Spin-dependence and coherent summations in two-nucleon transfer reactions

Nuclear Physics A218 (1974) 441 --460; ~ ) North-Holland Publishing Co., Amsterdam Not to be reproduced by photoprint or microfilmwithout written permissionfrom the publisher

SPIN-DEPENDENCE

AND COHERENT SUMMATIONS

IN TWO-NUCLEON

TRANSFER REACTIONS t

J. M. NELSON Department of Physics, University of Birmingham, Birmingham, England and W. R. FALK Cyclotron Laboratory, Department of Physics, University of Manitoba, Winnipeg, Canada Received 18 October 1973 Abstract: The differential cross sections of the 19F(p, 3He)170 and 19F(p, t)tTF reactions leading to the ground and first three excited states have been measured at an incident energy of 42.4 MeV. The polarization analysing powers of two of the (p, 3He) reactions have been measured at a higher energy of 49.5 MeV with a polarized incident beam. The experimental results have been analysed in terms of conventional DWBA theory, and the importance of interference terms arising from the proper coherent summations contained in the definition of the differential cross section and analysing power, have been investigated. It is found that the interference term arising from the coherent summation over the transferred spin S is significant and should not normally be neglected. NUCLEAR REACTIONS 19F(p, 3I~e), 19F(p, t); E = 42.4 MeV, measured a(0); XgF(p, 3He), E = 49.5 MeV, measured polarization analysing powers, polarized proton beam. Teflon targets.

1. Introduction I n a p r e v i o u s p a p e r 1), a study was m a d e o f the sensitivity o f the t w o - n u c l e o n t r a n s f e r r e a c t i o n t o the presence o f s p i n - o r b i t forces in the o p t i c a l potentials. T h e r e a c t i o n s were carefully c h o s e n so as t o avoid, as far as possible, the occurrence o f m u l t i p l e a n g u l a r m o m e n t u m transfers. This analysis is here e x t e n d e d to the general case w h e r e m u l t i p l e transfers o f all three a n g u l a r m o m e n t u m quantities J, L a n d S are allowed. T h e differential cross sections a n d p o l a r i z a t i o n analysing p o w e r s have been m e a s u r e d for the 19F(p, 3 H e ) t 7 0 r e a c t i o n l e a d i n g to the g r o u n d a n d first excited state o f 170. T h e differential cross sections for o t h e r (p, 3He) t r a n s i t i o n s in this r e a c t i o n have also b e e n m e a s u r e d t o g e t h e r with the c o r r e s p o n d i n g (p, t) transitions a n d these will be discussed fully in a l a t e r p a p e r d e a l i n g with shell m o d e l wave functions. The (p, t) reactions have u n i q u e values o f the a n g u l a r m o m e n t u m transfers a n d thus p r o v i d e a useful c o m p a r i s o n . T h e analysis o f t w o - n u c l e o n transfer r e a c t i o n s is generally c o m p l i c a t e d b y the existence o f m o r e t h a n one value for the t r i a d o f t r a n s f e r r e d a n g u l a r m o m e n t a t Work supported in part by the Atomic Energy Control Board of Canada. 441

442

J.M. NELSON AND W. R. FALK

(JLS). These are the total, orbital and spin angular momenta, respectively, of the transferred pair of nucleons. In (aHe, p) reactions, for example, two, three and more values of this triad are possible for a single transition. It is well known that the spectroscopic amplitudes relevant to a particular triad configuration of the transferred particles are required before a two-nucleon transfer calculation can be carried out. These amplitudes are obtained from the overlap of the wave functions of the initial and final nuclei. In some cases, the value of the amplitude for a particular triad may be small in comparison with that for another, and on this basis, the former triad configuration is ignored. In this way, the number of (JLS) triads may be reduced. The criterion for this simplification is the magnitude of the relevant spectroscopic amplitude, but since the nuclear wave functions, and hence the spectroscopic amplitudes, are themselves under test, this leads t o a simplification of the analysis which, by its circular nature, prevents a reliable examination of the wave functions from being made. The problem is essentially one of numerical inconvenience and concerns the method by which the various transition amplitudes connected with the (JLS) triads are summed. The differential cross section in the Born approximation for the transfer of two nucleons coupled so that the total, orbital and spin angular momenta are given by J, L and S, respectively, is given by dtr dr --

J

XtXXX ~ ' ~'s ~JLsl 2 ,

~s

~

where fl~Ls is the transition amplitude. The quantity ~ s is the spectroscopic amplitude relevant to the transfer of two nucleons in a configuration ?, =- n~ llj~ ; n212j2 coupled to J and S. Its formal definition and the method by which it is calculated, will be given in sect. 4. The numerical complication implied by the coherence of the summation of L and S is usually avoided by neglecting it. That is to say, the summation of L and S is taken incoherently, along with the summation of 3". The coherence of the sum, mation of 7 is crucial to the mechanism of the two-nucleon transfer reaction and cannot be removed, although the number of terms appearing in it may sometimes be reduced by approximations in defining the nuclear wave functions. The summation of L and S becomes incoherent only when the optical potentials do not contain spin-orbit terms. The existence of spin-orbit coupling in the proton channel, however, is well defined, the strength being about 6 MeV. The strength of the 3He (and triton) spin-orbit coupling is uncertain. Though non-zero, it is likely to be similar to, or less than the nucleon spin-orbit strength. From previous calculations 1) it would appear that as far as the summation of L is concerned, the magnitude of the interference term arising from the coherent summation is small in comparison to the direct terms. The significance of the interference term arising from the coherent summation of S has not been studied before. It is the purpose of this paper to discuss this point in the context of the 19F(p, 3He)l 7O reaction. Few approximations will be made in

TWO.NUCLEON TRANSFER

443

this analysis, and in particular, the simplifications described in the Opening paragraphs will not be made since a computer program is available which will carry out all the various summations, coherent and incoherent, as a normal operation. 2. Experimental methods 2.1. DIFFERENTIAL CROSS SECTIONS

Angular distributions for the 19F(p, t)tTF and 19F(p, 3I--[e)t70 reactions were measured using a 42.4 MeV momentum analysed beam from the University of Manitoba sector focussed cyclotron. Two detector telescopes, each employing a 200 #m AE detector, a 3 mm lithium drifted E-detector and a veto counter, mounted in a 71 cm diameter scattering chamber, were used in collecting the data. Particle identification was performed in a conventional manner using Ortec particle identifier units. The identifier pulses were fed i n t o multi-level discriminator units which provided the routing information for the PDP-15 computer interface. Spectra corresponding to deuterons, tritons, helions and alpha particles were stored for each detector telescope. The target material consisted of FEP fluorocarbon film (manufactured by du Pont) of 12.7 /~m thickness. The fluorine surface density was 2.19 m g . cm -2 as determined from the chemical composition of the film. During the course of the experiments it was found that, even at currents as low as 10 nA, deterioration of )

tO00

2000

igF{ 0 3He)'r 0

*~F(p,I )'rF E~ : 4Z.4MeV

1800

900

£p : 42.4MeV 81013 : 45 °

8101D= 45 ° 8OO

1600

700

t400

I)

J

z

1200

~

tooo

~Z

800

8

600

=,55oot ~4ool '

o.o

' ,

(D

.

600

~1%~'~

400

20O 0

'"

I 200

300 CHANNEL

NUMBER

•I 400

0 I~ 200

I 300 CHANNEL

400 NUMBER

Fig. I. Triton and SHe spectra f r o m t h e 1 9 F ( p , t ) l T F and *gF(p, SHe)17Oreactions at 4 2 . 4 M e V i n c i d e n t proton energy.

444

J . M . NELSON AND W. R. F A L K 19F(p,3He ) 170

l~F(p.t ) 17F

Ep = 42.4 MeV

300

Ep=42.4

!

MeV

°*% ,.

100

lol

II

g.s. 25-"

I

30

-

1

100

4

g.s. E 2

30

~flll•

!

'

i

lo

t

r, t

t

3e

l

0.871

b }

I/1

1"

T

•

3[

P

0.50

!

~o

t

10

&

ft P t

f I

t

t

t~

II

tt

I

1o

10

t~

it~

3.06

~-

II i

I

t! 1

Id

ililll

3.10

h ft

t

t 30

lO

t

t t

I Ii!}I

ttt 'It

t

3.85 ~-

tt .It +tl

tt

t

t t

t t

t

3.86

tt

tt | 40

.

l

,

J 80

.

. | 100

ec.m.

•

.

80

o

~

i

100

Fig. 2. Differential cross sections for the 19F(p, 3I--Ie)t70 and 19F(p, t)lTF reactions leading to the ground and first three excited states of the residual nuclei.

the film resulted. Thus the beam current was generally not permitted to exceed 10 nA, and the targets were changed frequently at intervals of several hours. All the targets were taken from the same sheet of fluorocarbon film. To provide a relative normalization between the different measurements, elastic scattering events were recorded with a N a I scintillation counter at an angle of 37.5 ° to the incident beam direction. A series of measurements to check the consistency of this normalization was then made by keeping one detector fixed at 75 ° . Comparison of the intensity of several prominent deuteron groups from both the 12C(p, d ) " C and ' 9F(p, d ) ' S F reactions with the recorded elastic scattering events revealed general consistency

TWO-NUCLEON

TRANSFER

445

within the statistical uncertainties except in a few cases. The sensitivity of the elastic scattering to slight changes in the beam position on the target readily explains the remaining discrepancies. The uncertainty in the relative normalizations between measurements was less than 5 ~. Energy spectra for tritons and helions recorded at a laboratory angle of 45 ° are shown in fig. 1. The angular distributions corresponding to the ground and first three excited states in each case are shown in fig. 2. The errors represent statistical uncertainties only. 2.2. P O L A R I Z A T I O N

ANALYSING

POWERS

These measurements were made at the slightly higher incident energy of 49.5 MeV with the polarized beam from the proton linear accelerator at the Rutherford High Energy Laboratory. The general techniques of asymmetry measurements of this kind have been described in detail elsewhere 2), and the particular experimental methods in refs. 1, 3). In brief, the reaction products were detected by two symmetrically placed counter telescopes, each consisting of three semi-conductor detectors. The first two were used in a particle identification circuit 4) and the third provided a signal to veto events where the reaction product did not come to rest in the second detector. The target was a 12.7 pm Teflon film. The polarization analysing power can be calculated from the equations E-1 PAy

=

E+I' L+R E, z

--

L-R

+ ,

where P is the polarization of the incident beam, and L +, R + a n d L - , R- are the 3He yields in left and right telescopes for spin-up and spin-down beams, respectively. The polarization of the incident beam was perpendicular to the reaction plane and a F (p,3He) 170 Ep = 49.5 MeV QS.

0.8"

0.871 MeV

g.s. ,

O J,'

]

o .E

g •~ -OJ,.

20

!

0.;.

[II If ~b

B0

ec.m"

2,0 ot]

-O.l,

Fig. 3. Polarization a n a l y s i n g powers for the XgF(p, 3H¢)tTO reaction leading to the g r o u n d a n d first excited state o f xTO.

446

J.M. NELSON AND W; R. FALK

could be inverted at the ion source. The beam intensity was monitored by counting protons elastically scattered from a gold foil placed just downstream from the scattering chamber. Two detectors were placed symmetrically on either side of the beam so that the beam position as well as the intensity were monitored. The incident charge was also collected in a Faraday cup and measured with a current integrator. The beam polarization was continuously monitored with a sampling polarimeter situated at the exit of the accelerator, The beam polarization was approximately 0.55, and the time averaged beam current about 70 pA at the target. Extensive measurements could not be made on account of the low beam current. The polarization analysing powers for the two transitions are shown in fig. 3. The errors shown are statistical and do not include the small uncertainty in the beam polarization. The angular acceptance in the reaction plane was approximately 3° .

3. Optical model analysis Elastic scattering data for the reactions p + 19F, 3He + 170 and t + 17F are required for a DWBA analysis of the (p, 3He) and (p, t) reactions. These data are not available and in their stead, proton elastic scattering data o n Z°Ne were considered 5) along with 3He scattering data on 160 [ref. 6)]; both sets of data were taken at approximately the relevant channel energies. The 3He potential was used for the triton channel. In the analyses of these data, spin-dependent potentials were not used and since such potentials are of importance in this analysis, the scattering data were re-analysed with a spin-orbit term included in the optical potential. Unfortunately no polarization data were available so the definition of the spin-orbit strength cannot be firmly established. The method of analysis will be outlined in a little detail since, by its use, a very rapid convergence was achieved t o a systematic set of potentials. The optical model code R A R O M P due to Pyle was used for the calculations 7). The existence of ambiguities i n the real central potential is very well known, especially in the case o f 3He particles s). The criterion for selecting a particular potential on the grounds o f its depth alone seems arbitrary in the light of recent developments in the understanding of both the optical model 9) and the systematics of these ambiguities 10). In this analysis, the volume integral per pair of interacting particles will form the basis of acceptance of a particular potential. It has been demonstrated for the case of nucleon-nucleus scattering that the volume integral should be approximately 400 to 450 MeV • fm 3 if the potential is to be related in a sensible manner to the nucleon-nucleon potential 9). Cage et aL ~o) in an investigation of the discrete and continuous ambiguities in the real central potential have made use of this concept to advantage in a discussion of potential families for p, d, 3He and 4He projectiles. The volume integral is defined as j = _ __1

IU(r)dr,

Ai ATd

TWO-NUCLEON TRANSFER

447

where U(r) is the real central potential and Ai and A T a r e the masses of the incident projectile and target nucleus respectively. For the Woods-Saxon form of U(r), U is given by Vorg [ 1 + ( n a o ~21

J =

\;77Tv J '

(1)

where Vo is the depth of the potential, r o and ao the radius and diffuseness parameters, respectively. In addition to the volume integral criterion, the rms radius of the real central potential was allowed to vary in a controlled manner. Greenlees et al. 9) show that for the case of nucleon-nucleus scattering, this radius is very simply related to the rms radius of the matter distribution. For a Woods-Saxon form of the potential, the rms radius is given approximately by (RE) ~r

/

F L1+

7 (rca0 ~2qt~r'

(2)

troA I J/

where it is related to the rms matter radius ( R ~ ) ~ by
+

(3)

where (r~) is the ms radius of the spin and isospin independent parts of the twobody potential. It is given the value 2.25 fm 2. It is clear that, from eq. (2), the geometrical parameters defining the real central potential cannot vary in an arbitrary fashion as is customary in many optical model analyses, if the physical significance of eq. (3) is to be considered. Although no relationship similar to eq. (3) has been shown to exist for the case of complex projectiles, nevertheless it would be reasonable to expect that the rms radius of the 3He real central potential should at least be constant for a given target nucleus. Eqs. (1) and (2) were used to determine starting parameters for Vo, ro and ao which provided an initial potential with the accepted value of the volume integral J together with a realistic rms radius of the potential. The radius parameter ro was not varied during the search and changes in r o were only made in deciding the starting values. The diffuseness parameter ao was allowed to vary in the last iteration of the search. The imaginary potential and its form factor were freely varied during the search. The strength of the spin-orbit potential was fixed at 0 and 5 MeV to obtain a series of spin-dependent and spin-independent potentials. The optical potential is defined as follows d V(r) = Vc(r~)- Vo f(ro , ao) - i Wv f ( r v , av) + 4i WoaD dr f ( r ° ' aD) a

+ where

f(rr~, aw) = {t +exp [(r-rwA~)/aw]}-1,

- Vs 1" a dr f ( r s , as), r

448

J. M. N E L S O N A N D W. R. F A L K

and Vc is the Coulomb potential due to a uniform sphere of charge of radius r c A ~, Vo is the real central potential strength, Wv is the volume absorption potential strength, WD is the surface absorption potential strength and Vs is the spin-orbit potential strength. The symbol ~ denotes the Pauli spin opzrator. In the proton scattering analysis, tVD was set to zero while for the 3He analysis Wv was set to zero since repeated calculations showed that consistently better fits were obtained with a surface absorption term rather than with a volume absorption term. The technique of applying eqs. (1) and (2), together with (3) in the case of proton scattering, leads to a rapid convergence of the search and allows a survey of Z2 in

20

Ne20 + p

E = 41.8 MeV

i

15 X~

1o

.

~

J-~400

(a)

3'.,-

~.s

3T6

3".?

3'.8

31m

~To


15

x' Io

(b)

1.1

1.2

1~

1.4 #rn

Fig. 4. (a) The variation o f Z 2 with the rms radius o f the real central potential for the family o f potentials with volume integral J .~ 400 M e V . fm 3. The point F I corresponds to the original potential o f Falk et aL 5). (b) The variation o f %2 with the radius parameter ro. The rms radius o f the potential is fixed. The results are for the J ~ 400 M e V . fm 3 family.

TWO-NUCLEON TRANSFER

449

the parameter space to be made with comparative ease. The goodness of fit parameter Z2 is defined as ~2 1 ~ (O'th--~_©xp~2, = Ni=l \

cSexv /

and it is plotted in fig. 4a against the rms radius of the proton real central potential. The rms radius of the matter distribution as calculated from the rms potential radius is 3.4 fm for 2°Ne. The fitting procedure showed a strong preference for the J ~ 400 MeV. fm 3 family of potentials, so much so that it did not prove possible to force the search to produce a family of potentials with a markedly different value of J. The variation of Z2 with the radius parameter ro, for a fixed rms radius of the potential is shown in fig. 4b. Several potentials corresponding to a rms radius of the potential in the region of 3.8 fm are presented in table 1. The potential to be used in the DWBA analysis is marked N1 and the original potential of Falk et al. 5) is labelled F1. TABLE 1 Proton optical model parameters (Ep = 42.6 MeV) Label

V a)

ro

ao

Wv

rv

av

Vs

FI d)

36.63 43.22 40.12 33.53 29.24

1.197 1.0 1.1 1.2 1.25

0.746 0.856 0.806 0.788 0.791

11.31 11.03 9.81 9.95 11.95

1.196 1.028 1.335 1.236 0.895

0.786 1.098 0.716 0.756 1.031

0 6.54 7.18 4.98 4.50

N1

a) b) c) d)

rs

1.29 1.1 1.2 1.25

as

rc

ja)

R c)

Z2

0.783 0.799 0.751 0.719

1.2 1.2 1.2 1.2 1.2

393.8 357.4 381.8 380.0 364.8

3.74 3.81 3.78 3.87 3.94

5.3 2.0 1.7 2.9 3.6

All energies are in MeV and lengths are in fm. The volume integral J is in M e V . f m 3. R denotes the rms radius of the real central potential. In addition, WD = 0.18 MeV, ro = 1.196 fm and at> = 0.786 fro. Quoted in ref. 5).

The 3He scattering data from 160 were re-analysed following the same method as outlined above for the neon case. The variation of ~2 with the rms radius of the potential is shown in fig. 5a. Potential families, grouped by the volume integral J, fall into easily defined sets lying on separate curves. Such a diagram, together with the choice of the J ,~ 400 MeV. fm 3 family, allows one some confidence in selecting potentials for the DWBA analysis. Potentials with a r m s radius in the region of 3.5 fm are listed in table 2 together with the potential SG which was the result of the original analysis of Sen Gupta et al. 6). The potential B is one used in an earlier analysis of the 19F(p, 3He)170 reaction and was also derived from the data of refs. 6, 11). The variation of ~2 with the radius parameter ro, shown in fig. 5b, does not show a clear preference for ro as do the proton potentials and several potentials with different values of ro were used in the DWBA analysis. The spinorbit strength is poorly defined in the absence of polarization data but the strengths

450

J . M . NELSON AND W. R. F A L K

ISO+ 3He

20

sG/

E = 29 MeV

/

15

X2

Q

10

lal

33

312

3T~

L3

10

Ls

3T5

3:7

38


X2 5

(b) 1.1

1.2

1.3

I./.

r, fm Fig. 5. (a) The variation o f Zz with the rms radius o f the real central potential for SHe scattering on 60. The two curves correspond to potential families with volume integrals J" m 400 MeV" fm 3 and J ~ 550 MeV. fm a. The point SG corresponds to the original potential of Sen Gupta et aL 6). (b) The variation o f Z2 with the radius parameter ro o f the real central potential from the J m 400 MeV. fm 3 family. TABLE 2 aHe optical model parameters (E3He = 28.9 MeV) Label

V a)

re

SG a) Jl J2 J3 J4 JS1 JS2 JS3 JS4 Bt)

64.8 140.0 125.83 112.9 108.0 139.1 124.5 111.2 106.4 170.0

1.6 1.0 1.1 1.2 1'.25 1.0 1.1 1.2 1.25 1.03

") b) ~) d) ~) r)

ao

0.58 =) 0.782 0.735 0.696 0.662 0.782 0.735 0.696 0.662 0.89

We

rD

65.6 1.6 13.06 1.702 12.27 1.666 11.70 1.603 11.63 1.517 13.06 1.706 12.27 1.666 11.70 1.603 11.63 1.517 15.0=) 2.06

aD

Its

0.58 0.562 0.595 0.635 0.679 0.595 0.635 0.679 0.765 0.51

0 0 0 0 0 5.19 5.45 5.32 4.32 0

All energies are in MeV and lengths in fm. The volume integral J is in MeV" fm 3. R denotes the rms radius of the real central potential. Potential quoted in ref. 6). Volume absorption. Potential quoted in ref. 12).

rs

1.0 1.1 1.2 1.25

as

rc

1.6 1.25 1.25 1.25 1.25 0.977 1 . 2 5 0.930 1 . 2 5 0.981 1 . 2 5 1.221 1 . 2 5 1.25

J ~)

R =)

Z2

443.8 380.6 394.5 412.8 420.5 378.2 390.5 406.6 414.5 560.3

3.79 3.50 3.47 3.49 3.47 3.50 3.48 3.49 3.46 3.86

19.5 3.4 3.8 4.4 5.2 1.5 1.5 1.5 1.4 62.1

TWO-NUCLEON TRANSFER

451

that were obtained are compatible with those derived from the analysis of polarization data in 3He scattering ~2). The optical model analysis illustrates the necessity of examining in a logical way, the overall behaviour of the real central potential. This is particularly true for the 3He analysis.

4. Spectroscopic amplitudes For the pick-up reaction A(a, b)B, the spectroscopic amplitude ~9~js is defined as

= FL2 c A*

q~ r

[~*JnTa(¢B)~j*j,j2JT(r 1

\IJATA

K

d, JATA [r-

X . f f M A N A ~ L ~ B :, /*1 ~

r2)d~B d r 1 d r 2

The initial and final nuclear wave functions were obtained from a shell model calculation which used the two-body matrix elements of Arima et al. for the s-d~ shell as input data t3). The shell model code used for this calculation was that of Towner and Davies ~4). The final nuclear wave function was taken simply as a single nucleon outside the closed ~60 core. The wave functions obtained are shown in table 3 together with the spectroscopic amplitudes calculated from them. The various possible (JLS) triads are also shown. This choice of wave function does not permit an overlap between the positive parity ground state of ~ 9 F and the negative parity second and third excited states of ~70 and ~7F which are quite strongly excited in these reactions. TABLE 3 Spectroscopic amplitudes for t9F(p, aHe)tTO and ~9F(o, t)tTF J

L

S a)

ld{. z

s~.d~

2sk z

State in ~70:

~+ (g.s.)

½+ (0.871)

State in tTF: ~+(g.s.) ½+ (0.495)

2 2 0 2 2 1 3 2 1 3 4 1 0 0 0 1 0 1 I 2 1

--0.416 --0.416 --0.58 0.41 0.41

2 0

0.416 --0.58

2 0

0 0

0.416

--0.648 --0.237 0.730 --0.699 0.699 0.699 --0.648

The nuclear wave functions used are b): 1 9 F = _0.39]d~35 --0.42[s½d~:(10)) +0.60[s½d~r2(01)) +0.55[s~35. XTF,.... 17Os.s. = 1.0Ida>. 17F15t ex~.,l~O;,t ,xc. ~ 1.012s-~. a) Angular momentum quantum numbers for the transferred pair. b) The figures in parentheses are (,IT) for the d 2 coupling.

--0.699

452

J'. M. N E L S O N A N D W. R. F A L K

This points to a weakness in the definition of the wave functions which can be overcome by more extensive shell model calculations in which the inert core is taken to be ~2C thus allowing the p½ shell to participate actively in the description of the wave functions. For the present, the interest in this reaction is to be limited to the problems of the reaction mechanism and the question of nuclear structure will be taken up at another time.

5. DWBA analysis The code of Nelson and Macefield 15) was used throughout. In similar calculations 1), it has been established that the finite range formalism provides the best basis for the description of the reaction compared to the possibilities of zero-range form factors as described by harmonic oscillator functions ~6), or a Woods-Saxon wave function expanded in an oscillator basis 17), or the simple form factor consisting of a product of Woods-Saxon wave functions is). Woods-Saxon single particle wave functions were used in the calculation of the form factor in the LEA approximation and the geometrical parameters of the appropriate potentials were chosen by means of eq. (2) to reflect an rms radius of the potential intermediate between those of the initial and final channel potentials. The depth of the potential was varied so that each single particle wave function shared equally the total binding energy of the transferred pair. A Gaussian form was chosen for the radial part of the interaction potential, that is ~

exp

i = 1, 2,

and the range parameter fl was given the value 1.69 fm. A Gaussian form was also used for the 3He wave function with a range parameter of 3.26 fm. The interaction potential, though of unknown strength, was taken to be spindependent. The formalism of Hardy and Towner 19) leads to a factor bsrD(S, T) which modifies the transition amplitude fl~Ls. Thus flJLS -'~

where

bsrD(S,

T)flJLS,

D(S, T) is given by FG

D(S, T) = ~,, (-)6U(½½~; FS)U(½~u~½;GT)Cr66r+G, 1, and where U(a, b, c, d; e,f) is normalised Racah coefficient. The quantity bsr is defined below. The spin and isospin of the transferred pair are denoted by S and T, respectively. The two coefficients C ol and C t o are related to the proportions of the force being used. Thus if the interaction potential Vp~is given by

Vp, = U(rp,)[W + BP~,- HP~,- MP~],

TWO-NUCLEON TRANSFER

453

where P ' , P~ and po, are the spin, isospin and space exchange operators, then C °1 = W + M + B + H

cl°=

= 1,

W-B-H+M.

The factor D(S, T) becomes, for the allowed values of S and T, D(0, 1) = 1 - 0 . 5 ( B + H ) , D(1, 0) = - 1 + 1.5(B+H). Two forces have been considered, a Wigner force with B = H = 0, i.e. a spinindependent force, and a Gillet force where B + H = 0.3 [ref. 20)]. The differential cross section now becomes da d--~

#.Pb

kb 2 S a + l

where bsr is a spectroscopic factor for the light particles and has the values bsr

fx/~(6 s , - 6 s . o ) = ~ 1" '

for for

(p, 3He) (p,t).

There are three important considerations of phase that must be dealt with before the calculations can proceed. Since the spectroscopic amplitudes are calculated separately but are part of the DWBA calculations, the phase conventions of the amplitude calculations must be the same as those used in the DWBA calculations. The complexity of the coherent summation contained in the above formula for the differential cross sections shows that it is especially important that a consistent phase convention be used throughout these two independent calculations. The phase conventions used are that the spherical harmonics are defined with phase i ~. The signs of all the two-body matrix elements were multiplied by (-)½(13+u-zt-z2) so that they conformed to this convention. In fact, this meant that only two matrix elements changed sign and these were (ddlVlsd) for J T = 21 and J T = 30. The results shown in table 3 include this phase convention. The angular momentum coupling convention defined by l + s = j was used throughout the shell model calculations and the form factor definition. Since the matrix elements do not depend on the phase of the single particle wave functions, this phase is undefined and both possibilities must be considered. The single particle wave functions used in the form factor calculations are always defined to be positive near the origin unless otherwise stated. The theoretical predictions are compared with (p, 3He) data in figs. 6 and 7, for a number of 3He optical potentials, both spin-dependent and spin-independent. All the 3He potentials were taken from the J ~ 400 MeV • fm a family. Changes in the proton potential resulted in comparatively insignificant alterations to the results. The (p, t) data are compared with DWBA calculations in fig. 8. Whereas the agree-

454

J. M. NELSON A N D W. R. F A L K

19F(p,3He)170

3He --..... ----

E = 42.4 MeV g.s.

0.871 MeV

~.

ld

z 0

potl. J1 J2 JS3 JS4

10:

~J lil u') u') 0 Pc o

•Ill|

pZ

,

b_

\.-.-i.x.

u__ 1( (:3

",

I

\

'\\',, i

20

i

t

~40

80

i

i

80

100

ec.m.

i

i

20

40

I

60

i

80

i

100

Fig. 6. A comparison of DWBA finite range calculations and data from the 19F(p, aHe)tTO reaction. Various aHe optical potentials are compared.

!gF(p 3He)170 Ep=49.5 M e V

0.B

0.4

t

0.871 MeV

g.S.

Ay ec'm' -0.l,

3He pott. J1 J2

JS3,JS&

Fig. 7. A comparison of finite range DWBA calculations and experimental polarization analysing powers from the 1OF(p, aHe)ITO reaction. SeVeral aHe potentials are compared.

TWO-NUCLEON TRANSFER

455

~9F( p, t ]~7 F

t

t

pott. JS3

Ep =/,2./, MeV

g.s.

0.5

MeV

6

.6 $

t

*

t

•

'

10 2

ol

o

t !

$

l

t

\

10

101

i

r

20

/.0

i

60

i

i

80

100

ec.m"

i

i

20

40

i

i

60

80

i

100

Fig. 8, A comparison of finite range D W B A calculations and data from the 19F(p, t)tTF reaction. The triton potential is 3,5;3.

19F(p 3He)170 Ep= 49.5 MeV

0.8

O.871 MeV

g.s. 0.L

Ay

I i [tt t t 20

, ec.m.

60

-0.4

~S

~L

Fig. 9. An illustration of the effect on D W B A calculations when the interference term arising from the coherent summation over S (ground state) or L (excited state) is omitted. The curves marked C and I are the results of coherent and incoherent calculations, respectively.

456

J . M . N E L S O N A N D W. R. F A L K

ment between the theoretical and experimental cross sections for the transitions is good, the experimental analysing powers for the ground state transition could not be matched by the theory. The discrepancies could not be removed by choosing different optical potentials, spin-orbit strengths or definitions of the form factor. It should be noted that the calculations are made for different incident energies in the case of the cross section and analysing power data. The consistency of the choice of 3He potentials is demonstrated by the fact that the results are very similar for a range of aHe potentials with well depth varying from 100 to 140 MeV. The critical results come from a study of the magnitude of the differential cross section and the shape of the analysing power distribution when the interference term arising from the coherent sum of L, or the sum of S, is deliberately omitted from the calculation. First, the coherent sum of L will be considered. It turns out that the magnitude of the interference term is small, the cross section being affected by less than 1 ~o when the interference term is neglected. The changes in the analysing power distribution are shown in fig. 9. These results are partly expected in view of the strengths of the individual (JLS) transitions. Other calculations have shown that even when the two competing L-components in the transitions amplitude are made to have approximately equal strengths by means of artificial spectroscopic amplitudes, the interference term is still very small. This result confirms the conclusions of an earlier analysis of the 1 6 0 ( p , 3He)t, N reaction 1). The interference term in the case of the sum of S is substantially larger. The shape of the differential cross section remains virtually unchanged when the interference term is omitted from the calculations but the magnitude is altered by 7 ~o when a Gillet force mixture is used for the interaction potential, and by about 11 9/0 when a spin-independent interaction potential is used. The effect on the analysing power distribution is shown in fig. 9. It can be seen from these results that the interference term arising from the coherent sum of S is considerably greater than that arising from the coherent L-summation. The effects on the magnitude of the cross section, though large, are not as large as the uncertainties in the theoretical predictions currently made with the distorted wave formalism. On the other hand, the effects on the analysing powers of the omission of the interference term are substantial. The choice of nuclear wave function can be examined by studying the ratio of experimental to theoretical cross sections for the two transitions. Since absolute cross sections cannot be predicted by two nucleon transfer theory, it is necessary to use the ground state ratio to determine the normalization constant for the DWBA calculations and then to apply this to the predictions for the first excited state. If the following ratio is used, then the normalization constant is not required: Ratio = (O'exP/O'th) . . . . . tate This ratio is shown in table 4 for (p, t) and (p, 3He) calculations performed with a number of 3He potentials taken from table 2. In each case the proper coherent

"I-We-NUCLEON T R A N S F E R

457

TABLE 4 Spectroscopic analysis ") aHe potential

Ratio b) (p, aHe)

JS1 JS2 JS3 JS3 JS4

0.53 0.47 0.52*

J1 J2 J3 ./4

0.69 0.76 0.51 0.48

Ratio ¢) (s-d)

Ratio b) (p, t)

0.17

0.69 0.59 0.49

0.18

0.45 0.65 0.55 0.48

0.64

0.44

1.46

") All calculations use the Gillet interaction (spin-dependent) except that marked with an asterisk which is with a Wigner interaction (spin-independent). b) The ratio is defined in the text and should be unity in the ideal case. c) In these calculations for the (p, aHe) reaction, the radial wave function used in the definition of the form factor is defined to be positive at infinity instead of near the origin.

summation of L and S was made where necessary as well as the incoherent sum over J. The interaction between the proton and the transferred particles is normally spindependent (Gillet mixture), but a comparison is also given for a spin-independent interaction (Wigner). In the ideal case, the ratio defined above should be unity. The results in table 4 are systematically less than unity, however. The consistency of the calculations is demonstrated by the agreement between the (p, 3He) and (p, t) results although, as the quantities in table 3 indicate, the (p, 3He) calculations are considerably more extensive and complex than the (p, t). It can be shown that this consistency will be lost if some of the (JLS) triads are omitted from the (p, 3He) calculation. The predictions made with the spin-dependent potentials (JSn)show that the results are largely independent of which 3He optical potential is used. The fourth column illustrates the effect of altering the phase convention in the definition of the single particle wave functions used in the calculation of the form factor. This phase is not often mentioned in analyses of two-nucleon transfer reactions and its importance may be overlooked. The following sum shows the configurations of two nucleons coupled to (JLS) = (220), and represents the summation of in the calculation of the differential cross section,

U(r) =

+0.41611d~)-0.6481ld~2s~.).

In the simple zero-range calculation, each term is represented by a product of Woods-Saxon wave functions, the radial part of which has hidden in it a dependence on the principal quantum number n. The wave function can either be defined to be positive at the origin, and thus behaves as ( - ) ~ - 1 asymptotically, or, defined

458

J. M. NELSON AND W. R. FALK

positive at infinity and thus having the sign ( - ) n - 1 near the origin. There is no reason to choose one definition over the other when spectroscopic amplitudes are calculated from wave functions generated from two-body matrix elements, which do not themselves depend on radial wave functions. Clearly, from the construction of the above summation, the s-d term involving wave functions with principal quantum numbers differing by 1, will have its sign altered if a different definition of the phase of the radial wave function is used. The coherent nature of the summation of = ( n l l l j l , r/z12J'2) implies that this must have a profound effect on the magnitude of the calculated cross section and possibly on the analysing power distribution as well. This is born out by the figures in column 3 of table 4 where the single particle wave functions have been defined positive at infinity instead of positive near the origin. From the fact that these results are even further from unity than the previous calculations, it can be supposed that the phase convention with the radial wave functions positive near the origin, is the appropriate one to choose. 6. Conclusions The magnitude of the interference term in the S-summation of the transition amplitudes has been shown to be considerably larger than that arising from Lsummation. Although the interference term affects the cross section predictions by only 10 ~ or so, spectroscopic information may be affected by up to 20 ~ or not at all since, in analyses of two nucleon transfer reactions, ratios of cross sections are considered. It is not possible to determine, a f o r t i o r i , the sign of the interference term for a given transition. The predictions of polarization analysing powers are clearly even more sensitive to the presence of the interference term than the cross section predictions. One must conclude that in analyses of two-nucleon transfer reactions where S = 0 and S = I transfers are possible for a given transition, the proper coherent summation of S should be made when calculating the differential cross section and polarization analysing power. This is of particular importance when polarization effects are being studied. The emphasis of the optical model analysis has been to remove ambiguities and arbitrary modifications of the optical potentials and the same considerations have been given to the construction of the form factor and the parameters used in the definition of the binding potential. The differential cross section and analysing powers have been calculated without making any simplifications regarding the complexity of the coherent summations. In a previous analysis of the differential cross sections of these two transitions, Cole et al. ~i) were unable to avoid making a number of simplifying assumptions and they concede that the results of their analysis may, therefore, be ambiguous. As can be seen from table 2, their 3He potential, labelled B, did not provide a good fit to the experimental scattering data. The ratio obtained in that analysis was 0.77 to b e compared with the ratio shown in table 4 of this analysis. Now that, in this analysis, no simplifying assumptions

TWO-NUCLEON TRANSFER

459

have been made, it still remains that theratios obtained are significantly different from unity and more consideration must be given to the nuclear wave functions since these are one of the two remaining aspects of this problem that have not been investigated. The nuclear wave functions were generated from two-body matrix elements pertaining to the s-d~ shell and must, by their nature, be a compromise for a particular nucleus. They do not account for the negative parity states which is a shortcoming, especially for 19F where the first excited state is only 109 keV above the ground state and has negative parity. These ratios should be examined again with wave functions which are more general .in their basis. Such an investigation is in progress for all the (p, t) and (p, 3He) transitions mentioned in sect. 2. The other aspect to this problem is one concerning the DWBA model itself. Recent studies have shown, for (d, p) and (p, d) reactions, that conventional DWBA theories can be improved 2~). Whether or not the conventional DWBA is the proper vehicle for analyses of two-nucleon transfer reactions will not be considered. It is very widely used at the present time, and the analysis that has been presented forms part of an attempt to remove arbitrary and ambiguous potentials as well as simplifying assumptions from the analysis of two nucleon transfer reactions within the framework of the DWBA so that the underlying uncertainties in the nuclear wave functions can be more confidently examined. The role of multistep processes in analyses of this type has been shown to be important 22). These processes can involve inelastic scattering which either precedes or follows the transfer. The two-nucleon transfer itself can also be considered to take place in more than one step. The possibility that the discrepancies between theory and experiment shown in this analysis may be resolved were these processes to be included explicitly is now being investigated. One of us (J.M.N.) would like to thank Professor K. G. Standing for the hospitality of the University of Manitoba Cyclotron Laboratory during a visit. References 1) J. M. Nelson, N. S. Chant and P. S. Fisher, Nucl. Phys. A156 (1970) 406 2) R. C. Hanna, Proc. Int. Symp. on polarization phenomena of nucleons (Birkhauser Verlag, Basel, 1966) p. 280 3) N. S. Chant and J. M. Nelson, Nucl. Phys. A U 7 (1968) 385 4) P. S. Fisher and D. K. Scott, Nucl. Instr. 49 (1967) 301 5) W. R. Falk, R. J. Kidney, P. Kulisic and G K. Tandon, Nucl. Phys. A157 (1970) 241 6) H. M. Sen Gupta, J. Rotblat, P. E. Hodgson and J. B. A. England, Nucl. Phys. 38 (1962) 361; J. B. A. England, private communication 7) G. J. Pyle, private communication 8) P. E. Hodgson, Adv. in Phys. 17 (1968) 563 9) G. W. Greenlees, G. J. Pyle and Y. C. Tang, Phys. Rev. 171 (1968) 1115 10) M. E. Cage, A. J. Cole and G. J. Pyle, Nucl. Phys. A201 (1973) 418 11) R. K. Cole, R. Dittman, H. S. Sandhu and C. N. Waddell, Nucl. Phys. A91 (1967) 665 12) W. S. McEver, T. B. Clegg, J. M. Joyce, E. J. Ludwig and R. L. Walter, Phys. Rev. Lett. 24 (1970) 1123

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J. M. NELSON A N D W. R. F A L K

13) A. Arlma, S, Cohen, R. D. Lawson and M. H. MacFarlane, Nucl. Phys. AI08 (1968) 94 14) I. S. Towner and W. G. Davies, Oxford University Nuclear Physics Theoretical Group report no. 43 (1969) unpublished 15) J. M. Nelson and B. E. F. Macefield, Oxford University Nuclear Physics Laboratory report 18/69 (1969) unpublished 16) N. K. Glendenning, Phys. Rev. 137 (1965) BI02 17) R. M. Drisko and F. Rybicki, Phys. Rev. Lett. 16 (1966) 275 18) J. R. Rook and D. Mitra, Nucl. Phys. 51 (1964) 96 19) J, C. Hardy and I. S. Towner, Phys. Lett. 25B (1967) 98 20) V. Gillet and N. Vinh-Mau, Nucl. Phys. 54 (1964) 321 21) J. D. Harvey and R. C. Johnson, Phys. Rev. C3 (1971) 636 22) G. R. Satchler, Prec. Int. Conf. on nuclear physics, Munich 1973, vol. 2, to be published