Coulomb contributions to nuclear radii and energies

Nuclear Physics @ North-Holland

A367 (1981) 215-236 Publishing Company

COULOMB

CONTRIBUTIONS TO NUCLEAR RADII AND ENERGIES

M.A.

JADID

and

H.A.

Physics Department, American Received

MAVROMATIS University, Beirut, Lebanon

13 February

1981

Abstract: The effect of including the isospin non-scalar Coulomb interaction (also taking into account the finite proton size) in binding energy and rms radius calculations for the closed shell nuclei 4He, I60 and ‘%a is discussed in detail. Using the saturating Sussex matrix elements and including 0 + 2hw excitations it is found that the major Coulomb contribution is in the first order, and that the pure second-order Coulomb contribution to the energy, near the saturation value of the size parameter for each nucleus, is more important than the mixed second-order contribution. Finally, for third order and above one can safely neglect the pure Coulomb contributions compared to the mixed ones which are in turn small compared to the pure nuclear contribution, in that order.

1. Introduction Recently 1,2), the Goldstone graph component technique 3, was extended to In particular, the include scalar operators other than the nuclear interaction. perturbation series up to third order in the nuclear interaction and up to 4hw excitations

for the one-body

shell nuclei 4He, I60 and diagonalization and inversion than energy

mass radius operator was calculated for the closed 40Ca. In addition some non-perturbative methods, 4), were extended

and some exact linked

results

obtained for the mass radius operator. In this paper we extend the above

to include

scalar operators

in the space of 0 + 2ho

techniques

to include

excitations

non-scalar

other were

operators.

In particular, we calculate via these techniques the contribution of the isospin non-scalar Coulomb interaction to the binding energies and radii of the closed shell nuclei 4He, 160 and 40Ca in the space of 0+ 2hw excitations. These calculations are performed using a harmonic oscillator basis and the results are displayed as a function of the oscillator size parameter. The nuclear interaction used is the saturating Sussex matrix elements 5, with a core radius c = 0.3 fm. In sect. 2 we give some details about the residual interaction and the approximate proton form factor used. In sect. 3 we show how to extend the Goldstone graph component technique to incorporate the isospin non-scalar Coulomb interaction and give some calculational details. In sect. 4 we give a summary of the procedures used for checking these calculations. In sect. 5 we give the diagrammatic representation of the energy and radius matrices used for obtaining the diagonalization and inversion 215

216

results.

M.A. Jadid, HA.

Finally,

Mavromatis

/ Coulomb contributions

in sect. 6, we list and discuss

the results

obtained

and give some

conclusions.

2. The residual interaction Following

the notation

of ref. ‘), the residual v=

interaction

can be written

as

v,+vo

(1)

given by eq. (4) of ref. ‘) and where V, = psaturating is the nuclear interaction is the Coulomb interaction. For point-like protons, vc=

c vii=

where

LEL

1

(2)

i
i
r,j = ]ri - rj] and 4; = e (k + tiL). Explicitly

Vc

one can write (3)

To find the two-body

matrix

of Vc we have,

elements

tl, +

tjz

=

Tz

m

Therefore v,, =

-g-& ;zX+l)+(~-+ r 0 I, [

which gives, upon acting on a two-body

1

t? + ti”

>I

(4)

,

isospin

coupled

state 1TMT),

(5) Thus the two-body

matrix

elements

([ab ]A,TjV&cWj

of Vc are given by = ([abILl] V& IICd]J’l)Sr,lG&.l

,

(6)

where V& (= e2/4mor) is a scalar. In order to incorporate the finite charge extension of the protons we use a Coulomb potential modified at short distances. Basically, this modification may be deduced from electron-proton scattering experiments. The results of these experiments may be summarized in an empirical formula for the proton form factor, G,(q*) = (1 + &r;q2)-2 where

the proton

rms radius

has the value

2: 1 - $r;q* + * * ’ , rp =0.83 fm. The modified

(7) Coulomb

217

M.A. Jadid, H.A. Mavromatis 1 Coulomb contributions

interaction, factors,

which includes

the finite proton

size for both protons,

apart from isospin

is then,

V:(q)= G,(q2)4T;;q2 In the coordinate

representation

Gp(q2)=e(&,r, 47Wl

it can be written

VE (r) In our calculations (8), that yields

=

as 6),

f& [l + gF(r)l .

we use the approximation

g”(r)

+. . .) .

q

= _

given by the first two terms

So .

fr;

(10)

r

The two-body i.e.

matrix

elements

are still given by eq. (6) where however

e*

I+-+--

1

--y,

l

2

47~~ [ r

S(r) 2 r

in eq.

Vc + V&

(11)

I ’

It can be easily verified that in a harmonic oscillator basis the relative matrix elements of Vz needed to evaluate the two-body matrix elements are given by,

XL&m M’n(n,n’) f(n x

where b = G function.

is the

c

T(n

r=O

+-&(n’-r

-r -r

+

l)r(n’-

8&t,o

f(n

+;)f(n’+$)

3rb3

f(n

+ l)T(n’+

oscillator

d

size parameter

+i)f(r+

I+ 1)

r + l)T(r

+

1)

(12)

1) I ’ and

f

is the

3. Diagrammatic representation of the perturbation and computational procedure

usual

gamma

series

In two previous publications we have shown that using the Goldstone ,graph component technique one can perform perturbation calculations up to fourth order in perturbation theory with a great economy in the number of graphs needed to build up the perturbation series, especially third and fourth orders. Here we extend

MA.

218

Jadid, H.A. Mavromatis / Coulomb contributions

systematically the above technique to include the isospin non-scalar interaction into this scheme. The five types of Goldstone graph components needed to calculate the binding energy and the mass radius operator series up to fourth order in perturbation theory are given in refs. lV3).F ormally one still needs only these five types of components to build up the energy and radius series with the provision that the nuclear interaction V, is now replaced by the residual interaction V given by eq. (1). In practice, however, the representative graph components have to be calculated with a definite interaction at each vertex since V, is a scalar while VC is not an isospin scalar. Thus each first-order graph component in V will yield two graph components (one pure nuclear and one pure Coulomb) and each second-order graph component in V will yield four graph components (one pure nuclear, two mixed, and one pure Coulomb). All these graph components have been calculated separately for simplicity. Diagrammatically a concise way of representing these graph components Pr h, Pz “2 -_~---~-i

9otr(if

f,,(i)

Fig. 1. First-order linked graph components in V corresponding to type I(i) = Vi0 where cy= n (nuclear), C (Coulomb).

is shown in figs 1 to 4. Analytically the expressions for the various graph components are obtained by coupling the intermediate state Ii) to J = 0, since the residual interaction is a scalar in the angular momentum space, and to T which may be different from zero because the residual interaction may not be an isospin scalar. h,

pi Pz h2 pl h, hz ‘3

h,

S

J-q

P2

h

9

h2

p2

4

“I

P2

hz

c i

f2sp fi)

f,,,(i)

P

h

9

t,pfi)

Fig. 2. Second-order

f,,p(i)

P

h

Q,,$)

fsO:p (i)

frcp(i)

h

P

93,$ (if

P

P

h

5+3(i)

f,,pfi)

fsplg(i)

h

%q3(‘)

linked graph components in V corresponding to type II (i) = ~~Vi~/~“i over j # O), where n = n, C and B = n, C.

(summed

219

M.A. Jadid, H.A. Mavromatis / Coulomb contributions P

h

a,(l) Fig. 3. Linked

graph

component

corresponding

to type III(i) = 0,0, where

13is the radius

operator

‘).

Thus the J-parts of any two identical graph components with different interactions are identical and have the same structures given in refs. I**). On the other hand, the T-part is obtained as follows: (a) For the Ii) = Ilp- lh) intermediate state the coupling is (t&,)= where T = 0,1. (b) For the Ii) = ]2p-2h) intermediate state the coupling is [(t,,t,,)T1(th,trt,)rz]T, whereT = 0, 1,2.

I

bIdi 1 h

P

P

a,,(l) Fig. 4. First-order

The analytic obtained with In terms of to the various

bza (i) h

h

a*,(l)

linked

bjm(l)

P

P

a,,(l)

b,,(l)

P

h

h

a5ti (i)

graph components in V corresponding to types IV(I) = B,,V,,JE~, V/3,,/&,, (summed over i # 0) where (Y= n, C.

and V(i) =

expressions for this part of the various components can then be the help of eq. (6). These expressions are given in appendix A. these graph components one can write the linked graph contributions orders of the perturbation series up to 2Aw as follows:

(i) Energy series. (a) Up to first order El =E,

(corrected (0)

+lzo,

(1)

for c.m. motion) (1)

+_!?o, -

gl”

220

MA.

Jadid, H.A. Mavromatis 1 Coulomb contributions

(b) Second order Eai=f[(~f000)2+(fXUa(~)]2]/~g~. (c)

Third order

(ii) Radius series

(a) Zeroth order (corrected for c.m.)

(b) First order

(c) Second order

(d) Third order

where the summations over (Y,p, y and S extend over the nuclear and Coulomb indices n and C respectively, and the superscripts in E$’ and 6$’ represent the pth order in V. 4. Checking procedures Normally, for calculations that include and their computer programs, checking the calculational procedure. Here we checking procedures developed in refs. with Coulomb vertices.

considerable complexity in the expressions procedures become an essential part of make use of the R* and the v+ crR* rs7) and extend them for checking graphs

M.A.

By analogy

Jadid, H.A.

Mavromatis

/ Coulomb

contributions

221

with ‘)

define

R++Znm2

&/ZJ2

withZ=:A.

To summarize the checking procedures; if E?'(V,,VC)and Sb"'(V,,, VC)are the pth order energy and radius in V then up to 2Ftw it can be shown that: (i) For pure nuclear or Coulomb graphs

Eb"(R~,O)=Ea'(O,R~)=fhw, Eb2'(R:,0)=E;')(0,R;)= -$iw, E;3'(R;,0)=Eb3)(0,R;)=&hw, Ef'(R,2,0)=E:)(O,R::)= -&fzw, eb"(R~,0)=861'(0,R~)= -$$, s:"(R:,0)=8:2'(0,R:)=~, ,$j3'(R;,0)=8~3'(0,R;)= -g. (ii) For pure and mixed graphs

As an additional checking procedure these perturbation orders were also calculated independently from the energy and radius matrices (see sect. 5) and the results were found

to be identical.

5. Matrix methods In refs. lm3) matrix diagonalization and inversion were used to calculate the exact ground-state energy and radius series to all orders for the closed shell nuclei ‘He, 160 and 40Ca in the space of 0+2hw excitations where only Op-Oh, lp-lh and 2p-2h states contribute. In the presence of the Coulomb interaction two complications arise from the isospin coupling of the intermediate states: (i) More involved expressions for the various diagrams appearing in the energy and radius matrices. (ii) A larger number of intermediate states contribute.

222

M.A. Jadid, H.A. Mavromatis

f Coulomb contributions

h

P

0 V

L

Ql

-a (HkH,,I

+

Q12

Fig. 5. Diagrammatric representation

The matrix

dimensions

in this case increase

of the energy matrix.

to 13(1,2,10)

for 4He, 107(1,6,100)

for I60 and 471(1,10,460) for 4oCa; where the numbers to (Op-Oh), (lp-lh) and (2p-2h) subspaces, respectively. In our calculations we use graphical radius matrices in an oscillator basis. The symmetric fig. 5, where

energy

matrix

techniques

is represented

a = -2hOS,i’

the symmetric

mass radius

cally as shown in fig. 6, where

in setting

correspond

up the energy

diagrammatically

and

as shown

,

QK=QK~+QKC, Similarly

in brackets

operator

K=l,..., 12, matrix

is represented

diagrammati-

in

223

M.A. Jadid, H.A. Mavromatis f Coulomb contributions P

(t3)=0,,1 -t

I

\

h

0

Fig. 6. Diagrammatic

representation

of the mass radius

operator

matrix.

The analytic expressions for the various graphs arising in figs. 5 and 6 are given in appendix B. Using these matrices one can also obtain the various orders in perturbation theory for the energy and radius. This we have done and obtained results up to sixth order in V for the energy and up to fifth order in V for the radius.

6. Results, discussion

and conclusions

In the tables and figures below, b and the rms radius are in fm; the radius (i.e. mean square radius) results are in fm* and the energy results are in MeV. The subscripts n, C, m and t stand for the nuclear, Coulomb, mixed and total contributions, respectively. The total contribution in any order is the sum of the nuclear, Coulomb and the mixed (if any). The fifth and sixth energy orders in V and the fourth and fifth radius orders in V are obtained using the energy and radius matrices.

6.1.

FIRST-ORDER

To first order

ENERGY

the Coulomb

AND

RADIUS

contribution

RESULTS

reduces

the binding

energy

by 10% for

4He, 33% for I60 and 45% for 40Ca near the saturation values of b, which are 1.4, 1.7 and 1.8 fm for 4He, 160 and 40Ca, respectively, as can be seen in fig. 7. These rather large percentage contributions are partly due to the near cancellation of @ho’ -&iw) and Eb’,’ (tables 1, 3 and 5). Moreover the pure first-order Coulomb contribution incorporating the finite size of the protons varies slowly with b and does not change the shape of the energy versus b graphs. The effect of the finite size of the protons is twofold: Firstly it makes the first-order pure Coulomb energy less sensitive to b than for point-like protons since E$ varies like A/b -B/b3 (with A = 1.63, 33.72, 200.69 and B = 0.75, 5.78 and 22.68 for 4He, 160 and 40Ca, respectively) where the inverse cube term is the finite size contribution. Secondly

224

M.A. Jadid, H.A. Mavromatis / Coulomb contributions 15 1

0

1.6 I

1.7 1.6 19 I, 1

2 0 f

b(fm)

-2

Elt

-6

-10

_

/

lEln

4"

(a) 0

-20

-40

j

_

-60

bEli -

I

E’n ,6

0

(b)

0

-100

-200

-L -L

E,t

E In 40

Ca

Fig. 7. Binding energy corrected to first order as a function of b.

since B is positive it also cuts down the Coulomb contribution due to point-like protons by 20% for 4He, 6% for I60 and 3.5% for 40Ca (near the saturation values of b), respectively. The relative contribution of the finite size term to the point-like term (i.e. B/Ab2) becomes less important for heavier nuclei because only relative s-states contribute to B. To first order the Coulomb contribution increases the size of the nucleus smoothly as a function of b and does not change the shape of the relevant graphs (see fig. 9). In first order the relative importance of the Coulomb versus the nuclear contribution to the radius is very sensitive to b. The Coulomb contribution always increases the size of the nucleus whereas the nuclear contribution changes sign (see

M.A. Jadid, H.A. Mavromatis

1

TABLE

4He perturbation,

inversion

6

and diagonalization

225

/ Coulomb contributions

results for the energy

in the space of 0 + 2hw excitations 1.9

2.0

1.5

1.6

1.7

1.8

83.340

73.248

64.884

57.875

51.943

46.879 -53.769 0.719

-82.914 0.833

-74.052 0.804

-66.377 0.775

-59.332 0.746

-2.425 -0.001 -0.003

-1.853 -0.001 0.004

-1.509 -0.001 0.011

-1.325 -0.002 0.019

-1.223 -0.002 0.028

-1.203 -0.002 0.037

m

0.691 -0.000 -0.007

0.524 -0.000 -0.002

0.351 -0.000 0.003

0.177 -0.000 0.009

0.003 -0.000 0.013

-0.146 -0.000 0.018

(4, : EO m

-0.266 -0.000 0.003

-0.219 -0.000 0.004

-0.189 -0.000 0.004

-0.176 -0.000 0.005

-0.173 -0.000 0.006

-0.195 -0.000 0.008

0.129

0.102

0.072

0.039

0.001

- 0.030

-93.210 0.862

Eo

(3)

z

(5, a Eat ) p5’

a

)

-0.072

-0.057

-0.045

-0.037

-0.033

-0.037

E;;

-1.952

-1.499

-1.303

-1.291

-1.378

-1.550

E I””t ?

-1.925

-1.478

-1.289

-1.285

-1.383

-1.565

E d,agt ?

-1.851

-1.427

-1.239

-1.220

-1.287

-1.418

0t

;

,=2

“) Pure Coulomb contribution second order onward.

tables

zero.

2, 4 and 6). Near the saturation

is more important @b:’ is considerably

6.2.

is practically

SECOND-ORDER

b, Pure 2ho

values

contribution

of b the Coulomb

to all orders,

contribution

than the nuclear contribution 0::. For all three nuclei, smaller than the zeroth order near saturation.

ENERGY

AND

RADIUS

i.e. from

0rG

however,

RESULTS

Fig. 8 shows the various second-order contributions to the energy for 160 and 40Ca. The overall contribution of the Coulomb interaction is to shift the secondorder binding energy graph towards larger b. The pure second-order Coulomb energy Eb2 is negative for all values of b and for all three nuclei thus partly compensating for the relatively large positive Coulomb contribution in first order. The mixed term E$, however, changes sign and hence the relative contribution of these two terms (Eb’d and Eb2A) is very b-dependent. In fact near saturation the pure Coulomb contribution is bigger than the mixed contribution for 160 and 40Ca.

226

M.A. Jadid, H.A. Mavromatis / Coulomb contributions TABLE 2

4He perturbation,

inversion and diagonalization results for the radius operator in the space of 0 + 2hw excitations

b

1.5

1.6

1.7

1.8

1.9

2.0

Bb”’- 3h2,‘2A

2.531

2.880

3.251

3.645

4.061

4.500

s&l’ { :

0.043 0.006

-0.075 0.009

-0.203 0.013

-0.357 0.016

-0.526 0.021

-0.730 0.026

n (2)i C m

0.078

0.032

-0.020

-0.083

-0.155

-0.237

0.000

0.000

0.000

0.000

0.000

0.000

0.002

0.003

0.003

0.004

0.005

0.006

n f3h31 i c m

-0.039 0.000 0.001

-0.049 0.000 0.001

-0.061 0.000 0.001

-0.077 0.000 0.000

-0.095 0.000 0.000

-0.116 0.000 -0.000

0.024

0.021

0.016

0.009

-0.003

-0.117

60

oh;’ “) 0;:’ “)

-0.015

-0.017

-0.017

-0.017

-0.016

-0.016

f. 0;; r-1

0.099

-0.075

-0.268

-0.504

-0.768

-1.185

0 Ill”I “)

0.106

-0.069

-0.263

-0.500

-0.765

-1.074

@diaat

0.095

-0,063

-0.238

-0.444

-0.660

-0.892

b,

“) Pure Coulomb contribution is practically zero. b, Pure 2hw contribution to all orders, i.e. from first order onward.

Like in first order, the second-order pure Coulomb contribution is much less than the nuclear second-order contribution. Tables 2 and 4 show that the second-order pure Coulomb contribution to the radius is negligibly small but positive for all values of b for 4He and 160. The mixed second order contribution to the radius 8 h*Ais always positive and its relative importance compared to f&z is b-dependent.

6.3. HIGHER

ORDER

ENERGY

AND RADIUS

RESULTS

Tables 1 and 3 show that the pure Coulomb perturbation corrections to the energy beyond second order are always negative but very small and thus can be neglected. Moreover, although the mixed contributions in a certain order are more important than the pure Coulomb contribution in that order they are in turn small compared to the nuclear contribution in that order near saturation. Tables 2 and 4 show that the pure Coulomb perturbation corrections to the radius beyond second order are always positive but very small and thus can be neglected. Moreover, although the mixed contributions in a certain order are more important than the pure Coulomb contribution in that order, they are in turn small compared to the nuclear contribution in that order.

%

perturbation, b

inversion and diagonalization results for the energy in the space of 0 + 2hw 1.6

I.7

1.8

561.568

497.445

443,708

-685.023 20.764

-615.662 19.660

-553.265 18.656

-15.561 -0,124 -1.122

-9.641 -0.133 -0.490

4.552 -U.c)Ol -0.359 -4.974 -0,000 0.050

1.5 638.940

1.9

excitatians 2.0

398.231

359.404

-498,988 17.739

-447.614 16.902

-407.610 16.135

-7.532 -0.140 -0.049

-7.084 -0.147 0.332

-7.363 -0.153 0.628

-8.342 -0.158 0.919

4.137 -0.002 -0.229

3.401 -0.002 -0.108

2.535 -0.002 0.024

1,460 -0.002 0.150

0.397 -0.002 0,282

-3.520 -0.000 0.056

-2.519 -0.000 0.066

-1.915 -0.000 0.091

-1.524 -0.000 0.122

-1,507 -0.ooa 0.166

4.161

2.856

1.961

1.371

0.853

0.527

-4.248

-2.739

-1.760

-1.170

-0.730

-0.568

-17.622

-9.703

-6.678

-5.970

-6.556

-8.284

-15.477

-8.356

-5.833

-5.422

-6.244

-8.098

-11.722

-6.996

-5,100

-4.700

-5.111

-6.020

“1 Pure Coulomb contribution is practicalty zero. b, Pure 2ho contribution to all orders, i.e. from second order onward.

6.4. DIAGONALIZATION

AND INVERSION

ENERGY

AND RADIUS

RESULTS

4He and 160 dittgonalization and inversion results for the energy are listed in tables 1 and 3, and those for the radius in tables 2 and 4. The difference between the d~agona~izatio~ and inversion results in either case is due to the presence of unlinked graphs in the d~agona~~2ation resuft. The above fact can be explained easily a5 follows: Since inversion in the space of 0 f 2fiw excitations is equivalent to summing up the linked series to all orders ls3); while diagona~i2at~on is equivalent to summing up the Rayleigh-Schriidinger (RS) series to all orders; and since it is known ‘) that the all-order RS series in a restricted excitation space contains unlinked graphs that can be cancelled only if all higher excitations are considered; then this difference is due to these unlinked graphs. This fact is easily verified if we compare the linked perturbation result to sixth order for the energy and to fifth order for the radius with the corresponding inversion and diagonalization results; where it is clear that this finite-order linked series is closer to the inversion than to the diagonalization result. It is also interesting to compare the all-order inversion

228

M.A. Jadid, H.A. Mavromatis

/ Coulomb contributions

TABLE 4 160 perturbation,

b I?$‘-3b2/2A (1)

&

1nC

inversion

and diagonalization

results for the radius excitations

operator

in the space of 0 + 2hw

1.5

1.6

1.7

1.8

1.9

2.0

4.852

5.520

6.232

6.986

7.784

8.625

0.615 0.066

0.306 0.083

0.021 0.102

-0.312 0.124

-0.662 0.149

-1.100 0.176

0.232 0.002 0.024

0.167 0.002 0.027

0.097 0.003 0.031

0.002 0.004 0.035

-0.113 0.005 0.040

-0.256 0.006 0.042

-0.034 0.000 0.009

-0.050 0.000 0.011

-0.070 0.000 0.012

-0.102 0.000 0.013

-0.138 0.000 0.013

-0.188 0.000 0.011

0.110 -0.116

0.101 -0.101

0.088 -0.088

0.072

0.047

0.025

-0.081

-0.071

-0.071

0.908

0.544

0.196

-0.245

-0.730

-1.353

0.976

0.602

0.244

-0.205

-0.703

-1.331

0.547

0.395

0.169

-0.147

-0.450

-0.743

“) Pure Coulomb contribution is practically zero. b, Pure 2hw contribution to all orders, i.e. from first order

onward.

TABLE 5 40Ca perturbation b

1.5

results for the energy 1.6

in the space of 0 + 2ho 1.7

1.8

excitations 1.9

2.0

EbO’ - qho

2149.620

1928.865

1708.614

1524.042

1367.839

1234.474

Eo(1)i : C”)

-2336.265 127.076 133.796

-2129.601 119.897 125.434

-1936.767 113.439 118.055

-1762.895 107.608 111.497

-1591.167 102.322 105.628

-1455.075 97.512 100.347

-78.968 -1.496 -2.075 -12.849 -15.039

-35.274 -1.561 -2.075 -7.048 -8.050

-19.128 -1.616 -2.075 -3.052 -3.394

-14.248 -1.662 -2.075 0.426 0.534

-15.784 -1.702 -2.075 3.233 3.623

-21.899 -1.737 -2.075 5.933 6.536

;: EL” 1 C”) m m “) “) Contribution

with point-like

protons.

M.A. Jadid, H.A. Ma~ro~atis

40Ca perturbation b

results for the radius operator in the space of 0 +2ho excitations

1.5

t’b”‘-3b2/2A

229

/ Coulomb contributions

1.6

1.7

1.8

1.9

2.0

6.666

7.584

8.562

9.599

10.695

11.850

1.512 0.173 0.203

0.990 0.214 0.247

0.512 0.261 0.296

-0.069 0.314 0.351

-0.707 0.372 0.413

-1.511 0.441 0.482

“) Contribution with point-like protons. 25

f

(2)

,,/,‘/-$

0

-25

-5

(2)

Eat

(2) E Oil

-50

-10

~

-15

-75

a)

‘$

Fig. 8. Second-order

energy cor;ections as a function of b,

b(tm)

230

MA.

Jadid, HA.

Mavromatis

3oc

/ Coulomb contributions

3.s

275

2.50

2.25

2 75

16 a)

2.00 15

1

1.6

i

1.7

0

1

I

1.8

19 b(tm)

t

2.0

2 5( 15

I

I

I

t

t

1.6

1.7

18

1.9

2.0

b(fm)

Fig. 9. Rms radius up to zeroth order (0) and corrected to first order (1) as a function of b.

results with and without the Coulomb interaction. The difference between these results is just the contribution of the linked graphs involving the Coulomb interaction, i.e. pure Coulomb plus mixed contributions to all orders. For I60 the pure 2hw inversion results with the nuclear interaction are *) -5.548 MeV and 0.09 fm* for the energy and radius, respectively, for b = 1.7 fm. Thus the Coulomb interaction results in -0.3 MeV additional binding and a positive 0.15 fm’ contribution to the expectation value of the radius operator 8.

h4.A. Jadid, H.A. ~a~romatis j Coulomb c~~t~but~o~s

231

6.5. CONCLUSIONS

From the preceding discussion we arrive at the following conclusions: As a general feature, introducing the Coulomb interaction results in a nonnegligible overall contribution that reduces the binding energy and increases the rms radius as should be expected. Because of the smallness of the coupling constant of the Coulomb interaction its major contribution is in first order in VC for both energy and radius. In second-order perturbation calculations the pure Coulomb contribution to the energy is more important than the mixed contribution near the saturation value of b. The pure Coulomb contributions in higher orders (beyond second order) are small compared to the mixed terms which are in turn small compared to the pure nuclear contribution and hence both can be neglected to a good approximation. This suggests that in perturbation calculations if one includes higher excitations (e.g. 4hw, . . .) or considers open shell systems [e.g. the Nolen-Schiffer anomaly 9‘“)] one need not go beyond second order in the Coulomb interaction (without however neglecting the pure Coulomb as compared to the mixed contribution in this order). This effect of the finite size of the proton on the Coulomb contribution is non-negligible and reduces the results obtained with point-like protons to values compatible with the empirical calculations of refs. ‘,l”). Appendix A

The analytic expressions for the graph components foa (cy = n, C), fifla (i = 1, . . . ,7; a,p=n,C), goa(CYZ&C), gi,@(i=1,..*,5; ac,p=n,C), Uo, Uia(i=1,*.*,.5; a!= n, C) and bi,(i = 1,. . . ,4; (Y= n, C) are given below. Exchange graphs arising from particle (hole) exchanges at the same interaction vertex are accounted for by using antisymmetrized two-body matrix elements. The following notation is used throughout: &=&X+1, &zb) = (1+&)-r, e

N(ab) = [~(~*u~)A(~~~2)1

F’(dcd) = (&, + E”(d) = (E, -

&b

-

-Ed)-’ ,

E,

.&

112 ,

,

c$(kt) = (kkk -k/TO), K(abcdJT)

= ([“~lLTI va

IIC~l~.T~.mtisymm 9

M.A. Jadid, H.A. Maoromatis / Coulomb contributions

232

where b is the oscillator size parameter

and A is the nucleon number. for lp - 1h states

(p, h; T) (i)= (pl,

fdi)

~2,

hI, h2; I, Tl, T2, T)for 2p-2h

states

i i

’

= -~~~~,T,GroN(ph)V,(P1P2hlh2IT1),

f&i) = -~(lT)~T11S=~1N(ph)VC(P1P2hlh211)9 F lnp-flkph)

C

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

PSGP4

fd4

=

@I~T~&-oFI~~,

flap(i)

=

4U

f2d(i)

=

t-1)

F3ap(v, e)-

T)ST~I~T~IFI~~

-JiN(ph)

Tl+TYlmp

where (Yand p are not both n ,

,

6)

withpk*hk(k=1,2,3,4),TlwT2,

,

C

twh3Jk

F3nn(77,0)

,

H3n~(i)~~2(-l)Tz’1~(1T)CF3cn(l,e)S(e), e H3~“(i)‘~2(;(-l)~~+l~(lT)

CFX(V, rl

H~cc(~)‘~(~T)ST~~ST~~F~CC(~, f301p(i)

=

H3d

l&(v) ,

1))

G) + (-1) ‘“;Cip2+‘+T1H3aS(i(p1C,p2))

+(-I)

ihCih*+r+T~H3mp(i(hl f* h2))

+(-I)

ip,+i~+ih,+1hz+Tl+T2H3ap(i(plc*p2,

hlt,h2))

,

233

H4cn(i)~(b(1T)ST11~T21F4Cn, H~cc(~)~~~(~T)ST~I~T*,F~CC, ‘p,+‘“+r+T,Ef4nB(i(P1”P2))

f4aa(~)=hfs,,G)+(-l)

T’+T’+1f4up(i),

fsa&)

= (-1)

f6*&)

= -f4aaG)

cein(i)=&-o; gdi) g4uPW

9

with ~3 + h3, EI(ptp3htb)

+ Efpm)

~~G3”“(T~),

=‘&,&k3,,,U), =

withPktfPk(k=l,2),P3-,h3,I;~Trz,

-g3apG),

where ac and p are not both n ,

withp-h,pk++hk(k

= 1,2),

,

234

MA.

ox(i)

fadid, H.A. ~avrQrnatjs

/ Coulomb co~i~ib~~ions

= t~)3’24(~T)c,cc(1) .

The graph components a~, arm and bi, are obtained from the previous expressions by using the linear replacement method of ref. I), i.e.

ha 6) = f4nn(i) , baa(i)

=fscrnG),

b3uG)

=feinaG),

64u

0)

ao(Q

ala(i)

=f7nu

with Y,(ac) +X(ac)

6.1 9

=

go(i)

,

=

g1anG)

(9

,

with Y&PI) -,X(PPI)

,

with Y~(~~~)~X(~~*),

a2a

0)

=

gaan

a3a

(8

=

g1,a (9 ,

with Ydp&) + Xbh)

,

,

h(i)

= g2ncr(i),

with Y&h11 + X(phd ,

aY(i>

=gsncxO~

with Y”(~~~~)+Xhhd

I

Appendix

.

B

Analytic expressions for the energy and radius graphs appearing in the matrix entries of figs. 5 and 6.

Qh =gOn ti) ,

QL = fan

6)

,

Q3c=b(~T)&W3c, Q4or = -Q3,,

withp++h,p’t*h’,

M.A. U&(Tl)

=

Jadid, H.A.

Mavromatis

/ Coulomb

contributions

-Sh,h~S,~,,,.~i,‘)-l/zN(ph)Vo(P1P*P~h2~~1)

@a = (-1)T1+T2+1&,

US,

=

&,

@,

Q8C

=

6 T2T$8

235

,

withplt*hk(k

TI*T2,

= 1,2),$-h’,

8 pzp:8hlh:Shzh:N(ph)N(p’h’)Y,(plp;),

USC,

U9a~-Sj,,jh;G,,,~S,,,;Sh,,;N(ph)N(p’h’)Y,(h,h~), Q9n =

~T1T;STzT$7T’U9n,

Q9c =

aT,T;S9U9C,

ulo,(Tl)~Gh,h:Sh,h:N(Ph)N(P’h’)V,(P,P,P~P;lT,),

Qlon = ~TIT;~TzT;~TT’UIO,(Tl)

,

Q~oc=ST~~~T~~~T~~ST~~~(~T)~(~T')UIOC(~), QII,= Qloa, una(7)s

withp,Hhk,pI*h;(k=1,2),

-s,,,:sh,h:N(ph)N(p’h’)Jjj(-l)~hl

Tl@Tz,

‘% +I+‘+’ c k^ k

TI

T,

T

T;

T;

1I

T;t*T;,

236

MA.

Jadid, HA.

Maoromatis

X

/ Coulomb c~~trjbz~tions

(TjOO)gO)( T'jOOllO)

,

The above expressions represent the direct terms of the diagrams depicted in figs. 5 and 6. Particle and/or hole exchange graphs at different interaction vertices (if any) are implied thoughout.

RI = ao(4 , R,+R,=

2b2 A &i’ P

References 1) M.A. Jadid and H.A. Mavromatis, Nucl. Phys. A317 (1979) 399 2) H.A. Mavromatis and M.A. Jadid, Phys. Lett. 81B (1979) 273 3.) H.A. Mavramatis, Nucl. Phys. A257 (1976) 109 4) H.A. Mavromatis, Nucl. Phys. A295(1978) 269 5) E.A. Sanderson, J.P. Elliot, H.A. Mavromatis and B. Singh, Nucl. Phys. A219 (1974) 190 6) N. Auerbach, J. Hiifner, A. Kerman and C.M. Shakin, Rev. Mod. Phys. 44 (1972) 48 7) H.A. Mavromatis and B. Singh, Nucl. Phys. Al78 (1971) 30 8) M.A. Jadid, Ph.D. Thesis, Am. Univ. Beirut (1978) 9) A. Barroso, Nucl. Phys. A281 (1977) 267 10) S. Moszkowski, Phys. Rev. C2 (1970) 402 11) K.C. Tam, H. Miither, M. Sommermann, T.T.S. Kuo and A. Faessler, Nucl. Phys. A341 (1981) 412