Application of cluster model wave functions to the 12C(d, p)13C stripping reaction

i

1.D.3

1 !

Nuclear Physics A142 (1970) 87--99; ~ ) North-Holland Publishing Co., Amsterdam Not to be reproduced by photoprint or microfilm without written permission from the publisher

A P P L I C A T I O N OF CLUSTER M O D E L WAVE F U N C T I O N S T O T H E 12C(d, p)laC S T R I P P I N G REACTION M. EL-NADI and O. ZOHNI Department of Physics, Faculty of Science, Cairo University, Cairo, U.A.R. Received 29 September 1969 Abstract: The ~-cluster structure of 12C was studied by the reaction 12C(d,p)~aC (g.s.) at different

incident deuteron energies in the range 2-28 MeV. Plane wave calculations based on the cluster model description of the initial and final nuclei are compared with plane wave calculations using the shell-model wave functions. The low-energy results show a marked preference for the shell-model description which agrees with the theoretical calculations using Faddeev's theory. An estimate of the rms radius of ~2C using the cluster assumption is also made. 1. Introduction The cluster model was found to explain several properties of the low-lying states of several light nuclei 1) successfully. Attemps to treat 12C as a system of three ~-clusters have yielded rather contradictory results. Using a variational parameter approach and assuming an ~-cluster structure, Sheline and Wildermuth 2) obtained a good fit for the low-lying states of this nucleus. Treating 12C as composed of three rigid ~-particles, Darriulat 3) has recently used a modified variational method and found that the lowest three-~ bound state for 12C is the 7.656 MeV excited level. The Faddeev theory which provides an exact solution of the three-body problem has also been u s e d 4 - 7 ) . With a Yamagouchi potential for the s-wave ct-~ interaction, Harrington ~) obtained a single eigenvalue near the ground state, but no excited states were found. Duck 5), taking a separable form for the two-body t-matrix, found no bound state at all. Hebach and Henneberg 7), using separable potentials for the s-wave and d-wave ct-~ interactions, found that ground state of 12C cannot be described by the three-body model, whereas the 7.656 MeV 0 + state of 12C probably can be. This shows that the cluster structure of 12C is still an open question. One expects cluster structure to show itself in some nuclear reactions 8-12) such as the (d, p) reaction. In the present work the plane wave Born approximation 13) (PWBA) is used in the calculation of the transition amplitude of the reaction 12C(d, p)13C, assuming an ~-cluster structure for ~2C. It is assumed that the neutron of the incident deuteron may be captured by a one of the three ~-clusters constituting the ~2C nucleus. Further, in the present work only the cluster structure of the ~2C ground state will be considered. 87

88

M. EL-NADI AND O. ZOHNI

The results of the cluster-model calculations for the reaction 12C(d, p)'3C (g.s.) at different incident deuteron energies in the range 2-28 MeV are compared with the results obtained, using the shell-model wave functions (with no clustering) for the target and residual nuclei. At low energies, the angular distributions based on the cluster model assumption are found to differ greatly from those obtained using the shell-model picture, while the latter agrees well with the experimental data. This result agrees well with the findings of Duck ~), Hebach and Henneberg 7) discussed above. At higher incident energies, the angular distributions obtained become rather insensitive to the type of wave functions used, and the cluster model agree with those derived from the shell model. An investigation of the rms radius of 1zC nucleus under the ~-cluster assumption has also been carried out. Using parameters which fit high-energy data in the range considered, its value was found to agree with that obtained from electron scattering using a shell-model wave function 14). In sect. 2 the cluster-model wave functions are chosen and the different parameters involved are specified. The formulation of the problem is carried out in sect. 3 and the transition amplitude is then derived in sect. 4. Results and discussions are presented in sect. 5. 2. Choice of cluster-model wave functions

Assuming a three-c~ cluster model for the ~2C target nucleus, one may write for its cluster wave function 1) oT('2c) =

= cp(~l)~o(~2)~o(%)((R)g(p ).

(2.1)

Here ~(R) is the wave function of relative motion between the ~-clusters two and three of the 8Be nucleus, and Z(P) is the wave function of relative motion between the ~-cluster one and the c.m. of the two other clusters. The internal wave functions of the three ~-clusters are represented respectively, by ~p(~), q~(~2) and q~(~3). Let now consider the final 13C nucleus. We assume that as the deuteron approaches the target nucleus, its neutron will be captured by one of the a-clusters. Hence one may write for the cluster wave function of ~3C the expression t q~r('3C) = q(SHe)q~(SBe)~(r/)

= ~p(n)cp(cq)qg(~2)~p(%)f(p.)C(R)¢(~l).

(2.2)

Here, the neutron is represented by ~o(n), and f(p,), ~(R) and ~(~/) are, respectively wave functions of relative motion between the neutron and the cluster ~ , between the clusters c~2 and c%, and between the c.m. of the SHe nucleus and the c.m. of the clusters ~2 and ~3. A schematic diagram for the coordinates involved in the reaction 12C(d, p)~3C is shown in fig. 1. Evidently, it is assumed here that the SHe nucleus is t It is to be understood that angular momentum coupling is to be properly taken into account.

12c(d, p)13C STRIPPING REACTION

89

formed of an a-cluster plus an outside neutron; a representation suitable for the description of its ground state 2). Furthermore, it may be noted that no antisymmetrization between different nucleons in the wave functions (2.1) and (2.2) is taken into account. This antisymmetrization will lead to a multiplying factor 1,15) in the cross section. As we are not concerned in the present work with the absolute magnitudes of the

[l 2 9,

~.

t

/

et /

/

-'

n

£f

P

Fig. I. Schematic diagram for the coordinates involved in the reaction 12C(d, p)13C. cross sections, we shall neglect this antisymmetrization. On the other hand, the antisymmetrization between the incident deuteron and the target nucleus will lead to exchange effects which would cause a relative backward rise in the angular distribution, as shown for example by Henneberg's 15) calculations of the 4He(d, p)SHe reaction assuming a cluster structure for the final nucleus. In the present work, we are mainly concerned with the forward part of the angular distribution. Moreover the data considered are generally characterized by pronounced forward peaks. So, the wave functions in (2.1) and (2.2) (along with neglect of deuteron-target antisymmetrization in the cross-section calculation) may suffice for the purpose of our investigation. Let us now consider the detailed forms of the wave functions entering the expressions (2.1) and (2.2) for q~x(12C) and ~pR(13C), respectively. The internal wave function of the a-particle may be taken to be of the Gaussian form 15) 1 4 q~(~) --- - - exp { - 2 # ~ ~ N~

(r,-R~)Z}Z°(~),

i= 1

(2.3)

90

M. EL-NADi AND O. ZOHNI

X°(a) being a spin wave function and N~ a normalization factor given by N~2 = (½)~"

.

(2.4)

The coordinate rl stands for the absolute position vectors of the nucleons and R= is their c.m. coordinate 4

R~ = ¼ ~ r,.

(2.5)

i=1

Neglecting the antisymmetrization between the neutron and the a-cluster, the cluster wave function of 5He may then be written in the form 1,1 s. 16) rP(SUe) = N~ {go(a)q~(n)f(p.)} 4-

_

1 ~(½m.,

1M-m.l~M) exp {-2/7=Y,(r~-R,) 2}

] ~ h mn

i=1

×xo(a)p° exp (--apn)X..n ¥1M-,.n(Pn)"

(2.6)

Here, X°(a) and Z~=. are spin wave functions for the a-particle and the neutron, respectively, and the Clebsh-Gordan coefficient (½m., 1M-md~M) couples the spin of the neutron and the angular momentum of the relative motion to the total spin ~. The normalization factor Nh is given by N~ -

9~ 5 (2a)~(4/7,) ~

{1 +o(a,/7,)},

9(a'fi=)=30"45(3"6'+2"S6fi-z~+~)a "

(2.7)

It remains now to define the wave functions of relative motion that appear in eqs. (2.1) and (2.2). These wave functions are taken to be of the oscillator form z(R) = R%xp ( -

½fiR2)Ylm(g).

(2.8)

The order of n and l has to be determined from shell-model configuration and from energy and angular momentum conservation requirements. Let us now consider the wave function Z(P) in eq. (2.1) for the relative motion between the a-cluster and the SBe in the 12C nucleus. Following arguments similar to these used by Perring and Skyrme lSa) and Wildermuth et al. 1,2, 18b) in the investigation of the cluster structure of some light nuclei, one may take n = 4 for this relative wave function, since then the cluster wave function has the same oscillation energy as the corresponding single-particle shell-model oscillator wave function allowed by the Pauli principle 1)t. The orbital angular momentum l is determined t R e f . 1) p p . 78 ft.

12c(d, p ) t a c STRIPPING REACTION

91

from the parity and angular momentum conservation laws. In our case this gives the values 1 = 0 and m = 0, i.e., one then has a 3S oscillation for the relative motion considered. The wave function )~(p) can therefore be expressed as X(P) = p4 exp (__~p2),

(2.9)

where ~ = Mco/h is the oscillator width parameter, with o being the oscillation frequency and M the nucleon mass. The wave function ~(~) of relative motion between 5He and aBe clusters in 13C, in eq. (2.2) can be treated in a similar way. In order that the wave functions be equivalent in the single-particle and the cluster-model oscillation representation 1), one may also assign n = 4. From angular momentum conservation one can assign a value of l = 2 corresponding to a 2d oscillation for the relative motion considered. Hence, the wave function ~(t/) takes the form ~(t/) = t/4 exp (--~aflt/2)y2vrM(~),

(2.10)

where, again, fl = Mco'/h is the oscillator width parameter, with co' being the oscillator frequency and M the nucleon mass. The cluster model wave function of 13C may then be expressed as ~R(13C) = y~ (~M, 2 ~ f - MI½~f)~0QHe)~(SBe)~(~),

(2.11)

M

where the Clebsch-Gordan coefficient (~:M, 2/~t-MI½/~t) couples the spin of the SHe nucleus and the angular momentum of relative motion to the total spin-½ corresponding to the ground state of 13C. Writing eq. (2.6) in equivalent form tP(SHe)

1

= N-h E,,o(½m,, 1M-- mnl~M)q~(~)q~(n)f(p.),

(2.12)

using eq. (2.2) and inserting eq. (2.12) into eq. (2.11), one finally gets tpR(lSC) = Y'. (½mn, 1 M -

mnlSM)(aM, 2/¢f-MI½/ct)

rata, M

x ¢p(n)~o(Ctl)CP(~2)tp(~s)f(p.)~(R)¢(tl).

(2.13)

3. Formulation The cross section for the stripping reaction 12C(d, p)13C may be written as

da_ d~

mimt kt 1 ~ ]Tnl2, (27ch2) 2 k i

(3.1)

3 ,.a, m~,

where mi, mf and kl, kf are the reduced masses and the wave numbers, respectively, in the initial and final channels, and ma, rap, #i and/~t stand for the magnetic quantum numbers in the initial and final channels. In the PWBA, the transition amplitude Tfi

92

M. EL-NADI A N D O. Z O H N I

is expressed as Tfi = (~flVpn[~i).

(3.2)

Here, Fpn is the interaction potential responsible for the stripping process, and qSi and I # f stand for the initial and final plane wave states q5i = (2n)-~oT(lZC)~0(d) exp (iki" ri),

(3.3)

q5e = (2re) -~ ~OR(13C)z~p exp (ik e • re),

(3.4)

where qgT, ~o(d) and q~Rare the internal wave functions of the target nucleus, the incident deuteron and the residual nucleus, respectively, and 2 ~ is the spin wave function of the outgoing proton. Putting eqs. (2.1), (2.2) and (2.11) into eqs. (3.3) and (3.4), one gets for the transition amplitude Tfi = (27z)-3 E ({M, 2/~f--MII/tr) M

x (¢p(SHe)q~(~l)~p(~z)ff(R)~(t/)Z~p exp (ikf. re) ]Vpn[ x ~p(el)9(e2)cp(c~a)((g)x(p)q~(d ) exp (iki" ri)).

(3.5)

Carrying out the integrations over the coordinates of the spectator c~-clusters, one immediately gets: Tfi = ~ (~-M, 2/~f-M[½pf),-C/(ki, ke), (3.6) M where the matrix element Jg(ki, kf) is given by: •.~(ki, kf) = (270- a(9(5He)¢(t/)Zm&p exp (ikf" re) [Vpn[ X q~(~)tp(d)z(p) exp (iki" r~)>.

(3.7)

4. The stripping cross section

Let us take for the deuteron internal wave function ineq. (3.7)the Gaussian form a 5) 1 ~Pd(r) = - 1- exp(--ktdr 2 )Z,,d,

(4.1)

Nd

where r = r n -rp, Z ~ is the deuteron spin wave function and Nd is a normalising constant given by

The interaction potential Vp~ will also be taken to be of a finite-range Gaussian form Vpn(r) = - Vo exp ( -

(4.3)

yr2).

Inserting these into eq. (3.6), using eqs. (2.3), (2.6), (2.9) and

(2.1o), then

squaring

93

1 2 c ( d , p)13C STRIPPING REACTION

Tfl and carrying out the summation over all magnetic quantum numbers, one finally gets for the differential cross section the expression d a _ 3mimf kfA2f2(q ,Q) dO (2~h2) 2 ki (-)*° [/2]~[/~]~(20, 110[/2 0)(20, l;OIl~ O) X ll, 12', 13, l'1,1'2

xO-110, l;Ol130) h 1-/1 lot2 ×

2 12 1-11

l~,t B(l~ 1213Ii l;)D(l~ l~Ql; l;),

Ill

(4.4)

where the summations over ll, 12, 13, l~ and Ii run through the values 0 < l~ < 1,

12-111 < 12 < (2+/1),

o
[1-/1-/21 < 13 < (1-l~+12),

12-/il=
and 5 ÷ 7z A = 1 3 ( 3 ) 3 (~-~) 11oNc (~-~7)

f(q,Q)=Qexp.{

( ~---~----t~' \#~ + fi# (1"3Q)2 t q2 4[--~s0@exp{ 4(/~:~y)}'

B(I t 1213l'~/~)= (-)r2-v1(i)¢° (23ll) ½ [_Fctt(ah++~-%cO1 v'~ d

(4.5) (4.6)

I 1+ l'3cq~ h+r2

x ~2 ½(`'+':+''' +,'~)-lt [{(3 -2/,)(3-2/;)}~ 1 213 ~ d -}F(I1 + 12+ 11) -}F(l'l + 1'2+ 11) , x (4.7) t 3

r(h+~)

r(t~+~)

D(Ix 12Q l'~ l~) = Q ~° 1F~(½(l~+ 12+ 11), 12+ 3; _ ~3 Q2) ×1F1(½(1~+1,2+11),,/2+~,-ctaQ 3. 2 ),

(4.8)

with the factor [l] = 2l+ 1 and qSo = l 2-l~ + I f - l;. The braces { } represent 6jsymbols. 5. Results and choices of parameters

The differential cross section for the reaction 12C(d, p)i3C was computed* at different incident energies using the expression (4.4) in the range from 2 to 28 MeV. Calculated values of the differential cross section are normalized by matching the theoretical maximum with the maximum point of the experimental data 23,, 2,). * The calculations were performed on an IBM 1620 computer at Alexandria University.

94

M. EL-NADI AND O. ZOHNI

During the course of the analysis, the width parameters a, = and fl characterizing the cluster wave functions of the 5He, 12C and 13C, respectively, were set as free parameters, t t The search for these parameters converged to the values given in table (1). The observed increase of the width parameters a, ~ and fl with increasing incident energy may be visualized as corresponding to a relative decrease in the mean radius of the target nucleus and consequently of the respective clusters 1). The angular distributions obtained with the parameters given in table 1 are shown in fig. 2. At low energies, they are characterized by narrow forward peaks which fit the experimental data rather badly. With increasing energy the quality of fits becomes more satisfactory, within the limits of the plane wave treatment. It may be noted that the width parameters given in table 1 and used for getting best fits to the experimental data, are considerably less than those obtained from a clustermodel determination of the binding energies of the levels of the corresponding nuclei lSb). This may be due to a common feature 19-21) of stripping reactions in the PWBA, the interaction radius is uniformly larger than the radius of nuclear matter 22). Fig. (2) represents the results of the plane-wave cluster-model (PWCM) treatment and the Butler cut-off theory 23b) (BT) using shell-model wave functions for the target and residual nuclei. It may be noticed that while the results of the clusterand shell-model calculations agree at high energies, they do differ at the lower energies considered. Hence, as far as the plane wave calculations can be trusted, the stripping reaction results at low energies depend sensibly on the models assumed for the target and residual nuclei. For low energies, the shell-model curves are characterized by broader angular distributions which fit the experimental data more satisfactorily than the PWCM curves. With increasing energy, the results of the PWCM treatment are seen to approach those of the shell model. One may conclude from these results that the cluster picture for 12C as indicated by the low-energy 12C(d, p)13C reaction (ground state) is not very probable. For higher energies, the calculations are not sensitive to the model wave functions used for the target and residual nuclei. The information so far obtained supports the results found by Duck 5), Hebach and Henneberg 7) in their investigation of the three e-particle model of 12C using the Faddeev's three-body formalism, and by Darriulat 3) in his variational treatment of the same problem; that the ground state of aZC cannot be described by a three-body cluster model. It would be of interest to extend the analysis done in the present work to higher excited states, which may then throw some light on the cluster configuration of the excited states of ~2C. One remark may be added about the use of the plane-wave approximation in the present calculations. Presumably, one would inquire whether the use of distorted waves may affect the results obtained. Distortion effects, however, tend particularly to enhance the cross section in the backward direction ~7, 26). Moreover, plane-wave ~t The parameters/~d and7 characterizing the deuteron wave function and the interaction Vpnwere kept constant during the calculations. However their variations will not cause much change. 28)

Z2c(d, p)Z3C STRIPPING REACTION Ed : 2 " 6 7 a-

~" 20 "E ~- '~,'/ / "~

- ~I,I ;,o l O V i

E d:9.26

0.001

"

:t/,\

90

a =0.001

a :0.01S

,5

e--0.0o9

°

'

60

~

;,o

,

30

MeV

..o 20 --E ~

" - '-.. "

i

0

MeV.

O~.0.01S ,e= 0.009

t Lt

95

Jl

120

30

ec.m.

I

:

|

60

90

120

ec.m.

E d : 9 MeV ..~ 25 20

t

--

"

,/\

E 15 .o 10

I'l

a = 0.002

\

¢x = 0.02

,.,

~ .o.ol

E

\

-

20

40 0 c.rn

I/

tS

I~t

¢ = 0.03 A = 0.02

~

,

O0

E d : i 0 . 2 Met,/. a = 0.003

0 I'-.~F~

~ 2

o V

60

,\\,

,

30

60 90 120 e c.m.

,

15 25

Ed :12.4 MeV.

~" 2 0 15

Ed =14.7 MeV.

a = 0.003

.... ~ 10 E ~ .,0

cx= 0.03 ~l~b'

10

j,= 0.02

a.0.003

-o o

,....

o

ee

• , so%

0

30

60

,

90

"

0

120

0 c.m.

0 8

Ed=28 a=

E

°e~

60

90

'

120

MeV.

0.01

o(= 0.09 ,8 = 0.10

4

-=

30

ec.m.

?,

,,~ .D

~=0.04 B=0.02

~'~,

.I

0

,,"';:',

,

~

9 c.m,

Fig. 2. Best fits obtained for the angular distributions of the 12C(d, p)X3C reaction at different incident deuteron energies. Separate points represent the experimental data, solid curves represent the calculations of the present work, while dashed curves represent the Butler theory calculations using shell-model wave functions. Solid curves are obtained withFd = 0 . 0 8 fm -2 and~ = 0.416 fm -2.

96

M. EL-NADI AND O. ZOHNI TABLE 1

Cluster model oscillator width parameters a, ct, and fl corresponding to cluster wave functions of SHe, 12C and laC, respectively, for different incident deuteron energies

E~

a

(MeV)

(fm- 2)

(fm- 2)

~ (fm- 2)

2.67 3.26 9.00 10.20 12.40 14.70 28.00

0.001 0.001 0.002 0.003 0.003 0.003 0.01

0.015 0.015 0.02 0.03 0.03 0.04 0.09

0.009 0.009 0.01 0.02 0.02 0.02 0.10

theory for deuteron stripping has been observed 27) to yield very good fits for light nuclei with A < 40 for deuteron energies of the order of 8 to 20 MeV, a range which is comparable with that considered in the present work. One may thus be tempted to consider the use of plane waves in the present work as reasonable.

6. Root-mean-square radius of 12C Using the alpha-deuteron cluster model wave function to calculate the Coulomb disintegration of accelerated 6Li ions, Hansteen and Kanestrom z s) found that the degree of clustering was lower than that determined from the calculation of the 6Li energy levels. A similar situation is also met here, where on the basis of the cluster model, the degree of clustering for 12C, given in table 1, is likewise low as compared with the value 0.43 fm obtained by Wildermuth and Kanellopoulos 18) from an c~-SBe cluster model determination of its energy levels. However, an estimate, of the rms radius of 6Li made by Hansteen and Kanestrom has yielded a value consistent with the shell-model value 14) extracted from electron scattering experiments 25). Likewise it would be interesting in the present work to make an estimate of the rms radius of 12C on the basis of the cluster model, and to compare it with the shellmodel value obtained from electron scattering experiments 14). Adopting a definition for the rms radius 16, 25) of the ground state of 1 2 C similar to that used 6) for Li, one may write t (R2) ~ =

q~*(2C) (Cop~o(12C)dz ,

(6.1)

where N 2 is a normalization constant and the operator is given by 12

% = A 2

i=l

=6(Zp

8

j=l

12

Z r,)2

i=1 12

+Z Ok2 + x, R , ),

(6.2)

/¢=9

T Effects of antisymmetrization are neglected between the constituent clusters of the 12C nucleus.

12c(d, p) 13C STRIPPING REACTION

97

ri being the position vector of the ith nucleon and where p j ----- r y - - R a B e ,

Pk = r k - - R ~ ,

(6.4)

g = Rsae-R~, 8

(6.3)

12

R*se = -~ Z r j,

R, = ¼ Z rk"

j=l

(6"5t

k=9

If one neglects the internal structure of the constituent clusters of the 12C nucleus, one may then write for the operator in eq. (6.2). ~op ~ 1 R2"

(6.6)

9

Using the relative wave function in (2.9) for ~o(12C) in eq. (6.1), one obtains for the rms radius of 12C the expression (R21½ ~ ( 55 ~ . \144~1

(6.7)

TABLE 2 Values of the rms radius of 12C for different parameters that fit the data at different incident deuteron energies

Ed

g

fg2~½

(MeV)

(fro -2)

(fm)

2.67 9.00 12.04 14.70 28.00

0.015 0.02 0.03 0.04 0.09

5.05 4.37 3.57 3.09 2.06

These values are obtained using formula (6.7) in the text which neglects the internal structure of the a-clusters. TABLE 3 Values of the rms radius of 12C using formula (6.9) in text which taken roughly into account the internal structure of the ~-clusters

Ea

~

(R21{

(MeV)

( f m - 2)

(fm)

2.67 9.00 12.40 14.70 28.00

0.015 0.02 0.03 0.04 0.09

5.30 4.66 3.92 3.48 2.56

98

M. EL-NADIAND O. ZOHNI

Inserting into this expression the values for the width parameter a given in table 1, one gets the values shown in table 2 for the rms radius. Knowing the rms radius of the a-particle to be 14) (r2)~ = 1.61 fm, (6.8) one may then roughly include the internal structure of the a-particle in formula (7.6) by putting (R2)~2c = {(R 2) + (r2)~} ½. (6.9)

1

°o

[

,'o ,;

I

2's

I

Ed (MeV)

Fig. 3. Values of the rms radius of 12C for different values of the parameters that fit data at different incident deuteron energies. The dashed line represents the shell-model approximation value of 2.41 fm.

Making use of eq. (6.9), we get in table 3 the value of the rms radius of 12C whose behaviour is illustrated graphically in fig. (3). Apart from small uncertainties in the computations, they are observed to decrease smoothly with energy from the value of 5.30 fm at Ed ----2.67 MeV to the value 2.56 fm at Ed = 28 MeV. The last highenergy value is comparable with the shell-model value 2.41 fm of the rms of ~2C quoted by Elton ~4) and extracted from electron scattering experiments. This is in harmony with the previous agreement found above between the shell-model and cluster model pictures in stripping reactions at high energies. In conclusion, one may note that the low a-parameter values, i.e., the low degree of clustering, deduced for ~2C in the present work do not lead to unreasonable values for the rms radius. The authors are thankful to Dr. W. Wadie for valuable comments and to Dr. A. Y. Abul-Magd for useful discussions. Thanks are also due to the staff members of the 1620-IBM Computer Centre at Alexandria University for their facilities. References 1) K. Wildermuth and W. McClure, Cluster representatior~ of nuclei (Springer-Verlag, Berlin, 1966) 2) R. K. Sheline and K. Wildermuth, Nucl. Phys. 21 (1960) 196

12c(d, p) 13 c STRIPPINGREACTION 3) P. Darriulat, Nucl. Phys. 76 (1966) 118 4) D. R. Harrington, Phys. Rev. 147 (1966) 685 5) I. Duck, Nucl. Phys. 84 (1966) 586 6) J. R. Fulco and D. Y. Wong, Phys. Rev. 172 (1968) 1062 7) H. Hebach and P. Henneberg, Z. Phys. 216 (1968) 204 8) J. J. Leigh, Phys. Rev. 133 (1961) 2145 9) V. E. Dtmakov, JETP (Soy. Phys.) 16 (1963) 1571 10) M. E1-Nadi, A. Rabie and H. Sherif, Nucl. Phys. 48 (1959) 569 11) M. E1-Nadi and T. H. Rihan, Nucl. Phys. 57 (1964) 466 12) A. Y. Abul-Magd, H. E1-Nadi and G. L. Vysotsky, Nucl. Phys. 71 (1965) 606 13) N. L. Glendenning, Ann. Rev. Nucl. Sci. 13 (1963) 191 14) L. R. B. Elton, Nuclear sizes (OUP, London, 1961) p. 31 15) P. Henneberg, Z. Phys. 193 (1966) 23 16) L. D. Pearlstein, Y. C. Targ and K. Wildermuth, Phys. Rev. 120 (1960) 214 17) W. Tobocman, Theory of direct nuclear reactions (OUP, London, 1961) 18a) J. K. Petting and T. H. R. Skyrme, Proc. Phys. Soc. A69 (1956) 600; b) K. Wildermuth and Th. Kanellopoulos, CERN Report 59-23, Geneve (1959) 19) R. Huby, Progress in nuclear physics (Academic Press, New York, 1953) vol. 3 20) J. S. Blair and E. M. Henley, Phys. Rev. 112 (1959) 2029 21) W. W. Dahnick and J. M. Fowler, Phys. Rev. 111 (1958) 1309 22) R. Hofstadter, Rev. Mod. Phys. 28 (1956) 214 23a) R. E. Benenson, K. W. Jones and M. T. McEllistrem, Phys. Rev. 101 (1956) 308; J. S. Green and R. Middleton, Proc. Phys. Soc, A69 (1956) 28; E. W. Hamburger, Phys. Rev. 173 (1961) 619; b) M. E1-Nadi and I-I. Fahmi, Arab. J. Nucl. Sci. Appl. 2 (1969) 1 24) H. J. Eramuspe and R. J. Slobodrian, Nucl. Phys. 49 (1963) 65 25) J. M. Hansteen and I. Kanestrom, Nucl. Phys. 46 (1963) 303 26) S. T. Butler, N. Austern and C. Pearson, Phys. Rev. 112 (1958) 1227 27) N. Austern, Fast neutron physics, part II (Interscience, New York, 1963) 28) H. C. Newns, Proc. Phys. Soc. A66 (1953) 477

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