DOCSLIB.ORG

Ab Initio Calculation of Electric-Field-Gradient Tensors Of

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Ab Initio Calculation of Electric-Field-Gradient Tensors Of American Mineralogist, Volume 81, pages 545-549, 1996 Ab initio calculation of electric-field-gradient tensorsof forsterite B.rdnN WrNxlrnrr Pnrnn BllHn,2.q,No KARLHTTNzScuwl.nz2 'Mineralogisch-PetrographischesInstitut der Christian-Albrechts Universitiit, Olshausenstrasse40, D 24098 Kiel, Germany 'zInstitutfiir TechnischeElektrochemie. Technische Universitiit Wien. A-1060 Vienna, Austna Ansrn-lcr Ab initio band-structure calculations basedon the density functional theory have been performed for forsterite to obtain, with a parameter-freemodel, electric-field gradientsfor all nuclei. Calculations basedon the generalizedgradient approximation yield the ratio of the largest components of the calculated electric-field gradients within l0loof the experi- mental value for 25Mgand within 2oloof the experimental value for '7O. The absolute values differ by about 5ol0,depending on which nuclear quadrupole moment is used in the conversion. The asymmetry parameters are also in good agreement with experimental data. Values obtained with a gradient-correctedexchange-correlation potential are better than those based on the standard local-density approximation. The calculated anglesbe- tween the principal axes of the quadrupole coupling tensor and the crystallographicaxes agreevery well with the experimental data. For the Ml site the maximum deviation is 1.8",and for the M2 site the maximum deviation is 0.6". The current calculationsallow an evaluation of three sets of experimental data for therTO electric-field gradient. They also confirm a proposed assignmentof the measuredelectric-field gradient tensors to spe- cific O atoms. INrnooucrroN V,", which varies between0 (axial symmetry) and 1. Here we follow the sameconventions as in the work of Schwarz In computational mineralogy, predictive microscopic et al. (1990). models are increasingly used to interpret experimental However, QCC and 4 cannot be derived from NMR data for which conflicting phenomenological explana- spectradirectly but must be determined by spectral sim- tions exist. Although reliable quantum mechanicalmodel ulation, which can be problematic for broad, overlapping, calculations require large computational resources,they or weak features. Conventionally, EFGs have been ex- offer the possibility of calculating rather subtle physical plained by point-chargemodels, which contain adjustable properties, including those due to electronic effects. parameters, such as formal chargesor the Sternheimer Therefore, they can be applied to a large variety ofprob- antishielding factors. The predictive power of such par- lems and yield better results than conventional static lat- ameterizedcalculations is limited, and the results are of- tice-energy-minimization calculations. Here we use pa- ten ambiguous. Hence, attempts have been made to de- rameter-free energy-band-structurecalculations to velop parameter-freecalculations. For example, Tossell compute electric-field gradient (EFG) tensors,which are and coworkers [e.g., Tossell (1992) and referencescited experimentally determined by nuclear magnetic reso- therein] usedHartree-Fock calculationson clustersto ob- nance (NMR), nuclear quadrupole resonance,or, M<iss- tain NMR parameters. These calculations required the bauer spectroscopy. use of several approximations owing to limitations in NMR has proven to be an invaluable tool for the in- available computing power. With a different method, vestigation of local structural details, and, because of which could copewith periodic lattices,Blaha et al. (1985) progressin the experimental technique, many nuclei rel- showed that full-potential, Iinearized, augmented plane- evant for the earth sciencescan now be used.An instruc- wave (FP-LAPW) calculations can be successfullyused tive review of applications of NMR spectroscopyin the to obtain EFGs. The crucial quantity that determines an earth scienceswas given by Kirkpatrick (1988). Structural EFG is the nonspherical chargedistribution close to the information can be obtained from NMR measurements nucleus,which itself is directly related to chemical bond- in which the interaction betweenthe electric quadrupole ing and different occupations of the valence electrons. moment, eQ, and the local electric-field gradient, eq : During the last decade,this method was used to inves- V,", is determined in terms of the quadrupole-coupling tigate severalhcp metals (Blaha et al. 1988) and many constant, QCC, givenby QCC : eV""Q/h. The EFG de- other substances,and its feasibility for complex struc- pendson the nonsphericalcharge distribution around the tures, for example, was demonstrated by successfulcal- nuclei, and its deviation from axial symmetry is defined culations for a high-2" cuprate superconductor(Schwarz by the asymmetryparameter, 4, given by ,t: (4" - V-)/ et al. 1990). From these calculations it can be deduced, 0003-004x/96l0506-0545$05.00 545 546 WINKLER ET AL.: ELECTRIC.FIELD GRADIENTS OF FORSTERITE TABLE1. Atomic coordinates used in the calculations and the proximation (LDA) or the generalizedgradient approxi- corresponding remaining forces mation (GGA). A recent in-depth summary can be found Fujino in the book by Singh (1994).For such calculations,one Hazen Forces Brown Forces et al. Forces Forces of the most accurate schemesis the LAPW method, in (1e76) (GGA) (1e70) (GGA) (1981) (LDA) (GGA) which the unit cell is divided into spherescentered at the Mg2x 0.9915 0.6 0.9896 1.0 0.99169 0.0 0.4 atomic positions and an interstitial region. In the latter -0.1 -0.4 -0.3 -0.4 Mg2y 0.2774 0.2776 0.27739 the basis set consistsof plane waves that are augmented Si x 0.4262 6.1 0 4226 48 6 0.42645 -1.3 2.2 Si y 0.0940 -0.9 0.0945 -4.5 0.09403 -1.9 1.4 by atomic-like solutions (numerical radial functions mul- 01 x 0.7657 18.6 0.7667 -3.4 0.76594 17.0 17.4 tiplied with sphericalharmonics) inside the spheres.The -0.4 01 y 0.0913 2.2 0.0918 1.7 0.09156 1.0 potential within the sphereis not restrictedto have spher- 02 x 0.2215 6.6 0.2202 11 3 0.22164 41 4.7 02 y 0.4474 -21.6 0 4477 -25.9 0.44705 -5.4 -14.8 ical symmetry, as in the older "muffin-tin" calculations 03 x 0.2777 -7.4 0.2781 -13 5 0.27751 -4.5 -5.9 but is allowedto be general,i.e., a full potential without y 03 0.1628 11 .9 0.1633 14.4 0.16310 2.5 8.5 It is well known that 03 z 0.0331 -15.6 0.0337 -23.9 0.03304 -6.0 -12.9 any shape approximation is used. DFT calculations in general give structural parameters given : Nofer Forces in mRyd/au;1 Ryd 13.6 eV, 1 au: 0.529A. within 1-20loof the experimental data. In the presentstudy, the full-potential, linearized, aug- mentedplane-wave package, WIEN95 (Blahaet al. 1995) that in generalFP-LAPW calculations yield EFGs within was used.[WIEN95 is an improved and updatedversion approximately 100/oand asymmetry parameters within ofthe original copyrightedcode by Blaha et al. (1990).1 -0. I of the experimentalvalues. WIEN95 is maintained at the Technical University of Consideringthe widespreaduse of NMR spectroscopy Vienna and includes the capability to employ local or- and the recent progressin quantum mechanical calcula- bitals (Singh 1991) and to calculateresidual forces on tions it is worthwhile to compute electric-field-gradient atoms (Yu et al. l99l: Kohler et al. 1996).The calcula- tensorsfor silicates,which are characterizedby their broad tion of the EFGs is based on the work of Blaha et al. range of bond types and bond strengths. We note that (1985); a more detailed description can be found in NMR spectroscopyis a local technique, and therefore it Schwarzet al. (1990).The precisionand accuracyofthe is often used to investigateshort-range order or local dis- calculationsare controlled by only a few parameters.For tortions. Ab initio calculations, however, exploit the pe- the expansion ofthe chargedensity (potential) in the in- riodicity of a perfectly ordered structure, so that solid terstitialregion a Fourier serieswith lG | = t Z wasused. solutions and the effectsofshort-range order are currently For the wave functions inside the atomic spheresangular approximated rather crudely. Also, the present calcula- momentum componentsup to /: 12 wereincluded, and tions are restricted to ground-state(0 K) properties. up to 18 k points in the irreducible wedgeof the Brillouin For fully ordered structures,theoretical studiesare pos- zonewere used. This correspondsto nearly 150 sampling sible and can be used for the interpretation of measured points in the whole Brillouin zone. The cut-off for the spectra,which is sometimes difficult using experimental plane-wavebasis set was chosenas k-u* : 4.5 ait and results only. Reliable calculations allow evaluations of was checkedby additional calculations,which proved that proposedexperiments, which canbe costlywhen isotopic the results presentedhere are well converged. enrichment is required, becausethey can predict NMR parametersand specify the necessaryresolution. For the Rnsur,rs interpretation of spectraof silicates,correlations between structural and NMR parameters have been established Structure (Sherritretal. l99l; Smith et al. 1983).These studies rely The all-electron ab initio techniques currently in use on available data, which may be scarcefor unusual iso- do not yet allow a constant-pressurerelaxation of low- topes. In theoretical studies, however, structures can be symmetry structures.In fact, even at constant volume a easily deformed, and thus NMR parametersas a function relaxation of an orthorhombic structure containing 28 of structural parameters can be calculated. Also, theory atoms using an all-electron approach would at present can help in evaluating sets of experimental data if the consumea prohibitive amount of computer time. Hence, latter are not in good agreement. in the present study all structural parameterswere fixed Forsterite,MgrSiOo, is a model systemin which the at their experimental values. To investigatethe influence same isotopes occupy different structural environments of small structural distortions on the calculated physical 2sMg and in which the EFGs for both and "O have been propertieswe usedthree setsof internal structural param- studied in detail. The small differencesin the Mg and O eters in our calculations.
Load more

Recommended publications
  • Electric Quadrupole Interaction in Cubic BCC Α-Fe
    Electric quadrupole interaction in cubic BCC α-Fe A. Błachowski 1, K. Kom ędera 1, K. Ruebenbauer 1* , G. Cios 2, J. Żukrowski 2, and R. Górnicki 3 1Mössbauer Spectroscopy Division, Institute of Physics, Pedagogical University ul. Podchor ąż ych 2, PL-30-084 Kraków, Poland 2AGH University of Science and Technology, Academic Center for Materials and Nanotechnology Av. A. Mickiewicza 30, PL-30-059 Kraków, Poland 3RENON ul. Gliniana 15/15, PL-30-732 Kraków, Poland *Corresponding author: [email protected] PACS: 76.80.+y, 75.50.Bb, 75.70.Tj Keywords: cubic ferromagnet, electric field gradient (EFG), Mössbauer spectroscopy Short title: EFG in α-Fe Abstract Mössbauer transmission spectra for the 14.41-keV resonant line in 57 Fe have been collected at room temperature by using 57 Co(Rh) commercial source and α-Fe strain-free single crystal as an absorber. The absorber was magnetized to saturation in the absorber plane perpendicular to the γ-ray beam axis applying small external magnetic field. Spectra were collected for various orientations of the magnetizing field, the latter lying close to the [110] crystal plane. A positive electric quadrupole coupling constant was found practically independent on += 19 -2 the field orientation. One obtains the following value Vzz )4(61.1 x 10 Vm for the (average) principal component of the electric field gradient (EFG) tensor under assumption that the EFG tensor is axially symmetric and the principal axis is aligned with the magnetic hyperfine field acting on the 57 Fe nucleus. The nuclear spectroscopic electric quadrupole moment for the first excited state of the 57 Fe nucleus was adopted as +0.17 b.
    [Show full text]
  • ß-NM R Study of the Electric Field Gradient in the Metallic Intercalation
    ß-NMR Study of the Electric Field Gradient in the Metallic Intercalation Compound LiC6 P. Freiländer, P. Heitjans, H. Ackermann, B. Bader, G. Kiese, A. Schirmer and H.-J. Stockmann Fachbereich Physik, Universität Marburg, West Germany and Institut Laue-Langevin, F-38042 Grenoble Cedex, France Z. Naturforsch. 41 a, 109 -1 1 2 (1986); received July 22, 1985 In the layered compound LiC 6 polarized /?-active 8Li probe nuclei were produced by capture of polarized thermal neutrons. Nuclear magnetic resonance spectra of 8Li were recorded via the /5-radiation asymmetry. The 8 Li quadrupole coupling constant e2qQ/h, measured in the temperature range 7 = 5... 500 K decreases with increasing 7 from 45.2(8) to 22.8(4) kHz. Anomalies in the overall temperature dependence are discussed in terms of phase transitions proposed for LiC6. 1. Introduction graphite the carbon layers are stacked in direct registry with respect to each other. The lithium A characteristic feature of graphite intercalation atoms, being sandwiched between two carbon layers compounds (GICs) is the variety of ordered struc­ such that one-third of the C hexagons have Li tures and phase transitions observed. A GIC of atoms directly above and below, form a 2D hexa­ stage n is formed by a periodic arrangement of one gonal superlattice (j/3 x |/3 30° structure: the Li- layer of intercalant followed by n layers of graphite. superlattice vectors are measured in units of the Both intercalant in-plane order and layer sequence graphite lattice, their direction is obtained by 30° are known to undergo various phase transitions for rotation of the graphite lattice vectors).
    [Show full text]
  • Accelerator Glossary of Terms
    Accelerator Glossary of Terms Table of Contents The Obligatory Disclaimer 2 -M- 75 -Numbers- 3 -N- 83 -A- 5 -O- 87 -B- 13 -P- 89 -C- 23 -Q- 99 -D- 37 -R- 101 -E- 45 -S- 107 -F- 49 -T- 121 -G- 53 -U- 130 -H- 55 -V- 132 -I- 61 -W- 134 -J- 65 -X- 136 -K- 67 -Y- 136 -L- 69 -Z- 136 Glossary v3.6 Last Revised: 6/8/11 Printed 6/8/11 Accelerator Glossary of Terms The Obligatory Disclaimer This is a glossary of accelerator related terms that are important to the Beams Division, Operations Department, Fermilab National Accelerator Laboratory. In the course of the day to day working with the accelerator there is a bewildering array of "Accelerator Technospeak" used in the Operations Group and other areas of the Beams Division. This glossary is meant to be an attempt at helping the uninitiated understand some of the jargon, slang, and buzz words that are commonly used in everyday communication within the Beams Division and the lab in general. Since this glossary was created primarily for Operations Department personnel there are some terms included in this glossary that might be confusing or unfamiliar to those outside of the Operations Department. By the same token there are many commonly understood terms used by other divisions and groups in the lab that are not well known or non existent within the Operations Department. Many terms will not be found within this document. If anybody has any contributions of terms, definitions or sources of other glossaries to be included please feel free to pass them on, even if they are not related to the Beams Division.
    [Show full text]
  • The Augmented Plane Wave Method (APW)
    New approach to the problem of electron semicore states in LAPW band structure calculations А.V. Nikolaev (Skobeltsyn Institute of Nuclear Physics, Lomonosov Moscow State University and Moscow Institute of Physics and Technology) collaboration with D. Lamoen (EMAT, Department of Physics, UA) and B. Partoens (CMT group, Department of Physics, UA) 30 august 2016 Outline Electric Field Gradient (EFG) 1. Where needed 2. Definitions 3. Relations between the electric field gradient and the expansions of the full potential around a nucleus Ab initio solid state calculation of solids (LAPW method) 1. Augmented plane wave method (APW) 2. Linearized augmented plane wave method (LAPW). 3. Full potential (FLAPW). Treatment of semicore states in the FLAPW method. J. Chem. Phys. v. 145, 014101 (2016) Experimental measurements (TDPAC-spectroscopy): А.V. Tsvyashchenko, L.N. Fomicheva (Vereshchagin Institute for High Pressure Physics, RAS) G.К. Ryasnyj (Skobeltsyn Institute of Nuclear Physics Lomonosov Moscow State University) А.I. Velichkov, А.V. Salamatin, О.I. Kochetov, D.А. Salamatin, (Joint Institute for Nuclear Research, Dubna) M. Budzynski (M.Curie–Sklodowska University, Lublin) Time dependent perturbed angular correlation (TDPAC) method Two -detectors for Nuclear states Propagation vectors two directions of in the - cascade of two -quanta - correlations The emission probability of two -quanta TDPAC method : probe nuclei in crystal Time evolution of the intensity of the -emission in external electric or magnetic field The probe nucleus experiences
    [Show full text]
  • And Quantum Mechanical Behaviour Is a Subject Which Is Not a Rarefied and Moribund Field
    International Journal of Modern Engineering Research (IJMER) www.ijmer.com Vol.2, Issue.4, July-Aug 2012 pp-1602-1731 ISSN: 2249-6645 Quantum Computer (Information) and Quantum Mechanical Behaviour - A Quid Pro Quo Model 1 2 3 Dr K N Prasanna Kumar, prof B S Kiranagi, Prof C S Bagewadi For (7), 36 ,37, 38,--39 Abstract: Perception may not be what you think it is. Perception is not just a collection of inputs from our sensory system. Instead, it is the brain's interpretation (positive, negative or neutral-no signature case) of stimuli which is based on an individual's genetics and past experiences. Perception is therefore produced by (e) brain‘s interpretation of stimuli .The universe actually a giant quantum computer? According to Seth Lloyd--professor of quantum-mechanical engineering at MIT and originator of the first technologically feasible design for a working quantum computer--the answer is yes. Interactions between particles in the universe, Lloyd explains, convey not only (- to+) energy but also information-- In brief, a quantum is the smallest unit of a physical quantity expressing (anagrammatic expression and representation) a property of matter having both a particle and wave nature. On the scale of atoms and molecules, matter (e&eb) behaves in a quantum manner. The idea that computation might be performed more efficiently by making clever use (e) of the fascinating properties of quantum mechanics is nothing other than the quantum computer. In actuality, everything that happens (either positive or negative e&eb) in our daily lives conforms (one that does not break the rules) (e (e)) to the principles of quantum mechanics if we were to observe them on a microscopic scale, that is, on the scale of atoms and molecules.
    [Show full text]
  • Nuclear Hyperfine Interactions
    Nuclear hyperfine interactions F. Tran, A. Khoo, R. Laskowski, P. Blaha Institute of Materials Chemistry Vienna University of Technology, A-1060 Vienna, Austria 25th WIEN2k workshop, 12-16 June 2018 Boston College, Boston, USA Introduction I What is it about? Why is it interesting in solid-state physics? I It is about the energy levels of the spin states of a nucleus. I An electric or magnetic field at the position of the nucleus may lead to a splitting of the nuclear spin states I The energy splitting depends on the intensity of the field, which in turns depends on the environment around the nucleus =) The measured spin nuclear transition energy provides information on the charge distribution around the nucleus I Outline of the talk I Hyperfine interactions I Electric (isomer shift, electric-field gradient) I Magnetic (Zeeman effect, hyperfine field) I Nuclear magnetic resonance I Calculation of chemical shielding (orbital and spin contributions) Introduction I What is it about? Why is it interesting in solid-state physics? I It is about the energy levels of the spin states of a nucleus. I An electric or magnetic field at the position of the nucleus may lead to a splitting of the nuclear spin states I The energy splitting depends on the intensity of the field, which in turns depends on the environment around the nucleus =) The measured spin nuclear transition energy provides information on the charge distribution around the nucleus I Outline of the talk I Hyperfine interactions I Electric (isomer shift, electric-field gradient) I Magnetic (Zeeman effect, hyperfine field) I Nuclear magnetic resonance I Calculation of chemical shielding (orbital and spin contributions) Hyperfine interactions ! Shift and splitting of energy levels of a nucleus I Interactions between a nucleus and the electromagnetic field produced by the other charges (electrons and other nuclei) in the system.
    [Show full text]
  • Electric Field Gradient in Aluminium Due to Vacancy and Muon * M
    Electric Field Gradient in Aluminium due to Vacancy and Muon * M. J. Ponnambalam University of the West Indies Z. Naturforsch. 47a, 45-48 (1992); received July 29, 1991 The electric field gradients (EFG) in aluminium due to a monovacancy and the interstitial muon are evaluated. The valence effect EFG qv is calculated using perturbed electron density ön(r) values obtained from density functional theory in an analytic expression which is valid at all distances from the impurity. The size effect EFG qs is evaluated using a new oscillatory form for the near neighbour (nn) displacements. The numerical values of qs are computed using fractional nn displacements available in the literature. For the total EFG good agreement with experiment is obtained without the use of any parameter. Key words: Nuclear Quadrupole Interaction, Electric Field Gradient 1. Introduction they used the following oscillatory form for the dis- placement of the host ions: In a cubic crystal, the electric field gradient (EFG) u(r) = A cos (2/c r + </>) r/r3, (2) vanishes at the site of any ion due to cubic symmetry F [1]. The presence of an impurity in a cubic host locally where kF is the Fermi wave vector while A and </> are destroys the cubic symmetry and thus produces a non- constants. vanishing EFG at the site of the host near neighbours In the case of a muon and a vacancy in Al and Cu, (nn) - thus leading to a non-vanishing nuclear quadru- Tripathi et al. [8] and Schmidt et al. [9] have carried pole interaction of the host nuclei.
    [Show full text]
  • Electric Field Gradients in Metals -'Ą
    HYPERFINE INTERACTIONS PROCEEDINGS OF XVII WINTER SCHOOL February 19 - March 3, 1979 Bielsko - Biała, POLAND Edited by : R. Kulessa and K. Królas Cracow; September 1979 4 NAKŁADEM INSTYTUTU FIZYKI JĄDROWEJ W KRAKOWIE v UL. RADZIKOWSKIEGO 152 Wydanie pierwsze. Nakład 220 egzemplarzy Arkuszy wydawniczych 25,5. Arkuszy drukarskich 22,4. Papier offsetowy kl. III, 70 g. 70 x 100. Oddano do druku w październiku 1979 roku. Druk ukończono w listopadzie 1979 roku. Zamówienie nr 295/79. Kopię kserograficzną, druk i oprawę wykonano w INSTYTUCIE FIZYKI JĄDROWEJ W KRAKOWIE SCHOOL HOSTS Institute of Nuclear Physics, Kraków Institute of Physics, Jagellonian University, Kraków SCHOOL ADDRESS Ośrodek Wdrażania Postępu Technicznego w Energetyce, Bielsko-Biała, ul. Brygadzistów 170 '\ OHSANIZIWG COMMITTEE Chairmen: R. Kulessa A.Z. Hrynkiewioz Members: E. Bożek M. Lach E. GOrlioh M» Marszałek H. Kmieć B. Haydell J. Kraczka K. Tomala K. Królas J. Wrzesiński Secretary: Z. Natkanieo 1. Electric Quadrupole Interaction ELECTRIC PIELD GRADIENTS IN METAIS G. Schatz 3 ELECTRIC FIELD GRADIENTS IN CUBIC ALLOTS K. Krdlas 21 SOME HYPERFINE INTERACTION STUDIES IN OXIDE GLASSES R. Kamal .............. 37 Panel Discussion: RESULTS OP QUADHUPOLE INTERACTION MEASUREMENTS IN METALS E. Bodenstedt ........ .. 48 REMARKS ON QUADRUPOLAR ELEMENTARY EXCITATIONS IN NONCOBIC METALS G. Schatz 61 2. Magnetic Dipole Interaction HYPERFINE FIELDS IN HEUSLER ALLOYS I.A. Campbell 67 MOSSBAUER SPECTROSCOPY IN INTERMETALLIC COMPOUNDS AND ALLOYS OP RARE EARTHS Z. Tomala ........ ....... 71 INVESTIGATIONS OP HYPERFINE INTERACTIONS IN NICKEL- FERRITE-ALUMINATES BY THE MOSSBAUER EPPECT METHOD J. Bara 93 MOSSBAUER STUDIES ON IRON BASE ALLOYS E. YJieser 105 MAGNETIC MOMENTS OP ns-ISOMERS IN 105Ag AND 1O3P<3 L.
    [Show full text]
  • Determination of Mossbauer Electric Field Gradient
    Revista Brasileira de Física, Vol. 10, N. O 3, 1980 Determination of Mossbauer Electric Field Gradient V. K. GARG Departamento de F/sica e Quimica, Universidade Federal do Espírito Santo, 29W Vitdria, ES, Brasil Recebido em 12 de Maio de 1980 There are severa1 reports of the electric quadrupole interac- tions available in the literat~re?-~The present discussion is a short survey, introducing the electric quadrupole up to the experimental pola- rised studies. Existem várias publ icaçÓes sobre interações de quadrupolo elé- tri co na 1 i teratura -" A presente discussão é um resumo, introduzindo o quadrupolo elétrico até o nivel de recentes estudos experimentais. 1. INTRODUCTION, (t STANDARD FORM OF EFG The magnitude of the quadrupole splitting is pronortional to the Z componerit of the electric field gradient (EFG) tensor which inte- racts with the quadrupole moment of the nucleus. For I = 3/2, the qua- drupole spl itting can be expressed as Where Q is the quadrupole moment of the nucleus eq = V = - the Z component of the EFG ZZ e = proton charge and 0 is the asymmetry parameter Either q or Vzz is referred to as the field gradient. The EFG tensor has nine cornponents. The potential at the Mtlssbauer nucleus (located at the ori- 1/2 gin) due to point charge q at (z,z~,z)a distance r = (x2 + y2 + z2) from the origin is given by V =q/r ( 3) The negative gradient of this potent ial, -$V, is the elec- -i tric field, E, at the nucleus, with components ++ the gradient of the electric field at the nucleus, TE, is given by the second partia1 der i vati ve tensor components are Vxx = q ( 3x2 - r2)r-' , V =q(3y2 - r2)r-5 , YY vzz = (3z2 - r2)r-' , - 5 v =v =3qq r> , w Y*C - 5 VXZ = vzx = 3 qxz r2 , - 5 v =Vzx=3qyz r .
    [Show full text]
  • Atomic Quadrupole Moment Measurement Using Dynamic Decoupling
    Atomic Quadrupole Moment Measurement Using Dynamic Decoupling R. Shaniv,1 N. Akerman,1 and R. Ozeri1 1Department of Physics of Complex Systems, Weizmann Institute of Science, Rehovot, Israel We present a method that uses dynamic decoupling of a multi-level quantum probe to distinguish 2 2 small frequency shifts that depend on mj , where mj is the angular momentum of level jji along the quantization axis, from large noisy shifts that are linear in mj , such as those due to magnetic field noise. Using this method we measured the electric quadrupole moment of the 4D 5 level in 2 88 + +0:026 2 Sr to be 2:973−0:033 ea0. Our measurement improves the uncertainty of this value by an order of magnitude and thus helps mitigate an important systematic uncertainty in 88Sr+ based optical atomic clocks and verifies complicated many-body quantum calculations. Increasing the coherence time of quantum superposi- sure of its deviation from perfect spherical symmetry. tions is an ongoing research topic. From the quantum While ground state S orbitals are spherically symmet- information point of view, prolonging the coherence time ric, higher levels, such as states in the D manifold allows for the performance of a larger number of coherent have a finite electric quadrupole moment. This electric operations, making more complex quantum computation quadrupole moment couples to an electric field gradient algorithms possible [1] as well as longer-living quantum across the atomic wavefunction. The resulting energy memory [2]. From a metrology perspective, decoherence shift is usually small in comparison to other shifts, e.g.
    [Show full text]
  • Lecture 21: Hyperfine Structure of Spectral Lines
    Lecture 21: Hyperfine Structure of Spectral Lines: Page-1 In this lecture, we will go through the hyperfine structure of atoms. Various origins of the hyperfine structure are discussed The coupling of nuclear and electronic total angular momentum is explained. Page-2 When individual multiplet ( JJ→ transitions) components are examined with spectral apparatus of the highest possible resolution, it is found that in many atomic spectra each of these components is still further split into a number of components lying extremely close together. This splitting is called hyperfine structure. − The magnitude of the splitting is ~2cm 1 . Hyperfine structure is caused by properties of the atomic nucleus. Isotopic effect: Heavier isotopes present 1 in 5000 in ordinary hydrogen. Therefore, different isotopes of same element have slightly different spectral lines. Consider 1H (hydrogen) and 2H (deuterium): 1 71− RRH = ∞ =1.09677 × 10 meter 1+ m M H 1 71− RRD = ∞ =1.097074 × 10 meter 1+ m M D The wavelength difference is therefore: λD ∆=λλHDH − λ = λ1 − λH R =λ 1 − H H R Figure shows the Balmer D line of H & D This is known as isotope shift. observed calculated −−11 ()cm ()cm Hα 1.79 1.787 Hβ 1.33 1.323 Hγ 1.19 1.182 Hδ 1.12 1.117 Page-3 A quantitative explanation of the isotope effect is not simple, exception H atom. For the heavier elements the effect is traced back to the change of nucleus radius with mass. Sm150− Sm 152 is double that of Sm152− Sm 154 . Usual increase is not from Sm150→ Sm 152 .
    [Show full text]
  • The Predictive Power of Different Projector-Augmented Wave Potentials for Nuclear Quadrupole Resonance
    crystals Article The Predictive Power of Different Projector-Augmented Wave Potentials for Nuclear Quadrupole Resonance Jaafar N. Ansari 1, Karen L. Sauer 2,3,* and James K. Glasbrenner 1,3,† 1 Department of Computational and Data Sciences, George Mason University, Fairfax, VA 22030, USA; [email protected] (J.N.A.); [email protected] (J.K.G.) 2 Department of Physics and Astronomy, George Mason University, Fairfax, VA 22030, USA 3 Quantum Materials Center, George Mason University, Fairfax, VA 22030, USA * Correspondence: [email protected] † Current address: Data Analytics Department, MITRE Corporation, 7515 Colshire Drive, McLean, VA 22102, USA. Received: 8 August 2019; Accepted: 25 September 2019; Published: 28 September 2019 Abstract: The projector-augmented wave (PAW) method is used to calculate electric field gradients (EFG) for various PAW potentials. A variety of crystals containing reactive nonmetal, simple metal, and transition elements, are evaluated in order to determine the predictive ability of the PAW method for the determination of nuclear quadrupole resonance frequencies in previously unstudied materials and their polymorphs. All results were compared to experimental results and, where possible, 14 to previous density functional theory (DFT) calculations. The EFG at the N site of NaNO2 is calculated by DFT for the first time. The reactive nonmetal elements were not very sensitive to the variation in PAW potentials, and calculations were quite close to experimental values. For the other elements, the various PAW potentials led to a clear spread in EFG values, with no one universal potential emerging. Within the spread, there was agreement with other ab initio models. Keywords: nuclear magnetic resonance; nuclear quadrupole resonance; electric field gradient; efg; density functional theory; projector-augmented wave method; paw 1.
    [Show full text]