A NEW EXPRESSION FOR RADIAL DISTRIBUTION FUNCTION OF NUCLEAR MATTER.

Transcription

1 Mathematcal and Computatonal Applcatons, Vol. 16, No., pp , 011. Assocaton for Scentfc Research A NEW EXPRESSION FOR RADIAL DISTRIBUTION FUNCTION OF NUCLEAR MATTER Fatma Mansa 1, Atalay Küçükbursa 1*, Kaan Mansa 1 and Tahsn Babacan 1 Dumlupınar Unversty, Department of Physcs, Kütahya, Turkey Celal Bayar Unversty, Department of Physcs, Mansa, Turkey atalay@dumlupnar.edu.tr Abstract-A Varatonal Monte Carlo Method (VMC) s used for the calculatons of radal dstrbuton functon of nuclear matter. Urbana v 14 potental s used for the nucleon-nucleon nteractons n the calculatons. The new expresson for radal dstrbuton functon of nuclear matter s obtaned by fttng of Monte Carlo smulatons results to a functon. Key Words-Nuclear matter, radal dstrbuton functon, Monte Carlo method. 1.INTRODUCTION The radal dstrbuton functon (RDF) s very mportant n the theory of N-body systems because t gves the number of the partcle between r and r+dr about a central partcle at the orgn of r and t can also be thought of as the factor that multples the bulk densty,, to gve a local densty ( r) g( r) about some fxed partcle. Another mportant pont s that RDF provdes a suffcent nformaton n calculatng all the thermodynamc functons of the system. The most relable calculatons of the radal dstrbuton functon are molecular dynamcs or Monte Carlo calculatons. One advantage of these calculaton methods s that the nter-partcle potental s known. Therefore, these calculatons serve as expermental data whch s used to test varous theores [1]. For hard spheres, the molecular dynamcs radal dstrbuton functon was frst gven by Alder and Hecht [], and Barker, Watts and Henderson [3]. It has been calculated for the Lennard-Jones potental by Verlet [4], and for an nverse 1 potental by Hansen and Wes [5]. Moreover, Barker, Fsher and Watts [6] have presented extensve Molecular Dynamc and Monte Carlo calculatons wth three body forces. They are useful n the perturbaton and transport theores [7]. Recently, molecular dynamc smulatons have used the radal dstrbuton functon for Lennard-Jones fluds [8, 9], and mxng rules for bnary Lennard-Jones chans have been tested by performng Monte Carlo smulatons [10]. The radal dstrbuton functon can also be determned by X-ray dffracton studes [11-13]. However, varous methods and technques have been used to obtan the radal dstrbuton functon [14-16]. In ths paper, the radal dstrbuton functon for nuclear matter s calculated by usng the varatonal Monte Carlo method (VMC) wth varous sospn asymmetry parameters, at dfferent denstes. Moreover, the new expresson for radal dstrbuton functon of nuclear matter s also obtaned by usng Monte Carlo smulatons results. Ths paper s organzed as follows: the nteracton potental and symmetrc, asymmetrc nuclear matter s defne and s determned n Secton and n Secton 3,

2 F. Mansa, A. Küçükbursa, K. Mansa and T. Babacan 415 respectvely. In secton 4, Monte Carlo smulatons are descrbed. The results for the applcaton of the Varatonal Monte Carlo method to the calculaton of the radal dstrbuton functon of nuclear matter are gven n secton 5. Then we conclude n secton 6.. INTERACTION POTENTIAL The Hamltonan operator of a system consstng of N partcles can be wrtten as H= V. (1) m where V s a two body nteracton potental. Here, we have used only the frst four terms of the Urbana V 14 potental [17] for two nucleon nteracton n our calculatons. Because the contrbutons of latter terms of the Urbana V 14 potental are much smaller than those of the frst four terms n our nuclear matter calculatons, 14 operator components of the Urbana V 14 potental are necessary to have a good ft to the expermental data. Thus, we have used the two-nucleon nteracton c V V V. ) V (. ) V (. )(. ). () ( where V c, V, V, and V depend only on the dstance between the nucleons and. Each term n Eq. () has three parts V V V V (3) I S representng the long-range nteractons ( V ), the ntermedate-range nteractons ( V ), and the short-range ( V ) nteractons. ( V ) s nonzero only for and s gven by V S r e r (1 e cr ) 1 where 0.7 fm s the nverse compton wavelength for pons. The ntermedate and short range parts are and V ( r) I I r ( r) e r r cr 1 e, V S S ( r), (5) ( rr) a 1 e / respectvely. The values of the potental strengths I and S and the parameters c, R, a are gven n Table I. Table 1:. Parameters of the Urbana V 14 nucleon-nucleon potental. I I S c c=0. fm -, R=0.5 fm, a=0. fm (4) I

3 416 A New Expresson for Radal Dstrbuton Functon of Nuclear Matter Another mportant pont s that three and more body nteractons are very mportant n nuclear matter calculatons. Therefore, we use the phenomenologcal approach assumng the densty dependent term to be proportonal to the short ranged part of the Urbana potental, and assume that the total nteracton, ncludng the many body effects, s of the form V14 TNI v vi vs vs ( ), (6) where s the number densty of nucleons. and n the above equaton are free parameters. 3. SYMMETRIC AND ASYMMETRIC NUCLEAR MATTER Nuclear matter s an dealzed system of nteractng nucleons (protons and neutrons). It s not matter n a nucleus, but a hypothetcal system consstng of a huge number of protons and neutrons nteractng by only nuclear force and no Coulomb force. Volume and partcle numbers are nfnte, but the rato of these quanttes s fnte. Infnte volume mples no surface effects and translatonal nvarance (only dfferences n poston matter, not absolute postons). A common dealzaton s a symmetrc nuclear matter whch conssts of equal numbers of protons and neutrons. The energy per partcle of the asymmetrc nuclear matter can be expanded about symmetrc nuclear matter n a Taylor seres n terms of the sospn asymmetry parameter 4 E(, ) E( ) S1( ) S ( )..., (7) If the fourth order and hgher order terms n are gnored, then the total energy per partcle of asymmetrc nuclear matter can be wrtten as: E(, ) Es ( ) S( ), (8) where E () s the energy per nucleon of symmetrc nuclear matter, s N N n n N N the sospn asymmetry parameter (where N n and N p are the numbers of neutrons and protons) and S () s called as the symmetry energy. A detaled explanaton and nformaton for the propertes of the symmetrc and asymmetrc nuclear matter can be found n our prevous publcatons [18, 19]. 4. MONTE CARLO SIMULATIONS We wll consder a system of nucleons confned n a cube of sde L wth perodc boundares. The sze of the cube determned from the densty and number of partcles was chosen so as to correspond to a closed shell n k-space representng all the symmetres of the ground state. One must use the fully occuped closed shells of plane waves for both neutrons and protons n order to preserve the sotropy of the system. The number of spn-sospn degeneracy of the spatal states s denoted by g (g= for proton and neutron matter). Number of spatal states n each shell s gven n Table 1. For our wave functon, n order to represent all symmetres of the ground state, the number of spatal states (I) should be chosen from the seres of I=1, 7, 19, 7, 33, 57, 81,..., so that p p s

4 F. Mansa, A. Küçükbursa, K. Mansa and T. Babacan 417 a complete shell n momentum space must be flled. Thus, the number of neutrons or protons must be chosen from the set of,14,38,54,66,114,. Then the total number of nucleons n nuclear matter becomes N=g(I n +I p ). The chosen I n, I p and the total number of nucleons are gven n Table. Table : Number of the spatal states n each shell. The number of nucleons were chosen n accordance wth ths table. n Allowed values n, n, n x y z ,0,0 1,0,0 1,1,0 1,1,1,0,0 0,1, 1,1, 0,1,0 1,0,1-1,1,1 0,,0 1,0, 1,,1 0,0,1 0,1,1 1,-1,1 0,0,,1,0,1,1-1,0,0 1,-1,0 1,1,-1 -,0,0 0,,1-1,1, 0,-1,0 1,0,-1-1,-1,1 0,-,0 1,,0 1,-1, 0,0,-1 0,1,-1-1,1,-1 0,0,-,0,1 1,1,- -1,1,0 1,-1,-1 0,-1, -1,,1-1,0,1-1,-1,-1-1,0, 1,-,1 0,-1,1,-1,0 1,,-1-1,-1,0 0,,-1 -,1,1-1,0,-1-1,,0,-1,1 0,-1,-1,0,-1,1,-1 0,1,- -1,-1, 1,0,- -1,1,- -,1,0 1,-1,- 0,-,1-1,-,1 1,-,0-1,,-1 -,0,1 1,-,-1 0,-1,- -,-1,1-1,0,- -,1,-1 -,-1,0,-1,-1 0,-,-1-1,-1,- -1,-,0-1,-,-1 -,0,-1 -,-1,-1 States n Ths Shell States up to Ths Shell Table 3: The numercal values used n Monte Carlo smulatons n p I p Number of proton n n I n Number of neutron Number of total nucleon

5 418 A New Expresson for Radal Dstrbuton Functon of Nuclear Matter Monte Carlo smulatons for nuclear matter wth varous sospn asymmetry parameters were carred out usng the Metropols algorthm [0] and a cubc box of sde L contanng N nucleons wth perodc boundary condtons. The tral wave functon used s a Jastrow type wave functon n the form ( R ) f ( r ), (9) where s the many partcle wave functon for the system of non-nteractng partcles and R s a 3N dmensonal vector representng the coordnates of partcles, whle f s the two partcle correlaton functon. Jastrow suggests that n general, ths correlaton functon can be an operator functon [1]. Howeve, n most applcatons f s assumed to depend only on the nterpartcle dstance, r k. r r r. One can use plane waves ( r ) e for the sngle partcle wave functons of the nucleons n bulk matter. Because we consder nucleons restrcted to a cubc box of sde L, k n / L and n s an nteger vector. In order to conserve the rotatonal nvarance of bulk nuclear matter, the number of neutrons and protons n the box s restrcted to completely flled shells only. Under these condtons, the many partcle wave functon n eq.(9) becomes P P N N ( R) D D D D (10) P P N N where D, D, D and D are the slater determnants of sngle partcle wave functons for the correspondng spn, sospn state wth s D det( d s )., (11) where s d (( r, s) ). (1) The nuclear forces are short ranged and saturate very quckly, thus the radal dstrbuton functon s not expected to have long range correlatons. Therefore, for the two partcle correlaton functon f n eq.(9), a functon n the followng form s used t 1 f ( r) ( r r ) / a 1 e 0, (13) where t, r 0 and a are the varatonal parameters. A pseudo potental u (r ) for practcal reasons s defned such that our varatonal wave functon( f r ) exp( u( r )) ) becomes the 3N dmensonal space wth the probablty dstrbuton ( P P N N exp u ( r ) D D D D. (14) ( R) dr ( R) Is sampled by usng a random walk created by the usual Metropols method. Ceperley et.al.[] have descrbed the most effectve way to handle the rato of the determnants n ths wave functon. (15)

6 F. Mansa, A. Küçükbursa, K. Mansa and T. Babacan 419 The VMC smulaton methodology has been reported before n [18, 19, ] and the detals can be found n those references. 5. RESULTS In ths secton, the results obtaned from Monte Carlo smulatons for nuclear matter wth varous sospn asymmetry parameters are presented. The radal dstrbuton functons of nuclear matter at denstes between 0.0fm -3 and 0.0fm -3 n 0.0 steps for each sospn asymmetry parameter and a new expresson for radal dstrbuton functon of nuclear matter from these data have been obtaned. The new expresson for RDF of nuclear matter s found by fttng the results obtaned from VMC smulatons to a functon: c g(r) a[1 exp( br)]. (16) In Fg. 1, we plot the selected radal dstrbuton functons of nuclear matter wth varous sospn asymmetry parameters at denstes between 0.0fm -3 and 0.0fm -3, calculated wth VMC method. All parameters a, b and c at each densty n Eq. (16) used n fttng. are gven n Table 4. It can be seen from Fg.1 that radal dstrbuton functons rapdly approach zero at shorter dstance than 0.3 fm. Ths effect caused from short range repulsve parts of nucleon-nucleon nteractons. However, all of radal dstrbuton functons approach to asymptotc value of 1 fm at shorter than 5 fm value. Radal dstrbuton functons reach to ths asymptotc value n short range so long as the densty ncreases. Also, radal dstrbuton functons almost dsplay the same behavor between 0 and 1 fm range. Fluctuaton that s between 1.5 and 3 fm range n radal dstrbuton functons takes smooth forms as so to be approached to symmetrc nuclear matter ( =0).

7 40 A New Expresson for Radal Dstrbuton Functon of Nuclear Matter g(r) 1, 1 0,8 0,6 0,4 0, r(fm) (a) g(r) 1, 1 0,8 0,6 0,4 0, r(fm) g(r) 1, 1 0,8 0,6 0,4 0, 0 (b) r(fm) (c) Fgure 1. Nuclear matter radal dstrbuton functons for =0.9 (a), = (b) and =0 (c). The densty values ncrease from bottom to top wth the lowermost curve correspondng to 0.0fm -3 and the uppermost one correspondng to 0.0fm -3.

8 F. Mansa, A. Küçükbursa, K. Mansa and T. Babacan 41 Table 4. Coeffcents of the expresson for RDF of nuclear matter YP=0.5 Symmetrc nuclear matter 3 ( fm ) a b c 0.0 0,9818 1,187 1, ,9810 1,735 1, ,9834,151 3, ,983,4556 3, ,9854,6501 3, ,9868,9113 4, ,9915,8519 4, ,9914 3,981 6, ,99 3,3361 6, ,9945 3,3575 6,6010 YP= ,9791 1,890 1, ,9805 1,6631 1, ,9818 1,9883, ,9837,34, ,9883,476, ,9947,3457 3, ,9907,910 5, ,9959,7497 4, ,9948 3,561 7, ,9956 3,3178 6,9899 YP= ,9764 1,615 1, ,9761 1,7750, ,9774,144, ,984 1,9617 1, ,9819,6704 3, ,9865,516 3, ,9897,753 4, ,9900 3,1357 5, ,9949,9434 5, ,9945 3,4697 8,371 YP= ,9801 1,137 1, ,9811 1,6596 1, ,9849 1,887, ,9865,0459, ,990,034, ,9901,453 3, ,991 3, , ,9895,8963 3, ,9948,8415 4, ,9973 3,1100 5,7366 YP= ,9879 1,1419 1, ,9877 1,4989 1, ,9888 1,8653, ,9893,1974, ,990,700 3, ,9931,4636 3, ,9990,4068 3, ,9975,753 4, ,9978,9100 5, ,001,939 5,745

9 4 A New Expresson for Radal Dstrbuton Functon of Nuclear Matter YP= ,9905 1,0494 1, ,9857 1,4598 1, ,9884 1,8147, ,9908,0471, ,991,934 3, ,9933,5413 3, ,9957,6499 4, ,9977,7733 4, ,999,98 5, ,9993 3,1976 7,1059 YP= ,9858 0,9743 1, ,9834 1,4495 1, ,9814 1,8316, ,9880,0863 3, ,9900,1030, ,9878,4669 3, ,9906,7497 4, ,997,6860 4, ,9996,8356 5, ,9985 3,95 7,5611 YP= ,9437 1,3460 1, ,949 1,894, ,9456,1444, ,9515,361 3, ,9561,617 4, ,0131,0387 3, ,9634 3,1603 6, ,9873 3,077 7, ,9754 3,5081 8, ,9803 3,4800 8,001 YP= ,9999 0,858 1, ,9950 1,3840 1, ,9938 1,5590 1, ,9960 1,877, ,9974,0096, ,9981,556 3, ,0001,453 4, ,9999,848 5, ,0017 3, , ,008,9581 6,047 YP= ,0141 0,7503 1, ,9963 1,630 1, ,0006 1,4073 1, ,0003 1,700, ,9910,1383 3, ,0175 1,943 3, ,004,453 4, ,0058,503 4, ,0039,9181 6, ,0159,4694 4,0736

10 F. Mansa, A. Küçükbursa, K. Mansa and T. Babacan CONCLUSION The frst applcaton of the Varatonal Monte Carlo methods n quantum mechancal systems was looked at by McMllan [3] nvestgatng bosonc systems. VMC methods were extended to study smple Ferm systems [,4] and were developed for fnte nucleon systems, and these methods have successfully been appled to few nucleon systems [5,6]. In order to obtan propertes of many body systems such as nuclear matter, a non-deal gas etc. the many body problem can be reduced a two-body problem, a three body-problem, etc. Such a decomposton s not applcable to a N-body system snce each partcle n a N body system s n constant nteracton wth large number of ts neghbours. Therefore, one have to use the methods that are more sutable for dense systems lke nuclear matter. One of these methods s Varatonal Monte Carlo method. In ths study, as an applcaton of the Varatonal Monte Carlo method, we have calculated radal dstrbuton functons of nuclear matter. In the calculatons, we have used the Urbana V 14 potental for the nucleon-nucleon nteracton and ncluded the three-body nteractons. We have also presented a smple expresson for nuclear matter RDF. Ths expresson nvolves 3 adustable parameters. One mportant pont s that the new RDF expresson presented n ths study may use to obtan the thermodynamcs propertes of nuclear matter n future studes. REFERENCES 1. D.A. McQuarre, Statstcal Mechancs, Harper Collns Publshers, New York, B.J. Alder and C.E. Hecht, Studes n Molecular Dynamcs. VII. Hard-Sphere Dstrbuton Functons and an Augmented van der Waals Theory, J. Chem. Phys., 50, p.03, J.A. Barker, R.O. Watts and D. Henderson, Monte Carlo values for the radal dstrbuton functon of a system of flud hard spheres, Mol. Phys., 1, p.187, L. Verlet, Computer "Experments" on Classcal Fluds. II. Equlbrum Correlaton Functons, Phys. Rev., 165, p.01, J.P. Hansen and J.J. Wes, Radal dstrbuton functons for nverse-1 ( soft sphere ) fluds, Mol. Phys., 3, p.853, J.A. Barker, R.A. Fsher and R.O. Watts, Lqud argon: Monte carlo and molecular dynamcs calculatons, Mol. Phys., 1, p.657, B.J. Alder and T. Wanwrght, In transport processes n statstcal mechancs, edted by I. Prgogne, Academc, New York, A. Morsal, E. K. Goharshad, G. Al Mansoor and Mohsen Abbaspour, An accurate expresson for radal dstrbuton functon of the Lennard-Jones flud, Chem. Phys. 310, p11-15, M. Bamdad, S. Alav, B. Naaf and E. Keshavarz, A new expresson for radal dstrbuton functon and nfnte shear modulus of Lennard-Jones fluds, Chem. Phys. 35, p554-56, N. von Solms, K.Y. Koo and Y.C. Chew, Mxng rules for bnary Lennard-Jones chans: theory and Monte Carlo smulaton, Flud Phase Equlbra, 180, p71-85, B.E. Warren, X-Ray Determnaton of the Structure of Lquds and Glass, J. Appl. Phys., 8, p.645, 1937.

11 44 A New Expresson for Radal Dstrbuton Functon of Nuclear Matter 1. N.S. Gngrch, The Dffracton of X-Rays by Lqud Elements, Rev. Mod. Phys., 15, p.90, R.E. Kruh, Dffracton Studes of the Structure of Lquds., Chem. Rev., 6, p.319, J. Largo and J.R. Solana, A smplfed perturbaton theory for equlbrum propertes of trangular-well fluds, Flud Phase Equlbra, 167, p.1, Y. Tang, C. Benamn and Y. Lu, On the mean sphercal approxmaton for the Lennard Jones flud, Flud Phase Equlbra, 171, p.7, C. X L, Z. Hao Wang, Y. Gu L and J. Fang Lu, Two body ntegrals for hard sphere flud base on Tang-Lu RDF expresson, Flud Phase Equlbra, 01, p.37, I.E. Lagars and V.R. Pandharpande, Phenomenologcal two-nucleon nteracton operator, Nucl. Phys. A, 359, 331, K. Mansa, U. Atav and R. Oğul, VMC calculatons of the ground state propertes of nuclear matter, Int. J. of Mod. Phys. E, 14, 55, K.Mansa, Ü. Atav and S. Sarıaydın, Equaton of state of asymmetrc nuclear matter: a VMC study, Cent. Eur. J. Phys., 8(4), , N. Metropols, A.W. Rosenbluth, M.N. Rosenbluth, A.H. Teller and E. Teller, Equaton of State Calculatons by Fast Computng Machnes, J. Chem. Phys. 1, 1087, R. Jastrow, Many-Body Problem wth Strong Forces, Phys. Rev. 98, 1479, D. Ceperley, G.V. Chester and M.H. Kalos, Monte Carlo smulaton of a manyfermon study, Phys. Rev. B16, 3081, W.L. McMllan, Phys. Rev. A, 138, 44, M. A. Lee, K. E. Schmdt, M. H. Kalos and G. V. Chester, Green's Functon Monte Carlo Method for Lqud 3 He, Phys. Rev. Lett., 46, 78, J. Lomntz-Adler, V. R. Pandharpande and R. A. Smth, Nucl. Phys. A, 361, 399, J. Carlson, Alpha partcle structure, Phys. Rev. C, 38, 1879, 1988.