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Binding Energy of Nuclear Matter with Pure Hard Core Interaction.

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A N N A L E N D E R PHYSIK
7. Folge. Band 35. 1978. Heft 4, S. 241-320
Binding Energy of Nuclear Matter with Pure Hard Core
Interaction
By 31,I‘. M. HASSAXand 5. S. MONTASSER
Physics Depnrtmcnt., PiLcrilty of Science, Cairo IJnivrrtiity, Cairo, A. R. Egypt
A 4 b ~ t r a c The
t . calculation of ground state energy of nuclear matter with nentron CXCCSS, which
has been done up to thc second order in (k,rc) (whcre k, is the Fermi momentum and r,. is the hard
core radius), is extended hwe t o include the third order term. By applying Rrueckner theory in the
low density limitwe calculate this term and thenwe expand the energy in terms of a (s= ( N - Z ) / A )
up to the fourth order to get the volume and symmetry terms of the Weizsiicker serniempcrical mass
formula. We also calculate thc volume and symmetry parts of the compressibility up to (kfrc)3.
Hindungsenergic! der Kerninat,erie rnit reiner hard -core -Wechselwirkung
I nhal t s u b e r s i c h t . Die Berechnung der Gruniizustandsenergie der Kernmaterie mit ffberschubneutronen, die bisher bis zur 2. Ordnung jn (kfrJ (worin kr don Fermi-Impuls und rc den hard-coreRadius bozeichnen) durchgefuhrt wurdc, wird hier bis znr 3. Ordnung erweitert. Wir berechimi
tliescn Term durch Anwendung der Rriickner-l’heoric im Grenzfall kleiner Dichten. Danach entwickcln wir die Energie nach dem Parameter a (wobei x = ( N - Z ) / A )bis zur 4.Ordnung, um die
Volumen und Symmetrieterme in der Iialbempirischen Weizsiicker-Formel zu erhalt.en. Ferner herechnen wir die Volumen und Symmetrieantcile der Komprewihilitat bis ziir Ordnung (l~jr,.)~.
1. Introduction
The expansion of the ground state energy of a hard core fermion system in powers of
(kfr,.)has been investigated by several authors [ 1- 111. In our calculation we use Hrueckner theory. This theory assunies that the correlation hetwecn more than two particles
can he neglected in coinparison with the correlation between any two particles. This
seenis to be a good approximation for nuclear matter. The nuclear potential energy will
then he just the sum of the interaction energies of each pair of nucleons which can be
determined in terins of the two particle scattering properties. The hard core problem
can be solved by replacing the niatrix elemcnts of the singular nucleon-nucleon potential
by the K-matrix which reprcsents a summation over leddcr diagrams rcprcsenting the
particle-particle scattering.
Our procedure of calculation is very similar to that of DABROWSKI
and HASSAX
[ 121
(here after rcfered to by DH). We express the reaction K-matrix in terms of the free
space reaction matrix. Then we calculate the matrix elements of the reaction matrix up
to third order in re so that we can calculate the binding energy up t o third order in
(lyre).DH have calculated it up to first order since they aimed to calculate the binding
c:nergy up to Recond order in ( k q C )We
. also extended our calculation and thst of DH t o
calculate thc fourth order coefficient of the symmetry energy in terms of a (a = ( N -Z)/
A ) . Finally we calculatc the volume and symmetry parts of the compressibility up to
the the third order term in (Icfr,).
IF
hiin l ’ l i \ ~ i k i
I-ol:c, nd. 75
11. Y. $1. Hassax and S.S.NORTASSER
242
3. Method of Calculation arid Results
2.1. K-matrix Calculations
The K-niatrix considered in the following is the Brueckner K-matrix [ 131 with unperturbed single particle energies in the intermediate states, suitable for the low density
limit. We have to distinguish hetuwn the K,,, matrix for neutron-neutron scattering,
the K P Fmatrix for proton-proton scattering, and the K,, matrix for neutron-proton
scattcring. In general, we shall use the notation K , , with p = n, p and Y = n, p. The KJLv
matrix is defined by the equation
(Pip;I K,vI PIP21 = (P;Pc 12'1 P1P2)
+ c ( P ; P c I v l kIk2)
kinks
where M is t,he nucleon mass, 2' is t,he nucleon-nucleon interact,ion, and the exclusion
principle operator G,l, is defined by:
1 for k,> x
0 othcrwisc,
and k,> x
I for k,> A and k,> A
QP,I(kl' k2) =
{ 0 otherwise ,
1 for kl > x
0 othcrwise.
and k,
>1
Here x anda are the Fermi momenta for neutrons and protons respectively (in units of ti).
I n our calculations, we consider neutrons and protons to occupy all the plane wave
states, with spin up and down, below the respective Fernii momenta Rx and R I and only
pure hard core interaction is taken into account. Since v and Q,,, are in our case spin
independent, the K-matrix is diagorial in thc spin variables, and we shall indicate the
dependence of K,, on the nucleon momenta only. The states denoted by Ip) are plane
waves normalized in the volume 52.
Now, we shall develop an equation for the K-matrix for pure hard core interaction v,,
denotcd by K,. For this special case, eq. (I) can be written in a compact form as follows:
+
(34
K,(4 = 8s V , W KA4.
In the above equation, we h a w dropped the suffix pv, for simplicity, and introduced the
quantity L(z):
where
A special case of K , is the free space react,ion matrix K," defined by:
K%) = v s
whero
+ V , L 0 ( 4 K%),
B i d i n g Energy of Nuclear Matter with Hard Core Iiiteractiori
243
and P denotes the Cauchy principal value. K , can be expanded in terms of K: as follows:
K,- K,O
+ K,' + K$ +
(a)
.i
where
K j = K,O(L- Lo) K:,
(jb)
K: := K:(L - Lo)L:(K - Lo)K,"
( 5c)
and the dots reprcsent higher order terms in K : . The first two ternis in eq. (5a) were
considered by DH and others. However, we shall write them here for thc sake of completeness. In the following we shall consider the s- and p-wave parts of the hard core
interaction.
For t,hc s-wave part of K: we have
and for the p-wave part,
where
K=ypI
+Pz, K ' = p ; - t P i ,
2k ;= PI - p 2 , 2k' = p i - pi,
&(/cr) is the spherical Uessel function of order one and
sin (kr,) cos (kr,)
Gf(rc, re) = -kr,
1
+ kr, sin (kr,) cos (kr,) - 2 sin2(kr,)
M
(7)
The 'f sign in equation (6b) corresponds to k = hk'and the dots in eq. (7) denote higher
powers in 7,. By expancijng equations (6a) and (6 b) in terms of rc till third order, we have
2 n fi2
( p i p ; IKi.1 Pip.-)$= 6 K , K' -i=~ rc(2 $(k2 - k'2/3)}
+
and
2.2. Calculation of the Binding Energy of Nuclear Rlatter with Neutron Excess
By following the method applied by DH, we can calculate the binding energy using
the resu1t.s of the K-niatrix calculation. We start with the unperturbed state of 2 protons and N neutrons ( N Z = A ) in a periodicity box of volume 0. Let us denote by
k the k'ernii momentum in the case 01 = 0 (with the same value of A ) , i.e.
kj = 3 n 2 A / ( 2 0 ) .
Then we have
+
x = kjyn
whcre
ya
1G'
=:
(1
-+
and 1. = kfy,,
,)*jS
and y p = (1 -
(9)
31. Y. N.HASSAY
and S. S. MOSTASSER
244
The unperturbed (kinetic) energy, E,, of our system is given by
The change in the energy due to hard core interaction, A E
ximately expressed in terms of K , as follows:
AE
= AE,,
+ AE,,
$-
AE,,,
= E - E,
can hc appro(1 1 )
where
(12b)
and
[ 2 {pl < x , p 2 < x )
for pv = ma,
The factors 4 and 2 in the above equations come from the sunmation over spin. Thc
summations in the above equat,ions can be changed into int,egrat,ion according to t,he
relation
,2.3. Results and Disciissions
By inserting eqs. ($a) and (8b) and (5a), (5b)and (5c),int.0 ( 3 2a) and (12b),kecping
terms up t.0 the third order in kfr,, we get the following expression for the third order
term of the binding energy per particle of nuclear matter with pure hard core interaction with neutron excess :
The first and second t e r m of ey. (13) are the contribution of the s-state part of the
hard core reaction matrix while the third tcrm is the corresponding one for the p-state.
The first and third terms of the same equation have been calculatcd analytically while
the second term has been computed niimerieally. If we include the other terms, calculated before, and expand all tcrms in powers of oc up to the fourth order only we get;
E j A = &,1
iC&X‘
+
E;:;,L~.~
(14)
Binding Energy of Nuclear Matter with Hard Core Interaction
where
In cq. (15) the term proportional to (kf.rc)consists of two parts. The first onc corrcsponds t o the sum of first and la& ternis in eq. ( I ) of EFIMOV
[ 7 ] .The second onc coincides with the second term of the Rame equation of EFIMOV.
The contribution of the
second term in cq. (13) to the symmetry energy is very small, conscquently we feel that
its contribution to thc fourth order term in a will be negligible.
Jn our work we considered only the pair correlation in the S and P states. The effect
of other correlated pairs on a given pair and that of triple correlations are not considered.
The compressibility K can bc calculated from the relation
1.35Fm-l,
Table 1 The contributions of different powers of (kfrc) aa well aa their sum a t k
rc = O.AFni, other theoretical calculatiom and the experimental vdues of + , l ,~'p':m,E L $ ~ ~ K @;
, ~&~, ,
and K:;),,,. The parameters of the effective lntoractions used in all these theoreticd calculations
are adjusted to give the cmpirical binding energy per particle of nucloar matter
The contribution of
EJA
(Icfr,) term
( L pc ) 2 term
(k,r,)*' term
Sum
Expcrimcntnl L14]
Theoretical [15]
Theoretical [I61
Theoretical [li]
ct0l
22.71
13.01
3.69
3.4'2
42.83
-16 to -16
-16.06
--17.78
&(1)
$2)
synr
12.62
-4.34
-0.23
1.88
9.43
28 to 32
32.72
44.89
41.68
0.47
0
-0.14
0.13
0.46
sym
-
1.7
1.67
Kvo~
46.42
78.08
44.23
68.42
236.15
70 to300
300
184:
261:
K(1)
sym
25.23
-26.03
- 2.76
37.54
33.98
-
-
68.21
K(2)
sym
0.48
0
-0.17
0.06
.37
-
Table 1 shows the contributions of different powers of (Icfr,.), their sum, and other
theoretical calculations of E , ~ ~d&&,
,
E~;L,
Kvol, Kit&, K&, (kf= 1.35 P;' and re =
0.4F,,),as well as their experimental values. From this Table we can see that the third
order term is not negligible. In the expression for evol,the second and the third order
terms are nearly cqual. In the case of Kvolthe third order term is even larger tlian the
second order one. From this we can see the importance of the third order tenn. To see
the relative importance of the hard core to the attractive part of the potential we quote
the results of the calculations which include the attractive part in Table 1. For the comparison of our results with the experiniental ones we recommend the inclusion of a
reasonable attractive potential in the calculations of the binding energy and the coinpressibility.
Refarences
[l] W. LEXZ,Z. Phys. 56, i78 (1929).
[2] I<. HUXAG
and C. N.YANG,Phys. Rev. 106, 767 (1957).
[3] T. D. LEE and C. N. YANO,Phys. Rev. 146,1119 (1957).
[4] C. m Doxih-IcIs and P. C. MARTIN,Phys. Rev. 103, 1417 (19.57).
[5] V. hi. GALITSKI,Zh. Eksp. Teor. Fix. 34,151 (1958).
[9J. S.LEVIXGER,11.RAZAVY,
0. ROJO, and N. WEBER,Phys. ltev. ll!),230 (1960).
[7] V. N. EFIYOV
and M.TA.AYCSIA, Zh. Eksp. Teor. Fiz. 47, 581 (1964).
181 V. N. EFIMOV,Phys. Lett. 16, 44 (1965), Zh. Eksp. Teor. Fiz. 49, 188 (1965).
[93 L. C. Gonms, T. D. WALECKA,
and V. F. Wmwcow, Ann. Phys. 3, 241 (1968).
[lo] J. S. LEVIXGEK,
Kucl. Phys. 3, 241 (1958).
[ll] W. Cevz and K. GOTTFRIED,
Ann. Phys. Yl, 47 (1963).
[12] J. D~BROWSKI
and 31. Y. 11. HASSAX,
Phys. Lett. l!),318 (196h).
[13] J. S. BELLand E. J. SQUIRES,
Adv. Phys. 10, 211 (1961).
[I41 J. D~BROWSKI.
Theory of Nuclear Structure, lAEA, Vienna (1970) p. 131.
1151 H. PLOC.ARD,
Proceeding of the International Conference on X'nclear Physics, vol. 1, ilunich,
Germany. 27. Aug.-1. Sopt. 1973, Amsterdam, North Holland, p. 40.
[lo1 I<. M. KHAKNA,
D. J A ~ R A T H ,and P. K. BARHAI,External Report of the International Centre for
Theoretical Physics, IC/75/18 (1978), Miramare, Trieste.
1171 31. Y. M. HASSAX,A.M. A. GHAZAL,
and K. hf. MAHMOUD,
(1977).
Bei der Rcdaktion eingegurigen am 22. hIan 1977.
(Hevidiertes Xanuskript eingcgangen am 13. Januilr 19%).
Anschr. d. Verf.: 31. Y. 31.HAss,is and s. s. hIoNTASSER
Physics Department, Paciilty of Science
Cairo University, Cairo, A. R. Egypt
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