Generalized optical model for elastic and quasi-elastic scattering of protons

Nuclear Physics A103 (1967) 513--524; ~

North-Holland Publishing Co., Amsterdam

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

GENERALIZED OPTICAL MODEL FOR ELASTIC AND QUASI-ELASTIC S C A T T E R I N G OF P R O T O N S Kh. MIJLLER

Central Institute for Nuclear Research Rossendorf near Dresden, DDR Received 19 June 1967

Abstract: The isospin dependence of the nuclear optical potential is studied using projection formalism. Exchange effects are taken into account. Instead of Lane's phenomenological potential V0+ Vzt" T, we find 1Io+Vl(2t3T3--t" T). The difference leads to a change of the relative sign between the wave functions p and n, describing the motion of the proton and a neutron, respectively. An explicit relation between the functions p, n and the full wave function is given. General expressions for the potentials Vo and V1are derived. They may serve as starting point for microscopic theorY~ 1. Introduction

,.

The success in describing quasi-elastic scattering of protons 1), reduced widths in direct reactions 2), isobaric-analogue resonances in elastic proton scattering 3) and other processes by the phenomenol0gical Lane equation' raises the problem of a microscopic understanding of this equation a n d , particularly, the meaning of the functions p, n and of the potentials involved. It was considered by Weidenmiiller 4) in a shell-model approach. We shall treat the problem in a more general way using Feshbach's method 5). For simplicity and clearness, we Consider here only the special case of target nuclei with closed proton shells and N neutrons above the same closed neutron shells. Furthermore we assume that the core of closed shells is not excited during scattering and that antisymmetrization between the outer nucleons and the nucleons of the core may be neglected. The antisymmetric ground state wave function of the target may then be written ~b ( 1 2 . . . N) and the wave function of the full problem q J ( 1 2 . . . N, 0). The index i = 1, 2 . . . N denotes space- and spin variables of the neutrons above closed shells, the index 0 belongs to the proton. We shall not use the isospin formalism. The crucial point of any optical-model treatment of scattering is the explicit definition of projection operators. We shall demand the following properties: ( i ) P~P ~ target ground s t a t e + p r o t o n , (ii)

Q~U .,~ analogue state+neutron, 513

514

Kh. MiJLLER

(iii) P, Q and therefore R = 1 - ( P + Q )

idempotent and orthogonal: p 2 = p, QZ = Q, R 2 = R, P Q = QP = P R = R P = Q R = R Q = 0. The function Rqj contains all other channels and also that part of the neutron channels not contained in QqL The operator P projects out that part of the state vector, which asymptotically describes an elastic scattering, whereas Q gives that part of the state vector, which asymptotically describes a quasi-elastic scattering. It should be emphasized that Q F does not contain all the possibilities for an outgoing neutron, but only those, in which the neutron left the nucleus in a state which is the analogue of the target ground state. This state is a superposition of many eigenstates of the Hamiltonian of the final nucleus 6). The operator Q, defined from quasi-elastic scattering will then be used also in the case of pure elastic scattering, i.e., if the neutron channels are closed, etc. (iv) P ~ and QqJ antisymmetric in all neutrons. The four properties give no unique definition of the operators under consideration, because in the interaction region they may be chosen in many different ways. From the point of view of an optical-model description of scattering it will be natural to define them to give the motion of an independent particle, the other particles being in the ground state or analogue state, respectively. This means, we extend requirements (i) and (ii) to the whole space of particle 0 or i, respectively. The subspaces P and Q therefore contain direct processes.

2. Construction of the projection operators To find the explicit form of the projection operators, we have to define some integral operators. The projection on ~ we denote by (ffl - f d l . . .

ON,b*,

(1)

d

where the integration runs over all variables contained in ~. Furthermore we need the integral operator ~ , which acting on a functionf(x) defines a new function F ( x ) F(x) = ~ f ( x )

= fOe f(x;

y)f(y).

(2)

The kernel 5U(x; y) is given by &~(x;y)-fdl...di-ldi+l...dNv(1...x...N)~*(1...y...N).

(3)

d

It is independent of the index i and related to the single-particle density matrix. We shall use the symbols i, j, k . . . for neutrons only; the symbols x, y may stand for neutron and proton, they replace in (3) the variable i.

OPTICAL MODEL

515

We shall now assume that the ground-state function ~k may be represented by a finite number of shell-model configurations. (In fact, this is no serious restriction of generality.) Then there exists a finite integer n . . . . so that all operators ~ " with integer n > nmax are linearly dependent on the JU n with 1 < n < nmax. The most general operator we may write down in terms of oU looks then like nmax

X = 1+ ~, ~,oU".

(4)

n=l

Now, it is possible to choose the ~, in such a way that

~x

= xx=

0.

(5)

The operator X becomes then idempotent nmax

X 2 = X + 2 ~n ~ r " X = X.

(6)

n=l

We can write down now the projection operators for the channels P, Q and R P - q~X(~O[,

(7a)

N

Q -- Y (Po,~)x<(Poiq,)l,

(7b)

i=1

R -- 1 - ( P + Q).

(7c)

They fulfill all conditions (i)-(iv). The Poi in (7b) substitute in ~k(1 . .. N) instead of i the index O. Orthogonality of PQ, for instance, may be shown in the following way: /g

PQ = ~kX(~bl ~ (Poi4,)x((Po,~,)l i=1

= ~bXi

dl...

= 0X

di

f

dNqJ*(PoiO)X(PoiO[

dl...di-ldi+l...dN(Poi~b)*X(P0i0i

N

= ~OX ~ ,~'S(~'lPio = O. i=1

The functions contained in P ~ and Q7j, respectively,

x ( o I V > - p(O), x ( ( P 0 , q 0 l ~ ' ) -=

n(i)

\/N'

(8a) (8b)

516

Kh. M/.JLLER

may be interpreted simply. For ro ~ 0% the contribution to p(0) from the integral operators in X vanishes, so that we have p(0) = S "

<0l~>.

(9)

ro--~ ~

Asymptotically p(0) therefore describes the motion of the proton, the N neutrons remaining in the target ground state. For finite values of 0 exchange terms are contained in p(0) which are expressed by the integral operators in X. An analogous discussion shows that n(i) describes, for rl --+ 0% a neutron which left the nucleus in its isobaric analogue state; for finite values of i the function n(i) takes into account exchange effects. 3. The coupled equations for n and p We shall now derive equations for p(0) and n(i). Using the properties (iii) of P, Q and R, we find from H~ = E~, (10) the two coupled equations ( P [ H + O ] P - E)P~P = - P [ H + O ]Q~P,

(1 la)

(Q[H-t- O ] Q - E)Qg-' = - Q[H + O ]P~,

(1 lb)

where O denotes the "generalized optical operator" 1

O-

HRE(+)

RHRRH.

(12)

The optical operator may be calculated by diagonalizing H in the subspace R. We write E (+) = E+ivl because R may contain open channels. Clearly, if we neglect in H the Coulomb part Hc then O becomes symmetric in all particles because of the symmetry of R and H - H c = H s. We can therefore split up O into a symmetric part 0 5, which does not change if H c --* 0, and an asymmetric part Oc, which vanishes if H c ~ 0. Here H c contains the Coulomb potential and the effect of the protonneutron mass difference. For the operator H ' = H + O, symmetric in the N variables 1 . . . N, we have to calculate now the four projections

P H ' P ~ = ~bXH't(O)p(O), Q H ' P ~u =

. (Po, O)X

P H ' Q T = x/NOX

"

d0~l(,,

(13a) 0)p(0),

" ' dt3/C2(On(i),

Q H ' Q T = ~ - Z(Po~O)X H i ( i ) n ( i ) - ( N - 1 )

'ST

(13b) (13c) dO3£Z;(i;O)n(O) .

(13d)

517

OPTICAL MODEL

The "weighted" operators are given by n~(o) = (~IH'I~>,

(14a)

H;(i) = ((Po,~k)lH- 'l(Po,~)),-

(14b)

JFa(z,O)=

'-

~(o, !

"

0 =

f dl...di-ldi+l...dN(Po,O)*H'O, .i

dl . . d i. - l .d ~.+ i .

t .. o~30, o) = ~j dl . . d .i - . l d. i +. 1

(14c) (14d)

dNO*W(Po,O),

dN(P, oO)*(Pkon')o,

k # i.

(14e)

Putting (13) into eqs. (11) and acting on (lla) with ($1 and on ( l l b ) with (P~o¢'l we find two coupled equations for p and n (xn'l(O) - E)p(O) =

- x/NX

fdl (O;

0~(0,

(XH'2(i) - E ) n ( i ) - ( N - 1) X f dO 3 ~ ( i ; O)n(O) = - ~/N X f dO ~ ; ( i ; O)p(O). In these equations the denotion of variables is meaningless, we may put therefore in the first one instead of 0 the index i. If similar as in (2) we introduce symbols for the integral operators, we get (XH'~(i)- E)p(i) = " ~ / N X ~ ' 2

n(i),

(Xn'2(i)-E)n(i)-(N- 1)X3¢~ n(i) = - ~/'-N X ~ l p (' i ) ,

(15a) (15b)

where

H;(i) -- P,oH'~(O) = ((Po,~P)I(P,oH')IPo,~) and the kernel of ~

(14f)

in (15a)

. . . 3¢t~2(,; 0) =. Pio3~2(0 , i) =

1 . .. d i -. l d i. + l .

dN(Po,~k)*(P, oH')~k.

(14g)

In (15) occur two local and three non-local operators. The differences in each of these two groups are caused by Coulomb effects only. This may be seen from (14). If we split up H ' into H' = H~ +O~, where H 's = Hs + Os is in all coordinates symmetric, we find H~l(i) = ((Po,~')l(Pion~)lPo,¢/) = ((Po,~)ln'~lPo,~,> = n~2 - H~,

(16a)

518

Kh. MtJLLER !

°.

~g~s3(/, 0) = f d l . . .

di - 1 di + 1 . . . dN(Po, ~k)*(PkoH~)ff

f d l . . d i. - l .d i +. l . !

..

o) =

dl...

di-ldi+l..,

dl...

di-ldi+l

dN(Po,~) *Hsff . =. ~ l.( z ; . 0) . - ~'~s(l," 0), (16b) dN(Po, O) (P, oHs)~k . . . dN(Poi~k)*H£~b = 3rt°sl(1, 0) - ~s(~, 0). (16c)

f

¢

-.

! ..

If we try to get Lane's equations from eqs. (15), we have to handle the Coulomb effects in an appropriate way. For this purpose we consider separately the contributions of H c and Oc to the operators in eqs. (15). The Coulomb operator Hc depends only on the coordinate 0 and consists of the two parts U'(0) [the Coulomb potential] and 6 [the neutron-proton mass difference: 6 = - (m.-mp)]. Putting this into (14) we find (17a)

Hcl(i ) = U'(i)+6, Hc2(i) = ((Po,$)lHciPo,¢')

!

= Ac,

~¢~c~P(i) = f dO 5/~'c,(i; 0)p(0) = jT-Hcp(i),

= fdo

.(i; o).(o)

= Hcg~'n(i),

d i - l d i + 1 . . . dN~b(Po~¢)*Uc n(0).

(17b) (lVc) (17d) (17e)

Here A~ is the Coulomb shift of the analogue state according to its definition as expectation value of Hc. We assume (i) that the Coulomb potential Hc(0) is nearly constant inside the nucleus and near the surface. Since the function ¢ vanishes exponentially outside the surface, we can then in (17e) approximately put

Jt~c3 n(i) ~ const. 5~n(i).

(17e')

(ii) The asymmetric part Oc of the optical operator does not depend upon the coordinates 1 . .. N, and the dependence on the variable 0 is very weak inside the nucleus and in the surface region. Then we find in analogy to Hc the relations Oc1(i) = Oc '

(18a)

Ocz(i) = d,

(18b)

OPTICAL MODEL

d 0 ¢ c 1 ( i ; 0)p(0) --

519

: Ocp(O,

(lSc)

ffd0 •c2(i; 0)n(0) = Oc dg'n(i),

(18d)

fd0/~c3(i; 0)n(0) ~ const. X'n(i).

(18e)

By d we have designated the expectation value of the asymmetric part of the optical operator in the state ~. Since on the right-hand sides of (15) and in the last term of the left-hand side of (15b) and at the top of the n-function the X-operator is contained, we get in these terms because of (17c-e'), (18c--e) and property (5) no contributions from H c and Oc, i.e. all three terms are built up with the same integral operator

-½V1 =- X ~ :

= X f dO(~s(i; 0)+~s(i; 0)).

(19)

The operators XH~(i) and XH~(i) from the left-hand sides of (15) may be simplified in the following way:

XH;(i) = XI-I:(i)+ XH'c,(O = XH',(O+ X(V'(i)+Oc(O+~ ),

(20a)

XH'2(i) = XH;(i) + Xn~2(i) = XH;(i) + X(A'c + a).

(20b)

The operators act only on functions, containing at the top an X-operator. Therefore, the X-operator in the last term of (20b) may be replaced by 1. If we use assumptions (i) and (ii), the contributions of the integral terms of X to the last term in (20a) vanish so that only the part proportional to 1 remains. Eqs. (20) then are

XH~(i) ~ XH;(i)+ U'(i)+Oc(i)+3 -- XH~(i)+ V(i)+~5,

(21a)

XH'2(i) = XH's(i) + A'c + d - XH's(i) + Ac.

(21b)

Here we have combined the Coulomb potential U'(i) and the asymmetric part of the optical operator to give an effective Coulomb potential U(i). The shifts At and d give together the effective Coulomb shift. Eqs. (15) may now be written in the following form:

(XH:(i) + U(i) + 6 - E)p(i) = ~/½To V1 n(i), (XH'~(i)+ac-E)n(i)+½(2To-1)Vln(i)

= x/½ToVlp(i).

(22a) (22b)

Here we used the usual notation N = 2To. For a comparison of our equations with Lane's phenomenological equation, we rewrite (22) as follows:

([XH~(i) + ½To V t ] - ½To V1 + U(i) + (~- E)p(i) = x/½To V1 n(i),

(23a)

(EXH's(i)+½ToV1]+½(To-1)Vl+dc-E)n(i)

(23b)

= x/½ToVtp(i ).

520

Kh. MULLER

4. The connection with Lane's coupled equations • Eqs. (22) have the well-known form of Lane's coupled equations but with one essential difference: The coupling terms show the opposite sign compared with Lane's equations• The functions nL and PL determined from Lane's equations, differ therefore from n, p from (22) in the relative phase, i.e. if

PL = P,

then

nL = --n.

(24)

On the other hand, the functions PL and n L underlie the same definition as the p, n. This may be seen from the form of the phenomenological wave function used in Lane's theory ~' ~ ~'roToPL(0)~(0)+ ~ o T o - 1 nL(0)V(0). (25) Here ~z and v are the wave functions of the proton and neutron states of the nucleon, respectively. In this representation the index 0 is not related to the proton. If we insert in (25) explicit expressions for ~IToToand ~IToTo_1, we get ~ ~(1... N)v(1)...

v(N)pL(O)rc(O)q-~b(1...N ) - 1

,/N

x (v(1)...

v(N-1)~(N)+

. . . + zt(1)v(2)..,

v(N))nL(O)v(O).

(26)

If from (26) we go over to the representation without isospin formalism, we find ~ ~ ( 1 . . . N)pL(0)+ ~ / ~

(Poi~k)nL(i).

(27)

Here again 0 is related to the proton. Comparing (27) with P ~ and Q~U we see that PL and n L equal p and n. This result contradicts conclusion (24). The origin of the contradiction seems to be the form of the isospin dependence of Lane's phenomenological equation. He used only a dependence on t • T, therefore assuming, that the problem under consideration possesses invariance under any rotation in iso-space. But in fact such a general invariance is not present. Indeed, among the possible rotations there are such rotations which change the neutron excess so that it becomes small or vanishes• It seems to be clear that in this case iso-spin is not a good quantum number and that also the spread of the analogue state over the states of the final nucleus becomes large. From this we expect that the model Hamiltonian should be invariant only under rotations around the third axis in isospace and under inversion of the third axis. An inversion changes the sign of the neutron excess, but the effects we have to deal with do not depend on this, at least in principle• The above invariance requirement is fulfilled by the operator t3T 3. From our argumentation we expect therefore that the phenomenological optical-model equation

OPTICAL MODEL

521

does not only contain a potential of type t . T b u t also a term t3T3 which destroys the high isobaric symmetry. Eq. (22) gives us the explicit form

Vl(2t3T3-- t. T).

(28)

In earlier work 7), it was shown that the optical potential should depend on symmetry number ( N , Z ) / A = ct and has the form W = Uo+aUl, where u 0 and u 1 do not depend on N or Z and the plus sign refers to the neutron and the minus sign to the proton. If Lane's potential Vo+(1/A)(t. T)vl is suitably averaged, the resulting potential has the form of W as was shown in ref. 1). The same holds also for our potential V0 + (2t3T3 - t , T) V1. The wave function of a nucleon and the target nucleus in its ground state with isospin T O = Mo = ½ ( N - Z ) may be written ~roUoX½m"Acting with the potential operator on this function, we get

(Vo+2mToV1)TJroMoX½m--(t • T)V, E(To½MomIT'M')TTo~T,~t,.

(29)

T'

In the state T' = T o + 3 the operator t . T has value ½To, in state T' = T o - ½ value - ½ ( T o + l ) . Therefore, in states T' = To+½ and T ' = T o - ½ the potential is given by VT,=To+~= Vo+(2mTo-½To)V 1 and VT'=To-~ = Vo+(2mTo+ ½(r o + 1)) V,. The states with T' = To+½ occur with weights (To½MomIT'M')2; the averaging procedure is defined 1) by

V = ~ (To½MomIT'M')2VT , .

(30)

T'

Then, for neutrons (m = ½), we get

V = Vo+¼a(AV,), and for protons (m = - 3 )

V = Vo-k~(AV,). The averaged potential V therefore also agrees in form with the potential W, which was used by several a u t h o r s .

5. Two-particle interaction The potentials Vo and V1 cannot be expressed in terms of the two-particle interaction completely. The reason is the complicated structure of the optical operator O. Nevertheless it is illustrative to calculate the direct contribution of H to the potentials Vo and V1. We use a two-particle interaction which contains "charge-exchange" forces. The symmetric part of H looks then N

Us = E T(i)+ T(O)+ ~ (V~k+ Wik(--P,k))+ E (V~o+ W/o(--P,o)). i=1

i
i

(31)

522

rda. MULLER

The operator (--Pik) has in our representation the same effect as the charge-exchange operator P~) which acts in iso-space, if isospin variables are used. Using definition (13a), we calculate first XH~(O)

SHs(O) = X(~I ~ T(i)+ ~ (V/k+ W~R(--P,k))I~k> i

i
+X(T(0)+(~b] ~ (V/o+ W~0(-Pio))]~)). (32) The first term contains the Hamiltonian for the ground state if, it may be replaced, therefore, by Xe, where ~ is the ground-state energy. Because all optical operators act on functions p and n, which have an X-operator at the top, we can drop the X at e [eq. (6)]. From the second term in (32), we get besides XT(O) a local potential V(0) x v ( o ) = xu(

(33)

lv/ol¢),

and a non-local potential f - XJdi

~(0;

i) = - X # ",

(34)

where the kernel ~ ( 0 ; i) is given by ~ ( 0 ; i) = N f d l . . .

d i - l d i + 1 . . . dN~k*W~o(Po,~b).

(35)

Collecting results we have

XHs(i) = 5+ XT(i)+ XV(i)- X'If'.

(36)

Now, we calculate X(d0~gt°s(i; 0) = X((Poi~b)]~., T(j)+ E (l/)k+ Wik(--PJk))[~) d

j

.i
+ X((PoI~k)]T(0)+ ~ (Vjo+ Wjo(-Pjo))[~b).

(37)

J

The first term vanishes, because of eXfdOoU(i, O) = eXoU = 0. From the second term, XJ{'T(O) vanishes. The remaining sum we split into X((Po, ~O)lV/oleO)- X((Po, ~b)lW~oP,olff) + X E ((Po,~0)lVjol~0)-X E ((Po,~)lW~oPjol~). (38) j~i

j~i

The first term gives 1

X~//.

N

1 X f d O ~e~(i; 0), N d

(39)

where in analogy to (35) we have 3¢~(i; 0) --- N f d i . . . di-ldi+ 1 . . . dN(Poi~b)*v/o~b. d

(40)

OPTICAL MODEL

523

The second term yields - X 1_ W ( i )

-

-

N

x((Pod,)l ~olPo#>,

(41)

which is similar to (33). The last two terms lead to the integral operators

X ( N - 1 ) I f dO f dl . . . d i - l di + l . . . dN(Po,~)*Vjo~, + fdOfdl...di-ldi+l...dN(Po,~b)*Wjo~,],

j # i. (42)

Defining the kernels

v(i; 0) - N f d l . . . d i - l d i + l . . , ~(i; 0) -

Nfdl...

di-ldi+l..,

dN(Po,~k)*Vjo~k,

j ~ i,

(43)

dN(P0,~k)*Wj0~k,

j ~ i,

(44)

we get instead of (42) the expression N-1 - N

(45)

X(u+w).

According to (19) we get for the operator 111 the expression

Vt = 2 X W ( i ) - 2 X ~ N

N

2(N-l)X(v+w)_2Xt~s" N

(46)

For the operator [XH'(i)+½ToV1] contained in our eqs. (23), we find [XH~(i)+½ To I/1] = ~ + (T(i)+ V(i)+½W(i))

+ E ~.~"'(T(i)+V(i)+½W(i)) - ( x ( ½ ~ + ~ ) + 3(2 7"o- 1)x(v +

+ X(Os(i)- To t~).

~)) (47)

The term e leads to a renormalization of the energy E. Because in the optical equations the core is excluded there remains only the energy ( E - e ) of the nucleon. The second term is the usual single-particle Hamiltonian. The third one may be related to exchange effects, whereas the fourth term contains non-local effects, which are caused not only by particle identity. The contribution of the optical operator is not specified here. In an approximation, which neglects exchange effects, all X-operators equal unity, the sum ~ , J 4 d " being zero.

524

Kh, M/.) LLER

For interesting discussions, I acknowledge Dr. J. Zimanyi and J. Revai from KFKI Budapest, Professor K. Fuchs and also the colleagues from the Nuclear Theory division in the ZfK Rossendorf, especially J. Slotta, K. Hehl and H. W. Barz. References 1) A. M. Lane, Nuclear Physics 35 (1962) 676 2) R. Stock and T. Tamura, Phys. Lett. 22 (1966) 304 3) D. Robson, Phys. Rev. 137 (1965) B535; J. P. Bondorf, S. J~igare and H. Liitken, Nuclear Physics, to be published 4) H. A. Weidenmiiller, Nuclear Physics 85 (1966) 241 5) H. Feshbach, Ann. of Phys. 5 (1958) 357, 19 (1962) 287 6) A. M. Lane and J. M. Soper, Phys. Rev. Lett. 7 (1961) 420; A. M. Lane, Nuclear Physics 37 (1962) 663 7) A. M. Lane, Revs. Mod. Phys. 29 (1957) 193; G. R. Satchler, Phys. Rev. 109 (1958) 429; A. E. S. Green and P. C. Sood, Phys. Rev. 111 (1958) 1147