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An absorption model for direct transfer reactions
PhysicsA157(1970)661-672; @ North-Holland Publishing Co., Amsterdam Not to bereproduced by photoprint or micro6lm without written permission from the publisher
Nuclear
AN ABSORPTlON
MODEL FOR DIREXX TRANSFER REACTIONS
J. DOBES Nuclear Research Institute of the Czechoslovak Academy of Sciences, Be%, Czechoslovakia Received 10 July 1970 Abstract: A simple model for direct transfer reactions is investigated. The transition amplitude is evaluated by means of the geometric mean (Sopkovich) formula with the sharp cut-off model for elastic scattering. An easy numerical computation removing a number of approximations adopted in previous papers is performed. The model is tested for some (d, p) and (p, d) reactions. Satisfactory agreement with experimental angular distributions and reasonable spectroscopic factors has been obtained.
1. Introduction
Direct nuclear reactions are analysed almost exclusively by means of the distorted wave Born approximation (DWBA) ‘). In spite of various uncertainties ‘) DWBA is very useful in obtaining spectroscopic information from experimental data. It implies, however, one disadvantage which is pointed out by some authors 3- “). DWBA is a rather complicated theory with many ambiguous parameters and does not give too much insight into the reaction mechanism that brings about relatively simple and well pronounced typical features of angular distributions. The shape of the angular distributions in direct processes exhibits a strongly diffractive character. This fact has stimulated a creation of diffraction models for direct reactions in the x (configuration) space. One assumes that both the incident and the outgoing particles are strongly absorbed by the target nucleus and the shadow arises in such a manner that the reaction can take place only in the limited region of the x-space, outside and near the nuclear surface. These models have successfully been used in elastic scattering “) and with less success also in the transfer reactions 3*‘). As it was pointed out in refs. ‘9 ‘) the localization in the 1 (angular momentum) space for direct processes is a more characteristic feature than the localization in the x-space. The contributions of lower partial waves are suppressed by a large number of competing exit channels and the main contribution comes from particles with angular momentum I z kR, where R is the target radius. This fact has been utilized for a derivation of simple formulas for inelastic scattering 4*lo) and transfer reactions 4.5.11-13 >* The present paper deals with the transfer reactions only. We adopt a simple model using l-space localization. The authors of the previous papers on this subject have tried to get analytical formulas for cross sections. For this reason they have introduced 661
662
J.
DOBE
a number of additional approximations. Instead, we have performed a simple numerical computation that enables us to overcome these approximations. Both the angular distributions and magnitudes of the theoretical and experimental cross sections have been compared. In order to make the features of the model explicit we use it for stripping (d, p) and inverse pick-up (p, d) reactions on target nuclei with closed subshells, for which the mechanism of the direct transfer is known best of all. The results of these calculations are in surprisingly satisfactory agreement with experimental angular distributions and, moreover, reasonable spectroscopic factors have been obtained. 2. Absorption models All the experimentally measurable quantities of the reaction A(d, p)B can be calculated provided that the transition amplitude, expressed in DWBA by BI” = <@‘(&
3Q#T(m)l~“plh3
4t++Ykl~ KID7
(1)
is given. Here, J/r’ and I++:-’denote the distorted waves and k,, Rd, k, and r,, are the wave vectors and relative coordinates of the centres of mass of the fragments in the initial and final channels, respectively. Furthermore, V,, stands for the interaction between the proton and neutron, $d is the internal wave function of the deuteron and &” is the wave function of the captured neutron with orbital momentum I and its z-projection m. in the l-space the decomposition of B;” can be written as
B;” = C ( -)“‘
Yfl'(Q.
(2)
It was suggested I42’ 5*‘) that the elements BFL’ of the transition amplitude in the I-representation can be approximated by the relations
(3) where Pf”’ are elements of the transition amplitude in the plane wave Born approximation (PWBA). More specifically, the matrix elements PF”’ are obtained from the plane wave amplitude P;” = (eiLp’Cp~;t(r~)l~“npl~deikd’Ih),
(4)
by decomposing it in the l-space p;” =
C LL’MM’
(-)“‘(L!-M’LMllm)P~L’Y,M*(f,)Yf’(kd).
(3
The quantities St,, ,Si which appear in eq. (3) are the elements of the elastic scattering matrix in the initial and final channels, respectively. Eq. (3) is often called the Sopkovich or geometric mean formula. It can be derived with the help of WKB approximation.
DIRECT
TRAMFER
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663
However, as it was remarked in refs. ‘*’ “) the formula (3) is not precise when used for low angular momenta. It is not, therefore, quite meaningful to use it with elastic phases calculated e.g. from the optical model. The application of eq. (3) is reasonable when the sharp cut-off model for elastic scattering r6) is applied, in which s, = 0,
L s L, z kR,
s==
L > L,.
1,
Here, k is the wave number and R is the target radius. Another formula exhibiting the absorptive corrections was proposed to which B,LL’ w a( 1 + S;)P,““‘( 1 + S;).
(6)
I’), according (7)
Eq. (7) has been derived recently by means of the three-particle formalism I*). It can be also easily obtained from the Heitler damping equation neglecting the effect of inelastic processes on elastic scattering and on the transfer reaction and approximating the K-matrix by PWBA. One runs into difficulties again when using the K-matrix approach (eq. (7)) with more realistic phases. The phases of elastic scattering depend on the Coulomb cut-off radius Rc [see e.g. ref. ’ 9)], this dependence, however, has to be eliminated in the final result. We have not succeeded in this elimination when eq. (7) was used. The difficulty disappears if the K-matrix is approximated by the Coulomb wave Born approximation. It was observed that the main difference between eqs. (3) and (7),rests in how the lower partial waves are dealt with. Eq. (7) was pointed out in ref. i5) to be unsatisfactory for these waves giving a non-zero contribution no matter how strong is the absorption and the competition from other inelastic channels.
3. Present calculations In what follows the sharp cut-off model for elastic scattering (eq. (6)) is adopted. Our main task therefore is to calculate the quantities PF”‘. A direct expansion of the plane waves in the initial and final channels into the partial waves was performed in previous papers 4* 5*“). The quantities PF”’ obtained in such a manner are rather complicated integrals. To calculate them one has to use the zero-range approximation for the neutron-proton potential Vnp. Further, in order to get analytical formulas for cross sections the asymptotical form of the bound neutron wave function has been used 4* 5), t h e smallness of the transferred angular momentum I compared to the values of the cut-off momental, and therefore the equality Py’ = Pf(L+L’)*f(L+L’) has been assumed, the adiabatic condition k, z k, has been adopted “) and, finally, the asymptotical expression for Clebsch-Gordan coefficients has been used. The validity of these conditions for the deuteron stripping on light nuclei, where the product kR is small and we have to deal with the low values of the cut-off momenta, is often violated.
664
3.
DOBEg
In order to remove this inconsistency we proceed in another way. We do not obtain an analytical formula but the numerical computation we have to perform is very easy. Once the plane wave amplitude P;” (eq.(4)) is known as a function of the angle of the final momentum direction fi, (z-axis is directed along the initial momentum s,) one calculates first of all the integrals IL =
s
p;“(&JXY’(&Jd~, -
Then using the orthogonality relation for Clebsch-Gordan quantities Pf” are easily obtained as the sum
(8)
coefficients the desired
In order to calculate the plane wave amplitude PI” we adopt the following assumptions. (i) The Hulthtn form of the internal deuteron wave function ‘“) is used & = N(e-w - e-““)/r,
(10)
with a’ = 7a (a is determined by the deuteron binding energy). (ii) We employ the separation energy prescription “) with a square well potential to obtain the wave function of the captured neutron. Eq. (4) with the wave function (10) becomes [see e.g. ref. 2’)]
4nh2 (a + a’)(a -
pp = _ -
P
1
a’)*N ~ ar2+K2
m~dr)h(~~)~2 s0
dr Y;“*(B),
(11)
where K = k,-3k,, q = k,- [A/(,4 + I)]&, (A is the target mass), p is the nucleon mass and jr is spherical Bessel function. The integration in eq. (11) is easily performed 22) (the radial function 41 is spherical Bessel or Hankel function inside and outside the well) s
o&jt(qr)r2dr
0
=
R2(1/[q2+(2m,/h2)E,1-1/Iq2+(2m,/h2)(E,- VIII x(hi;--44&~.
(12)
Here, m, is the reduced mass of the neutron in the final nucleus, En is its binding energy and V is the depth of the square well with the radius R. The calculation of the integral (8) and of the partial waves PI”’ from eq. (9) is then straightforward using the amplitude PI” from eqs. (11) and (12). Let us mention that we have no difficulties in considering the finite range of V,,,. 4. Results and discussion We have used the models with absorptive corrections for various reactions of the deuteron stripping (d, n) and pick-up Cp, d). The selected target nuclei have closed
aa 2* 2s) 26 2s;
17: 2s
12 25.9 12 15 12 27.5 27.5 12 12 2*
17
ref.
J&WV1
2.63 3.87 3.01 3.01 4.25 4.69 4.24 4.37 4.37
R = 1.25A+fm,R
kd R with
2.18 3.02 2.38 2.30 3.69 4.76 4.76 3.89 3.67
k, R with = 1.25A+fm
0.83 0.64 0.69 0.67 0.23 -0.69 0.66-1.82 0.33-1.72 0.36-0.61 0.35-0.50
present analysis
0.55 -- 1.38 0.32- 1.25 0.6 -0.8 0.5 -0.7
1.16 0.70 0.33 -0.60 0.62
DWBA
)
;
28
9 1
‘)
9
23
25
30)
23
ref.
Spectroscopic factors
0.85 0.82 0.9 0.65
0.61 0.61 0.75 0.80
___--
theory
32; 32 33 .?J
? 29; 31 3’)
ref.
.-
-
The values of products kdR and k, R which are important for determining cut-off momenta are given in the second and third columns. DWBA and theoretical spectroscopic factors are found in the last two columns of the table. Different parameter values give different spectroscopic factors in DWBA and present analysis in the range given in the table.
‘*C(d, p)‘3C(g.s.) 12C(d, p)‘%Jg.s.) 160(d, p)“O(g.s.) r60(d, p)“O*(O.87 MeV) ?Si(d, p)?Si(g.s.) 40Ca(p, d)39Ca(g.s.) 40Ca(p, d)s9Ca*(2.47 MeV) 40Ca(d, p)“Ca(g.s.) 40Ca(d, p)4’Ca+(1.95 MeV)
Reaction
1
The list of examined reactions (column 1) and obtained spectroscopic factors (column 4)
TABLE
5 ii z
:
2 5 :
3
tr 5i
666
J. DOB;5
subshells and one can use with confidence the separation energy prescription. The S-matrix of elastic scattering has been taken in the sharp cut-off model (eq. (6)) and the quantities PF”’ have been calculated as was described above, i.e. with plane waves in the initial and final channels. The Coulomb interaction is thus neglected and our description is valid for energies above the Coulomb barrier. For this reason we have restricted ourselves to the targets with the mass number A S 40. The examined reactions together with corresponding references are summarized in table 1. We consider it to be a rather representative set with regard to the angular
_ : \ 3 c 4 %
5-
24:
cL5-
0.2-
at Ed = 12 MeV. Experimental Fig. 1. Differential cross section for the reaction “C(d, p)‘%(g.s.) points are taken from ref. 23). The curves are calculated from eq. (3) as described in the text. The parameter values are for the solid curves rO = 1.25 fm, Lz = 3 and Lg = 2. The same parameters. are used for the dashed curve except for L: = 2.
momentum transfer and reaction Q-value. The deuteron energy is, except for one case,. in the region of 10-15 MeV, this being the most interesting region from the spectra-scopic point of view, in which majority of experimental data is available as well. We have tried both the geometric mean formula (eq. (3)) and K-matrix approach (eq. (7)). A poor agreement with the experimental data has been found with eq. (7).. All the following analyses concern the application of eq. (3). The results are shown in figs. 1-9 and the spectroscopic factors are summarized in table 1. We have at our disposal three parameters, the continuous one r. being related to the radius R of the square well for the neutron by R = r. A* and two discrete ones the values of the cut-off momenta Li and Lp in the deuteron and proton channels.
DIRECT
20
40
TRANSFER
667
REACTIONS
60
30
m
f20
a,,
Fig. 2. Differential cross section for the reaction ‘%(d, p)t”C (B.S.) at Ed = 25.9 MeV. Experimental points are taken from ref. 24). The curves are calculated from eq. (3) as described in the text. The parameter values are for the solid curve rb = 1.25 fm. 15: = 4 and Lg = 3. The same parameters are used for the dashed curve except for Lz = 3.
Fig. 3. Differential cross section for the reaction 160(d, p)“O (p.s.) at Ed = 12 MeV. Experimental points are taken from ref. 2s). Th e curve is calculated from eq.(3) as described in the text. The parameter values are re = 1.25 fm, Ld, = 3 and Lg = 2.
668
J.
DOB&
..*.: ..nr ,l-l l*
.
. .’
.
l
l
,
20
40
60
80
4W
00
Fig. 4. Differential cross section for the reaction 160(d, p)“0*(0.87 MeV) at & = 12 MeV. Experimental points are taken from ref. *‘). The curve is calculated from eq. (3) as described in the text. The parameter values are r0 = 1.25 fm, Lt = 3 and I$ = 2.
I-1
I
20
I
40
I
60
Fig. 5. Differential cross section for the reaction *%(d, p)2qSi(g.s.) at Ed = 15 MeV. Experimental points are taken from ref. 26). Th e curves are calculated from eq. (3) as described in the text. The parameter values are for the solid curve r0 = 1.25 fm, Lg = 5 and Lg = 3. The same parameters are used for the dashed curve except for Lg = 4.
Fig. 6. Differential cross section for the reaction *‘%a@, d)sgCa(g.s.) at En = 27.5 MeV. Experimental points are taken from ref. 17). The curves are calculated from eq. (3) as described in the text. The parameter values are for the solid curve rO = 1.25 fm, L,6 = 4 and L+ = 4. The same parameters are used for the dashed curve except for Ld, - 5. Decrease of rO to rO - 1.15 fm does not change the shape of the solid curve, spectroscopic factor, however, is increased as indicated in the figure. (ebl,,,ciJt&47WJ ---
I
I 20
I
I 40
I
I 60
L$=5 l-d=4
I
9
,
8o
452”
KV
s=172 s=a33.Kj=mfm (S=a52 c=7.%bJ
,
I
100
I
I
I
120
Fig. 7. Differential cross section for the reaction ‘&a(p, d)Wa+(2.47 MeV) at Er, = 27.5 MeV. Experimental points are taken from ref. “). The curves are calculated from eq. (3) as described in the text. The parameter values are for the solid curve re = 1.25 fm, Ld, = 4 and Lp = 4. The same parameters are used for the dashed curve except for L$ = 5. Decrease of rO to re = 1.15 fm does not change the shape of the solid curve, spectroscopic factor, however, is increased as indicated in the figure.
__.
-.
.~._
‘&,(~,&dg t
J
I 4=12 M.V
1=3
J
Fig. 8. Differential points are taken parameter values change the shape
cross section for the reaction *Wa(d, p)“Ca(g.s.) at Ed = 12 MeV. Experimental from ref. 2s). The curve is calculated from eq. (3) as described in the test. The are r0 = 1.25 fm. L.$ = 4 and Lg = 3. Decrease of r,, to r,, = 1.15 fm does not of the curve, spectroscopic factor, however, is increased as indicated in the figure.
Fig. 9. Differential Experimental points text. The parameter does tiot cliange the
cross section for the reaction q°Ca(d, p)41Ca*(1.95 MeV) at Ed = 12 MeV. are taken from ref. za). The curve is calculated from eq. (3) as described in the values are r,, = 1.25 fm, Lz = 4 and L; = 3. Decrease of r0 to r,, = 1.15 fm shape of the curvd, spectroscopic factor, however, is increased as indicated in the figare.
DIRECT
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REACTIONS
671
Only minor changes are exhibited in the shape of the angular distributions with the variation of r,,, the spectroscopic factors, however, are lowered with the increasing re. The choice of the cut-off momenta shows a major influence on the cross sections and there exists some uncertainty particularly for deuterons. In the second and third columns of table 1 the values of the products k,R and k,R with R = 1.25 A* are given. All the calculated curves have Lg equal to the largest integer smaller than the value k,R. For the reactions in figs. 1, 2, 5-7 two choices of Li have been used, one value being chosen larger and the other smaller than k, R. The larger value Ld, is slightly preferred for the reactions on “C and ‘*Si while for the reactions on 40Ca the smaller value is better. The parameter r. of the neutron potential well is 1.25 fm. Spectroscopic factors given by this value r. were found to be smaller than expected for the reactions on 40Ca. The agreement was improved by lowering r. to 1.15 fm, that was still within the reasonable range. The shape of the calculated angular distributions follows the experimental course very well. The worst agreement relatively exists for the reaction 40Ca(p, d)39Ca with I = 0, this transition, however, can be fitted by DWBA with difficulties as well “). The spectroscopic factors, particularly for the lightest nuclei, are in a good agreement with the DWBA analyses and the theory (DWBA and theoretical spectroscopic factors are given in the last two columns of table 1). The change of the parameter Ld, yields different values of the spectroscopic factors for nuclei Si and Ca. However, looking at the DWBA spectroscopic factors of the reaction 40Ca(p, d)39Ca in table 1, one finds that similar differences can occur also in the DWBA analysis. We have achieved a surprisingly good agreement of experiment and theory cutting off the small angular momenta from plane waves in the initial and final channels. The idea of neglecting contributions from the lowest partial waves is a very wellknown one and it has been utilized in the derivation of Butler theory for (d, p) stripping 34). One assumes that this neglecting and the absorption of the particles penetrating into the nucleus can be simulated by cutting off the radial integral. Then the integral in eq. (11) is not taken from zero but from the non-zero value of the cut-off radius. The fact that the Butler theory gives only a poor fit to experiment compared to the good results of the present calculations suggests that the radial cut-off is only a rough approximation to the I-space cut off. The very simple and easily computable model which we have examined in this paper explains the experimental data satisfactorily. The absorptive corrections are well exhibited in the geometric mean formula while the K-matrix approach was not found to be very successful. The model could be suitable in the first off-hand analysis of measured results and its simplicity could be useful in the theoretical study of effects the calculation of which is difficult in DWBA. The author would like to thank Dr. F. Janouch for many helpful discussions. Valuable conversations with Dr. J. Cejpek and Dr. P. Pajas are also gratefully acknowledged.
672
I.
DolIES
References 1) W. Tobocman, Theory of direct nuclear reactions (Oxford University Press, London, 1961) 2) R. J. Philpott, W. T. Pinkston and G. R. Satchler, Nucl. Phys. All9 (1968) 241 3) A. Dar, Nucl. Phys. 55 (1964) 305 4) A. Dar, Nucl. Phys. 82 (1966) 354; Phys. Rev. 139 (1965) Bl193 5) W. E. Frahn and M. A. Sharaf, Nucl. Phys. Al33 (1969) 593 6) S. I. Drozdov, JETP (Sov. Phys.) 28 (1955) 734; E. V. Inopin, JETP (Sov. Phys.) 31 (1956) 901; J. S. Blair, Phys. Rev. 115 (1959) 928 7) E. M. Henley and D. U. L. Yu, Phys. Rev. 133 (1964) Bl445; 135 (1964) Bl152 8) N. Austern, Ann. of Phys. 15 (1961) 299; Selected topics in nuclear theory, ed. F. Janouch (IAEA, Vienna, 1963) p. 17 9) M. B. Hooper, Nucl. Phys. 76 (1966) 449 10) N. Austern and J. S. Blair, Ann. of Phys. 33 (1965) 15; W. Bassichis and A. Dar, Ann. of Phys. 36 (1966) 130; F. J. W. Hahne, Nucl. Phys. 80 (1966) 113; J. M. Potgieter and W. E. Frahn, Nucl. Phys. 80 (1966) 434; A92 (1967) 84 11) A. Dar and B. Kozlowsky, Phys. Rev. Lett. 15 (1965) 1036 S. Varma, Nucl. Phys. A106 (1968) 233 12) A. V. Shebeko and E. V. Inopin, Sov. J. Nucl. Phys. 8 (1968) 1142 13) I. Borbey, E. I. Dolinsky and V. V. Turovtsev, Sov. J. Nucl. Phys. 8 (1968) 492; E. I. Dolinsky and V. V. Turovtsev, Sov. J. Nucl. Phys. 9 (1969) 515; 1. Borbey and E. I. Dolinsky, Sov. J. Nucl. Phys. 10 (1969) 299 14) N. J. Sopkovich, Nuovo Cim. 26 (1962) 186; K. Gottfried and J. D. Jakson, Nuovo Cim. 34 (1964) 735 15) L. Durand and Y. T. Chiu, Phys. Rev. 139 (1965) B646 16) H. A. Bethe and G. Placzek, Phys. Rev. 57 (1940) 1075; J. S. Blair, Phys. Rev. 95 (1954) 1218; 108 (1957) 827 17) M. H. Ross and E. L. Shaw. Phys. Rev. Lett. 12 (1964) 627; G. Kramer and K. Schilling, Z. Phys. 191 (1966) 51 18) E. 0. Alt, P. Grassberger and W. Sandhas, Nucl. Phys. Al39 (1969) 209 19) M. L. Goldberger and K. M. Watson, Collision theory, (Wiley, New York, 1964) ch. 6 20) L. Hulthen and M. Sugawara, Handbuch der Physik, ed. S. Fliigge (Springer Verlag, Berlin, 1957) vol. XXXIX, p. 1 21) N. K. Glendenning, Ann. Rev. Nucl. Sci. 13 (1963) 191 22) G. N. Watson, Theory of Bessel functions (Cambridge University Press. 1952) p. 133; R. M. May, Nature 207 (1965) 1348; D. C. Peaslee, Phys. Rev. 171 (1968) 1208 23) J. P. Schiffer, G. C. Morrison, R. H. Siemssen and B. Zeidman, Phys. Rev. 164 (1967) 1274 24) R. van Dantzig and L. A. Ch. Koerts, Nucl. Phys. 48 (1963) 177 25) J. L. Alty, L. L. Green, R. Huby, G. D. Jones, J. R. Mines and J. F. Sharpey-Schafer, Nucl. Phys. A97 (1967) 541 26) A. Blair, Unpublished doctoral dissertation. University of Pittsburgh (1960) 27) C. Glasshausser. M. Kondo, M. E. Rickey and E. Rost, Phys. L&t. 14 (1965) 113 28) K. K. Seth, J. Picard and G. R. Satchler, Nucl. Phys. A140 (1970) 577 29) S. Cohen and B. Kurath, Nucl. Phys. Al01 (1967) 30) R. van Dantzig and W. Tobocman, Phys. Rev. 136 (1964) B1682 31) G. E. Brown, J. A. Evans, D. J. Thouless, Nucl. Phys. 45 (1963) 164 32) L. B. Hubbard and H. P. Jolly, Phys. Rev. 164 (1967) 1434 33) W. J. Gerace and A. M. Green, Nucl. Phys. A93 (1967) 110 34) S. T. Butler and 0. H. Hittmair, Nuclear stripping reactions (Horowitz Publications, Sydney 1957)
Nuclear
AN ABSORPTlON
MODEL FOR DIREXX TRANSFER REACTIONS
J. DOBES Nuclear Research Institute of the Czechoslovak Academy of Sciences, Be%, Czechoslovakia Received 10 July 1970 Abstract: A simple model for direct transfer reactions is investigated. The transition amplitude is evaluated by means of the geometric mean (Sopkovich) formula with the sharp cut-off model for elastic scattering. An easy numerical computation removing a number of approximations adopted in previous papers is performed. The model is tested for some (d, p) and (p, d) reactions. Satisfactory agreement with experimental angular distributions and reasonable spectroscopic factors has been obtained.
1. Introduction
Direct nuclear reactions are analysed almost exclusively by means of the distorted wave Born approximation (DWBA) ‘). In spite of various uncertainties ‘) DWBA is very useful in obtaining spectroscopic information from experimental data. It implies, however, one disadvantage which is pointed out by some authors 3- “). DWBA is a rather complicated theory with many ambiguous parameters and does not give too much insight into the reaction mechanism that brings about relatively simple and well pronounced typical features of angular distributions. The shape of the angular distributions in direct processes exhibits a strongly diffractive character. This fact has stimulated a creation of diffraction models for direct reactions in the x (configuration) space. One assumes that both the incident and the outgoing particles are strongly absorbed by the target nucleus and the shadow arises in such a manner that the reaction can take place only in the limited region of the x-space, outside and near the nuclear surface. These models have successfully been used in elastic scattering “) and with less success also in the transfer reactions 3*‘). As it was pointed out in refs. ‘9 ‘) the localization in the 1 (angular momentum) space for direct processes is a more characteristic feature than the localization in the x-space. The contributions of lower partial waves are suppressed by a large number of competing exit channels and the main contribution comes from particles with angular momentum I z kR, where R is the target radius. This fact has been utilized for a derivation of simple formulas for inelastic scattering 4*lo) and transfer reactions 4.5.11-13 >* The present paper deals with the transfer reactions only. We adopt a simple model using l-space localization. The authors of the previous papers on this subject have tried to get analytical formulas for cross sections. For this reason they have introduced 661
662
J.
DOBE
a number of additional approximations. Instead, we have performed a simple numerical computation that enables us to overcome these approximations. Both the angular distributions and magnitudes of the theoretical and experimental cross sections have been compared. In order to make the features of the model explicit we use it for stripping (d, p) and inverse pick-up (p, d) reactions on target nuclei with closed subshells, for which the mechanism of the direct transfer is known best of all. The results of these calculations are in surprisingly satisfactory agreement with experimental angular distributions and, moreover, reasonable spectroscopic factors have been obtained. 2. Absorption models All the experimentally measurable quantities of the reaction A(d, p)B can be calculated provided that the transition amplitude, expressed in DWBA by BI” = <@‘(&
3Q#T(m)l~“plh3
4t++Ykl~ KID7
(1)
is given. Here, J/r’ and I++:-’denote the distorted waves and k,, Rd, k, and r,, are the wave vectors and relative coordinates of the centres of mass of the fragments in the initial and final channels, respectively. Furthermore, V,, stands for the interaction between the proton and neutron, $d is the internal wave function of the deuteron and &” is the wave function of the captured neutron with orbital momentum I and its z-projection m. in the l-space the decomposition of B;” can be written as
B;” = C ( -)“‘
Yfl'(Q.
(2)
It was suggested I42’ 5*‘) that the elements BFL’ of the transition amplitude in the I-representation can be approximated by the relations
(3) where Pf”’ are elements of the transition amplitude in the plane wave Born approximation (PWBA). More specifically, the matrix elements PF”’ are obtained from the plane wave amplitude P;” = (eiLp’Cp~;t(r~)l~“npl~deikd’Ih),
(4)
by decomposing it in the l-space p;” =
C LL’MM’
(-)“‘(L!-M’LMllm)P~L’Y,M*(f,)Yf’(kd).
(3
The quantities St,, ,Si which appear in eq. (3) are the elements of the elastic scattering matrix in the initial and final channels, respectively. Eq. (3) is often called the Sopkovich or geometric mean formula. It can be derived with the help of WKB approximation.
DIRECT
TRAMFER
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663
However, as it was remarked in refs. ‘*’ “) the formula (3) is not precise when used for low angular momenta. It is not, therefore, quite meaningful to use it with elastic phases calculated e.g. from the optical model. The application of eq. (3) is reasonable when the sharp cut-off model for elastic scattering r6) is applied, in which s, = 0,
L s L, z kR,
s==
L > L,.
1,
Here, k is the wave number and R is the target radius. Another formula exhibiting the absorptive corrections was proposed to which B,LL’ w a( 1 + S;)P,““‘( 1 + S;).
(6)
I’), according (7)
Eq. (7) has been derived recently by means of the three-particle formalism I*). It can be also easily obtained from the Heitler damping equation neglecting the effect of inelastic processes on elastic scattering and on the transfer reaction and approximating the K-matrix by PWBA. One runs into difficulties again when using the K-matrix approach (eq. (7)) with more realistic phases. The phases of elastic scattering depend on the Coulomb cut-off radius Rc [see e.g. ref. ’ 9)], this dependence, however, has to be eliminated in the final result. We have not succeeded in this elimination when eq. (7) was used. The difficulty disappears if the K-matrix is approximated by the Coulomb wave Born approximation. It was observed that the main difference between eqs. (3) and (7),rests in how the lower partial waves are dealt with. Eq. (7) was pointed out in ref. i5) to be unsatisfactory for these waves giving a non-zero contribution no matter how strong is the absorption and the competition from other inelastic channels.
3. Present calculations In what follows the sharp cut-off model for elastic scattering (eq. (6)) is adopted. Our main task therefore is to calculate the quantities PF”‘. A direct expansion of the plane waves in the initial and final channels into the partial waves was performed in previous papers 4* 5*“). The quantities PF”’ obtained in such a manner are rather complicated integrals. To calculate them one has to use the zero-range approximation for the neutron-proton potential Vnp. Further, in order to get analytical formulas for cross sections the asymptotical form of the bound neutron wave function has been used 4* 5), t h e smallness of the transferred angular momentum I compared to the values of the cut-off momental, and therefore the equality Py’ = Pf(L+L’)*f(L+L’) has been assumed, the adiabatic condition k, z k, has been adopted “) and, finally, the asymptotical expression for Clebsch-Gordan coefficients has been used. The validity of these conditions for the deuteron stripping on light nuclei, where the product kR is small and we have to deal with the low values of the cut-off momenta, is often violated.
664
3.
DOBEg
In order to remove this inconsistency we proceed in another way. We do not obtain an analytical formula but the numerical computation we have to perform is very easy. Once the plane wave amplitude P;” (eq.(4)) is known as a function of the angle of the final momentum direction fi, (z-axis is directed along the initial momentum s,) one calculates first of all the integrals IL =
s
p;“(&JXY’(&Jd~, -
Then using the orthogonality relation for Clebsch-Gordan quantities Pf” are easily obtained as the sum
(8)
coefficients the desired
In order to calculate the plane wave amplitude PI” we adopt the following assumptions. (i) The Hulthtn form of the internal deuteron wave function ‘“) is used & = N(e-w - e-““)/r,
(10)
with a’ = 7a (a is determined by the deuteron binding energy). (ii) We employ the separation energy prescription “) with a square well potential to obtain the wave function of the captured neutron. Eq. (4) with the wave function (10) becomes [see e.g. ref. 2’)]
4nh2 (a + a’)(a -
pp = _ -
P
1
a’)*N ~ ar2+K2
m~dr)h(~~)~2 s0
dr Y;“*(B),
(11)
where K = k,-3k,, q = k,- [A/(,4 + I)]&, (A is the target mass), p is the nucleon mass and jr is spherical Bessel function. The integration in eq. (11) is easily performed 22) (the radial function 41 is spherical Bessel or Hankel function inside and outside the well) s
o&jt(qr)r2dr
0
=
R2(1/[q2+(2m,/h2)E,1-1/Iq2+(2m,/h2)(E,- VIII x(hi;--44&~.
(12)
Here, m, is the reduced mass of the neutron in the final nucleus, En is its binding energy and V is the depth of the square well with the radius R. The calculation of the integral (8) and of the partial waves PI”’ from eq. (9) is then straightforward using the amplitude PI” from eqs. (11) and (12). Let us mention that we have no difficulties in considering the finite range of V,,,. 4. Results and discussion We have used the models with absorptive corrections for various reactions of the deuteron stripping (d, n) and pick-up Cp, d). The selected target nuclei have closed
aa 2* 2s) 26 2s;
17: 2s
12 25.9 12 15 12 27.5 27.5 12 12 2*
17
ref.
J&WV1
2.63 3.87 3.01 3.01 4.25 4.69 4.24 4.37 4.37
R = 1.25A+fm,R
kd R with
2.18 3.02 2.38 2.30 3.69 4.76 4.76 3.89 3.67
k, R with = 1.25A+fm
0.83 0.64 0.69 0.67 0.23 -0.69 0.66-1.82 0.33-1.72 0.36-0.61 0.35-0.50
present analysis
0.55 -- 1.38 0.32- 1.25 0.6 -0.8 0.5 -0.7
1.16 0.70 0.33 -0.60 0.62
DWBA
)
;
28
9 1
‘)
9
23
25
30)
23
ref.
Spectroscopic factors
0.85 0.82 0.9 0.65
0.61 0.61 0.75 0.80
___--
theory
32; 32 33 .?J
? 29; 31 3’)
ref.
.-
-
The values of products kdR and k, R which are important for determining cut-off momenta are given in the second and third columns. DWBA and theoretical spectroscopic factors are found in the last two columns of the table. Different parameter values give different spectroscopic factors in DWBA and present analysis in the range given in the table.
‘*C(d, p)‘3C(g.s.) 12C(d, p)‘%Jg.s.) 160(d, p)“O(g.s.) r60(d, p)“O*(O.87 MeV) ?Si(d, p)?Si(g.s.) 40Ca(p, d)39Ca(g.s.) 40Ca(p, d)s9Ca*(2.47 MeV) 40Ca(d, p)“Ca(g.s.) 40Ca(d, p)4’Ca+(1.95 MeV)
Reaction
1
The list of examined reactions (column 1) and obtained spectroscopic factors (column 4)
TABLE
5 ii z
:
2 5 :
3
tr 5i
666
J. DOB;5
subshells and one can use with confidence the separation energy prescription. The S-matrix of elastic scattering has been taken in the sharp cut-off model (eq. (6)) and the quantities PF”’ have been calculated as was described above, i.e. with plane waves in the initial and final channels. The Coulomb interaction is thus neglected and our description is valid for energies above the Coulomb barrier. For this reason we have restricted ourselves to the targets with the mass number A S 40. The examined reactions together with corresponding references are summarized in table 1. We consider it to be a rather representative set with regard to the angular
_ : \ 3 c 4 %
5-
24:
cL5-
0.2-
at Ed = 12 MeV. Experimental Fig. 1. Differential cross section for the reaction “C(d, p)‘%(g.s.) points are taken from ref. 23). The curves are calculated from eq. (3) as described in the text. The parameter values are for the solid curves rO = 1.25 fm, Lz = 3 and Lg = 2. The same parameters. are used for the dashed curve except for L: = 2.
momentum transfer and reaction Q-value. The deuteron energy is, except for one case,. in the region of 10-15 MeV, this being the most interesting region from the spectra-scopic point of view, in which majority of experimental data is available as well. We have tried both the geometric mean formula (eq. (3)) and K-matrix approach (eq. (7)). A poor agreement with the experimental data has been found with eq. (7).. All the following analyses concern the application of eq. (3). The results are shown in figs. 1-9 and the spectroscopic factors are summarized in table 1. We have at our disposal three parameters, the continuous one r. being related to the radius R of the square well for the neutron by R = r. A* and two discrete ones the values of the cut-off momenta Li and Lp in the deuteron and proton channels.
DIRECT
20
40
TRANSFER
667
REACTIONS
60
30
m
f20
a,,
Fig. 2. Differential cross section for the reaction ‘%(d, p)t”C (B.S.) at Ed = 25.9 MeV. Experimental points are taken from ref. 24). The curves are calculated from eq. (3) as described in the text. The parameter values are for the solid curve rb = 1.25 fm. 15: = 4 and Lg = 3. The same parameters are used for the dashed curve except for Lz = 3.
Fig. 3. Differential cross section for the reaction 160(d, p)“O (p.s.) at Ed = 12 MeV. Experimental points are taken from ref. 2s). Th e curve is calculated from eq.(3) as described in the text. The parameter values are re = 1.25 fm, Ld, = 3 and Lg = 2.
668
J.
DOB&
..*.: ..nr ,l-l l*
.
. .’
.
l
l
,
20
40
60
80
4W
00
Fig. 4. Differential cross section for the reaction 160(d, p)“0*(0.87 MeV) at & = 12 MeV. Experimental points are taken from ref. *‘). The curve is calculated from eq. (3) as described in the text. The parameter values are r0 = 1.25 fm, Lt = 3 and I$ = 2.
I-1
I
20
I
40
I
60
Fig. 5. Differential cross section for the reaction *%(d, p)2qSi(g.s.) at Ed = 15 MeV. Experimental points are taken from ref. 26). Th e curves are calculated from eq. (3) as described in the text. The parameter values are for the solid curve r0 = 1.25 fm, Lg = 5 and Lg = 3. The same parameters are used for the dashed curve except for Lg = 4.
Fig. 6. Differential cross section for the reaction *‘%a@, d)sgCa(g.s.) at En = 27.5 MeV. Experimental points are taken from ref. 17). The curves are calculated from eq. (3) as described in the text. The parameter values are for the solid curve rO = 1.25 fm, L,6 = 4 and L+ = 4. The same parameters are used for the dashed curve except for Ld, - 5. Decrease of rO to rO - 1.15 fm does not change the shape of the solid curve, spectroscopic factor, however, is increased as indicated in the figure. (ebl,,,ciJt&47WJ ---
I
I 20
I
I 40
I
I 60
L$=5 l-d=4
I
9
,
8o
452”
KV
s=172 s=a33.Kj=mfm (S=a52 c=7.%bJ
,
I
100
I
I
I
120
Fig. 7. Differential cross section for the reaction ‘&a(p, d)Wa+(2.47 MeV) at Er, = 27.5 MeV. Experimental points are taken from ref. “). The curves are calculated from eq. (3) as described in the text. The parameter values are for the solid curve re = 1.25 fm, Ld, = 4 and Lp = 4. The same parameters are used for the dashed curve except for L$ = 5. Decrease of rO to re = 1.15 fm does not change the shape of the solid curve, spectroscopic factor, however, is increased as indicated in the figure.
__.
-.
.~._
‘&,(~,&dg t
J
I 4=12 M.V
1=3
J
Fig. 8. Differential points are taken parameter values change the shape
cross section for the reaction *Wa(d, p)“Ca(g.s.) at Ed = 12 MeV. Experimental from ref. 2s). The curve is calculated from eq. (3) as described in the test. The are r0 = 1.25 fm. L.$ = 4 and Lg = 3. Decrease of r,, to r,, = 1.15 fm does not of the curve, spectroscopic factor, however, is increased as indicated in the figure.
Fig. 9. Differential Experimental points text. The parameter does tiot cliange the
cross section for the reaction q°Ca(d, p)41Ca*(1.95 MeV) at Ed = 12 MeV. are taken from ref. za). The curve is calculated from eq. (3) as described in the values are r,, = 1.25 fm, Lz = 4 and L; = 3. Decrease of r0 to r,, = 1.15 fm shape of the curvd, spectroscopic factor, however, is increased as indicated in the figare.
DIRECT
TRANSFER
REACTIONS
671
Only minor changes are exhibited in the shape of the angular distributions with the variation of r,,, the spectroscopic factors, however, are lowered with the increasing re. The choice of the cut-off momenta shows a major influence on the cross sections and there exists some uncertainty particularly for deuterons. In the second and third columns of table 1 the values of the products k,R and k,R with R = 1.25 A* are given. All the calculated curves have Lg equal to the largest integer smaller than the value k,R. For the reactions in figs. 1, 2, 5-7 two choices of Li have been used, one value being chosen larger and the other smaller than k, R. The larger value Ld, is slightly preferred for the reactions on “C and ‘*Si while for the reactions on 40Ca the smaller value is better. The parameter r. of the neutron potential well is 1.25 fm. Spectroscopic factors given by this value r. were found to be smaller than expected for the reactions on 40Ca. The agreement was improved by lowering r. to 1.15 fm, that was still within the reasonable range. The shape of the calculated angular distributions follows the experimental course very well. The worst agreement relatively exists for the reaction 40Ca(p, d)39Ca with I = 0, this transition, however, can be fitted by DWBA with difficulties as well “). The spectroscopic factors, particularly for the lightest nuclei, are in a good agreement with the DWBA analyses and the theory (DWBA and theoretical spectroscopic factors are given in the last two columns of table 1). The change of the parameter Ld, yields different values of the spectroscopic factors for nuclei Si and Ca. However, looking at the DWBA spectroscopic factors of the reaction 40Ca(p, d)39Ca in table 1, one finds that similar differences can occur also in the DWBA analysis. We have achieved a surprisingly good agreement of experiment and theory cutting off the small angular momenta from plane waves in the initial and final channels. The idea of neglecting contributions from the lowest partial waves is a very wellknown one and it has been utilized in the derivation of Butler theory for (d, p) stripping 34). One assumes that this neglecting and the absorption of the particles penetrating into the nucleus can be simulated by cutting off the radial integral. Then the integral in eq. (11) is not taken from zero but from the non-zero value of the cut-off radius. The fact that the Butler theory gives only a poor fit to experiment compared to the good results of the present calculations suggests that the radial cut-off is only a rough approximation to the I-space cut off. The very simple and easily computable model which we have examined in this paper explains the experimental data satisfactorily. The absorptive corrections are well exhibited in the geometric mean formula while the K-matrix approach was not found to be very successful. The model could be suitable in the first off-hand analysis of measured results and its simplicity could be useful in the theoretical study of effects the calculation of which is difficult in DWBA. The author would like to thank Dr. F. Janouch for many helpful discussions. Valuable conversations with Dr. J. Cejpek and Dr. P. Pajas are also gratefully acknowledged.
672
I.
DolIES
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