare parameters e0, DRG is the operator dµ=µ∂µ for fixed e0 expressed in the renormalized variables (13) DDRG ≡µ +∂+∂−ββγγγu u w w f DDD f −λλττ − .

Here we have written Dx=x∂x for any variable x, the anomalous dimensions are defined as γF ≡dµ lnZF and the β-functions for the two dimensionless couplings u and w from (13) are β≡duu =−− εγβ, ≡dww =−− ξγ. (14) uµ[ uw] µ[ w]

2 Using (10) and (12) one can find, that Z( u= Zg , and hence γu= 2γg) uR uR 5uR uR γγγγγ=1111,,,,, = =w = =− ϕϕτλ′ 3636f (15) uR1 w γγw=−=−−w,4,u( RRu21) 62 with corrections of order u2, w2, uw and higher. It is well known that possible large-scale regimes of a renormalizable model are associated with IR attractive fixed points of the corresponding RG equations.

~~The coordinates g* of the fixed points are found from the requirement that the~~

~~β-functions, corresponding to all renormalized couplings gi, vanish. The type of a fixed point is determined by the matrix ijΩ =∂βi /∂gj, where βi is the full set of~~

~~β-functions and gj is the full set of couplings. For IR attractive fixed points the matrix is positive, i.e., the real parts of all its eigenvalues are positive. In our case,~~

~~159 gi={u,w}. The functions βi, calculated in the one-loop approximation from (14) and the explicit relations (15), have the forms:~~

βεβξu=−++u[ Ru w/2,] w = w[ −− uR1 /6 + w] , (16) where we have introduced a new convenient parameter R=R1−4R2. The analysis of the functions (16) reveals four possible IR attractive points (coordinates of the fixed points and corresponding domains of stability):

~~1. u∗ = 0, w∗ = 0; IR attractive for ε < 0, ξ < 0;~~

~~2. u∗ = 0 (exact result to all orders), w∗ = ξ; IR attractive for ξ >2ε, ξ > 0.~~

~~3. w∗ = 0 (exact result to all orders), u∗ = ε /R; this scaling regime exists if R>0; IR attractive for ε >0, ξ <−εR1/6R.~~

4. u∗ = (12ε−6ξ)/(12R+R1), w∗ = (12Rξ+2R1ε)/(12R+R1); this scaling regime exists if R+R1/12>0. IR attractive for ξ<2ε, ξ >−εR1/6R if R>0 and ξ<2ε, ξ <−ε(6R+R1)/3R if R<0. The first fixed point is a Gaussian (free) one. In the scaling regime corresponding to the fixed point 2, the nonlinearityφ 2 in the stochastic equation (1) becomes irrelevant due to the exact relation u∗ = 0. Thus we arrive at the linear advection- diffusion equation for a passive scalar field φ. In turn, the effects of the velocity field become irrelevant in the third regime (fixed point 3). The isotropy violated by the velocity ensemble is restored and the leading terms of the IR behaviour coincide with those of the equilibrium dynamic model ATP. Finally, the last point represent a new nontrivial IR universality class, in which the both nonlinearities of the model are simultaneously important.

The critical dimensions ∆F of the IR relevant quantities F are given by the re- ⊥ || ω ∗ lations ∆F = dF +∆|| dF + ∆ω dF +γF with the normalization condition ∆⊥=1, here ⊥,||,ω ∗ dF are the canonical dimensions of F and γF =γF (u∗,w∗). Results for ∆||, ∆ω, ∆τ for all the universality classes are given in the following table:

R № FP1 FP2 FP3 FP4 (∆ =R+R1/12): R ∆|| 1 1+ξ/2 1 1+(6Rξ+R1ε)/12∆ R ∆ω 2 2 2+R1ε/6R 2+ R1(2ε−ξ)/12∆ R ∆τ 2 2 2+R1ε/3R 2+ 5R1(2ε−ξ)/6∆ One can easily see that the most realistic values of the model parameters (for example, d = 3 and the Kolmogorov exponent ξ = 4/3) belong completely to the region of stability of the most nontrivial fixed point 4 (for both physically interesting cases n = 2 and n = 0). More detailed presentation of this work can be found in [3]. The author thanks the Dynasty Foundation for the financial support. References 1. de Alcantara Bonfim O.F. et al. // J. Phys. A: Math. Gen. 13 L247 (1980). 2. Vasil'ev A.N. The field theoretic renormalization group in critical behavior theory and stochastic dynamics.- Boca Raton: Chapman & Hall/CRC, 2004. 3. Antonov N.V., Malyshev A.V. // arXiv:1111.6238v1. 160 Inertial-range behaviour of a passive scalar field in a random shear flow: Renormalization group analysis of a simple model

~~Malyshev Aleksei [email protected]~~

~~Scientific supervisor: Prof. Dr. Antonov N.V., High Energy and Elementary Particles Physics, Faculty of Physics, Saint-Petersburg State University~~

Infrared asymptotic behavior of a scalar field, passively advected by a random shear flow, is studied by means of the field theoretic renormalization group (RG) and the operator product expansion (OPE). The advecting velocity is Gaussian, d −1+ξ white in time, with correlation function of the form ∝δ(t-t′ )/k⊥ , where k⊥=|k⊥| and k⊥ is the component of the wave vector, perpendicular to the distinguished direction ("direction of the flow"). The structure functions of the scalar field in the infrared range exhibit scaling behavior with exactly known critical dimensions. It is strongly anisotropic in the sense that the dimensions related to the directions parallel and perpendicular to the flow are essentially different. In contrast to the isotropic Kraichnan’s rapid-change model, the structure functions show no anomalous (multi)scaling and have finite limits when the integral turbulence scale tends to infinity. On the contrary, the dependence of the internal scale persists in the infrared range. The problem of turbulent advection, being of practical importance in itself, has become a cornerstone in studying fully developed hydrodynamical turbulence on the whole. On one hand, deviations from the classical Kolmogorov theory – intermittency and anomalous scaling – are much stronger pronounced for a passively advected scalar field (temperature of the fluid or concentration of impurity) than for the advecting turbulent field itself. On the other, the problem of passive advection appears easier tractable theoretically. Most remarkable progress was achieved for Kraichnan’s rapid-change model: for the first time, the anomalous exponents were derived on the basis of a dynamical model and within controlled approximations. In Kraichnan’s model, the turbulent velocity field is modeled by the Gaussian distribution with the pair correlation function of the form −−d ξ vvi j ∝ D0δ ( t− t′) Pkij , (1) 2 where Pij=δij−kikj/k is the transverse projector, k ≡ |k| is the wave number, D0>0 is an amplitude factor, d is the dimension of the x space and ξ is an arbitrary exponent with the most realistic (Kolmogorov) value ξ = 4/3. The issue of interest is the behavior of the equal-time structure functions n Sr=θθ t,,,x − t x′ r =− xx′(2) n ( ) ( ) ( )

~~161 of the scalar field θ(x) with x ≡{t, x} in the inertial range l<~~

−−nn(2ξ ) ∆n 2 S20n( r)∝ D r( r/ L) ,∆n =− 2(1)/(2) nn −ξξ d + + O( ) . (3) Thus the functions (2) depend on the integral scale and diverge for L→∞ (the anomalous exponents Δn are negative), in contradiction with the classical Kolmogorov theory. In [1] the field theoretic renormalization group and operator product expansion were applied to Kraichnan’s model. The anomalous scaling for the structure functions emerges as a consequence of the existence in the corresponding OPE of "dangerous" composite fields (composite operators in the field theoretic terminol- ogy) of the form (∂θ)2n, whose negative critical dimensions are identified with the anomalous exponents Δn. In this work we apply RG+OPE to the model of a passive scalar field in a random shear flow. We show that the inertial-range behavior of this model appears essentially different from the isotropic Kraichnan’s model. The advection-diffusion equation for the scalar fieldθ (x) has the form 2 ∇tθνθζ =0 ∂ +, ∇=∂+ttiiv ∂,(4) 2 here ∇t is the Lagrangian Galilean covariant derivative, ∂t=∂/∂t, ∂i = ∂/∂xi, ∂ is the Laplacian, ν0 is the diffusion coefficient andζ (t,x) is a Gaussian random noise with zero mean and the pair correlation function

Dζ =ζζδ( t,,x) ( t′′ x) =( ttC − ′) ( rrxx ), =−′ . (5) The function C(r) is finite at r = 0 (we assume the normalization C(0) =1) and rapidly decays for r→∞; its precise form is inessential. The velocity ensemble is defined as follows. Let n be a unit constant vector that determines some distinguished direction ("direction of the flow"). Then any vector can be decomposed into the components perpendicular and parallel to the flow, for example,x = x⊥+nx|| with x⊥⋅n = 0. The velocity field will be taken in the form v = nv(t,x⊥). For v(t,x⊥) we assume a Gaussian distribution with zero mean and the pair correlation function of the form: dk vt,,x vt′′ x=−δ t t ′ D ⊥ exp ikxx⋅−′ k−d +−1 ξ ,. k =k ( ⊥) ( ⊥) ( ) 0 ∫ d −1 ( ⊥⊥⊥⊥( )) ⊥ ⊥(6) (2π ) Here and below d is the dimension of the x space, D0 is a constant amplitude factor, ξ is an arbitrary exponent (the most realistic Kolmogorov value ξ = 4/3). In order to ensure multiplicative renormalizability of the model, it is necessary 2 2 2 to split the Laplacian in (4) into the parallel and perpendicular parts ∂ →∂|| +f0∂⊥ by introducing a new parameter f0 > 0. In the anisotropic case, these two terms will be renormalized in a different way. According to the general theorem [2], the stochastic problem (4)−(6) is equivalent to the field theoretic model of the extended set of fieldsΦ= {θ', θ, v} with action functional

1 22 SD(Φ) = θ′′′ζ⊥ θ + θ −∇tv + ν0 ∂ +fS0 ∂ || θ + (v). (7) 2 { ( )} 162 The first few terms represent the De Dominicis–Janssen action functional for the stochastic problem (4), (5) at fixedv ; it involves auxiliary scalar response field θ'(x). All the required integrations over x≡{t, x} are implied. The last term in (7) corresponds to the Gaussian averaging over v with the correlator (6)

1 −1 −1 −1 2(1 −−d ) ξ Sv(v)=− dtddvtD xx⊥⊥′( , x ⊥) vv( x ⊥− x ′′⊥) vt( ,, x ⊥) DrDr( ⊥)∝ 0 ⊥ . 2 ∫∫ This formulation means that statistical averages of random quantities in the original stochastic problem coincide with the Green functions of the field theoretic model with the action (7), given by functional averages with the weight exp S(Φ). This allows one to apply the field theoretic renormalization theory and renormalization group to our stochastic problem. The action (7) corresponds to the standard Feynman diagrammatic technique. The role of the bare coupling constant (expansion parameter in the ordinary perturbation theory) is played by the parameter w0=D0/ν0 f0. Dimensional analysis shows that for our model superficial UV divergences, whose removal requires counterterms, can be present only in 1-irreducible Green function 〈θ'θ〉1-ir. The cor- 2 responding counterterm must contain two symbols ∂|| and therefore reduces to θ'∂|| θ. Inclusion of this counterterm is reproduced by the multiplicative renormalization of the action (7) with the only independent renormalization constant Zf: ξ −1 νν000====,,,.f fZf w wµ Z wwf Z Z (8) Here the reference scale μ is an additional parameter of the renormalized theory, ν, f and w are renormalized analogs of the bare parameters (with the subscript

"0") and Zi are the renormalization constants. Their relation in (8) results from the ξ absence of renormalization of the contribution with D0 in (7), D0=w0ν0f0=wμ νf. No renormalization of the fields is required. It turns out that in our model all the multiloop diagrams needed for calculating of the renormalization constant Zf vanish [3]. This means that Zf is given exactly by the one-loop approximation. This calculation is easily performed. In the MS scheme we obtain Zf =−1/,wξ (9) (d− 1) where we have absorbed the factor Sd−1/2(2π) into the coupling constant. The RG equation for the renormalized Green functions GR(e,μ...) is D+γ GeR ,µ , = 0, { RG G } ( ) e denotes the set of the renormalized counterparts for the bare parameters e0, DRG is the operator dµ=µ∂µ for fixed e0 expressed in the renormalized variables

~~DDDRG ≡+∂−µ βγw f f .~~

~~Here we have written Dx=x∂x for any variable x, the anomalous dimensions are defined asγ F ≡dµ lnZF and the β-function for the dimensionless couplings w is~~

β≡=−−dwwµ [ ξγw ]. Using (9) one obtains exact expressions (10) γγβξf=−w =www, =[ −+ ]. 163 It is well known that IR asymptotic behavior of the Green functions is governed by IR attractive fixed points of the RG equations, defined by the relations β(w*)=0 and β'(w*)>0. From (10) it follows that for our model

~~w*=ξβξ,′(w*)=.~~

~~This fixed point is positive and IR attractive for ξ>0. The critical dimensions ∆F of the IR relevant quantities F are given by the relations~~

~~⊥ω|| * ∆FF =dd +∆|| F +∆ω d FF +γ; ∆ ωθ = 2, ∆ = 1 +ξ / 2,~~

⊥,||,ω with the normalization condition Δ ⊥=1, here dF are the canonical dimensions ∗ of F and γF =γF (w∗). For example, Δθ = −1. The key role in the following will be played by the critical dimensions of certain composite fields. We begin with the simplest operatorsθ n. The analysis shows that n n such operators are UV finite and requires no counterterms: θ =Z[θ ]R with Z = 1. It then follows that their critical dimensions are simply given by the sum of the ∗ critical dimensions of the constituents: Δn=nΔθ=−n, because γF =0. In what follows we will be interested also in the critical dimensions of the operators with minimal canonical dimension (namely, dF = 0) that are invariant with respect to the shift θ→θ+const. In Kraichnan’s rapid-change model, the anomalous exponents (3) are identified with the negative critical dimensions of such operators. For that isotropic case, the scalar operator of the needed form is unique for any n given n: Fn=(∂iθ∂iθ) . In the case at hand one can construct n+1 different operators of the form (∂θ)2n, invariant under the residual symmetry: ks ⊥⊥ || || (11) Fks, =( ∂ i θ∂ i θ) ( ∂θ∂θ i i ) ,.ksn + = It was shown that in spite of renormalization mixing of the operators (11) it is possible to findexact expressions for their critical dimensions (for details see [3]). They have the forms:

∆ks, =22k + s ∆−θω ( ks + ) ∆ =ξ s . (12) In contrast to the results (3) for the isotropic Kraichnan’s model, the expressions (12) have no corrections of order O(ξ2) and higher and are positive for all k, s and ξ>0. The last results allow one to find the IR scaling behavior of the correlation functions. For generality, consider the different-time structure functions

S2n (τ,, r⊥ r|| ) = θ( t ,x) − θ( t′′ , x)  ; τ = t ′ − tr ,⊥ =xx ⊥⊥′ − , r|| = x || ′ − x || . (13) The function (13) is a linear combination of the two-point correlators 〈θ(t,x)kθ(t',x')s〉 with n the fixed k+s=2n. Due to simple exact relations Δn= −n for the operators θ , the critical dimensions of these correlators are all equal, Δk+Δs=−(k+s) =−2n =2nΔθ, and hence the function S2n in the IR range behaves as a single object. From the dimensional considerations −nn2 1/2 2 S2n = ν rQrwrfr⊥( µ ⊥,,/ ⊥ ⊥ ,ντ rrL ⊥⊥ ,/,) (14) where Q(...) are some functions of completely dimensionless arguments. Using (14) and RG equation the following asymptotic expression for the structure functions in the IR range (l<

D0=w0ν0 entering the velocity correlator (1), but not on the diffusivity coefficientν 0 and the coupling constant w0 separately; see (3). This fact is in agreement with the second Kolmogorov hypothesis about the independence of the correlation functions in the IR range of the parameters, related to the UV (dissipation) scale. In the case at hand, the UV parameters ν0, f0 and w0 survive in the IR asymptotic expression

(15) for the structure functions (they do not form the combination D0=w0ν0 f0 even if we set f0=1). Thus we may conclude that, in contrast to the isotropic case, the second Kolmogorov hypothesis is invalid for the shear flow. The asymptotic representations (15) hold in the IR asymptotic range, specified by the inequality l<

References 1. Adzhemyan L.Ts., Antonov N.V., Vasil’ev A.N. // Phys. Rev. E 58, 1823–1835 (1998). 2. Vasil'ev A.N. The field theoretic renormalization group in critical behavior theory and stochastic dynamics.- Boca Raton: Chapman & Hall/CRC, 2004.

~~3. http://www.springerlink.com/content/k7328k18m3745733/ 165 Effects of Stefan’s flow and concentration-dependent diffusivity in binary condensation~~

~~Martyukova Darya [email protected]~~

~~Scientific supervisor: Prof. Dr. Kuchma A.E., Department of Statistical Physics, Faculty of Physics, Saint-Petersburg State University~~

Introduction The subject of this paper is studying the problem of droplet growth in the atmosphere of two condensable vapors and noncondensable carrier gas. The results of such study are essential for fundamental and applied problems of the theory of decomposition of solid and liquid solutions, and the theory of phase transitions in the Earth atmosphere. A typical example of binary condensation under the conditions of the Earth atmosphere is condensation of water and sulfuric acid vapors in rain drops. Discussion We consider a spherical supercritical droplet of binary solution, isothermally growing in the diffusion regime. Let n1, n2 – the numbers of molecules of the first and second components of the vapor per unit volume, n3 – the number of molecules of carrier gas per unit volume, and ñ = n1+n2+n3 - total number of molecules per unit volume in the vapor-gas mixture. We introduce a spherical coordinate so that r is distance from the center of the droplet. Denote by R the radius of the droplet. The boundary conditions in this problem have the form:  = nri ()r→∞ ni0  (1) nr() = nx()  i rR= ii∞ ni0 – is the concentration of the i-th (i = 1,2) component of the vapor far from the droplet, ni∞( xi ) – equilibrium concentration of the i-th (i = 1,2) component of the vapor at the droplet of flat boundary, andx i – current value of the molar concentration of the i-th (i = 1,2) component in the droplet.

Denote by ji – flow density of the i-th (i = 1,2) vapor component. For the case of stationary diffusion, we have: R2 jr()= jR() (2) iir 2 Total flow densityj(r) of molecules of both components of the vapor is:

~~jr()=+jr12() jr() (3)~~

The density of a stationary flow of molecules of i-th component ji(r) taking into account the hydrodynamic flow of vapor-gas mixture with the radial velocityv(r) can be written as: 166 ∂nr() −Dr() i +=nr()υ()rj()ri(,= 12,)3 (4) i ∂r ii

Flow density j3 corresponds to carrier gas. The flow of noncondensable carrier gas is equal zero. So using (4) we have: ∂nr() −Dr() 3 +=nr()υ()r 0 (5) 3 ∂r 3 Because the sum of diffusion flows in the reference system moving with velocity v(r), is equal to zero, we obtain: ∂nr() ∂nr() ∂nr() −D 3 = D 1 + D 2 (6) 3 ∂r 1 ∂r 2 ∂r Expressing the density of the total flow through the flow densities of both components of the vapor through (3) and using (4) - (6), we have: jr()= nrυ() (7)

For accounting concentration-dependent diffusivity, we approximate Di(r) as follows:  nr1 () nr2 () Drii()=+D 011 εεii+ 2  (8)  nn  where Di0 – diffusion coefficient of molecules of i-th (i = 1,2) component vapor in a pure carrier gas with a concentration ñ and εi1, εi2 – numerical constants of order unity. Using (2), (3), (7), (8) we obtain from (4): 2 R  nr1 () nr21() ∂nri () jr( ))(+ jr2 ) jRii() 2 =−D 011++εεii2  + ni (9) r  nn  ∂r n In the lowest approximation for the concentration (linear approximation), we have: R2 ∂nr()1 () (10) jR()1 () =−D i iir 2 0 ∂r Integrating this equation and using boundary conditions (1), we obtain the following expression for the flows in the lowest approximation:

~~()1 Di0 (11) jRi ()=− []nnii0 − ∞ ()xi R Substituting (11) in (10), separating variables and integrating, we obtain the expression for the concentrations in the lowest approximation:~~

()1 R nrii()=−nn00[]ii− nx∞ ()i (12) r We return to equation (9), which contains terms both linear and quadratic for the concentration of vapor contributions and substitute flows in linear approximation for the concentration (11) and concentrations in linear approximation (12) into the quadratic terms. Then we have:

167 R 2 nn10 −−[]10 nx11∞ () R ∂nr1 () r jR1 () =−D10 −+D10 ()1 ε11 ⋅ r 2 ∂r n R nn20 −−[]20 nx22∞ () R r ⋅ 2 []nn10 − 11∞ ()xD− 10ε12 ⋅ r n R nn10 −−[]10 nx11∞ () R r R ⋅−2 []nn10 11∞ (x ) − D20 2 []nn20 − 22∞ ()x r n r R 2 nn20 −−[]20 nx22∞ () R ∂nr2 () r jR2 () =−D20 −+D20 ()1 ε22 ⋅ r 2 ∂r n R nn10 −−[]10 nx11∞ () R r ⋅ 2 []nn20 − 22∞ ()xD− 20ε21 ⋅ r n R nn20 −−[]20 nx22∞ () R r R ⋅−2 []nn20 22∞ (x ) − D10 2 []nn10 − 11∞ ()x r n r Integrating the last two expressions with the boundary conditions (1), we obtain the required expressions for the flow densities:

D10  nn10 + 11∞∞()x nn20 + 22()x jR1 ()=− ()nn10 − 11∞ ()x 1+ εε11 + 12 + R  2n 2n (13) nn10 + 11∞∞()x nn10 + 11()x D20 []nn20 − 22∞ ()x  + +  22n n D10 []n110 − nx11∞ ()

~~D20  nn20 + 22∞∞()x nn10 + 11()x jR2 ()=− ()nn20 − 22∞ ()x 1+ εε22 + 21 + R  2n 2n (14)~~

nn20 + 22∞∞()x nn20 + 22()x D10 []nn10 − 11∞ ()x  + +  22n n D20 []n220 − nx22∞ () As can be seen from the expression for the flow density of the first vapor component ji(r) (13), corrections to the flows obtained in the lowest approximation are caused by different mechanisms. Contributions with ε11 and ε12 come from concentration-dependent diffusivity. Fourth contribution describes the effect of Stefan’s flow, and the last - the influence on the Stefan’s flow of the second component vapor. Conclusions The equations (13) and (14) express the dependence of the flow densities caused by Stefan’s flow and concentration dependence of diffusion coefficients. The obtained results can be used to clarify the description of binary condensation which developed in [1-3]. 168 References 1. Kulmala M., Vesala T., Wagner P.E. // Proc. Royal Soc., Vol. 441, P. 589 (1993). 2. Kuchma A.E., Shchekin A.K., Kuni F.M. // Colloid Journal, Vol. 73, P. 215 (2011). 3. Kuchma A.E., Shchekin A.K., Kuni F.M. // Physica A, Vol. 390, Issue 20, P. 3308 (2011).

~~169 A matrix approach for dyadic Green's function in multilayered elastic media~~

~~Nikitina Margarita [email protected]~~

~~Scientific supervisor: Dr. Val’kov A.Y., Department of Statistics Physics, Department of Physics, Saint-Petersburg State University~~

Introduction The Green’s function method is widely used for studies of layered media [1, 2], in particular for analysis of the synthetic seismograms. Problems of this type have a long history going back to the articles of Kelvin and Stokes. They are extremely important for geophysical applications and a lot of papers in mathematical physics were devoted to them. Generals An elastic medium is described with the fieldu (r,t) which is the displacement of matter in point r at the time t. The motion equation (Navier-Cauchy) in an elastic medium can be written as 2 ∂ uα ∂σαβ ρ()rF2 =+β (1) ∂t ∂rβ where ρ is the mass density, σ is the stress tensor, and F is the body force per unit volume. The stress tensor is related to the strain tensor epsilon, defined for small displacements with the constitutive relations.

~~σ=αβC αβγς ε γζ , (2)~~

1  ∂uγ ∂uζ  εγζ =  +  (3) 2  ∂rζ ∂rγ  In the linear approximation the constitutive relations know as Hook law (1). And C is the stiffness tensor of 4-th-order (4) 2 CKαβγζ ()rr= () δδ+µαβ γζ () r δδαγ βζ +δδ αζ γβ − δδ αβ γζ (4) 3 Differential equation (1) must be added with proper boundary condition. We use two types of boundary conditions: • on the boundary S of two homogeneous medium the matter deformations u(r) are to satisfy two conditions of continuity, where n is the normal vector to the boundary. The first equation means the continuity of the displacement vector, and the second one corresponds to the equality of pressures on the opposite sides of the boundary. (5) ur()SS= ur ()+ , − σnn =σ ,. (6) SS− + 170 • in case of elastic medium boundary with the vacuum one requires only the pressure components on the boundary should be zero.

σ=n S 0 (7) Presently we consider displacement field harmonic in time. In this case the solution for deformations can be presented in form of the plane waves (8) where k is the wave vector and ω is the circular frequency. it()kr −ω u(,)r t= Ae (8) There are three wave modes, longitudinal one, with polarization vector A being parallel to wave vector k and two transverse ones with A perpendicular k. The wave numbers of these modes are determined by dispersion relationship, which you can see in Fig. 1. (a) (b)

Fig. 1. a) General information about longitudinal wave and direction of polarization vector and wave vector; b) General information about transverse wave and direction of polarization vector and wave vector.

~~The Green function (fundamental solution) G of equation (1) is defined by~~

The physical sense of Gγη(r,r’;t-t’) nis the γ-component of the displacement u(r,t) initiated by a source that is impulse of impact point force directed along η-axis, localized in point r’ and running at the moment t’. In addition G obeys to • Causality principle: Gˆ(r,r',t − t') = 0, t < t'. • Infinity- the Sommerfeld radiation condition • Medium-medium ∂∂GG ˆˆ γη γη G== G, nCβ αβγζ nCβ αβγζ rS∈ rS ∈+ − ∂∂rrζζ rS∈∈−+rS • Medium-vacuum ∂G nC γη = 0 β αβγζ ∂r ζ rS∈ 171 Method for arbitrary layered medium In this article we consider layered medium wherein the elastic modules depends on the z-axis only. Let the system occupies the region l1 ≤z ≤ l2 The Green’s function obeys to equation ∂∂2 Kˆ() z+ K ˆˆ () z + K () z Gˆ (q ,, zz ',)ω=δ Iˆ ( z − z ') 2∂z2 10∂z ⊥ (1) where matrices K2(z),K1(z),K0(z) depend on the mediums constants.

ˆˆ2 22 Kz2 ()= czIczt () +− ( lt () cz ())nn ⊗ , ˆ 22 Kz1 ( )= ( czlt ( ) − cz ( ))(Q ⊗+⊗ nn Q), (2) ˆˆ22 2 2 2 Kz2 ()=ω− ( czqIt ()⊥ ) − ( cz lt () − cz ())QQ ⊗ , The most generalized form of the third-type (Robin) boundary condition

~~ˆˆ∂ ˆ B10() ljj+ B () l G (q⊥ ,,',) zzω == 0; j 1,2 (3) ∂z zl= j~~

ˆ 2 B1 () z=µ () zQn ⊗ + Kz () − µ () z nQ ⊗ , 3 (4) ˆˆ1 B0 () z=µ () zI + Kz () + µ () z nn ⊗ . 3 At the same way we can write the equations for deformations and boundary condition. And there are six linear independent solutions of equation. Three of them satisfy one of boundary condition (with l2) uuu(1)(2)(3),,, (5) >> > and three another condition (with l1) (1)(2)(3) uuu<<,, < (6) For convenience we present these solutions as two 3×3 matrices ˆˆ(1)(2)(3) (1)(2)(3) Vz>()==( u >> , u , u >) , Vz < ()( u << , u , u <) (7) In these terms the Green’s function can be represented in this form ˆˆ−−11 ˆ V>>( zV ) ( zW ) ( z '), z≥ z ', Gzzˆ( , ') =  ˆˆ−−11 ˆ (8) V<<( zV ) ( zW ) ( z '), z< z ', where W can be understood as the generalized matrix Wronskian of the set of functions u. Wzˆ()=− KzVzVzVzVz ()(() ˆˆ′′ () ˆˆ () ()) << >> (9)

To obtain matrices V>(z) and V<(z) we express them in terms of six independent solutions of the wave equation (2), u(1)(z),…, u(6)(z). We arranged conveniently the 3×6 matrix ˆ (1)(2)(3)(4)(5)(6) Vz()= ( u (), zu (), zu (), zu (), zu (), zu () z) (10) 172 (1),(2),(3) (1),(2),(3) (1) Since solutions u< and u> are the linear combinations of vectors u (6) (z),…, u (z), we can present the columns of the matrices V>(z) and V<(z) as linear combinations the columns of matrix V(z). That’s why we can get this formula ˆ ˆˆˆ ˆˆ (11) V<<() z== VzJ () , V>> () z VzJ () where J1,2 are the 6×3 matrices of coefficients of the linear combinations. For obtaining J1,2 we use formulae (3,11) BlVlˆˆ' + iBlVl ˆˆ J ==0, j 1, 2 10( j) ( j) ( j) ( jj) Iˆ ˆ ˆ ˆ ˆˆ ˆˆ I J2=,A1=− ( iBlV 11 () '() l 1 + BlVl 01 ()()) 1 , ˆˆ-1 ˆ −A2( l 2) Al 12() 0 −Aˆˆ-1 l Al() 0ˆ ˆ 1( 1) 21 ˆ ˆˆ ˆˆ J1= ,A2=− ( iBlVl 12 ()'() 2 + BlVl 02 ()()) 2   IIˆˆ 

Method for multi-layered medium But the explicit analytical results for solutions of the homogeneous wave equation (i.e. for matrix V) can be obtained only for some specials cases. In what follows we consider one such case, which is important for applications. In case of piecewise – homogeneous medium we can write normal modes in the layer in the explicit form. In (q perpendicular, z)-presentation in particular we have

±±±λizjp ()q⊥ 22 uqjp (⊥⊥ ;)z= eqjp ( ) e ,λ=jpkq jp −⊥,  2 ±±qn⊥ ⊗ λjpqn⊥⊥ +q ± q ⊥±λjp n ee12pp==, ,, e 3 p = q⊥⊥qktp klp where j=1,2- are transverse waves, and j=3- is longitudinal wave. Here we can see the result for fields deformation matricesV < and V>. Iˆ −Aˆˆ-1 l Al() ˆ ˆˆ><ˆ ˆˆ 1( 1) 21 V><() z== Upp () zM ,V () z Upp () zM . ˆˆ-1 ˆ −A2( l 2) Al 12() I And after that I’ll show using formulae. ˆ ˆˆ+− Up() z= ( U pp () zU ()), z ˆ ±±= ±λiz123p()qqq⊥±±λ iz pp() ⊥⊥±±λ iz() Uzpp()(eq12 ()⊥⊥⊥ e ; eqp ()e ; eq 3p ()e )

Iˆˆ  ˆˆˆˆ±± ˆˆ ± ˆ ˆˆ0 Al1,2() 1= ( BlUl 1 () 1 1 () 1 Λ+ 1 BlUlE 0 () 1 1 ()) 1 1,2 EE12==,.  0ˆ Iˆ ˆ Uzp () Szˆ ()= , p ˆˆˆ ˆˆ BUz1pp()Λ+ p BUz opp () where Λp is diagonal matrix 6×6 with wave numbers ±λip , i=1,2,3 on the diagonal.

~~173  (M ... M )−1 , pN< , ˆ >  Np−1 M p =  ˆ ˆ ˆˆ−1 I,,pN= Mp= S p+1 ()() zSz p pp Ipˆ, = 1, ˆ <  M p =   Mp−11... Mp ,> 1~~

Discussion Let’s discuss finally GF in ther -representation. Performing the inverse Fourier- transform we obtain dq Gˆˆ(Rq ; zz , ')= G ( ; zz , ') eiqR⊥⊥ ⊥ ⊥⊥∫ 2 (2π ) There exists inputs to GF several types, such as • Far field: the stationary phase points vicinity • Poles (Rayleigh , Love, Stoneley waves) • Head wave (λ=0)

• Near-field (wide areaq ^ input). The final formula for the Green’s function is contributed by the body waves, transverse and longitudinal, and by surface waves generated by the poles of the matrix Wronskian. This formula has been used for numerical computation of the harmonic Green function in the layered media.

References 1. Aki K., Richards P.G. Quantitative Seismology: Theory and Methods, v.1, 1980; Freeman W.H., Co. Shikin A.M. et al. // Phys. Rev. B 62, pp. 13202–13208, 2000. 2. Wapenaar K., Fokkema J. // Geophysics, v. 71, Iss. 4, pp. SI33-I46 Suppl. (2006); Liu E., Zhang Z., Yue J., Dobson A. // Commun. Comput. Phys., v. 3, pp. 52-62, 2008.

~~174 Detectable effects in classical supergravity~~

~~Niyazov Ramil [email protected]~~

~~Scientific supervisor: Dr. Shadchin S.V., Department of High Energy and Elementary Particles Physics, Faculty of Physics, Saint Petersburg State University~~

Introduction Supersymmetric theories and, in particular, supergravity manifest a symmetry between fermions and bosons. Despite the popularity of such theories, they miss experimental evidences. That’s why it’s important to search for possible experiments to confirm these theories.

Supersymmetric corrections In N=1 supergravity exist two particles: quantum of gravitational field - gravi- ton - particle with spin 2 and its superpartner - gravitino - particle with spin 3/2, described by Rarita-Schwinger field. Supergravity Lagrangian is sum of Einstein Lagrangian (formulated in vierbein (tetrad) formalism) ab gµν()= x e µ () xe ν () xηab (for possibility of introduction spinors) so Rarita-Schwinger Lagrangian read [1] 11 LL=+ L = − ||g R − ieeµνρ e e ψγabcD ψ (1) E RS 42abc µ νρ where 1 ab Dµψ=  ∂ µµ + ωγab ψ 4

ab and ωμ – gauge fields called spin-connection introduced for invariance of Rarita- Schwinger Lagrangian under Lorentz transformations. Supergravity Lagrangian is invariant under three kinds of local symmetries [1]. • general coordinate transformations with parameters ξμ(x): aνν aa δG () ξeµ = −ξ ∂ νµ ee − ∂ µ ξ ν, (2)

~~νν δG () ξψµ = −ξ∂ψ νµ −∂ξψ µ ν. (3) • local Lorentz transformation with parameters λab(x) (λab= - λba):~~

~~a ab δL () λeeµµ = −λ b , (4)~~

~~1 ab δL () λψµµ = − λ γab ψ . (5) 4 c • local supertransformations with parameter εα(x) (ε =ε): 175 aa δQ () εeiµµ = − εγ ψ , (6)~~

δQ () εψµµ =D ε . (7) Follows from (1) that General Relativity can be seen as supergravity with vanishing gravitino so with the supersymmetry transformations applied to the Schwarzschild solution one can obtain solution in classical supergravity. ψ=0 +∆ψ ⇒ (8) µµ aa(0) a eeµµ= +∆ e µ ⇒ (9) gg= (0) +∆ g µν µν   µν (10) GR SUSY Supersymmetric metric calculated in paper [3]. Consider its in equatorial plane then 1 (11) g =,− rr r 1− g r

~~1 rg g=−− rk22 1, − (12) θθ 2 r~~

~~221 rg gϕϕ =−− rk 1, − (13) 2 r 2 rrgg1 gk=1−+ 2 , (14) tt r 8 r 4~~

~~3 rg ggA= =1− (15) ϕϕtt 2 r where~~

~~k ≡ εε A = εγ52 γ ε~~

Motion of a probe particle To calculate equation of the motion of a probe particle, we use approach developed in [2]. Consider Hamilton-Jacobi equation ∂∂SS gµν − mc22=0 (16) ∂∂xxµν where m is particle mass. Solution of this equation is sought as

S=− Et + L ϕ+ Sr () r (17) Hence for radial part of action we have 1 gtt E2 gϕt LE 1 2 S=2 m22 c −+ −gϕϕ L2 dr (18) r ∫ 2 rr cgc We introduce a new variable r' in such way that 2 . rrr(−gg )= r′′ , ⇒ , r ≈− rr /2

176 as Greek letter γ. We need the contravariant components of ׳Denote ratio rg/r the metric, we obtain from inverting (11). Introduce the non-relativistic energy E-mc2, expand the metric tensor up to terms of order γ2 and k2 and obtain≡׳E 22  EE′′rg  S =++ 2Em′′ 24 ++Em mc22 − r ∫  22  ccr′ 

 22 13 2Er′ g −−L2 m 222 c r − −4E′ mr 2 + 22 gg rc′ 2 (19)  GR corrections 1  2 22 2LEA2 k  E′ 22 + +−A 2 +2Em′ + mc  dr′. c 2  c    SUSY corrections  L+EA/c will remove the first-order supersymmetric in the=׳The substitution L constant redefinition of the angular momentumL . This is legitimate since actually We introduce constant B so .׳the angular momentum which we measure is the L B . In B leave in the order of с2 – 2׳has the form L 2׳that the coefficient of 1/r 3 E 2 B= mcr222+ k 2 (20) 2 g 2c2 Next we use the fact that the angle change during one rotation of a probe particle in its orbit is ∂ ∆ϕ= − ∆S ∂L r where ΔSr corresponded change in Sr. It expanded in powers of a small correction B then we obtain B ∂∆S (0) ∆SS=, ∆−(0) r (21) rr2LL∂ ′ where ∆S (0) corresponds to classical motion that is r ∂ − ∆S (0)= ∆ϕ (0) =2 π ∂L r As a result Bπ ∆ϕ=2 π + 2 . (22) L′ We distinguish the corrections relating to supergravity 1 B= mck22 2 SUSY 2 and to General Relativity 3 B= mcr222 GR 2 g and their ratio 2 BSUSY k (23) =.2 BGR 3rg Thus we can assert the smaller the gravitational radius of the black hole, the greater the supersymmetric correction. Also this phenomena can be observed in 177 the formula of the correction to angle changing at 2π by only supersymmetry with expressing the momentum through the focal parameter of the orbit L′22= m pGM

k 2π ∆ϕ =. (24) 2 prg Suppose that a probe particle rotates in the field of a black hole with mass M = 1 kg with a period T = 1 day, the eccentricity is e = 0.5 and the anomalous displacement of perihelion Δφ=1°. Focal parameter can be expressed through a rotation period of the particle. Note that the semimajor axis of the orbit about 0.25 meter. Then we can estimate the order of k:

~~23rrggπ kp= ∆ϕ − ~10−15 m. (25) π 2~~

Motion of a light ray For light ray we will modify our considerations assuming m = 0 and passing from particle energy E to the light frequency ω, from L to ρ=cL/ω. Having a similar argument as above (16) - (22) one can obtain:

∂∂(0) 2rRg ∆ϕ== − ∆ψrr − ∆ψ + .(26) ∂∂LL()ρ+AR22 − () ρ+ A (0) We limit R to infinity and note that the straight ray Δψr changing equal π so 2r ∆ϕ=. π + g (27) ρ+ A It is seen that the first-order corrections can be removed by redefinition of constant ρ. Besides such supplement, we can not measure.

Conclusions First order corrections (on a suitable parameter) are not observable and can be eliminated by redefinition of momentum and impact parameter for the particles, and for the light beam. Second order corrections exist only for the motion of probe particle. Such corrections appear in the anomalous displacement of perihelion of the probe particle. Thus effects of second-order supergravity can be detected while observation of probe particle perihelion displacement for the small black holes. To make effect measurable, some conditions should be imposed to the laboratory system. The size (according to calculations about 0.25 meter) and the weight (about 1 kilogram) of this system are quite human-natural, but complete isolation of the laboratory from external fields is required.

178 References 1. Tanii Y. Introduction to Supergravities in Diverse Dimensions, 1998. 2. Landau L.D., Lifshits E.M. Course of Theoretical Physics, V. 2. The Classical Theory of Fields. Butterworth-Heinemann.- М., 4th edition, 1975. 3. Baaklini N., Ferrara S. van Nieuwenhuizen P. // Lettere Al Nuovo Cimento (1971 – 1985), 20:113-116, 1977.

~~179 Calculation of characteristics of critical behavior in logarithmic dimensions~~

~~Artem Pismenskiy [email protected]~~

~~Scientific supervisor: Prof. Dr. Pis’mak Yu.M., Physical Faculty, Saint Petersburg State University~~

We study the asymptotic behavior of Green functions for the theories φ3 and φ4 in a critical point by means of renormalization group equation and also by means of self-consistent equation. In a critical point all the dimensional coupling constants become equal to zero, only dimensionless coupling constants remain. The action for the φ3 theory is of the form d 1 23λ S[ϕ] = d x ( ∂ϕ ()) x − ϕ (). x ∫ 2 3! Here φ(x) is a scalar field, d is the space dimension, λ is a coupling constant, d 2 µ ((∂=ϕϕxx)) ∑∂µ ()∂ ϕ()x . µ=1 For the φ4 theory, d  1 22g 2  Sd[]ϕϕ=∂xx (()) − ((ϕ x)) . ∫  24!  where φ(x) is n-component vector field, d is the space dimension, g is a coupling constant, d n n 2 µ 2 ((∂=ϕϕxx)) ∑∑∂µ ii()∂=ϕϕ()x , ∑ϕϕii µ==11i i=1 First, let’s consider the renormalization group equation [1]:  ∂ ∂  µ + βγ()g + mg()Wpm (),,µ g = 0  ∂µ ∂g 

Here μ is a scale parameter, g is a coupling constant, Wm is m-point connected Green function, p denotes a set of momentums, β(g) is a beta-function , γ(g) is an anomalous dimension of the field φ. δm W = lnGJ()|. m δJ m J =0 Here G(J) is generating functional of Green function:

~~GJ()= cD∫ ϕe−SJ[]ϕϕ+ cD−−1 = ∫ ϕe S[]ϕ~~

We are interested in two-point correlation function (propagator): D = W2 (m=2). (μ ∂/∂μ + β(g) ∂/∂g + 2γ(g) ) D(p,μ,g) = 0. There is other equation, which contains ∂/∂μ and ∂/∂p [1]: 180 (μ ∂/∂μ + p ∂/∂p – 2) D (p,μ,g) = 0. Combining these 2 equations we exclude ∂/∂μ.  ∂ ∂  p + β()g − 22³g()− Dp(),,¼g = 0�  ∂p ∂g  (1) We want to find the dependence of the propagatorD on the momentum p. The solution of (1) reads −2  p   gp()2γ ()x  Dp()= D0 exp − dx ,  p  ∫ β x 0  g0 ()  where D0, p0, g0 are constants and the function g(p) fulfills the equation: p g dx −=ln p ∫ β x 0 g0 () We study infrared (p→0) or ultraviolet (p→∞) asymptotic of the propagator when g→0. From the renormalization group equation we can obtain only one of two asymptotics. To understand which of them we can obtain we should consider the beta-function. For both theories φ3 and φ4 in logarithmic dimension (d=6 for φ3 and d=4 for φ4) the main approximation of the beta-function has the form 2 β(g) = b2g , and we receive the equation: p 11 1  −=ln  −  pb02 gg0 

3 2 For the φ theory it holds b2<0 and g~λ . If λ is real then g>0 and we obtain the ultraviolet asymptotic. Usually one considers λ to be imaginary then g<0 and we 4 receive the infrared asymptotic. For the φ -theory we have b2>0, g>0 and we obtain the infrared asymptotic. For the φ3 theory in the logarithmic dimension d=6 it holds [1]: β(g) = – 3g2/2 – 125g3/72 + … γ(g) = g/12 + 13g2/432 + … and we receive  p  −−2 19/ ln ln()    p   p  125 p0 Dp()= D0 ln 1+ ++… (2)     p  p0   p0   1458   ln()   p0  For the φ4 theory in logarithmic dimension d=4 we have [1]: β(g) = g2(n+8)/3 – g3(3n+14)/3 + … γ(g) = g2(n+2)/36 – g3(n+2)(n+8)/432 +… and we obtain  p  −2 lnnln( )    p  n + 2 12()nn++23()14 p0 Dp()= D0 1− − +… (3)   2 p 4 p  p0   28()n + 28()n + 2   ln ln ()   p0 p0  181 Now, let us consider the self-consistent equation. The full propagator D fulfills the Dyson equation D–1 = Δ–1 – Σ (4) Here Δ is the bar propagator, Σ is the mass operator. Σ is presented by infinite number of 1-irreducible diagrams. We study infrared asymptotic of the full propagator D. The problem is non- perturbative, because every next term in Σ is more significant than previous one and we need to take into consideration infinite number of terms. If D and Δ have different asymptotics then we drop out from the right-hand side of (4) the bar propagator Δ and we receive the self-consistent equation: D–1 = – Σ The mass operator Σ in 1-loop approximation represents a product of 2 propagators in the coordinate representation. λ2 First of all we consider the φ3 theory with ∑= 2 The line is the full propagator D. It is technically simpler to make calculations in the coordinate representation then the equation (2) in our approximation is written as 2 λ 2 Dx−1 ()=− Dx(). 2 For solution we use the ansatz A Dx()= . xx22a (ln)α In the logarithmic dimension d=6 we obtain a=2, α=1/3 and A Dx()= xx42(ln)13/ (5) 12 with λ23A = π6 and in the momentum representation it looks like

A1 Dp()= 2 13/ pp(ln/p0 ) where A1 is a function of A. This result does not coincide with the main approximation of (2). Now we consider the φ4 theory. Introducing auxiliary scalar field ψ 2 4  ψ  ecΛϕ =−Dψ exp + ψϕ2 ∫  4Λ  we obtain the system of 2 self-consistent equations [1]

182 The solid line is the full propagator of the field φ and the dashed line is the full the propagator of the field ψ. That is Dx−1 ()=−Dx()Dx()  ϕϕψ  −1 n 2  Dxψϕ()=− Dx()  2 In the logarithmic dimension d=4 we receive the following result A  n / 4  Dxϕ ()=+221 2  x  ln xx⋅ln ln  B Dxψ ()= xx42(ln)22⋅ln ln x where n AB2 = 2π4 In the momentum representation the propagator Dφ of the field φ looks like A  A'  =+1 Dpϕ () 2 1  (6) p  ln(/pp00)l⋅ nln( pp/)

A1 is a function of A and A' is a function of n. We see that results (3) and (6) coincide in the main approximation, but it is not the case for next corrections. Logarithmic corrections at d=4 were also considered by Y. Okabe [2]. He investigated characteristics of the system behavior in the neighborhood of the critical point, which we did not calculate.

Conclusions We have investigated the infrared asymptotic of propagators for the theories φ3 and φ4 by two different methods: renormalization group equations and self- consistent equations. For the φ4 theory the results coincide only in the main approximation, and for the φ3 theory the results disagree. The possible reason is that in the renormalization group equation we wrote the beta-function taking into account the renormalization of vertex. In the self-consistent equation we did not make this. To correct result one should insert the full vertexes into the mass operator

References 1. Vasil'ev A.N. The field theoretic renormalization group in critical behavior theory and stochastic dynamics. SPb, 1998. 2. Okabe Y. // Progress of Theoretical Physics, vol. 59, № 2, 1978.

~~183 Deal.ii library as a tool to study three-body quantum systems~~

~~Shmeleva Yulia [email protected]~~

~~Scientific supervisor: Dr. Yarevsky E.A., Department of Computational Physics, Faculty of Physics, Saint-Petersburg State University~~

Introduction Few-body quantum mechanical problem is the well-known challenging problem in quantum mechanics. Provided that interparticle interactions are known, it poses the eigenvalue or scattering problem. Their solutions should be calculated with high accuracy and can be compared with experiment so that physical model and numerical approach are tested [1]. Several successful approximation techniques have been developed for few-body problem, including the Hartree Fock method, finite difference methods and various variational approximations. The finite element method (FEM) has originally been used in technical applications like elastic and fluid mechanics [2]. The implementation of the FEM in quantum mechanics was rather rare in spite of its advantages. However, starting since 1985 some works applying the FEM to three-body problems have appeared and demonstrated good results in this area [1, 3]. One of the available implementations of the FEM is the Deal.ii library. Deal. ii is the powerful general purpose object oriented finite element differential equations analysis library. It provides a wide collection of tool classes such as adaptive meshes, a variety of finite elements, parallelization, support of various formats and many other routines [4]. Use of such a frameworks will significantly simplify the process of writing the program code and let the researcher concentrate on the physical or technical problem itself. The aim of the present work is to explore deal.II and to evaluate its efficiency, stability and accuracy for quantum mechanical problems.

Results and Discussion The helium atom with the zero total angular momentum has been chosen as a benchmark of the three-body Coulomb problem. The system of the helium atom is described by the six-dimensional Schrodinger equation. It is unpractical to be solved directly by FEM. After expanding wave function in terms of Wigner D functions and using their orthogonality relation one can obtain exact three-dimensional Schrödinger equation [1] 2  2 1  ∂2 ∂−(1 c ) ∂   −  ΨΨ ∑ 2  ri 2 r+i  +V (rr1, 2,c) =E ,  i=1 2r ∂r ∂cc∂   i  i  

184 22 1 Vr( 1,rc2, )= −−+ . rr 2 2 12 rr1 − 2r12c+r2 In this work the atomic units are used, i.e. Bohr radius for the length, Hartree for the energy and the standard Jacobi coordinate system (Fig. 1), where r1 and r2 are distances between electrons and nucleus and

c=cos( rˆ12 , r ) By integrating the Schrödinger equation over the inter-electronic angle (rrˆ12, ) one can obtain the two-dimensional equation for the S-wave model Fig. 1. Jacobi coordinate system.

~~2 1 ∂2 − ˆ ΨΨ ∑ 2 ri +V( r1, r 2 ) = E , i=1 2ri ∂ri~~

ˆ 22 1 V( rr1, 2 ) =−− −[|r1 − r 2 | − | r 1 + r 2 |.] r1 r 22 rr 12 Both two- and three-dimensional equations have been investigated using FEM techniques implemented in the deal.ii library. The three-dimensional space formed by r1, r2 and chas been divided into some number of rectangular boxes numbered by i. The problem has been solved on static (Figs. 2, 4) and auto- matically refined grids (Fig. 3). The wave function has been expanded in terms of finite-element basis functions such that Fig. 2. 2D problem, Fig. 3. 2D problem, adap- input grid. tive grid.

~~Ψ(rrc1, 2, ) =∑ aim f im ( rrc1, 2, ). im The Lagrange polynomials have been chosen as the basis functions, the polynomial degree varies from 1 to 5. The~~

~~coefficients aim and the energy E are obtained by minimization of the func- Fig .4. 3D problem, input grid. tional 185 ΨΨ||H~~

That leads to the eigenvalue problem Ha = ESa,  Him, jk =f im|| Hf jk ,  Sim, jk =f im|. f jk All matrix elements are computed by the standard Gaussian integration formula. Deal.ii library provides an interface to SLEPc library, which implements few eigenvalue solvers including the Krylov-Schur solver. This solver supports all types of eigenvalue problems. It can be used for searching any part of spectrum. In this work it has been applied to solve the generalized eigenvalue problem. The ground state energy has been calculated and compared to the high precision reference results. The Table 1 and Figs. 5, 6 show the relative error of the approximate eigenvalue for the ground state.

Table 1. FEM results for the ground state. 2D problem, 2D problem, 3D problem, static grid adaptive grid static grid Reference results -2,879194803 -2,879194803 -2,903724377... Obtained approximate -2,87901 -2,87903 -2,90358 results Number of DOF 1681 5645 10086 Number of GPs 64 x 64 64 x 64 16 x 16 x 40 CPU time T (sec) 30 77 ~14000 Here DOF – the number of degrees of freedom, GPs – the number of Gaussian points, T – the computational time.

~~Fig. 5. Error of the solution in 2D Fig. 6. Error of the solution in 3D problem. problem.~~

186 With the number of degrees of freedom increasing (achieved by increasing the number of elements or polynomial degree), the error of the solution becomes smaller. In both two-dimensional and three-dimensional problems the value of obtained error is 10-4 a.u. This non-zero limit might be a consequence of Krylov- Schur solver application. Further investigation of different eigenvalue solvers is in progress. The graph for the error in the three-dimensional problem (Fig. 6) does not reflect the limit of the error. The three-dimensional problem demands substantional computational resources such as CPU time and memory, and further increasing of degrees of freedom is rather problematic.

References 1. Elander N., Yarevsky E. // Phys. Rev. A, 57(4) (1998) 3119–3122. 2. Zienkiewicz O.C., Taylor R.L. The finite element method. The basis. Vol. 1.- Butterworth-Heinemann, Fifth Edition, 2000. 3. Levin F.S., Shertzer J. // Phys. Rev. A, 32(6) (1985) 3285–3290. 4. http://dealii.org/ (A Finite Element Differential Equations Analysis Library).

~~187 Second order effects in the hyperfine and Zeeman splittings in highly charged ions~~

~~Mikhail M. Sokolov [email protected]~~

~~Scientific supervisor: Dr. Glazov D.A., Department of Quantum Mechanics, Faculty of Physics, Saint-Petersburg State University~~

Introduction Development of experimental techniques for cooling and trapping of individual charged particles has lead to the measurements of the g factor of light hydrogen- and lithium-like ions with a relative accuracy of 10-9 [1-3]. In turn, it motivates for high-precision theoretical calculations [4]. In the boron-like systems, which are of particular interest presently [5], the non-linear effects in magnetic field become considerable. In the present work the second- and third-order effects for the ground state Zeeman splitting of the boron-like argon ion 40Ar13+ are investigated. Possible influence of the second-order effects in the hyperfine interaction on the observed transition energies between the fine-structure components is estimated as well. The case of boron-like argon is considered, where high-accuracy experimental values are available [6]. The results for various argon isotopes 33Ar, 35Ar, 37Ar and 39Ar are presented. Relativistic units (ћ = c = m = 1) and the Heaviside charge unit (e2 = 4πα) are used in the work.

Zeeman splitting High-precision experimental and theoretical investigations of the g factor of highly charged ions aim at stringent tests of QED in strong field domain. These investigations allow as well for accurate determination of the values of some fundamental constants such as the electron mass and the fine structure constant. The proposed experiment with boron-like argon is based on spectroscopic determination of the frequency of forbidden transitions between the Zeeman sublevels of an isolated ion placed in a trap. In the present case of boron-like argon ion 40Ar13+ the spin of the nucleus is zero and the total angular momentum of an ion is determined only by the electrons. The hyperfine structure is absent; the scheme of the energy levels is shown in Fig. 1. Energy shift of the Zeeman sublevels of the fine structure due to the interaction with external magnetic field is determined by the expression:

~~∆Eγ ,0M () B =µ g () B BM~~

Here, μ0 – Bohr magneton, B – magnetic field strength,M – projection of the total angular momentum of the ion, γ – quantum numbers defining the level of the fine structure. The function g(B) tends to the g factor at B→0. However, in any ex- 188 Fig. 1. Energy levels scheme for the 2p state in magnetic field. The notation:

∆E =− EE(0) (0) aE=∆ (12) aE=∆ ( ) FS (2p) 2p32 2p12 ,,1 2pM12 ,=± 12 2 2pM12 ,=± 12 aEbEbEbE=∆ (31) =∆ ( ) =∆ ( 2) =∆ (3) 3 2pM12 ,=± 12,,, 1 2pM3332 ,=± 12 2 2pM2 ,=± 12 3 2pM2 ,=± 12 perimental setup the value of B is small but finite. Thus, to accurately determine the g factor from the measured value of ∆Eγ,M the nonlinear terms in magnetic field might be relevant. We consider the perturbation theory expansion up to the third order, ∆E =∆ EEE(1) +∆(2) +∆ (3) 2,pM12 2, pM12 2, pM12 2, pM12 There is one electron in the 2p state over the closed 1s and 2s shells in the ground state of the boron-like ion. Therefore we employ the one-electron approximation. The corrections of the first, second and third orders are given by standard formulas of the perturbation theory, ∆=E(1) 2p ,MVˆ 2p , M 2p12 ,M 1 2 hom 1 2 2 ∆≈E(2) 2p ,MVˆ 2p , M E(0)− E (0) 2p12 ,M 1 2 hom 3 2 ( 2p12 2p32 ) (3) ˆˆ ∆≈E 2p3 2 ,MV hom 2p 3 2 , M− 2p1 2 , MV hom 2p 1 2 , M × 2p12 ,M ( ) 2 2 ˆ (0) (0) ×−2p1 2 ,MV hom 2p 3 2 , M E2p E 2p ( 12 32 )

We restrict the sums over the spectrum to the fine structure component 2p3/2, since it provides the dominant contribution. The omitted terms are smaller by three orders 189 of magnitude. The formulas involve the matrix elements of the relativistic operator, describing the electron interaction with the external magnetic field, ˆ 1 Vhom =2 erB ⋅( n ×α ), nr ≡ r where α – the Dirac matrices. The results of numerical calculations can be presented in the following form: ∆=E[eV] zBMzBMzB [T] 1 + [T] + ( [T])2 (*) 2p12 1 ( 2 3 ) The values of the coefficients calculated for hydrogen-like wave functions are presented in the second column of Table 1. In order to take into account in a simple way the interelectronic interaction with the 1s and 2s electrons we use the approximation of effective nuclear charge. The value of Zeff is taken as to reproduce the experimental value of the fine structure ΔEFS(2p) =2.810 eV within the Dirac equation for the nuclear charge Zeff. The results obtained with Zeff =15.7504 are presented in the third column of Table 1.

Hyperfine structure The hyperfine splitting in highly charged ions can serve as a tool to test QED in strong electromagnetic fields [7] and to probe the nuclear properties. Investigations of the hyperfine splitting of low- and middle-Z ions are also motivated by astronomical search in hot astrophysical plasma [8]. In particular, the most accurate up-to-date calculations of the hyperfine splittings in boron-like ions in the range Z=7–28, including QED, correlation and nuclear effects, were presented in [9]. In this paper, we consider the effects of higher orders in the hyperfine interaction that can affect both fine and hyperfine splittings. The relativistic operator of magnetic dipole hyperfine interaction is ˆ 1 −2 Vhfi =4π er µ⋅()n ×α where μ – nuclear magnetic moment operator. This interaction leads to the shift of the energy levels depending on the total angular momentum F. Due to the conservation of F and the large values of the nuclear excitation energies on the atomic scale, only the following matrix elements are considered as non-zero, ˆ γγ1FMFF I j1 V hfi 2 FM I j2 The state vector FM F γ=FMF I j∑ CIMjm IM γ jm Mm, describes the ion with the angular momentums of the electron shell (j), of the nucleus (I), and of the ion (F). Here

FM F CIMjm 190 Fig. 2. The hyperfine structure of the 2p level for a positive nuclear magnetic moment (the nuclear spin I≥3/2). The notation: ˆ AFF=γ=γ= FM I, j 12 Vhfi FMF I, j 12 ˆ (1) (1) BFF=γ=γ= FM I, j 32 Vhfi FMF I, j 32 DEF=−3 2, FF E 1 2, are the Clebsch-Gordan coefficients, M and m – the projections of the angular momenta. To the first order of the perturbation theory the energy shift due to the hyperfine interaction is given by the diagonal in γ and j matrix elements of Vhfi. Evaluation of this matrix elements with hydrogen-like functions leads to the known expressions for the hyperfine splitting. In order to take into account the higher-order effects for the 2p state we diagonalize the corresponding Hamiltonian matrix. Since the dominant contribution is provided by the 2p state with different j, we restrict the matrix to fine-structure components. It consists of the blocks of the form (1) EC11 1 22,II±± 2 (1) CE131 I ± 2 22,I ± on the diagonal. Here we introduce the following notation for the non-diagonal matrix elements and the first-order energies of the hyperfine components: ˆ CFF=γ=γ= FM I, j 12 Vhfi FMF I, j 32 (1) (0) ˆ 1 3 EjF, = E j +γ FMF I j Vhfi γ FMF I j j=22, 191 (0) where Ej – the fine-structure components Dirac energy. Only the levels with equal total angular momenta of the ion are mixed (see Fig. 2). The diagonalization of the Hamiltonian matrix leads to the following explicit expressions for the mixing- correction to the ground-state energy in case of the nuclear spin I≥3/2, (mix) (mix) 112 2 ∆=−∆=+−E31,F E,F 22 DF4 C FF D for FI =± 2 2 ( ) (mix) 3 ∆E3 =0 for FI=±2 2 ,F Here (1) (1) DEF=−3 2, FF E 1 2, In the case of the nuclear spin I=1/2 and I=1 the correction has a similar form. The centers of gravity of the hyperfine multiplets coincide with the corresponding fine-structure levels, when the hyperfine interactionV hfi is taken to the first order. The higher-order (mixing) corrections lead to the shift of the center of gravity,

(mix)  (mix)   ∆εj =∑∑ ∆EFjF, (2 + 1) (2 F+ 1)  FF  In principle, this effect can be observed as a shift of the transition energy between the fine-structure components. The Table 2 shows the calculated values for the mixing corrections and the shifts of the centers of gravity.

Acknowledgments. Valuable conversations with V.M. Shabaev are acknowledged. The work was supported by the grant of the President of the Russian Federation (Grant No. MK-3215.2011.2) and by RFBR (Grant No. 10-02-00450).

References 1. Hermanspahn N. et al. // Phys. Rev. Lett. 84, 427 (2000); Häffner H. et al. // Phys. Rev. Lett. 85, 5308 (2000). 2. Verdú J.L. et al. // Phys. Rev. Lett. 92, 093002 (2004). 3. S. Sturm et al., Phys. Rev. Lett. 107, 023002 (2011). 4. Mohr P.J., Taylor B.N., Newell D.B. // Rev. Mod. Phys. 80, 633 (2008), and refs. therein. 5. Vogel M., Quint W. // Phys. Rep. 490, 1 (2010). 6. Draganic I. et al. // Phys. Rev. Lett. 91, 183001 (2003). 7. Shabaev V.M. et al. // Phys. Rev. Lett. 86, 3959 (2001). 8. Sunyaev R.A., Churazov E.M. // Sov. Astron. Lett. 10, 201 (1984). 9. Volotka A.V. et al. // Phys. Rev. A 78, 062507 (2008). 192 Hamiltonian Mechanics in Spaces of Constant Negative Curvature

~~Stepanov Vasiliy [email protected] Scientific supervisor: Prof. Dr. Manida S.N., Department of High Energy and Elementary Particles Physics, Faculty of Physics, Saint-Petersburg State University.~~

Introduction Investigations of non-flat spaces are motivated by existence of non-nil term in Einstein equation, corresponding to cosmological constant. Non-nil cosmological constant means curved space, and at first spaces of constant curvature must be investigated. Lagrangian and integrals of motion have been introduced for spaces of constant curvature. The aim of the present work is to introduce Hamiltonian mechanics in the anti-de Sitter space (AdS), i.e. in space of constant negative curvature, described by radius R and velocity of light c. Impulse and Hamiltonian were deduced and 10 integrals of motion were also calculated due to Noether theorem and Lagrangian symmetries in terms of dynamical variables. At first all calculations have been carried out in Beltrami coordinates. In those coordinates free particle equations of motion are linear, but free particle Hamiltonian does not coincide with free particle energy. In order to make them match new coordinates are required. Such coordinates are derived in the present work and in those coordinates operation of time translation is trivial, and Lagrangian, Hamiltonian and energy are time-idependent.

Results and Discussion Lets consider anti-de Sitter space – it is a section by hyperplane of a cone in six-dimensional pseudo-euclidic flat space zμ with metric (+,+,-,-,-,-). Anti-de Sitter space is a four-dimensional space of constant negative curvature. Symmetry group in this space are produced by a rotations in hyperplanes (zμ, zν) where μ ≠ ν and consists of 10 movement and 5 conformal transformations. Usually Beltrami coordinates used for calculations:

~~Time translation is a transformation of particular interest:~~

~~193 Lagrange function is known for anti-de Sitter space, it writes as follows:~~

10 integrals of motion can be derived due to Noether theorem: energy E, im- → pulse P (this impulse is an integral of motion and does not coincide with dynamical → → variable p→ ), dual impulse K and angular momentum J. Dual impulse appears due to symmetry of Lagrangean with respect to boost transformations.

~~Hamiltonian can be deduced by Legendre transformation of Lagrangian.~~

~~where p→ – impulse equal to~~

Obviously Hamiltonian is time-dependent and consiquently not coincide with energy because energy is an integral of motion deduced by Noether theorem from Lagrangian. To make Hamiltonian time-independent new coordinates were introduced:

~~In these coordinates time-translation is trivial and Lagrangian, Hamiltonian, energy and other integrals of motions writes as follows:~~

194 → → It is interesting, that P and - K are rotated by time translation. Also it is known that anti-de Sitters space is described by two constants: light velocity (c), and radius of curvature (R). Minkowsky space is a flat limit of anti-de Sitter space (with R→∞). Galilei space is a non-relativistic limit of Minkowsky space (with c →∞). Lorenz-Fock space is a non-relativistic limit of anti-de Sitter space (with c →∞). ParaGalilei space is a flat limit of Lorenz-Fock space (withR→∞ ). Let c tend to infinity in order to expand Hamilton function in a Taylor series and consider the two highest terms:

~~Let R tend to infinityand consider two highest terms in the new expansion of Hamiltonian:~~

~~The most interesting term is~~

Although both R and c are infinite we will consider theirs ratio as finite. Lets 2 introduce new constant T=R/c and denote mc by self-energy Eself. Now the highest term of Schrodinger equation writes as follows:

~~The same equation can be obtained by the same expension but with other order of tension to infinity toR and c. This equation describes a harmonic oscillator:~~

~~with energy of and oscillation frequency~~

This solution does not coincide with well-known plane-wave solution of free particle Schrodinger equation in Galilei space. But plane-wave solution is a limit of T→∞ of the obtained one. Because T is a very big constant (about age of the Universe), its conribution is small, and neglectable in most cases. Therefore new solition of free particle Schrodinger equation is obtained due to non flat structure of the space.

Conclusions In this article hamiltonian function is obtained for anti-de Sitter space and integrals of motion are calculated. Due to time dependence of hamiltonian new coordinates are introduced. In these new coordinates some new facts appear: 195 1) New integrals of motion inherit the rotation symmetry of AdS. 2)Hamiltonian Taylor series in nearly flat and non-relativistic case show new Schrodinger equation for free paticles. Consequently new solution appears. 3) New solution desribes a harmonic oscillator with period about the age of the Universe instead of a plane wave. 4) New results does not contradict with old ones, because plane wave is a limit of obtained solution.

References 1. W. de Sitter. Monthly Notices of the Royal Astronomical Society, Vol. 71, p. 388-415, 1911. 2. Manida S.N. Additional Chapters of Physics. Mechanics, p. 57, Saint-Petersburg, Faculty of Physics printing establishment, 2007.

~~196 3D isotropic random walks with exponential distribution on free paths. Application to evaporation of a droplet at transient Knudsen numbers Telyatnik Rodion [email protected]~~

~~Scientific supervisor: Prof. Dr. Adzhemyan L.Ts., Department of Statistical Physics, Faculty of Physics, Saint-Petersburg State University~~

1. History and basic theory of random walks The classical problem of random walks (RW), first formulated in the letter to Nature journal in 1905 by Pearson [1, 2], who investigated the spread of mos- quito populations, is the question about the probability to find a random walker at certain distance from the start point after n steps, which can occur in general at random direction and random distance with given probability distributions (the simplest Pearson’s case is the isotropic walks with constant step length λ). Earlier in 1880 for similar problem of composition of n iso-periodic vibrations with unit amplitude and random phase in the limits n→∞ and λ→0 Rayleigh found Gaussian asymptotic solution [2] that reflects modern Central Limit Theorem (CLT) about the sum of random independent identically distributed values. In 1905 Einstein obtained [3] this asymptotic solution as Green function for diffusion equation, which he derived in the same limits from the recurrent equation for Markov chain of RW that expresses the full probability of a position by the sum of all conditional probabilities, also known as Kolmogorov-Chapman equation [4], first proposed by Bachelier in his Ph.D. thesis (1900) for modeling random prices in a stock market [1]. We generalize this equation for RW chain with k-steps links:      ρ(r ) = dr P ( r − r ) ... dr P ( r−ρ r ) ( r ) = dr P ( r − r ) ρ ( r ) (1.1) n n∫ n−11 n n − 1 ∫∫ 01 1 0 0 0 ∫ nk− k n nk − nk −− nk

Here ρn(rn) is the probability density function (PDF) for a particle to occur at point rn after n steps of RW, and Pk(rn-rn-k) is the k-steps transition probability density function (TPDF) for a particle to occur at point rn after k steps of RW from the place at point rn-k (in general TPDF can depend on step number n). In essence PDF represents the local concentration of particles normalized to their total amount in the system. If we make change r’= rn-rn-1 in (1.1) for k=1 and make Taylor expansion of ρn-1(rn-r’) near rn with Einstein’s condition |r’ ln ρn-1(rn)| << 1 of relatively small change of concentration on step distance r’, we will get equation, which we write for generality for d-dimensional space: ∇     1 d d ∂2ρ ()r   ρρ()rP= ()rr′ ()− rr′∇+ρ () rr′′ n−1 n + ... dr′ nn ∫ 11 nn−−nn1 ∑∑ αβ  (1.2)  2 α=1 β=1 ∂∂rrαβ 

Let P1(|r’|) be an isotropic one-step TPDF and let angle brackets <…> denote averaging with the TPDF. As it is normalized to unity (the probability to occur anywhere), the equation (1.2) in notation r≡rn turns to: 197 1  ρρ()rr=+() rr′ 22∇+ρ () ... (1.3) nn−−1 2d n 1

The expansion (1.3)by m-th moments of TPDF is known as Kramers-Moyal expansion [1,4]. Setting ρ(r,t=nτ)≡ρn(r) for n→∞, where τ is the mean time of RW step, and dividing by τ→0 one can obtain diffusional equation: ∂ρ(,rt)  1 2 1 =∆Drρ(,tD)l , ≡ im r′ (1.4) ∂t 2d τ→0 τ Rigorously we should expand (1.3) by times as well as by space [1, 5]. For d = 3 and = 2λ2 (e.g. for exponential TPDF, see (2.1)), where λ is mean free path of a particle, one can see usual expression for D = λ/3, where is mean velocity. So called anomalous diffusion happens if TPDF hasn’t finite moments.

For one particle flying out the origin of coordinates with ρ0(r0)=δ(r0) according to (1.1) PDF after n steps is essentially TPDF for n steps, i.e. ρn(r)=Pn(r), and it is described for n→∞ by Green function of (1.4) that represents CLT: r2  −  1 4Dt Prn ()==ρτ(,rt n )(→ d 2 er , ρδ0 )(= r) (1.5) n→∞ ()4πDt Corrections to the asymptotic (1.5), known as Gram-Charlier expansion [1, 6] (by Chebyshev-Hermite polynomials defined by the derivatives of Gaussian exponent), can be constructed by Laplace asymptotic calculation for Fourier transform (FT) of PDFs, i.e. by characteristic functions with their moments and cumulants in terms of probability theory. Indeed, chain equation (1.1) is the convolution, which is turned by FT to the algebraic expression:       ρ(r ) = ( P ∗ ... ∗ P ∗ρ )( r ) , ρˆˆ ( q ) = PPqqˆnn ( ) ρ ( ) , Pq ˆˆ ( )= Pq ( ) n1    10 nn1 0 1 (1.6) n FT of an isotropic function in spherical coordinates can be expressed with Bessel functions of first kindJ ν(z) [2, 7], e.g. for one-step TPDF: ∞ d 21−   d 2 1  Pˆ (| q |)= e−−iqr P ( r ) dq =π 2 rd 1 J( qr ) P (| r |) dr (1.7) 1 ∫∫1 ( )  d 21− 1 0 qr Formal solutions obtained by reverse FT of (1.6) can’t be evaluated in elementary functions for most cases of n, d and P1(r), e.g. for n-steps TPDF: ∞ d 21−  1  11  P(| r |) ==eiqr Pˆˆ n () q dq qd −1 J( qr ) Pn (| q |) dq nddd∫∫1 2  21− 1 (1.8) (22ππ) ( ) 0 qr

~~For isotropic RWs it is simpler to work with radial PDF (RPDF) σn(r) and radial~~

TPDF (RTPDF) Sn(r) expressed by volume density probabilities multiplied by the surface area of d-dimensional sphere with radius r (Γ(z) is Gamma function): dd−−12 dd12 σρnn(r )≡ (| rr |) 2 π Γ≡ ( d 2) , Srnn ( ) P (| rr |) 2π Γ ( d 2) (1.9)

In essence σn(r) is the normalized surface concentration in centrally symmetric system, and Sn(r)dr is the probability for a particle to occur after n steps from the origin of coordinates at the distances in the interval [r,r+dr]. Asymptotic (1.5) for 198 Sn(r) takes form of d-dimensional Rayleigh distribution. In spherical coordinates chain equation (1.1) hasn’t convolution form, it can be rewritten as:  ∞  σρ()r== S (, r r ) ( r ) dr S (,| r r |) σ ( r ) dr (1.10) n ∫∫k nk− nk −− nk nk − k nk− nk −− nk nk − 0

Here Sk(r,rn-k) is the probability, which we call spherical TPDF (STPDF), to fly out the point rn-k and to occur after k steps at the distance r from the origin of coordinates, at that Sk(r,0)=Sk(r) defined in (1.9). Although STPDF doesn’t depend on direction of the vector rn-k (from the symmetry of the problem) we keep its designation to remember its point-to-sphere transition nature.

2. Random walks with exponential distribution on free paths If a lot of particles experiencing RWs have constant mean free path λ (in effective sense for long-ranged forces), which is defined by surrounding system and independent on concentration of particles themselves (e.g. Brownian particles or vapor in passive gas), then their random free paths at mean free time τ have exponential distribution [8] that becomes RTPDF (1.9) for one step: r ∞∞ 1 − Sr()==eSλ , ()rdrr1 , mm==rS()rdrmλm ! (2.1) 11∫∫1 λ 0 0 Random free times have the same exponential distribution for fixed free path equal to λ if we substitute r=(λ/τ)t in the integral in (2.1), that’s how RW problem can be formulated in terms of continuous time RW (CTRW model of Weiss [1, 2]), which is widely used in RWs on lattices. Another than (2.1)distribution on free paths Tait derived in 1886 by another definition of averaging of free paths (by amount of collisions not in unit time, but from the chosen moment to the next collision, see differences in [9]). STPDF can be derived from RTPDF by means of the probability dPr to occur in spherical layer segment with center at point 2 r0 and volume dV’=dφ’sinθ’dθ’r’ dr’ (where r’=r-r0, see Fig.1) written in corresponding Fig. 1. RW in spherical coor- spherical coordinates that can be expressed dinates. from original ones:  2 2 rr′(,θθ|)rr0 =−2rr00cos + r , ϕϕ′ =  (2.2) sinsθθ′ =⋅in rr′ , cosscθθ′ =−()rros 0 r′

~~rr′(,θ) − ∂θθ′ ∂∂θ′ ∂r dV ′ 1 λ r sin θ dSPr = 1 ()r′ 2 = e ddϕθdr (2.3) 4πr′ 4πλ rr′(,θ) ∂r′ ∂∂∂θ rr′ ∂~~

Integral of dPr over φ and θ will give us S1(r,r0)dr. After evaluating Jacobian de- terminant in (2.3) with the help of (2.2) and integrating upon changing variable θ to ξ=-r’(r, θ)/λ [-(r+r0)/λ , -|r-r0|/λ] we will finally obtain STPDF: 199 ∈ −+()rr0 λ    1 r 11ξ r  rr+ 0   rr− 0  Sr10(,r ) ==v.p. edξ Ei  −  −−Ei    (2.4) 2λξr ∫ 2λλr    λ  00−−rr0 λ  

Principle-value integral in (2.4) (as r=r0 is logarithmic singularity, but we neglect the probability to remain on 2D surface after 3D RW step) can’t be taken in elementary functions and it is represented by exponential integrals Ei(x). Modulus in range of definition of ξ expresses the ambiguity of particle position, which is observed under angle θ (see Fig.1). From the chain equation with links of single steps and σ0(r0)= δ(r0) one can evaluate Sn(r)=Sn(r,0): ∞ ∞ ∞  Sr()==v.p. dr Sr(,rr)(δ )(v.p. dr Sr,)rd... rS(,rr)(δ r ) nn∫ 000 ∫ nn−−11 10∫ 11100 (2.5) 0 0 0 So, RTPDF (2.5) plays role of Green 0.25 RW simulation data function whose asymptotic for big S (r)=0.09 r2e xp(-r1.4/ 3.4) 10 0.2 Asymptotic (1.5) for t=n enough n (factually for n≥30) is described Formula (2.6) with a=0.07, d b =0.77, c=0.0055 by (1.5) with (1.9). For any n RTPDF 0.15 Sn(r) can be approximated within simula-

~~( r )~~

~~S tion data (Fig. 2) by the general function 0.1 ar2exp(-rb/c) with varying constants a,b,c 2 0.05 connected by normalization (power of r ahead of exponent is from space dimen-~~

~~2 4 6 8 10 12 sion, so it is fixed). RTPDF (2.5) can be~~

~~r ( = 1) roughly evaluated analytically near ap-~~

~~preciable singularities rk=rk-1 (k=1,…,n)~~

~~Fig. 2. Sn(r) for n=10 compared to its from (2.4) except the special cases rk close asymptotic. to zero with the help of asymptotes of Ei(x) in zero and infinity [10]. In this way for 2~~

~~200 R ∞  ρπ⋅=4 Rd2 Rdv.p. rdrS(,rr)(σ r ) + ∫ ∫ 01 000 (3.1) 0 R~~

While growing up on dR+ the droplet catches neighboring vapor particles resulting in the new increment accounting from iterative procedure similar to (3.1). Particles flying out the surface of the droplet with constant probability α (in the unit time; quasi-stationary case) with the same λ (temperature) and initial surface concentra- 2 + tion σ0(r0)=α ρ4 R (r0-R) have new surface concentration σ1 (r) like but with condition θ’ [0,π/2] in for the limiting tangent line (see Fig. 1 meaning r0=R) 휏 휋 훿 that results in new STPDF SR(r) in place of (2.4): ∈ −1 22 Sr()≡ 2λλrR⋅−Ei rR− −−Ei rR− λ  , rR> (3.2) R ()  ()( ())

++ 2 2 ρπ1 (| rr|)44==σα1 ()rRτρ π SrR () (3.3) 3 But such evaporation causes radius decrease R(t)=R0-αtl, where l is the mean volume of liquid phase attributed to one particle. TPDFs, i.e. RW steps, are defined only on time interval τ, so we can’t use (3.2) in principle if decrement dR-=-ατl is significant, nevertheless by substitutingR (t) into (3.3) and replacing τ by t (linear interpolation) and integrating (3.3) by t [0, ] we can obtain continuous interpolation between RW steps that smooths numerical noise and cancels logarithmic singularity r=R in (3.2). Stationary vapor∈ concentration휏 profile can be derived by differentiating by R the equation with change dR+ on dR=dR+-dR-=Const that leads to Volterra integral equation of second kind: R ∞  ρπ(| RR|)442 v.p. Sr(,Rd)(rS= v.p. Rr,)ρπ(| rr|) 2dr − 8πρConstR ∫ 1 ∫ 10 000 (3.4) 0 R

First iteration of (3.4) with constant ρ(r)=ρ∞ for Kn 1 gives Maxwell’s diffusive (0) (0) stationary solution ρ(r)=ρ∞+(ρR -ρ∞)R/r [12] with boundary values ρR and ρ∞, and for arbitrary Kn it gives required correction to this≪ asymptotic:

2r  1+ Lr() 1− LR() 1+ LR()  R λ  −  λ (3.5) ρρ(| r |) ~ ∞∞+  ρρR −  , Lr()≡−1 e  1− Lr() 1− Lr() 1− Lr()  r 2r   (0) Concentration (3.5) with its boundary value ρR=(ρR +ρ∞L(R))/(1-L(R)) exceeds the Maxwell’s one, that is known as Langmuir “jump” of concentration near drop’s surface. Other treatment based on kinetic Boltzmann equation is in [13].

~~This work was supported by the RFFI grant № 10-03-00903 and by SPbSU research effort № 0.37.138.2011.~~

References 1. Bazant M.Z. et al // MIT lecture notes №18. 325 (2001), №18. 366 (2006). 2. Hughes B.D. // Random walks and random environments, v.1, ch. 2 (1995). 3. Einstein A. // Annalen der Physik (ser. 4), v. 17, p. 549–560 (1905). 4. Risken H. // The Fokker-Planck Equation, ch. 2, 4 (1989). 201 5. Aranovich G., Donohue M. // Molecular Physics, v. 105, №8, p. 1085 (2007). 6. Gnedenko B., Kolmogorov A. // Limit distributions for sums of independent random variables, ch. 8 (1968, Russian edition in 1949). 7. Kingman J.// Acta Mathematica, v. 109, p.11-53 (1963). 8. Feynman R.P. et al. // The Feynman lectures on physics, ch. 43 (1963). 9. Whitman J.// Ph.D. Thesis, Johns Hopkins University, Baltimore (2010). 10. Gradshteyn, Ryzhik // Table of integrals, series and products, ch.8.2 (1980). 11. Gentle J.// Random number generation and Monte Carlo methods (2005). 12. Fuks N.// Evaporation and droplet growth in gaseous media, ch. 1, 5 (1959). 13. Lushnikov A. et al // Fizika aerodispersnyh sistem, v. 37, p.7 (2001).

~~202 Investigation of the dependence of the number of binary interactions and the number of participants on the class of centrality in ultrarelativistic heavy ion collisions~~

~~Vorobyev Ivan [email protected]~~

~~Scientific supervisor: Prof. Dr. Vechernin V.V., Department of High Energy and Elementary Particles Physics, Faculty of Physics, Saint-Petersburg State University~~

Introduction The investigation is connected with the activity of the SPSU research team, including into the ALICE at LHC and NA61/SHINE at SPS collaborations at CERN. The SPSU team searches for the long-range correlations, which can be considered as one of the signs of quark-gluon strings fusion in high energy heavy ions collisions. In the usual version of quark-gluon string model [1] there is no interaction between the strings. In hadron interactions the quark-gluon strings are formed and then in the process of their fragmentation the observable hadrons are produced. In the case of high energy heavy ion collisions one needs to take into account interaction between strings [2], which in turn changes their fragmentation. As quark-gluon string is an extended object in the rapidity space, i.e. contributes by fragmentation to wide rapidity range, one can expect appearance of the correlations between observable quantities in distant rapidity intervals (see, for instance [3]).

The number of strings Nstr formed in the high energy nucleus-nucleus collision depends on the number of binary nucleon-nucleon (NN) collisions Ncoll and the number of nucleon-participants Npart. This dependence can be parameterized as follows [4]: Nx=+NNNN ()1− xN strstr partcoll (1) NN where the parameter x and the number of strings per NN-interaction N str depends only on the energy of interaction. Usually for simplicity one supposes that the event by event fluctuations of the number of strings at fixed impact parameter value obey the Poisson distribution [4], which means that DN str = 1 (2) Nstr In present work it was shown numerically that the fluctuations of the number of binary interactions and the number of participants don’t obey the Poisson distribution, which affects the fluctuations of the number of strings. Using the C++ MC simulation code of AA collision, developed in the present work, the normalized dispersions of the numbers of wounded nucleons and binary interactions were calculated both at fixed impact parameter and at one corresponding to certain centrality classes.

203 Model of AA scattering In this work one uses Glauber model with the standard Woods-Saxon approximation for profile function of colliding nuclei [5]: rR− Ta()==dzρρ()rr,,22az+=2 ()r ρ (e1+ xp A )−1 (3) A ∫ 0 k 1/3 with RA=R0A , R0=1.07 fm, k=0.545 fm, ∫daTA(a)=1. For NN-interaction cross- section we use empirical formula [4]: NN 21− σin =−(.32 08 1.l574 n.EE+⋅0 6622ln ) 10 fm (4) where E is the energy in GeV per NN-interaction in the center-of-mass system. We also consider two alternative versions for the probability of NN-interaction, the so-called “black disc”: NN 2 σσ()rr=−Θ()Nirr, n = π N (5) r2 and the Gaussian one: − 2 rN NN 2 σσ()re==, in πrN (6) where r is the distance between the centers of colliding nucleons in the impact parameter plane, and rN is the radius of NN-interaction, which depends on the energy of collision (see (4)-(6)). For numerical modeling of heavy ion high-energy collisions the following algorithm was realized as C++ code. First, according to the nucleus profile function (5) we distribute certain number of nucleons (207 for Pb) for each of colliding nuclei in the impact parameter plane. Then taking into account the impact parameter we determined the distance r between all pairs of nucleons. If σ(r) has occurred more than some parameter t (t is the random parameter, evenly distributed in the interval [0, 1]), then we consider these nucleons to interact with each other. Next we calculate number of wounded nucleons and number of NN-interactions in each collision, repeating (up to 50000 times) this cycle for fixed impact parameter (or centrality class) and storing required average values. So it is possible to calculate various values for any AA collisions (it is not even necessary for colliding nuclei to be the same) at given profile functions of colliding nuclei and probability of NN-interaction. Verification of the code For this purpose let’s consider the simulation of Pb-Pb collisions for SPS energy (Super Proton Synchrotron, 17 Gev per NN-interaction in c. m. system) at fixed impact parameter, because similar calculations have been carried out earlier in [5]. NN 2 For E=17 GeV according to (4)-(6) we have σ in = 3.14 fm and rN =1 fm. We see in Fig. 1 that all our calculation results are in a good accordance with the results obtained earlier (Fig. 1). Subdividing into centrality classes In real experiment, of course, we can’t fix impact parameter, so one have to deal with impact parameter fluctuations of order about 2-3 fm. So in a real experiment the impact parameter range is divided into several parts, so-called centrality classes. One classifies events by centrality classes and then in each class we calculate necessary values and average them on the events. There are different methods using in a real experiment for dividing events into centrality classes, but usually 204 it can’t be done with accuracy better than 3 fm, at best 1.5 fm. We will consider both these cases: 3 fm and 1.5 fm.

Fig. 1. 17 GeV Pb-Pb collisions at fixed impact parameter. Left – doubled correlator between the numbers of wounded nucleons in colliding nuclei vs impact parameter; right – relative dispersion of NN-collisions number vs impact parameter. Results of our code marked as line with triangles For a generation of random impact parameter one uses a following formula:

bb= max y (7) where bmax is a constant, at which the probability of nucleus interaction is insig- nificant and y is a random parameter, evenly distributed in range [0, 1]. It is easily to see from (7) that we have: 2 2 bmax db= dy (8) 2π which corresponds to uniform distribution in impact parameter plane.

Calculations results Let’s consider results for Pb-Pb collisions simulation at SPS energy (1 GeV per NN-interaction). As we can see in Fig. 2, relative dispersions both for number of participants and number of collisions don’t equal to 1 as it would be for Poisson distribution, so it is obvious that in real experiment the distributions of these quantities considerably differ from Poisson one. Moreover, with the size of centrality class grows up, dispersions also increase. In Fig. 3 similar results are represented for the case of LHC energy (E=5.5 TeV NN 2 per NN σ in = 6.75 fm , rN = 1.45 fm), and again we can see huge difference between Ncoll and Npart relative dispersions and Poisson case.

Comparing results at different energies With the energy increases from 17 GeV to 5.5 TeV per NN interaction, the radius of this NN interaction according to (4)-(6) enlarge from 1 fm to 1.45 fm, that is, nucleons become enlarged (expanded). One can explain this fact something like this way: with the energy increases, additional virtual quark-antiquark pairs are formed a bit further from nucleon’s center of mass in impact parameter plane, and nucleons effectively become larger. Number of wounded nucleons in one nucleus increases insignificantly, which is quite natural, as well as relative dispersion of this value (Fig. 4).

~~205 black_disc_1.5fm black_disc_1.5fm black_disc_3fm black_disc_3fm gauss_1.5fm 30 gauss_1.5fm gauss_3fm 10 gauss_3fm bla c k _ fix b bla c k _ fix b gauss_fixb gauss_fixb pois s on pois s on 20~~

~~5 10~~

0 0 0 8 16 0 8 16 impact parameter b, fm impact parameter b, fm Fig. 2. 17 GeV Pb-Pb collisions for different centrality classes. Left – relative dispersion of number of wounded nucleons vs impact parameter; right – relative dispersion of NN-collisions number vs impact parameter. Poisson case is represented as straight line equals to 1 at the bottom of graph.

~~black_disc_1.5fm black_disc_1.5fm black_disc_3fm black_disc_3fm gauss_1.5fm gauss_1.5fm gauss_3fm gauss_3fm 60 bla c k _ fix b bla c k _ fix b 10 gauss_fixb gauss_fixb pois s on pois s on~~

~~30 5~~

0 0 0 8 16 0 8 16 impact parameter b, fm impact parameter b, fm Fig. 3. 5.5 TeV Pb-Pb collisions for different centrality classes. Left – relative dispersion of number of wounded nucleons vs impact parameter; right – relative dispersion of NN-collisions number vs impact parameter.

~~bla c k _ L E 6 gauss_LE 200 bla c k _ H E gauss_HE bla c k _ L E gauss_LE bla c k _ H E gauss_HE~~

~~3 100~~

~~0 0 0 8 16 0 8 16 impact parameter b, fm impact parameter b, fm~~

Fig. 4. Pb-Pb collisions at different energies - 17 GeV and 5.5 TeV. Left – Number of wounded nucleons in incoming nucleus vs impact parameter; right – relative dispersion of number of wounded nucleons in both nuclei vs impact parameter. Results for higher energy are marked as black squares and circles, for lower energy as white ones. All results are represented for case of 1.5 fm centrality class. But for number of collisions picture is completely different (Fig. 5). For all of centrality classes number of interactions increases two times, which is a consequence of nucleons “swell” – now each nucleon interact with greater amount of nucleons in other nucleus, and general number of collisions increases significantly. 206 30

~~bl a c k _ L E 2000 gauss_LE bl a c k _ L E bl a c k _ H E gauss_HE gauss_LE 20 bl a c k _ H E gauss_HE~~

~~1000~~

0 0 0 8 16 0 8 16 impact parameter b, fm impact parameter b, fm Fig. 5. Pb-Pb collisions by different energies - 17 GeV and 5.5 TeV. Left – Number of NN-collisions vs impact parameter; right – relative dispersion of number of wounded nucleons in both nuclei vs impact parameter. Results for higher energy are marked as black squares and circles, for lower energy as white ones. All results are represented for case of 1.5 fm centrality class. As consequence, relative dispersions of this value also differ of one another quite considerably. One can explain this result as follows: number of wounded nucleons increases insignificantly only thanks to nucleons on the periphery of interacting area of nucleus. But as nucleons become thicker, each nucleon inside the interacting area now interacts with greater amount of nucleons in other nucleus, and general number of NN-interactions increases significantly (about two times in each centrality class). Conclusions a) Relative dispersions of number of wounded nucleons and number of NN collisions differ from Poisson one even for case of fixed impact parameter. Moreover, as we allow the impact parameter to fluctuate, relative dispersions increase even more, so in real experiment distributions of Ncoll and Npart (therefore of Nstr as well) are far away from the Poisson one.

b) There is week dependence on energy for number of participants Npart. By contrast, number of collisions (as well as its relative dispersion) enlarges quite significantly. c) There is almost no difference between results obtained with using of different versions for the probability of NN-interaction - the Gaussian and the “black disc” at condition that the total cross-sections are the same. References 1. Kaidalov A.B. // Phys. Lett. B 116, 459, (1982); Kaidalov A.B., Ter-Martirosyan K.A. // Phys. Lett. B 117, 247, (1982); Capella A., Sukhatme U.P., Tan C.-I., J. Tran Thanh Van // Phys. Lett. B 81, 68 (1979); Phys. Rep. 236, 225 (1994). 2. Braun M.A., Pajares C. // Phys. Lett. B 287, 154 (1992); Nucl. Phys. B 390, 542; 549 (1993). 3. Braun M.A., Kolevatov R.S., Pajares C., Vechernin V.V. // Eur. Phys. J. C 32, 535 (2004). 4. Vechernin V.V., Kolevatov R.S. // Phys. of Atom. Nucl. 70, 1797; 1809 (2007). 5. Vechernin V.V., Nguyen H.S. // Phys. Rev. C 84, 054909 (2011); Vechernin V.V. // Relativistic Nuclear Physics and Quantum Chromodynamics, vol.2, JINR, Dubna, 2008, pp.88-94; hep-ph/0702141. 207

~~H. Biophysics Application of Surface Plasmon Resonance for Detection of DNA Immobilization on Gold Surface~~

~~Fironov Alexander [email protected]~~

~~Scientific supervisor: Prof. Dr. Kasyanenko N.A., Department of Molecular Biophysics, Faculty of Physics, Saint-Petersburg State University~~

Introduction Nanoplasmonics is a rapidly developing branch of science. Phenomena studied by nanoplasmonics have found their application for wide variety of tasks, such as creation of biosensors, therapy and visualization of tumors and much more. Creation of a thin metal film modified by DNA is the first step to in-depth investigation of DNA interaction with various compounds using surface plasmon resonance (SPR). Functionalized self-assembled monolayers (SAMs) have been employed to immobilize DNA on a gold surface, based on covalent bonding attachment. For example, 11-mercaptoundecanoic acid (MUDA) SAMs were employed to immobilize DNA on a gold surface for AFM imaging, based on carbodimide covalent coupling. Divalent cations, such as Mg2+, Ni2+, Zn2+, and so on, have been widely employed to immobilize DNA on a mica surface for AFM imaging This method is based on a ‘‘salt bridge’’ effect mediated by divalent cations between the negatively charged mica surface and the negatively charged DNA. By choosing the appropriate cation as a bridge ion, a weak electrostatic attachment to mica could be obtained. In this paper an attempt to immobilize DNA on gold surface using 2+ thioglycolic acid (HSCH2COOH) and Mg ions as connecting agents was made. The thioglycolic acid (TGA) was first self-assembled onto a gold surface to produce a negatively charged surface. Then DNA was attached onto this surface via the divalent cation bridge. The technique based on a surface plasmon resonance, which is very sensitive to modification of the metal surface was used for the detection of DNA biding. SPR phenomenon connects with the conduction electrons oscillation in the metallic lattice. This collective oscillation when excited by light with specific wavelength at the angle of total inner reflection, TIR (the light frequency must be the same as for inner oscillation of nonvalent electrons in metallic lattice) produces the surface plasmons. Known as a surface plasmon resonance (SPR), this phenomenon results in unusually strong scattering and absorption properties. In the reflected light the SPR frequency disappears. One can see the darkness in spectra. The device fixes the intensity dependence on the angle for the determination of TIR angle which depends on reflecting constant and, therefore, on mass of adsorbed substrate. In addition to data obtained from SPR method, gold surface was studied with atomic force microscope (AFM). 210 Results and Discussion Gold chip was immersed in TGA for 24 hours in order to obtain TGA-modified sample. After sample was taken out of TGA and rinsed with distilled water. Then during another 24 hours sample was reacted by (covered by) NaCl solution of DNA with ionic strength of 0,005M and DNA concentration of 0,0135%. The measurement of SPR curves was carried out with Nanospr8 model 481 device. The dynamic of interaction can be observed in the experiment. Eight different combinations of reacting compounds were taken into consideration. In the Figs.1-3 a 24 hours dynamics of relative shift of SPR angle is presented.

~~Fig. 1. Dynamics of gold surface modification with TGA.~~

~~211 Fig. 2. 24 hours SPR measurement for non-modified gold surface.~~

~~Fig. 3. 24 hours SPR measurement for modified gold surface.~~

212 According to the information presented on these figures we suppose that TGA interacts with gold and forms a film on its surface. Also we can observe large shift of SPR angle for DNA with Mg2+ ions interaction with TGA-modified gold surface. But AFM image, obtained with NanoScope 4a (Veeco), of this sample doesn’t confirm our suggestion.

~~Fig. 4. AFM image of DNA with Mg2+ ions on TGA-modified gold surface.~~

~~Fig. 5. Dependence of SPR angle shift on different ionic strength of the solution.~~

213 Conclusions According to experimental data the modification of gold surface with thioglycolic acid and DNA is observed, but further investigation of mechanics of interaction is suggested. Subsequent consideration of the possibilities of SPR technique for DNA examination and the improvement of measurements should be carried out.

~~References 1. Yonghai Song, Zhuang Li, Zhiguo Liu,Gang Wei, Li Wang, Lanlan Sun // Microscopy Research And Technique 68:59–64 (2005). 2. Klimov V.V. Nanoplasmonics.- Moscow: Fizmatlit, 2009.~~

~~214 DNA Interaction with Palladium Compound K2[PdHGluCl2] in vitro~~

~~Kozhenkov Pavel [email protected]~~

~~Scientific supervisor: Prof. Dr. Kasyanenko N.A., Department of Molecular Biophysics, Faculty of Physics, Saint-Petersburg State University~~

Introduction The coordination compounds of metals from Platinum group play the important role in antitumor therapy. Amazing results were reached with the using of Platinum drugs, but only for certain kinds of tumor. Most widely used and successfully applied drug cisplatin prevents DNA replication in tumor cells. Unfortunately, cisplatin generates the serious side-effects (toxicity and non-specific influence). This circumstance stimulates the synthesis of new compounds for the selection of non-toxic medicaments with high antitumor activity. The coordination compounds of other metals – palladium, titan, ruthenium, etc. are tested for antitumor activity via the binding with DNA in a solution.

The influence of Palladium complex K2[PdHGluCl2] on DNA conformation in vitro was investigated in current report. The structure of Palladium complex was calculated. Methods and materials The change in the Palladium coordination sphere due to the aquation is analyzed. Quantum-mechanical calculations of a molecular structure of [PdHGluCl2] with two stages of aquation (the replacement of one and second chlorine atoms by water molecules) have been carried out with software packages HyperChem 8.0 and GAMESS (FireFly 7.1g). The unlimited Hartree-Fock method and bases SBKJC VDZ ECP for Palladium, DH for Hydrogen atoms and 6,31+G* for all other atoms have been used. DNA circular dichroism (CD) spectra were registered with Mark 4 (Jobin Ivon, France) autodichrograph. The absorption spectra of components in a solution were obtained with spectrophotometer SF-56 (Russia). Atomic Force Microscopic (AFM) images of DNA and its complexes with Palladium compound (DNA- Pd complexes) have been received by means of microscope NanoScope 4a (Veeco) in a taping mode on air. DNA fixation on a mica surface was carried out by the spontaneous adsorption of DNA molecules from a solution containing magnesium ions. DNA interaction with Palladium compound was studied in 0,005 M NaCl or 0,15 M NaCl with the variation of DNA and Pd concentration.

~~Results and Discussion~~

Fig. 1 shows the absorption spectrum of K2[PdHGluCl2] solutions in distilled water, 0,005 M NaCl and 0,15 M NaCl at room temperature. One can see that the 215 addition of salt into K2[PdHGluCl2] solution leads to the long-wave shift of the maximum. The observed changes in the absorption spectrum associated with the state of the coordination sphere of the complex ion.

Fig. 1. The absorption spectra of K2 [PdHGluCl2] solutions at different NaCl concentrations. Fig. 2 shows the experimental results obtained by circular dichroism method. CD spectra of DNA in complex with palladium compound in 0,005 M NaCl solution for the different concentrations of the tested palladium compound C(Pd) are presented. It is clear from the Fig. 2 that DNA interacts with compound. At low C(Pd)=3×10-5M we can not see any changes in DNA CD spectrum. At C(Pd)> 9×10-5M the similar CD spectra are registered. The state of DNA double helix is identical and differs from free DNA. It can indicate the filling of vacant binding

~~Fig. 2. CD spectra of DNA complexes with~~

~~K 2[PdHGluCl2]in 0,005 M NaCl at constant concentration of DNA and varying concentrations of 5 K2[PdHGluCl2]: C(Pd)×10 , М= 0 (0),25(1), 19 (2), 13(3), 9(4), 3(5).~~

216 positions for Pd in that kind of complex. The exceeding of Pd concentration up to 25×10-5M may be leads to the perturbation in DNA-Pd complexes due to alternative binding mode. The absorption spectrum of Pd compound has a band out of DNA spectrum. Fig. 3 demonstrates that this band modifies during interaction with DNA (blue shift of the maximum and drop in intensity). We can not see the significant change in band for free Pd and Pd in complex with DNA during the time. The comparative examination of the spectra showed that the maximum of K2[PdHGluCl2] in DNA solution (371 nm) does not shift within a week as well as a maximum of absorption band for free K2[PdHGluCl2] (378 nm). It should be noted that similar experiments were carried out in solution with distilled water and 0,15 M NaCl and the identical results were obtained (the shift of the maximum was observed). These data testify that Pd compound in complex with DNA is more stable than in solution without DNA. Fig. 4 indicates that in contrast to 0.005 M NaCl DNA does not bind with Pd compound in 0.15 M – the spectral properties of Pd does not change with increase in DNA concentration out of DNA band. The corresponding spectra were recorded in one day after preparation of the complexes.

~~0,3 after 15 min 0,3 after week 0,03~~

~~0,05 K2[PdHGluCl2] K2[PdHGluCl2] 0,2 complex DNA-Pd 0,2 complex DNA-Pd D D~~

~~0,1 0,1 7 nm 7 nm~~

~~0,0 0,0 330 360 390 420 450 480 330 360 390 420 450 480 λ, nm λ, nm Fig. 3. Absorption spectra of Pd-compound and its complex with DNA in time.~~

0,6 -4 0,8 C(PdGlu)=5*10 M; NaCl-0,005 M C(PdGlu)=5*10-4M; NaCl-0,15 M 0,7 0,5 C(DNA) = 0 % 0,6 C(DNA) = 0 % C(DNA) = 0.0001 % C(DNA) = 0.0001 % 0,4 C(DNA) = 0.0003 % 0,5 C(DNA) = 0.0003 % C(DNA) = 0.0004 % C(DNA) = 0.0004 % C(DNA) = 0.0005 % D D C(DNA) = 0.0005 % 0,3 C(DNA) = 0.0008 % 0,4 C(DNA) = 0.0008 % C(DNA) = 0.001 % C(DNA) = 0.001 % 0,3 0,2 0,2 0,1 0,1

~~0,0 0,0 240 270 300 330 360 390 420 450 480 240 270 300 330 360 390 420 450 480 λ, nm λ, nm~~

Fig. 4. Difference in interaction of K2[PdHGluCl2] with DNA in low and high ionic strength. 217 It can be suggested that the aquation process of Palladium atom in 0,15 M NaCl is absent. The interaction of K2[PdHGluCl2] with DNA in 0,005 M NaCl may be realized due to the coordination of Pd-atom to DNA or due to ligand binding with macromolecule. Fig. 5 shows the images obtained by atomic force microscopy in tapping mode. The study was performed for a circular plasmid DNA (Fig. 5a). The image in Fig. 5a shows the fixation of free individual molecules of DNA on a substrate.

~~b c Free DNA (С(DNA) = 0,75*10-4%)~~

~~-5 (b), (c) C(K2[PdHGluCl2]) = 10 М -6 (d), (e) C(K2[PdHGluCl2]) = 2·10 М~~

~~d e~~

~~Fig. 5. AFM images of DNA in complex with~~

~~K2[PdHGluCl2].~~

Fig. 6. Molecular structure of [PdHGluCl2] and two stages of its aquation. 218 In Fig. 6 the calculated structure of compound under the study in two stages of its aquation is represented ([PdHGluCl2], [PdHGluH2OCl] and [PdHGlu2(H2O)]). As the result of calculations the absolute values of energy for each molecule were obtained. The received values of energy can indicate that in aqueous solution the most probable process is the substitution of the chlorine atoms to a water molecule (aquation), as in this case the energy of the system is minimal. Atomic distances and angles between these atoms were individually obtained for each molecule. Conclusion It is shown that Palladium complex interacts with DNA. The binding causes change in DNA conformation. Complex formation is realized in solutions of low NaCl concentration (0,005 M), whereas under physiological conditions (in 0,15 M NaCl) DNA interaction with Palladium compound is not observed.

References 1. Kasyanenko N.A., Levykina E.V., Erofeev D.C. etc. // Journal of Structural Chemistry, 2009, V. 50, № 5. 2. Kozhenkov P.V. Quantum-mechanical calculation of the structures of potential anticancer coordination compounds using package HyperChem and GAMESS // Term work, Saint Petersburg State University, 2009. 3. Firefly (PC GAMESS) version 7.1.G, build number 5618. Copyright (c) 1994, 2009 by Alex A. Granovsky, Firefly Project, Moscow, Russia. 4. HyperChemTM release 8.0.9 for Windows. Copyright (c) 1995-2011 Hypercube, Inc. 5. Stevens W.J., Krauss M., Basch H., Jasien P.G. SBKJC VDZ ECP EMSL Basis Set Exchange Library K – RN, 1992. 6. Dill J.D., Pople J.A. // J. Chem. 6-31+G* EMSL Basis Set Exchange Library Li – Ne, 1975.

~~219 Studing of the UV radiation influence on the DNA in a solution in the presence of caffeine~~

~~Platonov Denis [email protected]~~

~~Scientific supervisor: Dr. Paston S.V., Department of Molecular Biophysics, Faculty of Physics, Saint-Petersburg State University~~

The absorption of UV-light by the nitrogenous bases of DNA (λmax=260 nm) can lead to changes in the structure of the macromolecule, such as hydration (occurs only in the single-stranded DNA) and the formation of pyrimidine adducts (stable products of addition of pyrimidine bases to other neighbour bases in the same DNA strand). Maximum quantum yield is observed for the reaction of thy- mines dimerization [1]. These damages in the DNA structure are the major cause of mutagenic and bactericidal action of ultraviolet radiation on a cell [2]. In the present investigation caffeine was chosen as a possible DNA protector from UV radiation (UVC range). This choice was dictated by the fact that the maximum of caffeine absorption spectrum is at λ=272 nm (near the maximum absorption of the DNA nitrogenous bases). Besides caffeine weakly interacts with DNA [3, 4], i.e. it is not a mutagen. Caffeine is a biologically active substance of plant origin, widely consumed in the world. It has a strong stimulating effect on the central nervous system [5], affects the cell cycle and the processes of repair of DNA damage [6]. In this study we used the sodium salt of calf thymus DNA of molecular weight M = 13,6 ± 0,6 MDa provided by D.Y. Lando (Institute of Bioorganic Chemistry NAS of Belarus), caffeine purchased from «Sigma». Ionic strength of all the investigated solutions were 0.003M NaCl. As a source of UV radiation a quartz mercury lamp of low pressure DRB-8 was used. The output of the lamp is 8 W,

λmax = 254 nm. DNA solutions were irradiated in quartz cuvettes a 0.2 cm thick. To prevent possible heating of the solutions in the way of UV light a quartz cell was set a 6 cm thick, filled with distilled water, absorbing thermal infrared radiation. At the same time the UV radiation transmission of the cell was 85% at λ = 254 nm. The concentration of DNA in all irradiated solutions was C = 0.011%, the distance to the source was 7 cm, which corresponds to the absorption intensity Iabs = 0.88*103 J/(kg*s). The exposure time varied between 12 min–2 h 15 min. After the UV-light exposure the alterations in DNA absorption (Fig. 1) and circular dichroism (CD) spectra (Fig. 2) are observed. The DNA absorption intensity increases at all wave lengths (Fig. 3) and the intensities of the positive and negative bands in the DNA CD spectrum decreases monotonously with the radiation dose growth (Fig. 4(a,b)). Also the shift of CD spectra to the long wavelength region is observed (Fig. 5). The alterations observed can be caused by the partial DNA denaturation and, possibly by the modification of nitrogenous bases. 220 Fig. 1. DNA absorption spectra after UV irradiation with the different doses.

~~Fig. 2. The DNA CD spectra after UV irradiation with different doses.~~

~~221~~

~~D D260 0,9 D 0,06 D310~~

~~0,8 0,05~~

~~0,7 0,04~~

~~0,6 0,03 D230 0,5 0,02~~

~~0,4 0,01 0 1 2 3 4 5 6 7 8 0 2 4 6 8 10 6 6 Dr, 10 J/kg Dr, 10 J/kg~~

~~Fig. 3. Dependences of the intensity of the absorption spectrum of DNA at wavelengths of 260nm, 230nm and 310nm on the dose of UV irradiation~~

~~1,9 −1 −1 -1,6 −1 −1 1,8 ∆ε275, Μ cm ∆ε240, Μ cm -1,8 1,7~~

~~1,6 -2,0~~

~~1,5 -2,2~~

~~1,4 -2,4 1,3 -2,6 1,2 -2,8 1,1~~

~~1,0 -3,0 0 2 4 6 8 0 2 4 6 8 Dr, 106 J/kg Dr, 106 J/kg~~

~~Fig. 4. The intensities of the positive (a) and negative (b) bands in the DNA CD spectrum.~~

~~262 λ, nm 261~~

~~260~~

~~259~~

~~258~~

~~257~~

~~256~~

~~255~~

~~254 0 2 4 6 8 Dr, 106 J/kg Fig. 5. The position of ∆ε = 0 in the DNA CD spectrum. 222~~

~~DNA DNA+caffeine (1.3*10-3mol/l) 1,0~~

~~0,8 r0 η / r η 0,6~~

~~0,4 0 3 6 Dr, 106 J/kg Fig. 6. Dose dependency of the DNA relative viscosity at C(DNA)=0.011%.~~

~~Table 1. Characteristic viscosity of DNA. 6 Ccaf, mol/l Dr, 10 J/kg [η] DNA, dl/g~~

~~0 0 135~~

~~0 0,64 82~~

~~0,0013 0,64 118~~

UV-irradiation also leads to the diminution of the volume of macromolecule in a solution. In our work it is obtained by the measurement of DNA intrinsic viscosity. The experiment shows that in the presence of caffeine in the DNA solution under the UV-light exposure the diminution of the DNA volume is smaller (Table 1). Investigation of the dose dependences of the relative viscosity of DNA solutions (Fig. 6) also reveals the photoprotective action of caffeine. We estimated the dose -3 reduction factor (DRF80) at Ccaf=1.3*10 mol/l according to the expression [7]: D ()DNA+caf DRF = r80 80 D ()DNA r80

(values Dr80(DNA+caf) and Dr80(DNA) are obtained from the Fig. 6). -3 It was found that for Ccaf=1.3*10 mol/l DRF80=13±3. The photoprotective action of caffeine may be explained by the fact that this substance and DNA absorb in the same spectral region (λmax(caffeine) = 272 nm) that is why the intensity of UV light in solution containing caffeine is lower then in a pure water. One can say that caffeine shields DNA from UV light.

223 References 1. Smith K.C., Hanawalt P.C. Molecular Photobiology. – Acad. Pr., New York and London, 1969. 2. Rubin A.B. Biophysics, v.2. – Moscow: High School, 1987. 3. Tarasov A.E. Study of the radioprotective properties of caffeine, catechines and aliphatic alcoholes in DNA solutions. Master's thesis, Saint Petersburg State University, Faculty of Physics, 2011 (in Russian). 4. Osipov N.D., Kondrat'eva O.P., Frisman E.V. // Vestnik LGU, № 4, 98 – 101, 1979 (in Russian). 5. Caffeine. PubChem public chemical database (http://pubchem.ncbi.nlm.nih. gov/summary/summary.cgi?cid=2519#Synonyms) 6. Conney A.H., Zhou S., Lee M.-J. et.al. // Toxicol. and Appl. Pharm., v. 224, p. 209 (2007). 7. Kudryashov Yu.B. Radiation Biophysics (Ionizing Radiations). – New York: Nova Science Publishers, Inc.; 2008.

~~224 Entropic sampling of thermodynamic and structural properties of polymer chains and stars within Wang- Landau algorithm~~

~~Silantyeva Irina [email protected]~~

~~Scientific supervisor: Prof. Dr. Vorontsov-Velyaminov P.N., Department of Molecular Biophysics, Faculty of Physics, Saint- Petersburg State University~~

Introduction Different types of numerical methods are used now for studying of polymers, such as molecular dynamics, entropic sampling [1, 2], numerical treatment in self-consistent field method [3]. Entropic sampling method allows us to obtain the density of states functions. Using these functions we can calculate canonical averages in a wide temperature range. In our work the entropic sampling (ES) combined with Wang-Landau (WL) algorithm is used for studying of polymer chains and stars. Polymer star is not an abstract model. Such molecules are synthesized (for example in University of Helsinki [4]) and used for transport of DNA and drugs into living cells. Results and Discussion The aim of our work is calculation of the density of states and the thermodynamic properties of lattice models of polymer chains and stars on 3D simple cubic lattice using WL algorithm. The semi-phantom random walk is used for generating models. The 6-arm polymer star with the length of arms Narm=5, 12, 20 segments is considered. So the overall number of segments is N = 30, 72, 120. In athermal case, when interaction between monomers is reduced to exclusion of intersections, the ratio of self-avoiding walk (SAW) conformations is calculated for stars with total number of segments up to N=720 (Table 1).

N Narm Ω0 N Narm Ω0 6 1 1 120 20 4.03E-05 12 2 5.67E-01 180 30 4.96E-07 18 3 1.64E-01 240 40 7.47E-09 24 4 1.06E-01 300 50 1.26E-10 30 5 5.19E-02 360 60 1.93E-12 36 6 3.24E-02 420 70 3.23E-14 42 7 1.81E-02 480 80 6.12E-16 48 8 1.12E-02 540 90 1.32E-17 54 9 6.71E-03 600 100 1.96E-19 60 10 4.21E-03 660 110 3.46E-21 72 12 1.49E-03 720 120 7.75E-23 Table 1. The ratio of SAW conformations among semi-phantom for stars. 225 a 0,2 30 b

~~0,15~~

~~0,1 C/ N 120~~

~~0,05 цепи~~

~~0 01 23 45 T~~

~~0,5 120 с 0,45 30 72 d 0,4~~

~~0,35 120 0,3~~

~~N 0,25 72 C/ 30 0,2~~

~~0,15~~

~~0,1~~

~~0,05~~

~~0 012345 T~~

~~0 0 f -0,1 -0,05 e -0,2 -0,1 -0,3 30 -0,15 -0,4 72, 120 N /N -0,5 S/ Δ -0,2 ∆S 30 72 -0,6 120 120 -0,25 -0,7 72 72 -0,8 -0,3 30 30 -0,9~~

-0,35 -1 0246810 01 23 45 β β Fig. 1. Specific energy (a, c), specific heat capacity (b, d) dependences on temperature and specific excess entropy (e, f) dependences on inverse temperature β for stars with overall number of segments N = 30, 72, 120 (thick lines) and for chains with the same number of segments (thin lines); ε>0(a, c, e), ε<0(b, d, f). Dashes on energy axis (a, c) denote limiting values of energy at T→∞.

226 Fig. 2. Mean square radius of inertia dependence on temperature; ε>0 (black line), ε<0 (grey line). Chain length N =30, 72, 120 segments. Mean square radius of inertia determines the size of the molecule. Horizontal dashed lines denote the limiting value of mean square radius of inertia at T→∞. 600

~~500~~

~~400~~

~~120~~

~~^2 > 300~~

~~200 72~~

~~100 30~~

0 0 2 4 6 8 10 T Fig. 3. Mean square end-to-end distance dependence on temperature; ε>0 (black line), ε<0 (grey line). Chain length N =30, 72, 120 segments. Horizontal dashed lines denote the limiting value of mean square end-to-end distance at T→∞. 227 Fig. 4. Square root of mean square end-to-end distance dependence on inverse temperature for chain with length N=30; ε>0 (diamonds), ε<0 (squares). These dependences are in agreement with data from work [1].

~~Fig. 5. Distribution of the mean square radius of inertia over number of contacts for stars. Total number of segments in stars N=30, 72, 120 (Narm =5, 12, 20).~~

In the thermal case we attribute an energy ε to each pair of non-neighboring monomers occurring at closest contact, ε>0 or ε<0. In this case the SAW conformations are selected and the distributions Ω0m over contacts between monomers are calculated. Using these distributions the equilibrium properties, such as conformational energy (Fig. 1a, c), heat capacity (Figs. 1b, d), entropy (Figs. 1e, f) are obtained as dependences on temperature. Results for chains and stars are presented in common figures for comparison. The temperature in all figures is in energy units. Also the mean square radius of inertia R< 2> and mean square end-to-end distance

dependences on temperature for chains (Figs. 2, 3) are obtained using equation: 228 mmax R2 exp−εmk/ T Ω ∑ m 0m RT22()==R m=0 m mmax −εmk/ T ∑ exp Ω0m m=0 2 where Ω0m – density distribution of SAW over contacts and m – distribution of mean square radius of inertia over number of contacts are obtained in our numeri- cal experiment. For chains in the attraction case (ε<0) the radius of inertia mono- tonically increases with temperature. In the repulsion case (ε>0) it monotonically decreases with temperature. The limiting value the radius of inertia at T→∞ can be calculated using equation mmax 22 lim RR= Ω0m T →∞ ∑ m m=0 At T→∞ the radius of inertia both for ε<0 and for ε>0 tend to the same limit (dashed line in Figs. 2, 3). The comparison of our results (Fig. 4) with that from work [1] shows good agreement. 2 The distributions of mean square radius of inertia m over contacts (Fig. 5) allow to obtain the corresponding temperature dependence for stars.
Conclusion As a result of our work we can make the following conclusions: 1) For repulsion case (ε>0) the specific heat capacity dependences on tem- perature have one maximum both for chains and stars. But for attraction (ε<0) the dependences are more complicated. They have several maxima. Maxima for stars are shifted to lower temperatures. 2) Heat capacity maxima denote the significant changes in polymer conformations. 3) Further studying of structure properties is required.
Acknowledgment. The work is supported by the grant RFBR 11-02-00084 a.
References 1. Vorontsov-Velyaminov P.N., Volkov N.A., Yurchenko A.A. // J. Phys. A, 2004. 37, pp. 1573-1588. 2. Volkov N.A., Yurchenko A.A., Lyubartsev A.P., Vorontsov-Velyaminov P.N. // Macromol. Theory and Simul. 2005, 14, pp. 491-504. 3. Birshtein T.M., Mercuryeva A.A., Leermakers F.A.M., Rud O.V. // Macromolecular compounds A, 2008, 50, № 8, pp. 1-18 (In Russia). 4. Anu Alhoranta, Julia Lehtinen, Arto Urtti, Sarah Butcher, Vladimir Aseyev, Heikki Tenhu. Book of abstract of the 14th IUPAC International Symposium of Macromolecular Complexes (MMC-14) August 14-17, 2011, Helsinki, Finland, p. 195.
229 Silver nanoparticles and their interaction with polymers in solution and on a surfaces
Varshavskii Mikhail [email protected]
Scientific supervisor: Prof. Dr. Kasyanenko N.A., Department of Molecular Biophysics, Faculty of Physics, Saint-Petersburg State University
Introduction Nanoparticles (NPs) are small objects with a narrow size distribution and diameter in the range of 10 – 200 nm. Optical and other physical properties of nanoparticles depend greatly on their size and shape, in contrast to a bulk material with the physical properties regardless of its size. Wide research activity takes place currently in the field of nanosensors based on NPs. For example biosensors based on surface plasmon resonance effect are used for biochemical tests for glucose and urea, for immunoassays of proteins, hormones, drugs, steroids, viruses, DNA testing and finally to study the kinetics of drug action in real time. In this paper the spectral properties of metal nanoparticles and their interaction with charged macromolecules in aqueous and aqueous-salt solutions were studied. CD-spectrophotometer Mark 4 (Jobin Ivon, France), UV-spectophotometers SF-56 and SF-2000 (Russia) were used for circular dichroism and UV-adsorption spectra studies. Visualization of the obtained structures was done using atomic force mi- croscopy (NanoScope 4a, Veeco) in tapping mode in air. Silver NPs was synthesized in National Technical University of Ukraine, Kyiv Polytechnic Institute by means of a new electric-spark dispersion technique. Calf thymus DNA (Sigma) with the molecular mass M=8×106 Da, determined from the value of the DNA intrinsic viscosity in 0.15 M NaCl solution.
Results and Discussion One of the most interesting phenomenon among the unique properties of metallic NPs is a local surface plasmon resonance (LSPR). The strong interaction of silver nanoparticles with light in this case is determined by the collective oscillation of conduction electrons within the metal. The “electron gas” motion due to quantum effects can be presented as the movement of quasiparticles plasmons. LSPR can be observed if the size of metallic particles is smaller than light wavelength as a result of light interaction with NPs. LSPR is realized when light frequency coincidents with the intrinsic plasmon frequency. The investigation of silver nanoparticles and their interaction with charged macromolecules is interesting for the different applications. It was shown that Ag NPs have a plasmon absorption band (peak is about 400 nm). This band does not overlap with the absorption spectrum of DNA. The absorptions spectra of different metal nanoparticles in water for two series of preparations are shown in Fig. 1. 230 Fig. 1. The absorptions spectra of metal nanoparticles in water for two series of preparations. Our experiments demonstrate that silver nanoparticles can interact with nega- tively charged DNA and positively charged polyallylamine (PAA). This follows from the change in the absorption spectrum of nanoparticles within the wavelength region where PAA and DNA do not absorb light. A solution of silver nanoparticles with polyallylamine is not so stable (the precipitation of NPs can be observed after 14 days). The solution of NPs with DNA is stable even after 2 weeks storage. The influence of salt concentration on the absorption spectrum of silver NPs was studied (Fig. 2a). The intensity of plasmon resonance peak (400 ± 1 nm in aqueous solution) increases and the maximum shifts to shorter wavelengths for 10 ± 1 nm with the increasing of NaCl concentration up to 0.1 M
Fig. 2. The absorption spectra of silver nanoparticles in aqueous salt solutions of different ionic strength (a) and the dependence of the intensity of the plasmon resonance peak on the salt concentration (b). 1 – 0 M; 2 – 0.001 M; 3 – 0.002 M; 4 – 0.003 M; 5 – 0.005 M; 6 – 0.01 M; 7 – 0.03 M; 8 – 0.05 M; 9 – 0.1 M; 10 – 0.15 M; 11 – 0.3 M; 12 – 0.5 M; 13 – 1 M NaCl.
With further increasing of salt concentration the intensity of plasmon peak decreases. The band changes the shape and shifts to the red region. Such changes may be associated with the precipitation of some NPs due to the aggregation at 231 high salt concentrations. We emphasize the essential difference in the effect of large (> 0.1 M) and low (<0.1 M) NaCl concentration on the spectral properties of nanoparticles. The shift of the maximum of the plasmon band is usually associated with the change in the size of the nanoparticles. For the systems under investigation this may be observed as a result of NPs interaction with ions. It was also found that silver nanoparticles in aqueous salt solution eventually precipitate over time. Fig. 3 shows the absorption spectra of silver nanoparticles in different so- lutions: 1) in water, 2) in the aqueous salt solution (0,005 M NaCl), 3) in the DNA solution (C(DNA)=0.005%). Maximum of plasmon peak shifts to shorter wavelengths in the presence of NaCl (0,005M) and DNA (0,005%). This result indicates the decrease in the size of silver clusters. Circular dichroism method showed Fig. 3. The absorption spectra of silver that silver nanoparticles are not opti- nanoparticles in water, in aqueous salt cally active in the range 220-480 nm solution (0.005 M NaCl), in 0.005% solu- (Fig. 4). In Fig. 4 one can see also tion of DNA. CD spectra of DNA in a solution with different amount of silver NPs. The increase in NPs concentration causes the great change in DNA circular di- chroism spectra. These data show us the interaction of silver nanoparticles with DNA. The influence of silver NPs on DNA conformation is observed. Atomic force microscopy was also used for the study of silver nanopar- ticles (Fig. 5). NPs were deposited on a mica substrate using a magnetic Fig. 4. Circular dichroism spectra of DNA stirrer. The sample was rotated about in the solutions with silver nanoparticles. 3 minutes. The size of NPs is about 30nm (Fig. 5a,b). The images contain the track of NPs under centrifugal force. Silver nanoparticles and DNA are not fixed on mica in the absence of MgCl2. In the presence of MgCl2 DNA and nanoparticles can be fixed on mica. For the DNA fixation from the solution with NPs one can observe knobs at the ends of DNA strands and crossing parts of DNA (Fig. 6a, b)). Diameter of the knobs is about 30 nanometers (Fig. 6c). So we can propose that NPs are localized at the ends of double stranded helix.
232 Fig. 5. Silver nanoparticles on a mica substrate.
(a) (b)
(c) Fig. 6. DNA-nanoparticles complex on a mica (a), (b). NPs section profile (c).
233 Conclusions • It was shown that the aqueous solutions of NPs are stable in time. • The increase of NaCl concentration in a solution influences on the state of silver nanoparticles. The low (<0.1 M) and high (> 0.1 M) NaCl concentrations cause the different changes in plasmon resonance peak. • The addition of NaCl into water solution provokes a partial precipitation of NPs in time. • The addition of DNA into NPs water-salt solution stabilizes the system regard- less on the concentration of NaCl. • The presence of anionic polymer (DNA) and cationic polyallylamine in NPs solution influences on the plasmon resonance band. • AFM images of the systems were obtained. It was shown that the size of nano- particles is about 30 nm.
234
I. Resonance Phenomena in Condensed Matter Phase transitions in magnesium: ab initio study
Klyukin Konstantin [email protected]
Scientific supervisor: Dr. Shelyapina M.G., Department of Quantum Magnetic Phenomena, Faculty of Physics, Saint- Petersburg State University
Introduction Magnesium-based hydrides stand as one of the most promising candidates for hydrogen storage due to its high hydrogen uptake (up to 7.6 w% in MgH2), large reserves and low cost. The main disadvantages of MgH2 are high hydrogen release temperature, slow sorption kinetics and a high reactivity toward air and oxygen. However, transition metals additives (TM), such as Ti, V or Nb, greatly enhance hydrogen sorption kinetics [1]. Numerous theoretical and experimental investigations have been made in order to understand the mechanism of hydrogen uptake by magnesium. But there is no clear conclusion. Moreover, in situ X-ray diffraction experiments have shown that penetration of hydrogen into Mg occurs through TM “gates” [2]: at first, a TM hydride appears and after that hydrogen penetrates into magnesium. Therefore, investigation of the interface border of magnesium with TM could be helpful to understand the role of TM additives on the hydrogen sorption process in magnesium. In our previous theoretical investigations of Mg/Nb thin films we have found that near the interface Mg keeps the bcc structure of Nb [3], that means that Nb layers (or particles) favor the bcc structure of Mg. Since Mg layers deposited on a Nb-(011) surface may keep the bcc stacking mode, we assume that there are two pathways for Mg → MgH2 transformation. The first one, corresponds to the direct
Mg (hcp) → MgH2 (rutile) transition. But even small additions of Nb or V with bcc structure can lead to the the second two-step hydrogenation scheme: the first step is the Mg(hcp) → Mg(bcc) transformation, which occurs near the Mg/TM interface border, the second step is that upon hydrogenation of the bcc magnesium a MgHx → MgH2 transformation occurs. In this contribution we report on the results of our theoretical study of phase transitions in metallic magnesium.
Method of calculation The electronic structures of all systems were calculated within a DFT full- potential linearized augmented plane-waves (FLAPW) method using the Perdew– Burke–Ernzerhof GGA exchange and correlation potential. In all calculations, self consistency was achieved with a tolerance in the total energy of 0.1 mRy. The number of k-points in the irreducible Brillouin zone was equal to 1000. For Mg and Nb the radius of nonoverlapping muffin-tin spheres was chosen equal to 2.0 236 a.u, for hydrogen – 1.1 a.u. The calculations were carried out using the package WIEN2k [4]. Results Hcp-bcc phase transition. To describe the hcp-bcc transition we have consid- ered the distortion model proposed by Burgers [5]. The simplest path of transition from bcc lattice into hcp structure includes two independent processes (see Fig. 1): • a shear deformation from the bcc (110) plane to the hexagonal basal plane; • a slide along the (110) planes.
Fig. 1. Scheme of hcp-bcc phase transformation of magnesium.
To describe the transformation we used a two-dimensional parameter space
(λ1, λ2), where λ1 represents the shear deformation and λ2 represents the slide dis- placement. The hcp and bcc structures correspond to (0,0) and (1,1), respectively. The lattice parameters of the primitive cell can be written as
=−λλ⋅+⋅= ⋅+λλ⋅− ⋅ aa()1311hcpbabcc ,( 1121 ) a   c   c =−()1 λ ⋅ +⋅2 λ ⋅ a 1   1  a opt  with Mg atoms at positions
111  511  0,+ ⋅λ22 , , and 0,+ ⋅λ , 362  662  In Fig. 2 we represent the counter plot of potential energy surface for magnesium hcp-bcc transition. The bcc structure is located at an unstable point. However, the energy difference between hcp and bcc phases is only 1.41 kJ/mol per atom. As it is seen in Fig. 2, during the hcp-bcc phase transition, at first, the shear deformation dominated, but at the end of transformation the slide displacement becomes stronger. 237 Fig. 2. Counter plot of potential energy surface (hcp-bcc transformation).
Bcc-fcc phase transition. However, bcc Mg structure is unstable and is held only by presence of Nb. So, tetragonal distortion of free bcc Mg cell leads to fcc Mg structure. As it is seen in Fig. 3, the energy curve has two minima. The first one cor- responds to metastable bcc Mg structure, the second one to stable fcc Mg structure.
Fig. 3. Bcc-fcc phase transition of magnesium.
As the normal state of magnesium hydride is the rutile phase, we also considered rutile structure of magnesium with hydrogen vacancies. The relative stability ∆E of all studied structures (relative to the hcp structure) was estimated by using the following expression:
∆=EE(Mgstructure) tot( Mg structure) – E tot ( Mghcp ) (1) The results, together with the equilibrium lattice parameters and total energy per atom are listed in Table 1. 238 ∆E Structure a (Å) c (Å) E (Ry/atom) hf tot (kJ/molatom) Mg (hcp) 3.218 5.108 -400.6672 0 Mg (bcc) 3.5763 3.5763 -400.6650 1.41 Mg (fcc) 4.5190 4.5190 -400.6663 0.56 Mg (rutile) 4.2526 2.8382 -400.6531 9.26 Table 1. Equilibrium lattice parameters, total energy and heat of formation.
(a) (b)
(c) (d) Fig. 4. The total, s- and p-states resolved DOS’s calculated for different magne- sium structures: (a) – hcp, (b) – fcc, (c) – bcc, (d) – rutile. The Fermi level cor- responds to E = 0 and is marked by a vertical solid line.
For better understanding of the nature of stability (or instability) of different magnesium structures we have studied their electronic structure. The calculated electronic density of states (DOS) are shown in Fig .4. As it is seen in all considered magnesium structures the DOS near the Fermi level (EF) is formed by the delocalized s- and p-states. For the hcp structure (Fig. 4a)
EF falls into a local minimum of both s- and p-DOS that characterizes a stable state, whereas for the rutile structure (Fig. 4d) EF corresponds to the local maximum of the p-states and a rather high population of the s-states. That, by-turn, character- 239 izes an unstable structure. The fcc and bcc structures are in the intermediate posi- tion: they have the Fermi level in the minimum of one state and in the maximum of another (Figs. 4b and 4c). This result is in agreement with total energy values listed in Table 1 and confirms that the rutile structure with hydrogen vacancies is the most unstable.
Conclusions The FLAPW calculations of different phases of magnesium, namely, hcp, bcc, fcc and rutile structure of MgH2 with hydrogen vacancies, have shown that the hcp phase is the most stable, whereas the rutile structure with hydrogen vacancies is the most unstable. However, as soon as one creates a bcc phase, for example by deposing Mg on a Nb-(011) surface, the bcc structure may relax into the hcp structure. At the beginning of the bcc-hcp phase transition the the slide displace- ment dominates, but at the end of transformation the shear deformation becomes stronger. Nevertheless, the bcc phase may be transformed into a fcc phase as well. The energy difference between the bcc and fcc phases is only 0.85 kJ/mol per atom. Hence, the fcc-Mg can be an intermediate phase during the bcc-hcp transition. Calculations of the fcc-hcp phase transition in magnesium are under evaluation.
References 1. Charbonnier J., P. de Rango, Fruchart D. et. al. // Alloys Compd. Vol. 383, p. 205, 2004. 2. Pelletier J.F., Huot J. // Phys. Rev. B, Vol. 63, Is. 5, 052103, 2001. 3. Klyukin K., Shelyapina M.G., Fruchart D. // Solid State Phenomena 170, p. 298-301, 2011. 4. Blaha P., Schwarz K. and Luitz J. Computer code WIEN2k,Vienna University of Technology, 2000. 5. Burgers W.G. // Physica, Amsterdam 1, 561, 1934.
240 NMR study of spin relaxation
Nefedov Denis [email protected]
Scientific supervisor: Prof. Dr. Charnaya E.V., Department of Solid State Physics, Faculty of Physics, Saint-Petersburg State University
Introduction The influence of size effects on various physical properties of materials, includ- ing atomic mobility, relaxation phenomena, and liquid fluidity, in confining systems has been recently attracting considerable interest. NMR is known to yield valuable information on dynamics in condensed media and has found wide application in studies of the mobility of liquids inserted into nanosized pores. A decrease in atomic mobility, first observed to occur in liquid gallium incorporated in nanosized pores [1], manifested itself in an increase in the nuclear spin relaxation rate by a few times due to enhancement of the role of the quadrupole contribution. In [1] only nanosized gallium particles were studied, which did not permit one to reveal the characteristic dimensions at which size effects in atomic mobility become notice- able in a gallium melt. The aim of this work was to process the data obtained in the study of the relax- ation times of the nuclei of isotopes of gallium (69Ga and 71Ga) in confined geometry with the use of NMR. The samples were isolated particles of gallium the size of about 50 microns, thin film of gallium on an opal the thickness of about 10 microns, gallium injected into the pores of the artificial opal and porous glass.
Results and discussion The gallium relaxation process can be described by the relation [2]: Mt() 4 Ctττ 1 Ct t =− −cñ + − − 1b  exp22 exp 22 exp MT015514+ ωτ00cc 1+ωτ m Here M(t) and M0 are the time-dependent (t) and equilibrium magnetizations, respectively; 1 – b is the relative magnetization immediately following the invert- 2 ing pulse; ω0 is the Larmor frequency; C is a constant proportional to Q and τc is the correlation time. The magnetization restoration process is described by a single exponent, and
C·τc is equal to the inverse quadrupole spin-lattice relaxation time, −11 C⋅=τ cq TR1 = q. Since ω0τc<<1, the relation of magnetization restoration process takes the form
Mt() t t t t =−1b exp − exp − =− 1bb exp − =− 1 exp − M TT  T  T 0 11qm  1  1
241 According to this formula the exponential approximation of experimental points is performed.
Designations : 71 71 T1 - the longitudinal relaxation time Ga 69 69 T1 - the longitudinal relaxation time Ga 71 71 71 71 R1 - the longitudinal relaxation rate Ga; R1 =1/T1 69 69 69 69 R1 - the longitudinal relaxation rate Ga; R1 =1/T1 71 71 R1q - the quadrupole contribution in longitudinal relaxation rate Ga 71 71 R1m - the magnetic contribution in longitudinal relaxation rate Ga 69 69 R1q - the quadrupole contribution in longitudinal relaxation rate Ga 69 69 R1m - the magnetic contribution in longitudinal relaxation rate Ga
It is necessary to find out contributions of magnetic and quadrupole relaxation rates to the overall rate.
Plan of calculations For the sample 1. Using the conditions 69 69 69 69 R1q Θγ22R1m 71=( 71 ) ≈ 2,51 (), 71=≈( 71 ) 0,62 (**) RR11qmΘγ And knowing, that: 71 71 71 69 69 69 R1 = R1m +R1q ; R1 = R1m +R1q ,
71 71 69 69 69 form the system of this four equations. R1 =1/T1 and R1 =1/T1 and T1 are obtained from the processing of experimental curves. Having solved this system 71 71 69 69 of equation we obtain R1q , R1m , R1q , R1m .
For the other three samples: Let’s consider Korringa ratio [3]:
2 T1ms TK=γ const/( K )
Here T is the temperature; KS is the Knight shift; K is a factor which takes into account the correlation effects. The experiment revealed, that the Knight shift did not vary from sample to sample. The temperature in this experiment was constant.
Thus we can make the assumption, that T1m remained constant during the transition from sample to sample. Then, after obtaining overall relaxation rates R1, we find 71 71 69 69 contributions R1q , R1m , R1q , R1m :
71 71 71 69 69 69 R1q = R1 +R1m ; R1q = R1 +R1m .
71 71 69 69 Results R1q , R1m , R1q , R1m must satisfy the conditions (*) and (**).
242 Results 1) Sample 1 (isolated particles of gallium, the size of about 50 microns). The dependence of the longitudinal magnetization on the time. 69 -6 69 -1 T1 = 680·10 (c) R1 = 1480 (c )
71 -6 71 -1 T1 = 530·10 (c) R1 = 1870 (c ) By solving the system of equations: 69 69 69 69 R1q Θγ22R1m 71=( 71 ) ≈ 2,51;71 =≈ ( 71 ) 0,62 RR11qmΘγ 71 71 71 69 69 69 R1=+ R 1mq RR 11; =+ R 1 mq R 1
We obtain: 69 -1 69 -1 R1q = 420 (c ) ; R1m = 1060 (c )
71 -1 71 -1 R1q = 170 (c ) ; R1m = 1710 (c )
Results of all samples:
1) 50 microns 2) Thin film 3)Opal 4) Porous glass
71 -1 R1q (c ) 170 170 2940 3730
69 -1 R1q (c ) 420 420 4790 9700
71 -1 R1m (c ) 1710 1710 1710 1710
69 -1 R1m (c ) 1060 1060 1060 1060
71 -1 R1 (c ) 1870 1870 4650 5440
69 -1 R1 (c ) 1480 1480 5850 10750
243 Conclusions Magnetic relaxation dominates in first two samples, and quadrupole relaxation in the two last ones. Magnetic contribution to the relaxation rate is constant from sample to sample. 69 69 R1m γ 2 71=≈( 71 ) 0,62 R1m γ It is confirmed experimentally within error. As a result of processing the experimental data and calculations magnetic and quadrupole contributions to the spin-lattice relaxation rate in liquid gallium were obtained. Reduction in the size of liquid gallium sample is followed by: 1) Increase in the quadrupole relaxation rate; 2) Decrease in the atomic mobility. 3) Increase in the correlation time.
References 1. Charnaya E.V., Loeser T., Michel D. et al. // Phys. Rev. Lett. 88, 097602 (2002). 2. Tien C., Charnaya E.V., Sedykh P. and Kumzerov Yu.A. //Physics of the Solid State, V. 45, N. 12, 2352-2356. 3. Слиткер Ч. Основы теории магнитного резонанса.- М: Мир, 1981.-173 с.
244 Magnetic properties of cubic magnetite Fe O : a density functional theory study 3 4
Irina Shikhman [email protected]
Scientific supervisor: Dr. Shelyapina M.G., Department of Quantum Magnetic Phenomena, Faculty of Physics, Saint- Petersburg State University
Introduction
Magnetite is the earliest discovered magnet. Extensive studies of Fe3O4 have been carried out over the past 60 years. Because of interesting electronic and mag- netic properties as well as potential industrial applications in magnetic multilayer devices and spintronics, magnetite has still attracted much attention. Theoretical studies can be very helpful to understand the nature of physical phenomena observed in magnetite, both bulk and multilayered. However, even for the bulk Fe3O4, the calculated properties, such as density of states, magnetic moments, magnetic anisotropy energy etc., are rather sensible to the method of calculation [1, 2]. The aim of the present work is to study structural and magnetic properties of magnetite by different density functional theory methods (GGA, LSDA, LDA+U).
It will be applied further to study Fe/Fe3O4 multilayes.
Method of calculations The calculations have been done within the framework of the full-potential linearized augmented plane waves (FLAPW) method that is one of the most ac- curate realizations of the DFT methods. We have tried several types of exchange- correlation potentials, such as GGA, LSDA and LDA+U. Non-overlapping “muffin-tin” spheres (RMT) were chosen equal to 1.75 a.u. and 1.55 a.u. for the
Fe and O atoms, respectively. For LDA+U method the parameter Ueff were chosen equal to 4.5 eV and 4.0 eV for atoms FeA and FeB, respectively. In all calculations, self-consistency was achieved with a tolerance in the total energy of 0.1 mRy. All calculations have been carried out using the Wien2k package [3].
Crystal structure Magnetite crystallizes in the inverse cubic spinel structure (Fd3m) above the so-called Verwey transition temperature (tV≈120 K). But at temperature lower than tV it takes the orthorhombic structure (Imma) [4 – 6]. Unit cells of the high temperature (HT) and low temperature (LT) phases are presented in Figs. 1a and 1b, respectively.
245 (a) (b)
Fig. 1. Unit cells of the HT (a) and LT (b) Fe3O4.
In both structures the iron atoms occupy octahedral and tetrahedral sites (the nearest neighboring of oxygen atoms forms an octahedron or a tetrahedron). The iron atoms of A-type (FeA) with valence 3+ held tetrahedral sites, whereas the iron atoms of B-type (FeB) with mixed 2.5+ valence held octahedral sites. Besides, in low temperature phase the B-site is separated into two positions FeB1 and FeB2 with different valences, 3+ and 2+, respectively. Positions of atoms in both structures are described in Table 1. Structural differences cause the difference in properties of magnetite, in particularly, conductivity of HT structure is higher than conduc- tivity of LT one. The experimental value of the lattice parameters [2, 6] and positions of atoms for the high and the low temperature phases of Fe3O4 are presented in Table 1.
High temperature phase Low temperature phase Space group Fd3m (227) Imma (74)
aexp (Å) 8.394 5.912
bexp (Å) 8.394 5.945
cexp (Å) 8.394 8.388
FeA positions 8a (0 0 0) 4e (0 1/4 1/8) 4b (0 0 ½) Fe positions 16d (5/8 5/8 5/8) B 4d (¼ ¼ ¾) 32e (u u u) 8h (0 2u u) O positions 8i (2u-¼ ¼ ¼-u) u 0.3798 0.2548 Table 1. Experimental lattice parameters and positions of atoms for HT and LT phases of Fe3O4.
246 Results
The properties of HT Fe3O4 were calculated within the framework of LSDA, GGA and LDA+U methods. The results of the geometry optimization and calculated atomic magnetic moments are listed in Table 2.
As is it seen from Table 2 all methods indicate that the magnetic sublattice of FeA and FeB atoms are ordered antiferromagnetically. As the number of FeB sites is two times greater than the number of FeA sites, the Fe3O4 is a ferrimagnetic. However the LSDA method leads to the extremely low values of the atomic magnetic mo- ments (the value of the magnetic moment on the FeA site is about 1 µB lower the experimental value). The gradient correction of the exchange-correlation potential improves the values of atomic magnetic moments, but the better agreement with experiment is achieved using the LDA+U method.
Method LSDA GGA LDA+U
aopt (Å) 8.1247 8.4018 8.1247
m(FeA) (µB) -2.86 -3.39 -3.71
m(FeB) (µB) 3.08 3.48 3.63
m(O) (µB) 0.07 0.07 0.06
Table 2. Optimized lattice parameter and magnetic moments of atoms in Fe3O4 calculated within different methods. In Fig. 2 we plot the total density of states (DOS) calculated within three dif- ferent methods. As it is seen, the different methods lead to rather different shape of DOS. For example, for the LSDA method the energy gap for the spin-up states is above the Fermi level (EF), whereas for the GGA method EF falls in the middle of a narrow gap of about 1 eV. And this gap becomes larger (of about 2 eV) for the LDA+U method. If we look at the site resolved DOS one can see that the down- spin DOS at the Fermi level is formed mainly by the FeB d-states.
The LSDA method leads to the well localized d-states for the both FeA and
FeB atoms. The up and down spin states are shifted relative to each other by about
3.5 eV in such a manner that the for the FeA atom the down-spin states are oc- cupied and the up-spin states are unoccupied. For the FeB atom, on the contrary, the up-spin states are almost completely occupied and the down-spin states are partly filled. The gradient correction does not change the DOS shape noticeably. However, consideration of LDA+U method results to dramatic changes in DOS. Namely, the shift between up- and down-spin states becomes much more important
(~ 10 eV); the up-spin states of FeB get delocalized, whereas the down-spin part remains almost unchanged. We do not plot the oxygen contribution to DOS, as it is of low intensity and is not visible in the picture. The analysis shows that the O p-states are strongly hybridized with d-states of iron atoms. 247 Fig. 3. Total spin-polarized DOS in the high temperature phase of Fe3O4 calculat- ed within different methods. The Fermi level corresponds to E = 0 and is marked by a vertical solid line.
Conclusions As a conclusion, according to our calculations the most accurate and suitable method to study Fe3O4 is LDA+U, and exactly this method will be used for study- ing of properties thin films Fe/Fe3O4.
References 1. Wenzel M.J. et al. // Phys. Rev. B, vol. 75, p. 214430 (2007). 2. Novak P. et al. // Physica B, 312-313, p. 785 (2002). 3. Blaha P. et al. // Comput. Phys. Commun. 59, p. 399 (1990). 4. Jeng H.-T. et al. // Phys. Rev. B, v. 65, p. 094429 (2002). 5. Verwey E.J.W. et al. // J. Chem. Phys., vol. 15, p. 174 (1947). Verwey E. J.W. et al. // J. Chem. Phys., vol. 15, p. 181 (1947). 6. Hamilton W.C. // Phys. Rev., vol. 110, number 5, p. 1050 (1958). 248
Table of Content
A. Chemistry...... 5
Solid-contact ion-selective electrodes with ion-to-electron transducer layer composed of nanostructured materials Ivanova Nataliya...... 6
Using of semiconductor oxide films for detection of volatile organic compounds in gases Lopatnikov Artem...... 11
Digital spectrographic analysis of human biological fluids for determination of microelements Savinov Sergey...... 15
Synthesis of condensed imidazole derivatives with a Nodal nitrogen atom - pyrido[1,2-a]benzimidazoles Sokolov Alexandr...... 19
С. Mathematics and Mechanics...... 23
Algebraic approximation of global attractors of discrete dynamical systems Malykh Artem...... 24
Taken’s time delay embedding theorem for dynamical systems on infinite- dimensional manifolds Popov Sergey...... 28
A two-phase problem arising from a microwave heating process in nonhomogeneous material Serkova Nadezhda...... 32
Lyapunov functions in upper Hausdorff dimension estimates of cocycle attractors Slepukhin Alexander...... 37
D. Solid State Physics...... 43
Intercalation of Al as a method of formation of quasifreestanding graphene Anna Popova, Alexander M. Shikin...... 44
Calculation of Sound Speed in Artificial Opal Andrey Uskov...... 49 250 Modification of spin and electronic structure of graphene by intercalation of Bi Evgeny Zhizhin...... 54
E. Applied Physics...... 59
Usage Pocket Comsol for the Numerical Nonstationary Nonlocal Plasma Modeling Burkova Zoya...... 60
Factorization of charge formfactors for clusterized light nuclei in reactions e+16O and e+12C Danilenko Valeria...... 64
Study of interaction forces between constant magnet and high-temperature superconductor Marek Veronika...... 68
Usage of stereoscopic 3D-visualization technologies Marek Veronika...... 73
Evaluation of the influence of readout cables in the CBM Silicon Tracking System Prokofyev Nikita...... 77
Application of graph theory to modeling of the complex hydraulic systems Strizhenko Olga...... 82
F. Optics and Spectroscopy...... 87
A modern implementation of Rozhdestvenski interferometer Agishev1 N.A., Medvedeva2 T.A., Ryabchikov1 E.L...... 88
Investigation of the two-photon induced fluorescence in Rb vapor excited by Ti:Sapphire femtosecond laser pulses Bondarchik Julia...... 92
The research of optical spectra of oil fraction in IR-area Chernova Ekaterina...... 97
Luminescence spectra of YVO4 and Y2O3 nanopowders Kolesnikov Ilya...... 102
251 Resonance grating based on InGaAs/GaAs quantum well Kozhaev Mikhail, Kapitonov Yury...... 106
Application of 2D-correlation spectroscopy method for interpretation of spectra and enhancing the spectral resolution Maximova Ekaterina, Lev Derzhavets...... 110
Observation of the fine structure for rovibronic spectral lines in visible part of emission spectra of D2 Umrikhin I.S., Zhukov A.S...... 114
G. Theoretical, Mathematical and Computational Physics...... 119
Conservation laws and energy-momentum tensor in Lorentz-Fock space Angsachon Tosaporn...... 120
Renormalization-group and ε- expansion: representation of anomalous dimensions as nonsingular integrals Batalov Lev...... 124
Surface states in semi-infinite superlattice with rough boundary Bylev Alexander...... 127
Modeling of thermal-hydraulic processes in complex domains by conservative immersed boundary method Chepilko Stepan...... 132
Development of functional integration techniques for drift-diffusion processes on Riemannian manifolds Chepilko Stepan...... 137
Multifractal generalization of the detrending moving average approach to time series analysis Ganin Denis...... 142
Propagation of photons and massive vector mesons between a parity breaking medium and vacuum Kolevatov Sergey...... 146
Analytical solution of two-dimensional Scarf II model by means of SUSY Krupitskaya Ekaterina...... 151
252 Effects of turbulent mixing on critical behaviour: Renormalization group analysis of the ATP model Malyshev Aleksei...... 156
Inertial-range behaviour of a passive scalar field in a random shear flow: Renormalization group analysis of a simple model Malyshev Aleksei...... 161
Effects of Stefan’s flow and concentration-dependent diffusivity in binary condensation Martyukova Darya...... 166
A matrix approach for dyadic Green's function in multilayered elastic media Nikitina Margarita...... 170
Detectable effects in classical supergravity Niyazov Ramil...... 175
Calculation of characteristics of critical behavior in logarithmic dimensions Artem Pismenskiy...... 180
Deal.ii library as a tool to study three-body quantum systems Shmeleva Yulia...... 184
Second order effects in the hyperfine and Zeeman splittings in highly charged ions Mikhail M. Sokolov...... 188
Hamiltonian Mechanics in Spaces of Constant Negative Curvature Stepanov Vasiliy...... 193
3D isotropic random walks with exponential distribution on free paths. Application to evaporation of a droplet at transient Knudsen numbers Telyatnik Rodion...... 197
Investigation of the dependence of the number of binary interactions and the number of participants on the class of centrality in ultrarelativistic heavy ion collisions Vorobyev Ivan...... 203
253 H. Biophysics...... 209
Application of Surface Plasmon Resonance for Detection of DNA Immobilization on Gold Surface Fironov Alexander...... 210
DNA Interaction with Palladium Compound K2[PdHGluCl2] in vitro Kozhenkov Pavel...... 215
Studing of the UV radiation influence on the DNA in a solution in the presence of caffeine Platonov Denis...... 220
Entropic sampling of thermodynamic and structural properties of polymer chains and stars within Wang-Landau algorithm Silantyeva Irina...... 225
Silver nanoparticles and their interaction with polymers in solution and on a surfaces Varshavskii Mikhail...... 230
I. Resonance Phenomena in Condensed Matter...... 235
Phase transitions in magnesium: ab initio study Klyukin Konstantin...... 236
NMR study of spin relaxation Nefedov Denis...... 241
Magnetic properties of cubic magnetite Fe3O4: a density functional theory study Irina Shikhma...... 245
254