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KFKMW2-12/A PREPRINT
I. LOVAS, L. MOLNÁR, К. SAILER, W.QREINER
PERIODIC STRUCTURE IN NUCLEAR MATTER
Hungarian Academy of Sciences CENTRAL RESEARCH INSTITUTE FOR PHYSICS BUDAPEST кпсммма/А BBCBSMftnr • ЩГШШшЯГШЩтШщ
PERIODIC STRUCTURE IN NUCLEAR MATTER
István Low and Lívia Molnár Derailment of Theoretical Phytkt Lajoi Koeuth University, H-4010 Debrecen, HUNGARY
Kornél Sauer* and Welter Greiner Institut für Theoretische Physik Johann Wolfeang Goethe UnivemtaX Frankfort am Main, BRD
' Present address: Department of Theoretical Physics, Lajos Kossuth University, H-4O10 Debrecen, Hungary
HU ISSN §3« 533* btvia Ura* LMe IMá, Катав Safer, Water Groaar: Periode Stncto a Nackar Matter КРИ-1992-ША ABSTIACT
Tar jnt|intiti »faárkai matin иг itaBrrt ia tar fraarwoih of Ihr qaaalaahaiaoiljanaii aa oMBesoafield, periodi c a space, a setf-consistent ut of «|«atkw к derived baKaarkldapproxiraatJoafw ÜKátocriptioaofaacleoasáa^ractá^vaoMacsm tioas have beea fouad: the baryoa density is constant ia spac^ lawcm the baryoa carrc* density is periodic. Taábif* density phase of »dear аШ1сгсш be prorfaced by аавиигорк external pressarc, ocuniagcg. ш rdatwistic heavy ioa reactions. The sdf-eoanstcat fields developing beyoad the iastabffity bait have a special screw symmetry. la the presence of sach an a» field the energy spectraa of the relativtttic aadcoas exUbits allowed and forbiddea baadssimilarrytoüxeaergycpectnmoftbeelectroosasoao^
Lovas István, МЫааг Lívia, Koran Sailer, Walter Grelaen Periodic Structure ín Nuclear MaUer KFKI-1992-12/A
KIVONAT
A maganyag tulajdonságait tanulmányozzuk a bantaaadrcdaamika keretei között Egy térben periodikus oMeret feltételezve szelfltoaaszlens egyenletrendszert származtatunk le átlag tér közelítésben a a- es омпегопок revén kölcsönható nukleonokra. Ezeknek az egyenleteknek megtaláltuk a megoldásait: a barion sűrűség térben állandó, de a barka áraaMŰrfiség térbea perk>duW. A maganyagmA ezt a nagy sűrűségű fázisát kfilsó anízoüópíkus nyomás képes létrehozni, oly^ mat anutyen a relstrtfsztilun előfordulni. Az ínstabilitásí küszöb után kialaku!ó szelíkonzísztens tér egy speciális csavar szimmetriával rendelkezik. Egy ilyen oMaezon térbea a relaüvisztíkas nukkoaok energiaspektna» hasonlóan a szilárd testek elektron- jaáakenergairxktniiiiáirazmegeageoVuéstA PERIODIC STRUCTURE IN NUCLEAR NATTER
István Lovas and Lívia Molnár
Department of Theoretical Physics
Lajos Kossuth University, H-4010 Debrecen, Hungary
Kornél Sailer and Walter Creiner
Institut für Theoretische Physik
Johann Wolfgang Goethe Universität,
Frankfurt aa Hain, BRD
ABSTRACT
The properties of nuclear satter are studied in the framework of the quantuahadrodynasics. Assuming an i*-meson field, periodic in space, a self-consistent set of equations is derived in mean field approximation for the description of nucleons interacting via «r-aeson and w-aeson fields. Solutions of these self-consistent equations have been found: the baryon density is constant in
Present address: Department of Theoretical Physics, Lajos Kossuth University, H-4010 Debrecen, Hungary
1
"••**•-*•••**№$«*(*•*&*,**.•, ^,*шм*шш»ыьи> sajSJM шваМиа**^^; .i space, however the baryon current density is periodic. This high density phase of nuclear Matter can be produced by anisotropic external pressure, occurring e.g. in relatlvistic heavy ion reactions.
The self-consistent fields developing beyond the instability limit have a special screw symmetry. In the presence of such an и field the energy spectrum of the relatlvistic nucleons exhibits allowed and forbidden bands similarly to the energy spectrum of the electrons in solids.
1. INTRODUCTION
In a great number of physical systems the homogeneous, isotropic distribution of matter looses its stability and at some specific values of the physical parameters a periodic structure is formed spontaneously [1]. In nuclear physics the pion condensation
[2] may serve as a typical example for such a phenomenon. In the last few years the instabilities of homogeneous nuclear matter leading to the formation of a periodic meson field have been discussed in a number of papers [3J-Í6J.
The possibility of a periodic structure associated with a periodic и field is expected for the following reason. The periodic structure corresponds to a specific combination of particle-hole excitations of the nucleons, which requires some excitation energy. The и field which mediates a repulsive interaction in the particle-particle channel gives rise to an
2 attraction in the particle-hole channel. If the energy gain due to this attractive particle-hole interaction overcoapensates the energy loss needed to excite th? particle-hole configuration, the formation of a periodic structure is favoured. A similar phenomenon can be observed in electron plasma (8), (9].
In this paper, working in the framework of the quantum hadrodynamics (QHD) [7), a periodic и field is assumed and a set of self-cons istent equations is derived in the relativistic mean field approximation. In this way the periodic structure can be studied also beyond the instability point of the homogeneous system.
It has to be noted that, for the moment, the reliability of the mean field approximation is not yet clarified.
Studying the possibility of periodic structures, Frlman and
Henning [4] have investigated the condition of the instability.
They have found the limits of the instability in mean field approximation. The instability, however, disappears when the quantum corrections in one-loop approximation are included. From this observation they have arrived at the conclusion that the in stability against the formation of a periodic structure found in mean field approximation is a spurious, non-physical effect. In the same time some other type of instability has been found by
Furnstahl and Horowitz [5] calculating the meson propagators on one loop level. Using the Random Phase Approximation, Price,
Shepard and McNeil have shown the possibility of a periodic structure which can be associated* with alpha particle clustering
3 [6]. The problem seeas to be even aore involved if we take into account the results of Ref. (13) where it was proved that the
perturbatlve corrections to the aean field approximation in terms of one, two', etc. loop contributions do not form a convergent
series. From these studles.lt follows that the results obtained in
the mean field approximation must be checked and corrected by non-
pert ur bat lve methods (14), e.g. by lattice QHD calculations [15].
2. THE MODEL
The field equations for the nucleon field ФЫ) and for the
(Г-meson and w-meson fields are given in the standard notation as
follows:
f? (íe^-g^íx)) •g(r »••>-•>*-a?» (2) 2 V ~ (3) I». и ц u v "or p For the sake of the flexibility of the model here we have introduced a self-interact ing given by U( The parameters of this potential (b,c) together with the masses 4 (a, a. , a ) and coupling constants (g , g ) are given in Table 1. In the spirit оГ the aean field approxiaation, we assuae that the aeson field operators can be replaced by their expectation values considered as classical fields. The density operators of the nucleon field, i.e., the source teres oi* the aeson fields in Eqs. (2) and (3), are also replaced by their expectation values. More specifically, we introduce the following Ansatz: «e(x) -> <«e(x)> - «* , (6) «Лх) -* »2(x) -* <»2(x)> « -u sinkz, (8) ы3(х) -* <»3(x)> - 0 , (9) ф Г ф -> <ф Г ф> . (10) Неге z denotes the x3 coaponent of the x fourvector and k(«k ) Is the only non-vanishing component of the wave number fourvector for the и field. By subtituting this Ansatz Into the field equations, we arrive at the following set of equations: 5 •Мшаii •/-^•-•:. . IIVII I llilTWIif \ I I 1 - 2 Г »,,** «J*V* «tt» (Hcoskz - 7 slnkz)-a*geel#(x)-0. (11) (12) 2-е _T_ e ^^ (12) I (kV)i coskz - gM<#arV . (14) 2 (kV)« slnkz - gw<#jr #> . (15) 3 (16) о - g0 • These equations determine the paraaeters of the assuaed mean fields, i.e., the values of v, w° and w. In the Ansatz given by Eqs. (5)-(9) we have assuaed that an и field periodic in space can be developed. It is easy to see that 1 the и field satisfies the "Lorentz" condition: a «/* - о (17) The field tensor defined as F"" - eV - eV (18) 4 I: » has only two non-vanishing elements: 6 * 0 J F13 - üc slnkz • H2 , (19) F23 • ük coskz • -H1 . (20) This "Magnetic" field Is perpendicular to the z axis and it rotates in the x-y plane as one goes along the z axis. The w field has a special symmetry. Namely, the и field is invariant against a screw transformation which is the combination of a translation along the z axis z • a (21a) and a simultaneous rotation around the z axis ы' «Вы; У» (21b) cos ka -sin ka 0 В sin ka cos ka 0 0 0 «J where the angle of rotation is proportional with the translation. Furthermore, there exists a discrete translational symmetry due to the periodic behaviour of the и field. Consequently, the solutions of the Dirac equation are Bloch waves: »px. Mp.x) - e ""•(p.z) (22) t 4f ^^ÍVATV)»W' * WHM**#« >«*<*twttjmmbä--* J-.-.с.:- WSMICT'^'I-I U58P»W*IMJLii4L\I твшшЛиШ Z3E \ where the function • aust be periodic with the "lattice spacing" 2ж/к: 1 2a\ • (p.z) « • (p.z+r-) (23) I « The quasl-nucleon states described by Eqs. (22)-(23) are plane waves aodulatod by the periodic и field. At a fixed value of the I aoaentua p, the function • (p,z) Is- written in the fora of a Fourier series: -Ink* Ф (p.z) « Ha ) u (p,n)e (24) I У . The blsplnor u (p.n) together with the energy eigenvalues E (p)(sp*) can be obtained by the diagonal isat ion of the Haailtonian H ,u(p,n') « Eu (p,n) , (25) , nn I v i i Z ы n where (26) and y(±) - (у1 ± iy2). According to this set of equations, we have to diagonal ize &'. the Haalltonlan (26) in a 4(2N+1) dimensional space. However, when we need only the energy eigenvalues, we can solve the equivalent eigenvalue problem given in the Appendix, which leads to a diagonalization in a 2(2N+1) dimensional space. The energy spectrum E (p) is somewhat similar to the spectrum) of crystalline solids having allowed and forbidden bands. t At every given value of p there exist two different states associated with the two spin degrees of freedom.. These states are degenerate when the amplitude of the periodic field и vanishes as It can be seen on the right hand side of Fig. 1. When the aaplitude of the periodic field и is nonzero the spin degeneracy is resolved. Allowed and forbidden energy bands are formed aaV it is shown in Fig. 1. The splitting of the degeneracy can be seen in Fig. 2 when the ы field amplitude is growing from zero to a finite value. The expectation values of the source terms of the field equations (12)-(16) can be expressed as follows: <^r *> » I I I e (27) •4» •a• +fk/*/22 d d , P, P, dp,e(Tp|»iE.(p))üi(p,n')T ui(p,ii) -* -* -t/2 The Fermi-Diгас distribution function в at a fixed temperature T and baryon chemical potential u is given by 9 etT.M.E.ip» - (l • •(Ei!B.,-|0/T)"\ (28) The integration on the first Brlllouln zone from -k/2 to +k/2 together with the summation on the complete set of states labeled by the index i Is equivalent with the integration on the full momentum space in the case of non-periodic systems. The three ' dimensional integration was performed by Gauss quadrature. The calculations were carried out by truncating the Hubert space at N-3. The results calculated at T-20 MeV are shown in Figs. 3-8. At a few values of the parameters we checked the rapidity of the convergence of the Fourier expansion. The calculations have been performed truncating the ort space at N«1,3 and 5. The differences between the numerical values of the physical quantities calculated at N«1 and N»3 are not negligible. However, the differences between N*3 and N>5 are already insignificant. We have found by direct calculation that the coefficients of e-i(n-n')kz ln the eource terms on the rlgnt hand slde of (12) and (13) were more than five orders of magnitude smaller ln the case n'*n than those with n'-n. At the same time, ln the source terms in (14) and (15) the only non-vanishing terms are n'«nil. That is, the solutions obtained with the Ansatz (5)-(9) and (22)-(24) are self-cons istent with a high accuracy. This means that the e',n z (n • 1, 2, .... N) type modulations of the plane wave states are 10 Vfc i »'Чг.иит 1,11 ШШШ*$.?*Ф*Ф* ОЯК№г*^.Ш'- ^ггГ^'.--,^ГЙ!*^1Г-^Ч1 I * not present in the nucleon current <ф i* ф > except for the components fi*l and M"2 which are Modulated only by coskz and -sinkz, respectively. In other words, there are no higher order harmonics in the nucleon current. To facilitate the physical Interpretation of the self- consistent solution, we reformulate the whole problem in the language of statistical physics in the next section. 3. THE EXTREHUM OF THE THERMODYNAMICAL POTENTIAL И Let us introduce the thermodynamical potential Й of the / system using the standard definition: T In 2 (29) h where the grand canonical partition function Z is defined as 1 usual: -(Н-цВ)/Т Z * Tr e (30) where the Hamiltonian of the system and the operator of the baryon 1 number are denoted by H and B, respectively. In this model it is assumed that the nucleons interact only #Ш via the mesoniс mean fields, consequently the system can be considered as a system of non-interacting quasi-nucleons ! i occupying the single particle states defined by Eqs. (22)-(24). The thermodynamical potential Я can be given expllcltely in 11 • it {щт^лим I x terms of the energy eigenvalues E.(p) of the quasi-nucleons and the parameters of the meson fields as follows: Q- Q • Q nocloon •••on 4 •k/2 »...,..„ - -\\ *J *J>4 '»(• • .-'W-"") • си» „fl 2 -2 , ,„-., ж 1 . 2^.2. -a " 1 2 -*2^ ••on ^2 «Г Zw 2 » J where the volume of the system is denoted by V. At a fixed value of ji, T and V, the necessary conditions of ft Í'. the thermodynamical equilibrium can be expressed by the following requirements: *.o . 55.-0- . S-o . g.o. (32) 1 V From the numerical studies it turned out that these equations can Л be satisfied simultaneously only if и • 0 (33) This means that thermodynamical equilibrium can be obtained only if the и field is constant in space and then the parameter к is irrelevant. Let us drop the requirement 12 ^ ж * ° • (34) Then the set of equations — • О , • О , — «О (3S) aí до aű does have a solution, which is identical with the solution of the self-consistent set of the equations given by Eqs. (11)-(16). This •eans that if the "wave number" к of the ы field is considered as a parameter defined by the external conditions and It is fixed then a self-consistent periodic и field with a non-vanishing anplitude w*0 can be formed. Let us investigate the stability of the self-cons istent solution. For this purpose we calculate numerically the second derivatives of the thermodynamical potential. The solutions can be classified into two groups. In a nuaber of cases ЕВ > о , ^0 > О (36) Ь9г du so there is a minimum in the theraodynamical potential, in other words the equilibrium is stable. However, in other cases Щ > 0 but ?-§ < 0 (37) which corresponds to a saddle point of (I, the equilibrium is not 13 «»»•»^^«^м^да^вв^^. •.. ^:.л^яетт. , ъ*ышйИ#!*ткы&мсяш*'' •У-'-*-" ^ •*••! ' ,j| , :,**Ъ*^*#:Р$9Щ*&&Ш stable. To clarify the situation, we calculate the anisotropy [11] of the system defined by the help of the pressure tensor components in the following form: P|-pl wpnr (38) I x where the pressure parallel and perpendicular compared to к are f denoted by P. and P^ , respectively. 4. RESULTS / In the framework of the quantum hadrodynamlcs, periodic и field has been found which is associated in a self consistent way with a periodic structure of the nucleon current while the density of the nuclear matter remains constant. This is in contrast with 1 other periodic structures discussed in the literature up to now, where the periodic structure is also present in the density. It was found that in the presence of a self-cons istent periodic и "ield, the pressure is anisotropic. With increasing amplitude u, the anisotropy A increases almost linearly up to a maximum, and afterwards it decreases again almost linearly (Flg.3). The amplitude и is a double-valued function of the anisotropy A for a given к value. Investigating the stability of the solu- 14 ,0:4» mm \ \ tlons. stable »inlaua in Q has been found on the decreasing branch of the А(ы) function while the increasing branch corresponds to saddle point In Q. (Broken lines designate stable periodic solutions and dotted lines designate unstable ones in the figures.) Looking at the field amplitude о as a function of the nucleon density, it is striking that the curve is very steeply rising i as far as the point where stable solutions appear (Fig. 4). This means that the density andli are strongly correlated. ! Let us denote by к the Fermi momentum defined as 1/3 / ',-(-4 (39) One can see in Fig. 5 that the wave number к of the w field increases almost linearly with the к values at which 1 stable solutions appear. Presumably in a phase transition the pressure anisotropy determines the field amplitude и and the nucleon density determines the "lattice spacing" 2*/k. In Figs. 6-8 the periodic phase solutions are compared to the normal phase of nuclea*» matter where o-(x) -» (40) w°(x) -» <м°(х)> « ű* are constant and «'« w2« «3» 0. (Full lines correspond to the 15 i 4 r J 0 I ft II» II и « щвпЛ^МШЬЯШвЯЯЯШЯЯШ OMpnmnffWK«» noraal phase in these figures.) For every prescribed value of k(«0) both the binding energy per nucleon E/N and the average pressure P»(P. • 2P^)/3 as a function of the nucleon density are smaller in the periodic phase. It can be seen that for higher к values (k«5, k=6) the field amplitude с and the average pressure are double-valued functions i of the nucleon density for th» branch of the unstable solutions but they are single-valued in the intervals corresponding to stable ones. It was found that the formation of the periodic structure depends on the softness of the nuclear equation of state. Using the parameters given by Walecka (b*0 and c*0), periodic self- consistent solutions are found above a critical density. Assuming a self-interacting с field giving rise to a soft EOS 112], we were unable to find any periodic self-cons istent solution even at much higher densities. The results shown in the figures were calculated by using the Walecka parameters. For the sake of simplicity of the presentation we have not mentioned the role of the antinucleons. Their contribution, how ever, has been checked and found to be negligible at low temperatures (below 50 MeV). At higher temperatures the contribu tion of the antinucleons must be included. ACKNOWLEDGEMENT The calculations were carried out on the Ml его VAX computer donated by the Alexander von Humboldt Foundation. 16 —ЕШЛЛ**1аШШьлл.шш..л^ ЩШШ£&а№&1т&ь, fwmmGwi •-»mpíwt ;чивн*н •Г rinlH I \ APPENDIX То calculate the derivatives of the theraodynaaical potential Q, the eigenfunctions are not needed. The energy eigenvalues can be obtained by solving Klein-Gordon type equations. The procedure is the following. Let us write the Dlrac equation as («s - •> x) « 0 , (Al) where / • i 10 - g « (A2) and я • я - gjr (A3) 1 Multiplying by [У*** • » 1, we get К - •**• t v%> • ° • <"' where F is the field tensor defined in (18) and - i M (AS) Writing ф as given in (22Ы24), and considering the "Lorentz" condition (17), as well as 17 ••В|рМАа.1/)»!' *W>5ffiSül,$f<£ A * 'J „•.ITA-* JZZL \ (A6) I* к кй • - к2 (A7) we get kV'(?> - 2 ikpV(C) • [p^- 2gjyA g*»/ #2^ (A8) Я • 2 *u |1PJ I where С s kz . (A9) Taking into account the "Ansatz" given by (6)-(9), the following set of equations can be obtained: E A ,u{p,n') * К u(p,n) , (A10) where the matrix to be diagonalised is given as A ,- ikV-2p3kn]e ,- S*"'« , - S,f)e , : (All) nn \ J nn n ,»»1 n fn-i S1*». ff [«pWl - kVtic»)] , (A12) 18 K-1 '*« '«.НИМИ МИНИ» "«Ц"Г' аь. ÜHWIWWWH" -*""W*-4 I \ 1 _ 23 .23 V • «• » Ijr у . (Al?) 2 31 3 ! «гае- • ijr у The eigenvalue К has the following fora: tf-2 -г\ p 2 28 8 u 2 -»а I - (A14) o.- в ' лЛ * Г'"4 • "J Froa here, the expression for the energy eigenvalue E can be obtained as follows: / — I 2 »2 2~ 2 (A15) E. • p » g ы 1VK *p •• +g и :i • 19 К I" с e m m b g m «T « ы "soft" nuclear -0.734 6.856 3.512 3.874 matter* 4.570 2.788 3.953 "stiff nuclear 0.0 0.0 9.573 11.67 matter * I. Lovas, J. Németh and K. Sailer, Nucl. Phys. A430. 731 (1984). ^J.D. Vfalecka, Ann. Phye. (N.V.) 83, 491 (1974). Table 1. The parameters of the mesonic fields 20 REFERENCES [1] С. Nlcolls and I. Prlgoglne, Selforganizatio n in Non- equilibriua Systems, J.Wiley, New York, 1977; H. Haken, Synergetics, Springer, Berlin, 1977. [2} A.B. Migdal, Zh. Eksp. Teor. Fiz. 6J., 2209 (1971); V. Ruck, И. Gyulassy and W. Crelner. Z. Phys. A277. 391 (1976); M. Cyulassy and V. Creiner, Ann. Phys. (N.Y.) 109. 485 (1977). (3) Yu.B. Ivanov, Nucl.Phys. A474. 669 (1987); Nucl.Phys. A474. 693 (1987). [4] B.L. Friaan and P. Henning, Phys. Lett. B206. 579 (198Ы); P. Henning and B.L. Frlaan, Nucl. Phys. A490. 689 (1989). (5) K. Li» and C.J. Horowitz, Nucl Phys, A501. 729 (1989); R.J. Furnstahl and C.J. Horowitz, Nucl. Phys. A485. 632 (1988). [6] C.E. Price, J.R. Shepard and J.A. McNeil, Phys. Rev.CJi, 1234 (1990); C.E. Price, J.R. Shepard and J.A. McNeil, Phys.Rev. £4£, 247, (1990). {71 J.D. Walecka, Ann.Phys. (N.Y.) 82. 491 (1974); B.D. Serot and J.D. Walecka, Adv. in Nucl.Phys. ifi, W. Negele and E. Vogt, eds. (plenum, N.Y. 1986). [8] A.W. Overhauser, Phys.Rev. Lett., 4, 415 (I960); A.W. Overhauser, Phys.Rev. 1ЗД, 1437 (1962). 21 [9] M. Noga, Nucl.Phys. B220. 185 (1982). [10] I. Lovas, J. Neaeth and K. Sailer, Nucl. Phys. A430. 731 (1984). [11] I. Lovas, Nucl.Phys. A367. 509 (1981). [12] J. Boguta and A.R. Bodaer, Nucl.Phys. A292. 413 (1977); I. Lovas and Gy. Wolf, Acta Phys. Hung. §&, 23 (1985). [13] R.J. Furnstahl. R.J. Perry and B.D. Serot, Phys. Rev. C40. 321 (1989). [14] S.A. Chin, Ann. Phys. (N.Y.) Ж. 301 (1977). [15] J. Frank, Gross properties of nuclei and nuclear excitations XVII. Hirschegg, 1989. Ed. H.Feldaeier. 22 ЩЩП&**?&} 1№^>емогР-,т,и$С*?Й^£*ЯЯ I FIGURE CAPTIONS: 4 Fig. 1. The lowest positive energy levels с * E-g w plotted I III О against momentum component p for a nucleon moving in a periodic и field (k - 4.0, p « p -2.0). Left side: w « 0.1. Right side: 0.0 Fig. 2. Splitting of the energy levels с * E-g и in the presence of a periodic ы field. k«6.0, p »p «0.0, p «2.5 (inside the first Brillouin zone) 2 Fig. 3. The anisotropy A defined in (38) as the function of the field amplitude и at various к values. Dotted lines: unstable solutions. Broken lines: stable solutions I ••I- Fig. 4. The field amplitude w as the function of the nucleon density p/p at various к values, p «0.193 fm -з is О О the density of normal nuclear matter. Doited lines: unstable solutions. Broken lines: stable solutions Fig. 5. The relation between the wave number к and the Fermi momentum к at which stable periodic solutions appear 23 »Illlll И—.и» -Я: The field amplitude The binding energy per nucleon E/N plotted against nucleon density p/p for various к values. Full о lines show the nornal phase solutions. Dotted and broken lines show the periodic phase unstable and stable solutions, respectively The average pressure P»(P. • 2Px)/3 plutted against nucleon density p/p for various к values. Full О lines show the nornal phase solutions. Dotted and broken lines show the periodic phase unstable and stable solutions, respectively 24 m^vXa** Ш•' i fri-• a* *T ii - ii 1 ;» ^JJw-^^iWl ДЖг« »-•wwiHrgwgJW«^ \ in CM "i 1 I T- I "1 Я in in CM см *_: / (tJ"*)3 ÍM г- МЭ -in 1 Vi (^ШЛЗ ^ »П1*|«1^»ц «чшшкм^1 .литш-чг-ялТДаквди•• ».«• ./—íkr.,, / I т 1 i г 0.00 0.02 0.04_ 0.06 0.08 0.10 со 1 Fig. 2 „•гйА. ЩЦМВД'- *ч***,.№яищт>и.*ш11111яж' L п»и áÁ I 0.16 •••• 0.14- 0.12- t I ik=6 // 0.10 •f i ; / % // % < 0.08 // k=5 0.06 \k=4 0.04- • Ф 0.02" ,:.- \k=3 \ JS \ 0.00 ££ T 0.00 0.05 0.10 0.15 0.20 0.25 0.30 Cü Fig. 3 тщщ^аит-- ••'••••^тг^ттШЯШВМЯт» • ».Ill — IJ . J^L •\ I\ 13 0.15 - у Q/QO 1 -* Pig. 4 i _»il.» H—il Hi. 'ШШШШЩ^^-? т^ъл?хж*жя&шям J-~ I \ 1 Г 1 1 1 1 г 2.0 2.1 2.2 2.3 2Л 2.5 2.6 2.7 2.8 2.9 / 41 1 Pig. 5 ftiilil »И „~V-, _>••-? -•;.. \ j \ 0.49 k=5U- .*~7 0.47- [=4 V? / 0.45 0ЛЗ- |ь 0.41- 0.39- 0.37- s I 0.35 Q/Qo 1 Fig. б •\ « J Гчм* >. .«.ufi mm rstf-f^ «-«warf* 1т*!ЯхеШШ •Ma ИИМИчрччмми J www»."» twi*u HPff' ' \ \ 4 I I О" • (A*W)N/3 1 # o» (A'W)N/3 -iV.fci' t0k»»i г, nii-u.^.. - •i—~....v~«~ \ -^>ч « n .%o -- in • ( .<Щ A*H)d е г - 1Л 1 *4 oo r» >om^m (CJUJ*A»H) d HiHIWW«» iUfT^" 1 №и*^й»а?>"*?«»'-*,"ЧШ(1 Á. ттчтм&ыччучщу**'*'* ** The Issues of the KFKI preprint/report s^fes ave classified as follows: A. Pamela and Nuclear Physics B. General RctatMty and Qravkatlon С Cosmic Rays and Space Research D. Fusion and Plasma Physics E. Seid State Physics F. Semiconductor and Bubble Memory Priysics and Technology 6. Nuclear Reactor Physics and Technology H. Laboratory, Biomedical and Nuclear Reactor Electronics I. Mechanical. Precision Mechanical and Nuclear Engineering J. Analytical and Physical Chemistry K. Health Physics L Vtoration Analysis. CAD, CAM M. Hardware aiidSoftwafe Development, Comput« N. Computer DestOACAMAC.OimpbtwC^Ttrc«^ Measurements The complete series or issues discussing one or more of the eub|ecU can be ordered; institutions are Mndry requested to cc«ita<4trw№^LK)rary,indMo4ieistrieMilK)rs. Tide and classification of the issues published this yean KFKMSS2-01/E ACzitrovszkyetal.:Laser analysisof aerosolparticles to calculate olstnbution of aerosol electric charges and size KFKMM2-02/A F. Glück et al.:Order-a radiativecorrection s for semfeptoote decays of poiarized baryons KFKMM2-03/A L DIóei: Coarse graining and decoherence translated Into von f4eurnann Language KFKMW2-04/E LDiósi: On the Caldein>Leggett Master Equation KFKHW2-05/B+M J. Kadlecs*: Tensor Manipulation Package for General RetetMryCetoUatkxis KFKHSftt-ofVB G.S.r^:SynimetriMlnGenen^RelatMry-ac KFKM W2-07/A LL Gabunte, T.Gémtsy, L Jer*, AI. KrwcMava, 8. Krasznovszky, V.N Penev. Gy.Pinter.AI 3r*kJvskaja,L8.Vertogrsdov; Production region rjt Identical pton» at38QeV/c д'-rnicWlrteractlone with high Pt partidét KFKMM24t/A К. Szlachényl, P. Vecsernyés: Quantum Symmetry and Braid Group Statistics и G-spln Models КПСММ24»/а Ar^a:E$tlrrielk^c40ussW>1ö<^fleecoylry KFKMtW-IO/A+B AFrstftetaiidF.KafcJyrMtopProormfnto >>-#•••! •чюам* iMCTi \ i KFKMM&11/A AHéw^L Y. Nojwt On th>lh>0Fy of atomic lotilHÉlonlolowifcig KFKMM2-12/A I.Uw^LMolnér^8al«r.W.eratar.PwlodtoainKluralnnuclMrinÉBw / Í 1 « J dtááb ж \ К \ г \ / 1 FvPV^PjvBPJP PJP ^ЧчРаЬр^чР»PeP W PAPP%PJPV VVWPMVVV^ РР PB^PB^PV FcMte ktadó: Lovas litván Szakmai lektor. WakJhaussr Béta NysM Mdor Kovács Tamás Példányszám: 315 Каком a PROSPERITÁS Kft. nyofndsuztmébMi Budapest, 1002. március hó -Á