A method for fast convergence of the reaction matrix

I

l*c

I

Nuclear Physics A184 (1972) 285-302;

@ North-Holland Publishing Co, Amsterdam

Not to be reproduced by photoprint or microfilm without written permission from:the publisher

A METHOD FOR FAST CONVERGENCE OF THEI REACTION MATRIX R. K. SATPATHY Department of Physics, Sambalpur University, Sambalpur, Orissa, India and SUPROKASH MUKHERJEE Saha Institute of Nuclear Physics, Calcutta, India Received 10 November 1971 The reference reaction matrix tRfor a local potential is expanded as a series in terms of the difference between this potential and an auxiliary potential. The latter is chosen to be a nonlocal separable potential, so that its matrix elements are summed up exactly. By suitably varying the parameters of the separable potential it is possible to get a fast convergence of the matrix elements of the series. Calculations are performed with a Gammel-Thaler potential to show the convergence rate. The method is an improvement over the convergence rate of the modified Born series of Bethe et al. and can be applied to nuclear matter calculations with more realistic potentials.

Abstract:

1. Introduction A good amount of work has been done in the last fifteen years to treat the reaction matrix ‘) accurately and to guarantee the convergence of the Brueckner-Goldstone series. Brueckner and Gammel “) (BG) solved the integral equation of the reaction matrix by transforming it into the coordinate space where the effect of the hard core [ref. “)I of the two-body potential is easily taken into account. The method has been criticised on the grounds that it is numerically complicated and many of the physical features are not clear. The separation methods of Moszkowski and Scott “) (MS) and the reference spectrum method “) (RSM) of Bethe et al. were therefore intended for simplifying the numerical calculations and to give more insight into the problem. In the former case the two-body potential is divided into a short-range part and a long-range part. The short-range part is treated without the exclusion principle and the long-range part in the Born approximation. The method is simple but it suffers from the drawbacks that it is not applicable in all partial waves, repulsive states in particular, while the separation distance is state and momentum dependent. The reference spectrum method transforms the integral equation of Brueckner and Gammel into a differential equation which is easier to solve. The correction term, i.e. the difference between the full reaction matrix and the reference reaction matrix, is generally small and can be treated systematically. But for cases 285

286

R. K. SATPATHY AND S. MUKHERJEE

where the potential is large just outside the core, it is necessary to combine the MS separation with the reference spectrum method (modified Moszkowski-Scott method MMS). Thus we have three different methods for treating the reaction matrix in nuclear matter. Comparatively, the reference spectrum method is preferable, because it reduces the size of the numerical computation and gives more insight into the subject. However, to overcome the difficulty of applying the MS separation to repulsive states and as an alternative method, we propose here a method for the calculation of the matrix elements of the reaction matrix t. The reference reaction matrix tR of Bethe et al. (BBP) for a local potential is expanded in a series on terms of the difference of this potential and an auxiliary potential. By suitable adjustments of the parameters of the latter, the series is expected to be rapidly convergent to any desired degree of accuracy. The main purpose of looking for such an expansion is to obtain a rapidly convergent expansion for the t-matrix. Essentially, it amounts to improve the convergent rate of the modified Born series of BBP. The use of an auxiliary potential, as discussed below, is equivalent to making a partial summation of the iterative solution of the modified Born series and is analogous to the method of condensation of a series used in mathematics. The use of a non-local separable potential is best suited for such purpose. Such an attempt is parallel to the quasi-particle method of Weinberg “) and is quite in line with similar methods used in some shell-model calculations [ref. ‘)I. 2. Formalism Our starting point is the operator identity of the reaction matrix t given by BBP t* = tB + L2J(D*- qj)OA + tp*

- I$$,.

(24

For PA = PB = l/8, this reduces to the form t,4 = tB + a&*

- V,)OA.

(2.2)

The index R stands for the reference spectrum. The two-body local potential II consisting of a hard core V, and external potential un may be written as U = a,+VE = (V,-f-V,- Vz)+ V, = V, + V,.

(2.3)

With 2)A= 21and zla = VI, eq. (2.3) becomes tR = ty + a:+ v, i-JR,

P-4)

where QR=

l-lfR, eR

i-27= l-

-1-t:. eR

(2.5)

FAST CONVERGENCE

OF REACTION

MATRIX

287

The wave operator QR through (2.4) and (2.5) has the expansion 1

fF=SZR--QRV52R 1 12 eR

=

a:- $i2:v261:+ $2yv2-I.y,a~+.. .. e

eR

(2.6)

Now, using (2.6) in (2.4) for OR we obtain tR = t’:+m;+V2Q;-@+v2

1 G$tV,Q;+ eR

....

(2.7)

The potential V, may be again thought of consisting of two parts v, and (Us- V2) like VI and V, as in the case of ZIso that reaction matrix ty can be again expanded in the same way as in (2.7) ty = tf + af+(v, - V,)@ - !sy(vE - v,) L Q;+(vE- V,)@ + . . . . eR Putting ty in (2.7), the expression is tR =

(24

c + @‘(v, - V,)@ - @‘(DE - V2) A- @(v, - v,)C$ + ... 1 I eR + [

~~tv2~~-~~tv2~~~'v2n~+... , 1 eR

(2.9

where t,” is the reference reaction matrix corresponding to the hard-core potential v,. The purpose of writing down tR as in (2.9) is as follows: t,” can be treated exactly as done by BBP. The other terms in first square bracket involve the difference (us - V2) with .G’foperating on both sides. The operator B: acting on a plane-wave state generates the hard-core wave function $t whose analytic form is known (2.12). Hence by a suitable choice of V,, the matrix elements of those terms, i.e. terms of the first bracket of (2.9) except those oft,“, can be made small as we like and we can guarantee the rapid convergence of the series in the first bracket to any degree of accuracy. On the other hand, the second bracket contains terms involving V2 with G$ operating on both sides. If V2 is a non-local separable potential, these terms can be summed up in a convenient manner due to factorization of the terms. This will be shown later in this section. Partial-wave expansion. The antisymmetrized, unperturbed two-particle wave function @ is written as

288

R. K. SATPATHY AND S. MUKHERJEE

iP * R

=-

where

e

(2.10)

xi,,p,T, are the spin and

isospin wave functions, j, i ml

is the Clebsch-Gardan

i m

j.2 m2

1

coefficient and

Here S2is the nuclear volume and the index /I stands for the spin and isospin quantum numbers. The relative and c.m. variables have be&n de%sed through r = PI---Pg,

R = $(r, -t-r,),

k = $(k, -k,),

B = k,+k2.

In (2.10), &,&> is defined as

In the similar manner, the hard-core wave function (omitting index R throughout)

with $ given by BBP u;( kr) = $&)

- St(kr),

(2.13)

where ,$1 and X1 are defined as gZ;,(W = W,(kr), (2.14) where HE(-)(‘r) is related to the usual Hankel functions Hj-)(yr) =

i"+"(Yiyr)hil'(f

iyr)

=E '-('*l)(t_iyr)hS2)(+iyr).

(2.15)

FAST CONVERGENCE

OF REACTION

MATRIX

289

In terms of reference spectrum parameters A and m* for states nz and n in the Fermi sea, y2 = 2Aks- k2 where k = $(k,,,--k,,) and kF is the Fermi momentum. Following again BBP, the non-diagonal matrix element oft, is written as S

J

ms

M

111 1 I’

S’

J’

ml, msf M’

Here a denotes the quantum numbers J, S and T; C is the radius of the hard core of the two-body potential. The next two terms in the first bracket of (2.9) in the operator form are given by s2L(uE- V,)sZ, and O~(ZJ,-- V2)(l/e)Q~(~,-- V2)sZ,. The nondiagonal matrix elements of these will be given as shown in appendix A to be

x i”-“l;(,:(i?)Y~,(~)

y@,““] a,.&6,,,,,

2

(2.17)

(2.18) Combining (2.16), (2.17) and (2.18) we obtain

c;?yr;p, ii’@‘) =

(@I$[;,

zs3

s ;]

[; I’

ms

iz’-zY;;(ff’)Y;,(~),

290

R. K. SATPATHYAND S. MUKHERJEE

(2.20) Next we consider the terms in the second bracket of (2.9) (2.21) with

(2.22) e

The non-diagonal matrix elements of S2ff28, are given according to the appendix A to be

x d3rld3k”‘d3k”d3~

+P:!,!(k, k’)[l+G,(k’)]-M,(k’)[P:;&k, Finally we combine

k’)+P$(k,

k’)-J/&(k’).

(2.19) and (2.24) to get the full partial-wave

(2.24)

expansion of


k')[l

k’)[l +&(k’)]

+ G,(k’)] -M,(k’)[P::&

k’)+P:;@,

k’)]>/Da(k’).

(2.25)

The matrix elements of t in (2.25) for the diagonal case can be used through “)

(B.l)--(B.6)

to calculate the single-particle energy and binding energy per particle matter. The calculation of the contribution of the correction term (t - tR) is exactly the same as discussed by BBP and will not be described here. in nuclear

FAST CONVERGENCE

OF REACTION

3. Numerical calculations

MATRIX

291

and discussions

The purpose of this section is to examine the usefulness of the method used in the

preceding section in the light of some numerical calculations. We use the GammelThaler potential as the two-body interaction and a non-local separable potential V, for subtraction. The functional forms of V, in different states, as given in this section, are chosen so as to match the matrix elements of z, up to 4 fm-I. For the results given in tables 2 and 3 the reference spectrum parameters were fixed at A = 0.6 and m* = 0.95. Choice of V,. The potential V, expanded in different partial waves in (A.4) consists of two terms ~~62’and g::’ which will be denoted here as gdl and hal respectively. With the aim of having a smaller number of parameters, an attempt was made in our calculation to choose a single form factor gal with a simple form in momentum space. The second factor h,, was considered in limited cases. The functional forms of g,,(k), h,,(k), &(r), i&r(r) for the various states are the following:

t%,(k)= 1

&[l+g$

gal(r) = 1 emfir+ p+

(p’e-“‘-

/?‘e-@)I

[

3s,:

1r.

g,,(k) and gal(r) are same as ‘So case 2 Uk)

=

Y (k2

:

c12)2

9

&(r) = ~(1-&xr)e-ar/r.

1

3P, :

*

3P,:

g,,(k) = 1

k

(k2 + p”)+ ’

B&9

= 2 a mw4~

b(k)

= Y

k (k2 + E”)* ’

292

R. K. SATPATHY AND S. MUKHERJEE

Lal(r) = y i uK,(ar).

‘D, and 3D,: k*

k= &dk)

=

g,,(r) =

'pZfp2)2

1+6k2+p2

$ [ (1 +j?r)eM8’-6

x (3+p4r4)

+(jg4)

1 '

( [(~2e-W-~2e-b’) (3+P4r4)

(P2-P2)2

e-!3r

r2

II

(P"-P") 2

+

(pe-"-/3e-8')

(p2:p2)2

[

3D, :

g&4

and s”d r > are same as ID,

+

Wr-1 (p-q)"

11II’

-fir

r

and 3D2 case.

2

Y (k2 : a2)2 9

kl(k)

=

/ial

= +y(l -I-cw) ‘G

,

3D, : LjgI(r) = k [(l +pr)e-p’-6(1

+/h-$/12r2)e-Br],

2 h&)

=

Rat(r) =

Y(k2ta2)2

:

.J

(1 + ar)e-“.

(3.1)

Here KI is the modified Bessel function. The parameters CI,6, y, 6, ~1and ,I are determined by matching the matrix elements of the Gammel-Thaler potential with that of the above potential through (A.7) for k and k’ up to 4 fm-‘. The values of these parameters are given in table 1. The ratios are approximately unity for diagonal matrix elements and for jk - k’l < 1. Such ratios in different states for k and k’ up to k, = 1.5 are given in table 2. The wave functions are already defined in (2.13) with the values of the y determined from the relation y2 = 2Aki - k*, where k is the relative momentum of the initial state.

FAST CONVERGENCE

OF REACTION

MATRIX

293

Using the parameters of table 1, matrix elements of the second-order term in (2.9)* in terms of the difference of the potentials (un- Y,) are also calculated in all the partial-wave states “) up to I = 2. Such matrix elements at four values of k within kF are put in table 3 along with the matrix elements of vE and - (uE- V.) for comparison. In these calculations y was calculated with k = k,, = -\/0.3k, to save computation time. TABLE1 Strengths and inverses of the ranges of the non-local separable potential Parameters of g(l) and gc2)

State

%I % lP1 3P0 3P1 3PZ II% 3D1 3Dz 3Da 3D1 -3S1

B

/I

a

1.3 1.3 1.1 3.6 1.1 1.2 1.3 1.13 1.13 1.9

4.95 4.55 9.55

6.24 5.0 1.73 23.92 1.60 1.41 1.57 2.23 2.23 0.57 8.32

9.55 9.95 4.95 5.75 5.75 1.0

Sign of g’l ),(I,

6

cc

Y

9.0 9.0 25.0

1.8

0.0

+

1.0

1.84

T-

2.1

0.0

0.9

0.57 7.44

-I-

40.0 50.0 7.5 7.5 4.0 6.5

g’2’g’2”

-

+ +

Parameters cc, 8, etc. have the dimension of fm-i, except for 6 which is dimensionless. Other parameters of the coupled case are same as 3S1 and 3D1 states.

It is evident from table 2 that it has been possible to make the ratios of the matrix elements approximately unity for k and k’ not differing widely. For ]k-k’l > 1 the ratios are less than unity because, off-diagonally, the matrix elements of the local potential fall more quickly than their counterparts in the non-local potential case. With different parameters or/and functional forms of gal and hbl given in (3.1), one can improve the ratios of the table 2 to be unity in a larger region. Examining table 3 we find that all second-order matrix elements are not small (say & of that of vE in all cases). Once the table 2 is improved this criterion can be achieved in all cases. However, with the parameters of table 1, it was observed that for A = 0.75, m* = 0.90, k, = 1.5 fm-’ and states upto 2 = 2, the contributions of the various terms of (2.9) to BE are found to be 34.6 MeV by the core, - 76.0 MeV by the 52: V, 52 term and only -3.0 MeV by the @(v,- v&2, term. The second-order term @(u,- V2)(l/e) Qd(% - V&2, contributes less than 1 MeV. The binding energy per particle calculated neglecting the contribution of the second- and higher-order terms in (v~-- V,) in the expansion of (2.9) and without the correction term (t -tR) for a few sets of A and m” are shown in fig. 1. It is seen that the present method will serve as an alternative way of treating the modified Born series of BBP and that its convergence rate can be improved by the use of the auxiliary potential V,. This, then gives a fast convergent

R. K. SATPATl-XY AND S. MUKHERJEE

294

TABLE 2 Ratio of the matrix elements of the local and non-local potential given in (A.7) for &.a with d = 0.6 and kF = 1.5 fm-‘. State

\ k k

0.25

0.57

0.92

1.44

IS*

0.25 0.57 0.92 1.44

1.04 1.00 0.92 0.67

1.01 1.00 0.96 0.77

0.93 0.97 0.99 0.91

0.71 0.78 0.90 x.12

jSi

0.25 0.57 0.92 1.44

1.09 1.09 1.05 0.90

1.09 I.10 1.08 0.96

1.06 1.09 1.10 1.05

0.91 0.96 1.03 1.14

IRi

0.25 0.57 0.92 1.44

1.31 1.11 0.84 0.53

1.11 1.07 0.93 0.65

0.85 0.93 1.00 0.86

0.53 0.65 0.86 1.10

3Po

0.25 0.57 0.92 1.44

1.14 0.88 0.57 0.15

0.89 0.87 0.71 0.24

0.57 0.71 0.84 0.49

0.16 0.24 0.50 1.50

3PI

0.25 0.57 0.92 1.44

1.46 1.07 0.72 0.51

1.07 0.98 0.80 0.61

0.73 0.80 0.86 0.79

0.50 0.60 0.78 1.02

3PZ

0.25 0.57 0.92 1.44

1.49 1.17 0.89 0.72

1.17 1.08 0.94 0.80

0.89 0.94 0.98 0.95

0.72 0.80 0.95 1.13

lb

0.25 0.57 0.92 1.44

1.31 1.19 0.97 0.65

1.19 1.14 1.02 0.75

0.97 1.02 1.04 0.91

0.65 0.75 0.91 1.07

3D1

0.25 0.57 0.92 1.44

1.37 1.06 0.69 0.35

1.06 0.98 0.77 0.45

0.69 0.77 0.81 0.65

0.35 0.46 0.64 0.87

3DZ

0.25 0.57 0.92 1.44 0.25 0.57 0.92 1.44

1.44 1.16 0.82 0.47 0.98 0.88 0.6X 0.11

1.16 1.10 0.93 0.62 0.88 0.94 0.81 0.19

0.82 0.92 1.03 0.90 0.61 0.81 1.01 0.47

0.47 0.62 0.90 1.27 0.11 0.19 0.47 0.14

0.25 0.57 0.92 1.44

1.11 1.11 1.07 0.95

0.79 0.91 1.Ol 1.03

0.51 0.64 0.88 1.17

0.38 0.59 -1.0 1.30

3D3

%i

--3si

Here k and k’ are given in units of fm-I.

FAST CONVERGENCE

OF REACTION

MATRIX

295

TABLE 3 Comparison of the diagonal matrix elements given by the quantities inside the curly brackets of (A.2), (2.17) and (2.18) multiplied by k2 and with the factor hZ/M taken out, designated as I, II and III respectively Matrix elements of

k fm-l

States

0.25

%I % lP1 3P0 3P1 3PZ IDZ QJ %

0.92

% % lP1 3P0 3P1 % IDZ? % 3Dz 3D3

1.44

‘SO % ‘P1 3P0 3P1 3PZ II& % 3Dz 3D3

I

II

Ratios III

I/II

I/III

-1.28~10-~ -6.72 x 1O-2 6.81 x 1O-3 1.22x1o-2 6.09~10-~ -2.52 x 1O-3 9.34 x 10-s -5.98 x 1O-4 1.47x1o-4

-7.03 x 1o-3 -6.47 x 1O-3 1.63 x 1O-3 -1.54x10-3 1.94x 10-s -8.31 x 1O-4 -2.25 x 10-5 -1.83 x lo-“ -2.03 x 1O-6

5.99 x 10-3 1.42 x 1O-2 4.01 x10-4 5.86~10-~ 3.21 x 1O-4 6.88 x10-5 7.44x10-6 5.11 x 1o-5 2.28 x 1O-6

18 10 4 8 3 3 4 3 72

21 5 17 21 19 37 13 12 64

-4.76 x10-l -2.76 x10-l 8.82 x lo-’ -1.16~10-~ 6.88 x 1o-2 -3.51 x10-2 --6.70~10-~ 2.82 x lo-’ -3.03 x10-z 6.42x 1O-3

-6.37x 1O-3 -2.69 x 1O-2 6.04~10-~ 1.66 x 10-Z 1.02 x 10-J 2.76x 1O-3 -8.53 x 1O-4 -4.96~10-~ -2.95 x.IO-~ -4.23 x 1O-3

1.88$( 10-Z 3.15x10-2 5.38~10-~ 6.30 x 1O-3 4.21 x 1O-3 1.04x 1o-3 5.61 x 1O-4 3.29x1o-3 2.91 x 1o-3 1.59 x 10-d

75 10 15 7 67 13 8 57 10 15

25 a 16 18 16, 34 12 9 10. 40

-7.72 x10-l -5.04x 10-l 2.69 x 10-l -2.32 x 10-l 2.17~10-~ -1.36x 10-l -5.32 x 1O-2 1.58~10-~ -1.78 x10-l 2.49 x 1O-2

6.88x1o-3 4.73 x10-2 -2.09 x 1O-4 4.38x10-’ 3.49x10-2 2.35 x 1O-3 -2.12 x10-3 3.59 x lo-= -5.30 x 10-3 2.64~10-~

2.09 x 1o-2 3.20 x 1O-2 1.29 x lo-’ 1.20 x 1o-2 9.08~10-~ 3.07 x 10-3 4.07 x 10-3 1.08 x 1O-2 1.34x10-2 1.17x 1o-3

112 11 1287 5 6 57 25 4 33 94

37 16 21 19 23 44 13 1.5 13 21

-8.26 x10-l -6.05 x10-l 5.49x10-l -1.01 x10-1 5.59 x10-1 -4.21 x10-l --2.27~10-~ 5.14 x 10-l -5.98 x lo- 1 -3.70 x10-3

-3.11 x10-2 -5.22 x lo’-’ 4.78 x 1O-2 -3.19 x10-2 1.13 x10-2 -4.91 x10-2 -1.56~10-~ -7.35 x10-2 -1.30x 10-l 2.14 x lo-*

2.00 x 10-Z 2.76 x lo-’ -1.70x10-3 1.88 x10-2 3.09 x10-2 8.36x 1O-3 1.01 x10-2 8.27~10-~ 1.24 x 1O-2 -2.65 x 1O-3

27 12 11 3 49 9 15 7 5 -17

41 22 322 5 8 50 22 62 48 1.4

y2 = (2A-0.3)kFz, A = 0.6 and m* = 0.95.

296

R. K. SATPATHY AND S. MUKHERJEE

-401

1.0

I.2

I.4 KF-IN

I.6

6.8

2

d

Fig. 1. The total energy per particle without Pauli and spectral corrections.

method for calculating the matrix elements of the reaction matrix t, both on and off the energy shell for a given range of momenta k and k’. Appendix A PARTIAL WAVE EXPANSIONS

Using the partial-wave expansions of&s(r)

and +iB(~) from sect. 2, we have


s

~,&~(r)‘C’;.(r, r’)$$,B,(r’)d3r d3r’

FAST CONVERSANCE

OF REACTION

MATRIX

The IO& potential part in the bracket above reduces to

where (v-&? is defined by as ~~~)~~= <~~(~)i>lu,(~)l~~~~~~>-

(A.3

The range of integration in form C to co because the core wave function vanishes inside the hard core. For the non-local potential part in (A.1 ), we make a partialwave expansion of V,(Y, Y’)following Tabakin “) and get

=

g,

Z~~~~(~)~ac~/~~~(~~~~~~~)l~~~>{~~~i,

@4

MrnT

where &(r) = 2 mgmz(k)jl(kz$k2dk z J’0

&z(r) =

~j~~~~(~)~~(~~)~‘d~.

(A.4a)

For each state specified by J, S, M, T and mT, thepotenti~ consists of in general two terms i = 1, 2. The value of E and I’ are consistent with J, S and the parity of that state. Using this expansion in (A.l) the non-local part simphfies to - 1 mCc C@W)i ~~)(~)~3(r’)~~(k’r’)~ dr r’dr’C& 6M&S,,,,, k-k’jsc c i=1 Now with the help of (A.2) and (AS), (A.1) becomes

The second-order term in the operator form is

.

298

R. K. SATPATHY AND S. MUKHERJEE

and is treated in the same way as the first-order term. The non-diagonal matrix elements of this are given by

XL!_ ___ l

~~~~,,(~“‘)[~~(~‘)s(rl-P”‘)-

(27~)~ e(k”, Y)

V,(Y”‘, r’)]

x r&&‘)d3p’d3r”d3kr’d3r”‘d3r.

(A.9

In the above equation the 4(r”) and $P(r”‘) arising out of intermediate states are not taken antisymmetric. We consider the integrand of (A.8) as the product of two factors. $$(r)[v&)6(r

- yl’) - &(r, r”)]&&“),

~&&_r”‘)[t~a(r’)S(r’-#“)-

VJr”‘, r’)]$~Tp(r’),

and treat them in partial-waves separately. The steps are similar to those from (A.l) to (A.7). The resulting partial-wave expansion becomes

where

and Bfi, has been defined earlier in (A.7). In above derivation J, S, T, M and mT are taken as good quantum numbers and the orthogonality of Clebsch-Gordan coefficients has been used. The partial-wave expansion of (k~lQ$,Q,lk’j?‘>. We write down
FAST CONVERGENCE

OF REACTION

MATRIX

299

the notation of Tabakin as = (2/n)

c iL’-L (&,(k”, JMS

k”‘>~~~(~“)~~~~(~“)l~~~><~~~l,

(All)

LLfTm~

and expand +&‘(r) in (2.23) in the notation of Brueckner and Gammel “) as (A.12) Here the suffix of I of U$ denotes the dominant wave (3S, state of the deuteron state) and I’ # 1 is the subsidiary wave (3D1 part of the deuteron). Now using the expansion given in (A.1 1) and expansions of 4 and +(I) from (2.10) and (A.12) in (2.23) we have

&?l(kr)jL~(k”r)(2/~)(?2)~~L~~(k”, k”‘) x jL,,,(k”‘r’)u~~~,,r,(k’r’)r2dr r’2dr’k”2dk”k”‘2dk’~‘~~~,S,,,,,+ . The quantity (f2)~CFL,,@“,k”‘) is evaluated as follows: 22 = V2-V&f12 e =

v2-v2(; -y)i,.

Taking the non-diagonal matrix elements

(A.13)

300

R. K. SATPATHY AND S. MUKHERJEE

(k”~l,~k”‘> = (k”lV.lk”‘>-(k”ll/, =
(f - 5 t, 3 &lk”‘)

c 1
x(k,B,l?zlk”‘)d3kld3k2,

(A.17)

and project out the 1 and 1’ partial waves to obtain

(I&@“, k”‘) = (v&@“, x @&,

k”‘)-(2/~)~~~~~V,);,,(k”, kd

, kz)(~&(kz >k”‘>k:% k”2 dk,

= [g,,(k”)g,,@“‘)+ h,,(k”)h&“‘)lx %&,

9

5,zs,” b&“)ga&)+

k&“h&)l

kz)(f&r(kz k’)k:dk, k: dk, -

(A.18)

9

w . The kernel of the above integral equaHere gal stands for gif’ and h,, stands for gal tion is separable and consists of two terms. Hence using the standard technique ‘), we obtain (f&o”,

k”‘) = [l +H,(k’)]g,,(k”)g,,,(k”‘)

+ [l + G,(k’)]h,,(k”)h,,(k”‘)

-M,(k’)[g,~(k”)h,,@“‘)+

h,,(k”)g,,(k”‘)]/D,(k’),

(A.19

where

.M,(k’) = f ~~~~Cg~l,(k~)h,,,(k~) +

h,,,(k~)galz(ka)l~~l,(k, 3k,)k:.dk kidkz364.20)

D,(k’) = [l+G,(k’)][l+N,(k’)]-@(k’).

(A.21)

These G, H, M and D are structurally similar to those of Tabakin [ref. *), eq. (2.21)]. The momentum k’ in G,, &, M, and D, refer to the momentum of the initial state of (kltlk ) through y in the energy denominator of t. Then using the expression for

FAST CONVERGENCE

(?z)&(k”, we get

OF REACTION

MATRIX

301

,I”) from (A.19) and making expansion up to first order in (z)~-- V,)

(A.22)

(A.23) Appen&x B In the calculation of the single-particle energy and the average binding energy per particle, one requires only the diagonal matrix element of I. Averaging over sixteens spin and isospin states of the interacting pair we write i% T <~*~I~~~~~> = 4% k>*

(B4

The single-particle energy E(km) for a state m inside the Fermi sea with momentum km is .E(km)= 5

+ jmN(km, k&k, k)dk, 0

(B-2)

302

R. K. SATPATHY

AND S. MLJKHERJEE

where N(k,, k) gives “) the probabi~ty that the particle with momentum k, has a relative moments k with respect to any particle in the Fermi sea: 0 < k < $(k,-km),

N(k,, k) = 36k2/n2, = 2 k;-(km-2k)2 n2

k2

+jk,-km1

< k < +(k,+k,),

’

kk,

x 0,

?l(kd

km) < k.

03.3)

The average binding energy per particle (BE) is given by RR2

!& 10 A4

s

kFf(k, k)~~k)~k

=~~+~~~~k,k),~-~~+~~~k’~k.

(R.4)

Here the density p is given by (B-9 and the normalized probability P(k) of two nucleons in nuclear matter is given by “)

We are grateful for the computing facilities provided by the Tata Institute of Fundamental Research, Bombay. One of us (R.K.S.) is indebted to the Saha Institute of Nuclear Physics, Calcutta, for the grant of a Research Fellowship during the period of work. References 1) K. A. Brueckner and C. A, Leviuson, Phys. Rev. 97 (1955) 1934; K. A. Brueckner, Phys. Rev. 97 (1955) 1958 2) K. A. Brueckner and J. L. Gammel, Phys. Rev. 109 (1958) 1023 3) J. L. Gammel and R. M. ThaJer, Phys. Rev. 107 (1957) 291; 107 (1957) 1337 4) S. A. Moszkowski and B. L. Scott, Am. of Phys. 11 (1960) 65 5) H. A. Bethe, B. H. Brandow and A. G. Petschek, Phys. Rev. 129 (1963) 225 6) M. Scadron and S. Weinberg, Phys. Rev. 133B (1964) 1589 7) M. H. Hull Jr. and C. Shakin, Phys. Lett. 19 (1966) 506 8) F. Tabakin, Am. de Phys. 30 (1964) 51 9) N. Azziz, Nucl. Phys. 85 (1966) 15