Single-particle states in non-local potentials and direct nuclear reactions

I

2.F

I

Nuclear Physics A101 (1967) 577--588; (~) North-Holland Publishino Co., Amsterdam

I

Not to be rel~roduced by photoprint or microfilm without written permission from the publisher

SINGLE-PARTICLE STATES IN NON-LOCAL POTENTIALS AND DIRECT NUCLEAR REACTIONS H. SCHULZ, J. WIEBICKE and R. REIF Central Institute of Nuclear Research, Rossendorf near Dresden, DDR

Received 28 April 1967 Abstract: The non-local model of Percy and Buck is extended to the bound-state problem. For 180 and 4°Ca, the single-particle states in the non-local potential are compared with those in a local Woods-Saxon potential and with experiment. Different approximations for the non-local wave functions of bound and scattering states are tested and found to be very useful. The influence of the non-locality in zero-range DWBA calculations of the 160(d, p)IT"O and 180(p, p') reactions is investigated. For the stripping reaction the differential cross sections are compared with those obtained using a sharp radial cut-off.

1. Introduction

In m a n y investigations o f the nuclear m a n y - b o d y p r o b l e m , a set o f single-particle states in a self-consistent average potential is introduced. In general, this singleparticle potential is expected to be non-local 1). Treating a local two-particle interaction in the H a r t r e e - F o c k a p p r o x i m a t i o n , non-locality o f the H a r t r e e - F o c k potential is a l r e a d y caused b y the exchange effects resulting f r o m the a n t i s y m m e t r i z a t i o n a c c o r d i n g to the Pauli principle. F o r c o n t i n u u m states, a n o n - l o c a l - p o t e n t i a l m o d e l for the elastic scattering o f nucleons by nuclei was f o u n d to be very successful in explaining the differential cross section a n d p o l a r i z a t i o n o f neutrons 2) a n d p r o t o n s 3) over a wide range o f mass n u m b e r a n d energy. W e extend the Percy-Buck m o d e l to the b o u n d - s t a t e p r o b l e m . In sect. 2, we discuss the single-particle states a n d wave functions o f such a m o d e l solving the integrodifferential e q u a t i o n by means o f an iteration procedure. F o r the nuclei ~60 a n d 4°Ca, the results are c o m p a r e d with calculations in a local potential and with experiments. In sect. 3, different a p p r o x i m a t i o n s to the b o u n d state a n d scattering wave functions o f the n o n - l o c a l p o t e n t i a l are c o m p a r e d a n d tested in the distorted wave t h e o r y o f direct nuclear reactions (sect. 4). I n the investigations o f the 160(d, p ) t 7 0 reaction, the non-local effect is c o m p a r e d with the influence o f a cut-off in the radial integrals.

2. Single-particle states

The S c h r 6 d i n g e r e q u a t i o n for a particle (reduced mass m) in a b o u n d state ~b,,lj(r) 577

H. SCHULZet al.

578

(quantum numbers n, l, j with the usual meaning) with a non-local potential is

[(hZ/2m)V 2 + E,t j - U~.o.(r)]~.tj(r) = i~dr I/(r, r ' )O.ij(r ' ).

(1)

In the kernel V(r, r') = VN(½lr+r'i)7c-~/? -3 exp [--(r--r')2/fl2 3

(2)

introduced by Percy and Buck 2), the potential UN has a Woods-Saxon shape with the depth Vr~, the parameter fl characterizes the range of the non-locality. For the sake of simplicity, we use a local spin-orbit term (diffuseness as, R = ro A} and pion mass M,) Us.o.(r) = - V~,o.(h/2M =C)2(asr ) - ' exp [ ( r - R)/a,][1 + exp { ( r - R)/a~}](a" i) (3) and consider neutrons only. After a multipole expansion of the potential (2) with coefficients g~(r, r'), the equation for the radial part u,l~(r) (5 dru2(r) = 1) of the wave function q*,ti(r) takes the form [(h2/2m)(d2/dra-I(l+l)/r2)+e.zj

- Us.o(r)]u.~j(r)

=

fo

dr'gl(r, r')u,,j(r').

(4)

In solving eq. (4) we use a method which combines the iteration procedure of Percy and Buck for the solution of the integro-differential equation and the iteration procedure for determining the energy eigenvalue in a local potential given by Brown, Gunn and Gould 4). A programme has been written for calculating either the depth of the non-local potential if the energy eigenvalues E.~i and the spin-orbit coupling strength is given, or the bound states for known potential depths. In the latter case the iteration procedure

[(h2/2m)(d2/dr 2 - l(l + 1)/r 2) + E(~j) - Us.o.(r)- UEL(r, ~nljFt'n)]q'laUnlj(m) "~-

f 0 ~dr'gt(r, r tx)u.tj (m- ')(r')- UEL(r, 1u(m-1)-i, (m--l)( "~ ~nlj ]"~lj k']

(5)

starts from the energy-dependent, equivalent, local potential UEL(r) determined by the equation VN(r) = UEL(r) exp [(2m/hZ)¼flZ(E,,j-- V~L(r))]

(6)

and a convenient value E ~ ) for the energy neglecting the right-hand side of eq. (5). With this programme the neutron single-particle states in the doubly magic nuclei °O and *°Ca are studied in the following way: we set the binding energy of the level occupied by the last neutron in ~60 and 4°Ca equal to the ground state separation energy of a neutron. This level and the level belonging to the same orbital momentum l but other ./-value are used to calculate the non-local potential depth VN and the strength Vs.o. of the spin-orbit term. With these potential strengths, all other bound

NON-LOCAL POTENTIALS

579

states are determined. The parameters r o = 1.22 fm, as = 0.65 fm and fl = 0.85 fm are taken from the non-local model for the scattering of neutrons. They are chosen to be fixed. For 160 and 4°Ca, we obtained a non-local potential depth of VN = 71.50 MeV and 69.3 MeV, respectively. The results agree with the value of VN = 71 MeV for the non-local optical model (table 1). The spin-orbit strength (Vs.o. = 10.l MeV and 8,98 MeV, respectively) differs from Vs.o.= 7.2 MeV used by Perey and Buck only by small amounts. TABLE 1 C o m p a r i s o n o f the scattering p a r a m e t e r s o f n e u t r o n s with those used for calculating the n e u t r o n single-particle levels Local

V Vs.o. ro as fl

(MeV) (MeV) (fm) (fm) (fro)

Non-local

160

4oCa

160

aoCa

56.3 10.46

54.7 9.82

71.5 lO.l

69.3 8.98

1.22 0.65 0.85

1.22 0.65 0.85

Non-local scattering parameters of Perey a n d Buck z) 71 7.2 1.22 0.65 0.85

In figs. 1 and 2, the bound states of these potentials are compared with experimental "single-particle states" and with the level scheme for a usual local, WoodsSaxon potential. The parameters V and V~.o. of the local potential are determined in the same way as in the non-local case. For both nuclei, the level ordering is reproduced correctly with the local and non-local potentials. For 4°Ca there is the effect that in the non-local model the binding energy of the unfilled levels decreases. This was already stated by Koshel 5). It may be seen that the non-local model does not improve the agreement between the calculated and measured single-particle energies. The main deficiency of the calculations is that they do not reproduce the correct level distances. This circumstance is not surprising. From an analysis ~) of the singleparticle states in the Ni region, it is known that in the concept of the effective mass taken from nuclear-matter calculations an effective mass larger than 1 is necessary for lower bound states. But the non-local model used here in principle yields an effective mass smaller than 1 only. Although the energy dependence of the potential well for bound states is not reproduced correctly by the Perey-Buck model, it seems worthwhile to study the wave functions of states in such a non-local potential at single energies. The comparison of the local and non-local wave functions for the same binding energy establishes the well-known Perey effect; inside the nucleus the non-local wave function is systematically smaller (10-20 ~ ) than the local one. Thus according to the normalization the non-local wave function in the external region is raised compared to the local one. Recently similar work was reported by Krell 6).

H. SCHULZ et al.

580

Loccff

/Von-/occd

Observed

i

E~O 2s~-32e

3.

/ / lc/~-~IE 5.0 / / " ", --'--,," "\.

~

5.86

i i

J

r

2ZP~

1,0~'2-2/7.

21,74.

Fig. 1. N e u t r o n levels in 160, LoccA

Observed

No,'~-(oco/ E-O

,//,y

1,'+* , \ , ~

z~,2D / f

./

l

I~73

Id3/2G~3

/R3/

r

2Z63

I~23

/

lcX?..~ 2/.6?

P/.~25

Fig. 2, Neutron levels in a~'Ca.

NON-LOCAL POTENTIALS

581

3. Approximations According to Percy 7) and Fiedeldey s), in a local energy approximation the nonlocal radial wave function u~(r) in the potential (2) may he obtained from the solution UEc(r) in the equivalent local potential UEL(r) from (6) by multiplying with a damping f a c t o r f ( r ) and a normalization constant C

U~(V'(r) = Cfe(r)(r) UEL(r).

(7)

Somewhat different expressions for the function f ( r ) are given in refs. 7,8)

fe(r) = (1 - (2m/h 2)¼fi2UEL(r))- 4,

(8)

fv(r) = exp [(2m/he)~-fl2VEc(r)].

(9)

If for a given energy eigenvalue the non-local potential depth is adjusted and the equivalent local potential is calculated with eq. (6), the binding energy of this level in the equivalent local potential is somewhat different from that used to fix the nonlocal potential. Therefore the functions UN(r) and UEL(r) have different exponential behaviour in the outer region. To remove this deficiency and also to avoid the difficulties of solving the integro-differential equation, we use a non-local potential depth VN according to UtN")(r) = u(Emc)(r)exp [(2m/h,.2 )~fl , 2(E,, s - UEc (m)( r ))3,

[(h2/2m)(d2/dr 2 - l( l + 1)/r z) + E,a; - U~.o.(r) - trf",)r.aq, V E L k . ] A . n l (m) j

(10)

= O.

We start with a convenient value for VN (°) and iterate (10) until the binding energy in the equivalent local potential coincides with the given one. This local potential and the corresponding wave functions are used in eqs. (7)-(9) for UEL(r) and UEL(r) to approximate the exact solution. TABLE 2 Maximum relative deviations of the approximated, non-local wave function from the exact, non-local ones for single-particle levels in 160 and 4°Ca Approximation of Percy

Level

Ap (Vo)

~60

4oCa

Approximation of Fiedeldey

AE (%)

E B (MeV)

nlj

inside

outside

inside

outside

21.74 15.60 4.15 3.28

lpk lP½ ld) 2s½

--0.8 --0.6 --2.5 --1.9

1.9 1.9 1.5 2.0

--2.4 --1.6 --4.3 --4.3

4.2 3.8 1.9 2.8

21.63 18.53 15.73 8.36 6.41 4.41

ldff 2s~ ld~ lf~ 2p~ 2pff

--1,7 --0,8 --0,8 --3.0 --2.2 --2.4

3.1 3.1 2.5 2.6 3.0 3.4

--3.1 --2.5 --I.0 --4.8 --4.4 --4.9

5.6 5.6 5.0 4.1 4.4 4.6

H. SCHULZ et al.

582

In the lower part of fig. 3, the relative deviation Af,(F ) = (Cfp(F)UEL--UN)/U N is shown as a function of the radial coordinate for both approximations (8) and (9). In table 2, the maximum values of the quantity A inside and outside of the nuclear radius are given for the approximated non-local wave functions for all levels in 160

•

O~lm/z-/eve! non -local local

"

"

7

tO-

"

-

7 ,!7

7 70-

1

2

3

G

7 "" .£,\

........................

L-c=--=~'-'~-==,-~l= ~

-2} ........... -

5

8

%t 4'- ] 0

4,

4

~

6

R

4'

--

5"

~

e

1

7 .

9

!

rl/my

~

8

- Ap(r)

M

~--x

9

I

|

.... aF¢~> ,'[t~

1

Fig. 3. N o r m a l i z e d non-local a n d local radial wave functions (upper part) a n d relative deviations o f the a p p r o x i m a t e d , non-local wave f u n c t i o n s f r o m the exact, non-local one (lower part) for the lp½ level in 1nO.

TABLE 3

M a x i m u m relative deviations o f the equivalent, local (AEL) a n d the a p p r o x i m a t e d , non-local wave f u n c t i o n s (dp, AF) f r o m the exact, non-local one for the scattering o f n e u t r o n s (7, 14 a n d 21 MeV) by 6aCu Incident energy (MeV) 7

14

21

1

Approximation o f Perey Ap (%)

Approximation o f Fiedeldey A F (%)

0

--1.5

1

1.6

--4.4 2.8

0

1

--3.0

1

1

--2.8

2 3 4

1 --2 --3

2.5 --4.3 3.5

0

t

--2.1

1

1

--2.0

calculated in sect. 2. The deviation is about 3 % for the Perey factor and with the Fiedeldey factor never more than 6 %. These investigations have been repeated for the scattering states in the non-local model with parameters used by Perey and Buck for the elastic scattering of 7, 14 and and 4°Ca

NON-LOCAL POTENTIALS

583

21 MeV neutrons on 63Cu. The results for the real part of some partial waves are shown in table 3. In the case of scattering states, the maximum deviation is of the same magnitude as for bound states, independent from the incident energy and the angular momentum of the partial waves. For charged particles the equation for the equivalent local potential is modified by the inclusion of the Coulomb force 3). With this equivalent local potential and the corresponding Perey factor, the same approximations to the non-local wave functions for protons and deuterons have been carried out successfully. For scattering as well as for bound states, the replacement of the equivalent local potential from (6) by an expression given in ref. s) U~L(r) = VEL(r)--~fl2d2UEL(r)/dr2--~fi2r-'(d/dr)UEL(r)

--(fi2/2m)6~4fl2E(d/dr)UvL(r)]2[1 --(h2/2m)(lfl2)UEL(r)]-i

(11)

does not improve the approximations. We may conclude that both approximations investigated above work very well for positive and negative energies. The results with the Perey factor (8) are in general somewhat better than those with the Fiedeldey factor (9).

4. Non-locafity and direct reactions Non-local effects in the DWBA theory of direct nuclear reactions have been discussed by the Oak Ridge group 9). To investigate the influence of the non-locality and to test the approximations discussed above in such precise calculations, we study the stripping reaction ~60(d, p) ~70 (E~ = 11.8 MeV) to the ground (ld~) and first excited state (2s~) and inelastic proton scattering by asO(Ep = 13.04 MeV) to the first 2 + and 4 + states. In a DWBA code with spin-orbit coupling written by Hehl, Reif and Slotta, we inserted the nonlocal, approximated non-local or the local functions for the distorted waves and the form factor and compared the differential cross sections. The potential of the distorted waves apart from the Coulomb term (R = rcA ~) consists of a real Woods-Saxon potential with the depth V, the radius parameter rv and the diffuseness a v, a surface absorption term of the Woods-Saxon derivative type with the parameters WD, rwD, and awo, and a spin-orbit potential dependent on Vs.... r .... and a~.o.. The values of the parameters are included in table 4. The stripping reaction is investigated in zero-range approximation. The local parameters of the deuteron channel are taken from a optical-model analysis of Fitz et al. lo) and of the proton channel from Rosen et al. ~2). The parameters of the corresponding non-local potential are chosen to fit the measured differential elastic cross section. The value of the non-locality parameter for the deuteron channel (fl = 0.36 fro) is taken from elastic scattering investigations performed by the authors and for the proton channel (fl -- 0.90 fro) from ref. 3).

584

H. SCHULZ e t al.

The results of the calculations are compared with the experiments by SchmidtRohr et al. 14), especially the spectroscopic factor S da/d(2 = S/((2Ji + 1)(2sd + I))aDWBA(0)

(12)

is extracted (Ji denotes the spin of the target nucleus and s d the spin of the deuteron). 4

TABLE

Optical parameters for the reactions l~O(d, p)170 and ~80(p, p')asO* 160(d, p)170 incident channel

1sO(p, p')180"

final channel

non-

local V

(MeV)

W D (MeV) Vs.o. (MeV) rv (fro) rWD (fro) rs.o. (fm) e (fro) r a V (fro) aWD (fro) as.o. (fro) /3 (fro)

local

118.0

incident channel

local

local

163.5

49.5 (Q = 1.05) 82.0 49.2 (Q = 1.92) 6.57 7.5 12.5 6.0 5.5 5.5 0.85 1.25 1.15 1.44 1.25 1.15 0.93 1.25 1.25 1.3 1,25 1.25 0,79 0.65 0.65 0.78 0.7 0.7 0.79 0.65 0.65 0.36 0.9

5.95 6.0 0.93 1.58 0.93 1,3 0.79 0.78 0.79

final channel

non-

non-

local

local

46.48

76.13

9.0 6.0 1,25 1.25 1.25 1.25 0.6 0.46 0.6

14.92 6.0 1.16 1.16 1.25 1.25 0.6 0.46 0.6 0.9

non-

local 47.47 (Q = - 1.98) 48.26 ( Q = - 3 . 5 5 ) 9.0 6.0 1.25 1,25 1.25 1.25 0.6 0.46 0.6

local 76.95 77.76 14.8 6.0 1.15 1.15 1.25 1.25 0.6 0.46 0.6 0.9

E,.b/s,-] k

i

,o

Q=104~? MeV 2S /le

If::% '1% '.\ ~ jf

i

- 01

Local

- -

-

I. . . . . . . 50

zOO

Per~i~c[or non(occz(

momtocct(/prL'nfac!or lifo O~rm.

Fig. 4. Differential cross section for the reaction 160(d, pl)170 (Q ~ 1.048 MeV, 2s~) at 11.8 MeV; local: form factor and distorted waves local, non-local: form factor and distorted wave, non-local, Perey factor: f o r m factor and distorted wave approximated with the Perey factor, non-local form factor: form factor non-local, distorted wave local.

NON-LOCAL POTENTIALS

585

The angular distribution is altered little if local functions for the distorted waves and the form factor are replaced by non-local ones (fig. 4). In the non-local case, the oscillation amplitudes are somewhat stronger and the cross section for the forward angles is raised. For both transitions, the spectroscopic factor is less than the singleparticle value of unity. The main effect of non-locality is to reduce the absolute spectroscopic factors considerably (table 5). Moreover, we carried out DWBA calculations using non-local wave functions approximated with the Perey factor. The cross sections obtained with non-local and approximated, non-local wave functions differ by small amounts only (fig. 4). Thus we may avoid the complicated iteration procedure to solve the integro-differential equation for distorted waves and form factors in nonlocal potentials. Calculations with the local distorted waves and the non-local form factor have shown that the main reduction of the spectroscopic factors is produced by the nonlocality in the form factor. In our examples this reduction becomes about 23 °/o in agreement with Satchler's results 9) for 4°Ca(d, p). The inclusion of the non-locality in the distorted waves yields an additional reduction of the spectroscopic factors (about 7 ~ ) contrary to the tendency usually expected. TABLE 5

Spectroscopic factors for the reaction 160(d, p)~70

160(d, Po) ~60(d, Px)

Local

Non-local

Approximation of Perey

Non-local form factor only

Cut-off

0.65 0.47

0.45 0.34

0.45 0.34

0.50 0.36

0.53 (1.95 fro) 0.49 (1.8 fm)

In order to test in which way non-local effects may be simulated by an appropriately chosen reduction of the contributions from the nuclear in:erior to the D W B A matrix element, we performed calculations with local wave functions and a sharp radial cut-off. For both transitions, no cut-off.radius could be found between 1 fm and 3 fm which reproduces the angular distributions as well as the spectroscopic factors obtained in the non-local case. Especially a cut-off near the nuclear radius yields large deviations. The best results are given in fig. 5 and table 6. For the inelastic scattering process 180(p, p,)l sO, the optical potential parameters are given in table 4, and the experimental data are from Stevens et al. 13). The reaction is treated in a shell-model description using a (spin-independent) real local, two-particle effective interaction Vio = vog(rio ) between the incoming proton and the two neutrons above the 160 core. The form factor in the D W B A matrix element for a transition 0 ~ J is a linear combination of radial integrals

Ij,jOo (J) . ) = f

dr1 ui.(t~l)f(J)(rl, r0)ui(ri),

(13)

H. SCHULZet

586

L tO ~)"~""V"t;"('~,II,'.

~. -: ~. O=2sl/2ZO4~ghleV

i,,}

15:_

'Li

~otd.p~ " O taw/.a

o.,

/J

\:

I

O= I..919MoV zCZSl2

',,,

5

al.

';

,.

•

.

:-~ . . . .

50

; z ~-.-,._- -_~.~,{i - : - ~

I00

ISO ~crn-

Fig. 5. Differential cross section for the reactions 160(d, p)l~O (Q = 1.919 MeV, ld~) and (Q = 1.048 MeV, 2s+) at ll.8 MeV. See fig. 4; cut-off radius --1.95 fin (ld}), 1.8 fm (2s@). TABLE 6 Potential parameters and binding energies for the calculation of the form factor in the reaction asO(p, p')180' Parameters

1.25 fm a s - 0.6 fm Vs.o. = 0 MeV /3 = 0.85 fm

nlj

-- E B (MeV)

V(local) (MeV)

V(non-local) (MeV)

8.068 7.199

62.6 61.4

82 81.5

ro

ld~ 2s½

w i t h f ( a ) ( r 1, r0) as t h e J t h - o r d e r coefficient in a m u l t i p o l e e x p a n s i o n o f g(rlo). W e use a t w o - p a r t i c l e p o t e n t i a l o f G a u s s i a n s h a p e g(rlo ) = exp [ - (rio~b) 2 ] (b = 1.85 fro) a n d restrict o u r s e l v e s t o the m a i n p a r t o f t h e f o r m f a c t o r in a m o r e a c c u r a t e shellm o d e l d e s c r i p t i o n o f t h e 1SO(p, p ' ) r e a c t i o n i n c l u d i n g c o n f i g u r a t i o n m i x i n g w i t h o u t c o r e - e x c i t a t i o n ~~).

NON-LOCAL POTENTIALS

60

.....

local

- -

tvon-

587

local

.7-2

x~

SO

,.~ y-Sl2,d" li2

#0 30 A~

20

Z

10

2

Fig. 6. Form factors

4-

for 3

I(.{!

8 ral [rn]

6

2 and J

=

: :

2

¢-

6"

B

4 tra ns i t i ons in the reaction 1sO(p, p')lsO*.

J J

d~ [mb/sr] dJ? i

L

i

i

i

i

,

Q- f.gBMeV 3=2 L.

O(p.p')

'eO"

" ...." ~..... ~

-

\~

Ep : I 3 O ~ M e V ,

{~

I

~

.....

- nonloccd . - _ Perex-foc/or I .......... m o m - l o c o l I

6~= 3'55 ~'eV

i

ffTvb/srj

t

laced

I© ~":-::'':';''':"

.......

;

SO

: ~. .- .

i

I

I00

~..i--'--~

1

'

' l,fO Ocr~

Fig. 7. Differential cross section for J = 2 and J ~ 4 t r a n s i t i o n s in the r e a c t i o n 180(p, p')180* at Ev = 13.04 MeV.

588

H. SCHULZ et al.

If in eq. (13), non-local, bound-state wave functions (potential parameters see table 6) are inserted, the integrals I~2~ and I ~ are shifted to larger radii (fig. 6) and the differential peak cross section increases (by 25-30 ~ ) while the angular distribution remains nearly unchanged (fig. 7). In the maximum of 1 (2) and I (4), the different aspects of the Perey effect in the bound-state wave functions are tested. The multipole J = 2 feels the reduction in the interior (I (z) is reduced by ~ 3 ~ ) while the increase in the external region reflects itself in 1 (4) (1 (4) increases by ~ 6 ~ ) . Therefore in this case the main effect of using non-local wave functions in the form factor is to increase the ratio of the cross sections in the excitation of the J = 4 + and J = 2 + levels. Taking into account the non-locality of the optical potential in the Perey a p p r o x i m a tion, this effect is strengthened further and amounts to ~ 20 ~ .

5. Summary The non-local model of Perey and Buck extended to the bound states does not improve the agreement between the calculated and measured single-particle energies in doubly magic nuclei remarkably. The approximations for the nonqocal wave functions work quite well. In refined D W B A calculations of direct reactions, one can avoid the solution of the integro-differential equation for the bound states and distorted waves. The main effect of the non-locality in direct reactions is to increase the cross section and to change the relative excitations while the angular distribution remains nearly unchanged. With a sharp cut-off, it seems impossible to simulate the influence of the non-locality by the Perey and Buck model. The authors wish to thank Mr. H. W. Barz for many stimulating discussions and comments on this work.

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