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CENTRAL RESEARCH INSTITUTE for PHYSICS BUDAPEST Кпсммма/А Bbcbsmftnr • Щгшшшягшщтшщ

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CENTRAL RESEARCH INSTITUTE for PHYSICS BUDAPEST Кпсммма/А Bbcbsmftnr • Щгшшшягшщтшщ HU33~4SÍ<^ 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 oMBesoa field, periodic 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 <r(x)-mU(x) » 0 , (1) »••>-•>*-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 <r field with the potential U(cr) given by U(<r) - fi bm • j cgdig^)3 . (4) 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: <r(x) -» <«-(x)> « 9 . (5) «e(x) -> <«e(x)> - «* , (6) «Лх) -* <w1(x)> • w coskz . (7) »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<h f> • 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).
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