Systematics of nucleon transfer between heavy ions at low energies

I

2-N

I

Nuclear Physics Al76 (1971) 299-320;

@ North-Holland

Publishing

Co., Amsterdam

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

SYSTEMATICS

OF NUCLEON

TRANSFER

BETWEEN HEAVY IONS AT LOW ENERGIES P. J. A. BUTTLE and L. J. B. GOLDFARB t The University,

Manchester

MI3 9PL

Received 26 April 1971 Abstract: A discussion is presented of the important dynamical factors influencing the yield in heavy-ion transfer reactions at energies below the Coulomb barrier together with illustrations of calculations for various targets and beams. It is found that the distances of closest approach of the ions before and after the reaction should be similar if cross sections are not to be extremely small. Consideration is given to the implications of this requirement as regards possible experimentation. Recoil effects are found to be important, especially for beams incident on very heavy targets and a technique is suggested for including these effects in the DWBA formalism.

1. Introduction With the promise of new accelerators for heavy ions and new techniques for the separation and detection of reaction products ‘), greater interest has been given to the possibilities associated with nucleon transfer processes initiated by heavy ions. Easiest to describe are the sub-Coulomb processes which are characterized both in the entrance and in the exit channel by motion that is predominantly Coulombic. These were first examined by Breit and Ebel “) who utilized a semi-classical approach in the study of the 14N(14N, 13N)r5N reaction and further elaboration taking into account quantum features has since been made by Breit et al. “). Parallel with this, the DWBA has been advocated “) and extensions have been made “) particularly to take into account transfers with arbitrary angular momentum, to allow for particular aspects of proton transfer “) and to extend the study to higher energies ‘, “). Our intention is to reconsider the process at sub-Coulomb energies. We ignore higher energies owing to the major uncertainties concerning the nucleus-heavy-ion interaction and the difficulties of taking this interaction properly into account. Our principal concern is to search for systematic features affecting the yield when different choices are made for the interacting target and projectile and for differing reaction Q-values. We shall show that these yields, in contrast to the usual transfer phenomenon, can deviate by an enormous range of magnitudes. The most obvious criterion for a high yield is that the nuclear transition be associated with states showing a high degree of overlap. In a sense, this is what we mean in demanding that the spectroscopic factor be large. The dynamics of the process, however, suggest even more rigid requirements. In particular, the yield should increase 7 A preliminary account of this work is to be found in ref. I)_ 299

300

P. J. A. BUTTLE

AND

L. J. B. GOLDFARB

the closer the energy is to the Coulomb-barrier energy, since there is then a greater probability that the Coulomb repulsion will be overcome. Equally important is the degree of similarity between the initial and final relative motions. For a high yield, the relative wave functions should have a similar radial form, at least for the most likely transfer radii. Expressed succinctly, there is a strong requirement that the classical distance of closest approach, a (= zZe2E-‘) be roughly the same for incoming as well as outgoing waves. Here, z and Z refer to the projectile and target initially and to the reaction product and residual nucleus finally and E denotes the relative kinetic energy. We can understand this condition from classical reasoning. The interacting nuclei move along relative hyperbolic trajectories. The transfer is most likely to occur on closest encounter, i.e., at relative distances Ui and a, before and after. If the classical view is to hold, we should expect the magnitude of the difference Idal = lUi-u,( to be less than the wavelength of relative motion; otherwise, the yield should be comparatively small. This feature is explored in greater detail in sect. 2 following a description of the basic results of DWBA as applied to heavy ions. Theoretical analyses have relied on the assumption that the mass of the transferred nucleon is small compared to that of either interacting nucleus. This leads to errors of the order of A - ‘, where A is the nuclear mass number. A semi-qualitative attempt has been made by Breit ‘) to assess the importance of recoil effects by consideration of a one-dimensional example involving a pair of &function potentials of equal strength. Dodd and Greider lo ) have also proposed a technique for including recoil effects; however this is designed to deal exclusively with high-energy processes. Apart from this, the full finite-range calculations of Kammuri and Yoshida “) represent the only treatment taking this feature properly into account. Their calculations cover energies above the Coulomb barrier except for one process, the 14N(r4N, 13N)15N reaction. We consider recoil effects in greater detail in sect. 3 and find them to be particularly important for very heavy targets. An approximate procedure enabling one to deal with these features is proposed. In view of the pronounced dynamic dependence of the reaction yield, we present in sect. 4 simple conditions to be met in an experiment and suggest favourable beams and targets taking into account these criteria. There is a natural separation according as to whether there is proton or neutron transfer. 2. The DWBA treatment We first summarize the DWBA treatment of heavy-ion reactions as described in refs. 50“). We consider the reaction: (Cl +t)+c, --) ci +(c,+t), al a2 where a nucleon t is transferred from the nucleus a, to the nucleus c2. The target may be either a, or c2. The letters a,, a2, cl and c2 are also used to denote the spins

NUCLEON

301

TRANSFER

of the nuclei, while the spin components are represented by the Greek equivalents. In the post representation the DWBA amplitude is a matrix element of the interaction AV,. Viewing c1 and c2 as inert cores, AVf can be written: AV, = V,,(r,)+

V,,(r)-

UTP’(r,)+Z1 Zte2r;l+Z1

Z,e2r-1-Z,(Z2+Z,)e2r~1,

(2.1)

where the vectors vi, Y and rf are those illustrated in fig. 1, VIt(r,) is the nuclear interaction between t and c1 , VI2 is the nuclear interaction between the cores c1 and c2 and U,Optis the optical potential between c1 and a2 used in computing the outgoing distorted wave xr(-) . The charge numbers of c1 , c2 and t respectively are Z, , Z2 and Z,. At energies well below the Coulomb barrier, the strong Coulomb repulsion keeps the cores well separated so that r and rt are significantly larger than the ranges of

Fig. 1. The coordinate

system.

from the second and third term of eq. (2.1) is therefore VI2 and UFp’. The contribution very small. The last three terms are also small but deserve some consideration and a procedure for handling them is given in ref. “). Thus, the DWBA amplitude is principally the matrix element of Vlt(rI). The prior representation involves the interaction A Vi which differs from A V, in that the suffices 1 and 2 must be interchanged and rf is to be replaced by ri. Similar reasoning shows that Vzt(r,) is the major term in A Vi. It can be shown that the post and prior DWBA amplitudes are rigorously equal: (&-‘lAVflxi+‘)

= ($‘lAVjx!+‘) 1,

.

(2.2)

Discrepancies arise when approximations are made in the evaluation of these amplitudes. For this, two approximations are made. To be definite, we deal with the post representation. Firstly, the vectors Yi and or are replaced by r and Y’ = mc,m&’ r respectively where m,, and mcI are the masses of nuclei a2 and c2, respectively. This approximation involves errors ot order A-l, where A is the mass number of the nuclei and we believe this approximation accounts for the main differences between the post and prior calculations. This will be discussed in sect. 3.

302

P. Je A. BUTTLE

AND

L. J. B. GOLDFARB

The second approximation is to repXace the radial bound-state uf2[rJ by a spherical Hankel function:

wave function

This is reasonable at energies well below the Coulomb barrier for neutron transfer since the presence of F;*(ri) in AV, constrains p1 to be of the order of the core radius Rr so that rz is considerably greater thau the core radius &. The quantity x2 relates to the bindingenergy Bz of the neutron to the core c,@?Xf = 2pIsB,). Eq. (2-3) also applies to proton transfer “) provided that ii coyers a restricted range of values of r, _ The quantities N2 and xz: are then taken to be adjustable parameters. The DWBA amplitude then involves the following integral:

where f(r) is a form factor described iu ref. “>. The nuclear part of d Vr (i.e., the contribution of Vlt(rl)) leads approximately to the form:

where Str, is an integraI ~escri~ in ref_ “)_ 1n additions there are the terms arising from the Coulomb con~ribntions in eq. (2,1), the expressions for which are given in ref. “). The differential cross section for the process is giveu by:

where pni and an, are the reduced masses9 initially and finally, and +!?l’ and Sq2’ are spectroscopic factors for the two nuclei. The integral Yll(0) is usually calculated by making a partial-wave expansion and using a conventional DWBA program. It is free of the usual ambiguities associated with the use of distorted waves- We are therefore provided with an attractive ~ossib~l~t~ of extracting nuclear ~~c~~~s~#~~~ factors. When several I-values occur in eq. (2,6) it is found that the largest I-value always gives the Xargest contribution to the cross section, usually by a factor of 2 or more. This is probably due to the slower fall-off of the Hankel function hl” for large f# The effect may atso be understood in terms of the fol~~~i~~ classical picture. The transferred particle is mast likely to be transferred if it is initially in some orbit such as the clockwise one iIIustrated in fig. 2. After transfer it will naturally tend to orbit the core c2 in the anticlockwise direction, so that the transfer of angular uromentum (i.e., that component norma to the plane of the paper) is I = I, +I,, the maximum value possible. Most of the sub-Coulomb transfer processes already studied are characterized by HOW Q-x&es. fnparticular, the r4N(1dN, rsN)‘%, raN(i4N, ‘3C)1S0, laN(l*% “g)

NUCLEON

303

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13N and 40Ca(‘4N, 13N)41Ca ground state reactions are associated with Q-values of 0.28, -0.26, -0.90 and -2.19 MeV respectively. That the reactions are strongly Q-dependent was already evident in calculations by Goldfarb and Steed ‘) for the

Fig. 2. Classical picture illustrating how the largest possible angular-momentum gives rise to the main contribution to the transfer reaction.

lo+ -2

f

I

4

/

I

I

I

I

-7

0

I

2

3

4

5

6

transfer I = /I +I2

;

Q-w&e in MeV Fig. 3. Variation of yield with Q-value for the reaction z5Mg(170, ‘60)Z6Mg. The binding energy in 26Mg is varied while the binding energy in I70 is fixed at 4.14 MeV.

14N+ 40Ca reaction leading to states of 41Ca. The experiment did not resolve final states of 41Ca, but calculations showed that the yield to the first excited state (which involves a Q-value of -4.2 MeV) decreases by a factor of more than a hundred. An example of an enormous Q-dependence is provided by calculations for the reaction 25Mg(170, 160)26Mg, the ground state transition being associated with a Q-value of 6.95 MeV. The yield at 180” which is plotted against Q-value in fig. 3,

304

P. J. A. BUTTLE AND L. J. B. GOLDFARB

shows a dramatic increase of nearly four orders of magnitude as Q falls to zero. This phenomenon runs counter to what is expected, for as the Q-value decreases, less energy is available to overcome the Coulomb barrier and the yield should decrease. An explanation might be offered in terms of the increased tail-wave function for the neutron in 26Mg, but this fails to explain why exactly the opposite effect occurs in 40Ca(‘4N, 13N)41Ca.

I

25Mg(170,'60)26Mg

f3

12MeV &=I0

-F

r in

r In

Q-695MeV lntegrand

Fig. 4. The radial part of the Coulomb wave functions and the corresponding integrand for the reaction 25Mg(‘70, 160)26Mg. The two top curves are for Q = -1.14 MeV while the two lower curves are for Q = 6.95 MeV.

A more convincing explanation can be given in terms of the initial and final distances of closest approach ai = 2qi/ki and a; = 2q,/k;, where ?i and qr are the usual Coulomb parameters (i.e., v] = zZe’/hv). These lengths affect the general shape of the Coulomb waves appearing in the integral (2.4) for r,(O). For small distances, the Coulomb functions are very small. At the distance of closest approach they rise abruptly to a broad maximum before settling down to their asymptotic

NUCLEON

TRANSFER

305

oscillatory behaviour. This contrasts with the form factor (2.5) which falls off exponentially so that the net result is a localization of the integrand around the region of the distance of closest approach, as one might expect from classical arguments. If z a; the two broad maxima contribute a coherent amount to the integral. On the ui other hand if ai and a, differ by very much, the broad maxima do not coincide and the integrand takes on an oscillatory nature which leads to a small amplitude. These effects are illustrated in fig. 4 for the reaction zsMg(‘70, 160)26Mg for the partial waves Zi = lr = 10. We suggest therefore that one criterion for a large yield is that a;. Naturally for neutron transfer the equality of a, and a; is roughly equivalent ai z to having a small Q-value. We should also want the waves in both entrance and exit channels to have similar wave lengths, i.e., equal or near equal values of k, so as to ensure maximum overlap and larger yields. An analytic treatment of integrals such as (2.4) for Coulomb stripping has been proposed by Ter-Martirosyan and others 11) to deal with (d, p) reactions and applied by Breit 12) to heavy-ion reactions. This treatment involves hypergeometric functions and is valid only if: (a) the form factor r -‘f(r) is a spherical Hankel function and (b) the transferred angular momentum I is zero. Since previous experimental work has been mainly ccncerned with low-Q reactions, it was reasonable to make the additional approximation that ki = k, and vi = nr. We are however interested in reactions where this is not so and it is therefore convenient to summarize the relevant formulae for the general case. At the same time we wish to extend the technique in an approximate way for arbitrary values of I. Some extensions of the Ter-Martirosyan treatment along these lines have already been proposed 13). These make use of a modified form factor; however the resultant expressions are fairly elaborate. We prefer to adopt a simpler procedure, similar to that already used 14) for (d, p) reactions below the Coulomb barrier. The form factor r -‘f(r) can usually be approximated extremely well in the region of interest by a Hankel function Fhy’(ip) where F and x are adjustable parameters: r-If(r)

w Fhr’(ip-).

(2.7)

These parameters can be evaluated by a separate computation. If necessary, one could do even better with a series of terms: xi FihF’(iXir), but this amount of effort is not usually needed. The spherical harmonic Y,*,(P) in eq. (2.4) is a slowly varying function and the main contribution 14) to the integral come from one region of space r z R along the recoil direction. If we replace Y:(P) by Y:(i), it can be taken out of the integral and the sum over it in eq. (2.6) is easily performed. The cross section then becomes: g = dS;!

4n

mimf _kf 2a2+1 ____ S(r)S(‘)Ni

(2~~2)2

ki 2c,+

1

F F2(j1 5, iZ -!$O)21T,o(0)12,

(2.8)

306

P. J. A. BUTTLE

AND L. J. B. GOLDFARB

where in the post representation:

(For Coulomb waves, xi-) IS ’ a function of (kf . Y’) = (kf * m,,ma<‘v) = (k; - P) where k; = RFZ,~~,~&). It is perhaps worth remarking that both F and x depend on the i-value, since the form factor r-If(r) depends “) on 1. The expression (2.9) is particularly convenient as it can be evaluated in terms of a h~ergeometric function:

T00 =

q1 + i&)T(l i- iqf) [(kf_ki)2+xZ]l*i~i+iqr

x

where -~ 4ki kr lo = (k;-ki)‘+X2

[ = co sin2 to,

(2.11)

’

and r is the gamma function. Sommerfeld has shown 1“) how the hy~rgeometric function may be approximated for large values of vi and uf in terms of exponential functions. One finds that:

ITool =

8n2rli

exP [2rli(~-~)-2rlf(J/‘-_‘)1

Vf

x”[(kf-ki)’

+x2]’

ro + CM@)

(2.12)

’

where s(e) =

+E4rli?f

tan #J =

2Xki k; -ki2 -X2 ’

cos J, =

t4i-qfX+2tli %%(l+

tan (p’ = cos

r))

(2.13)

l-‘-fVi-11f)21”,

+F =

’

2& kf --,$2 +x2 ’ f?i-Vf)i-2qf ---

(2.14) ’

%f(1+5)*

The four angles 4, +‘, $ and $’ all lie in the range 0 to 7~.This approximation is reasonably accurate for large values of vi and qf as long as 5 > V- ‘. Application of some tedious trigonometry leads to a slightly more concise form for ITo0(8)12: stool”

=

8X-_

r7i%

I2

x2 [(k; - ki)’ +x212 i(i;fl)so

(2.15)

’

where I2 = exP [ -2(qi+qf)

(arctan ki:k;

+arctan qzj

k;- ki I --2(tli - ?lf) iarctan - x +arctan ‘%I

j .

(2.16)

NUCLEON

307

TRANSFER

The four angles appearing as inverse tangents are in the range --)n to +$T. The cross section is obtained by inserting this expression for (~00(0)1” in eq. (2.8). The corresponding prior form of l’ir,,(e)l ’ is obtained by replacing ki by ki = m,,(m,,)-‘k, and k; by k, in eqs. (2.15) and (2.16). In addition, the parameters F and x will be different since the form factor (2.7) is different.

o/, , , , 40

I

,

I

50

,L”px,, 60

L

1

I

I

I,

76

‘0

Fig. 5. Variation of the total cross section and the 180” cross section as a function of J%,,,,,,in a conventional DWBA calculation for the reaction z0sPb(‘70, 160)209Pb. The dashed lines show the result of a calculation using the Ter-Martirosyan calculation.

Calculations comparing these expressions with a conventional DWBA partial-wave analysis indicate that an agreement to within 5 % or better is obtained for 1 = 0 cross sections. The disagreement becomes worse (as expected) for larger l-values, rising in an approximately linear way to about 50 y0 for 1 = 7 transitions. We have not done exhaustive tests on this however and individual cases might be outside these limits. The expression (2.16) is difficult to interpret in any simple way but the following observations seem to be valid. The first term in the exponential is responsible for the well-known energy dependence and backward peaking of the cross section, while the last term usually contributes in a negative way so as to reduce the cross section. The best one can do is to reduce its effect as much as possible. For example if Ui = a;, one can show that this last term vanishes at 180” and the expression for I* reduces to:

I2 = exp

- 8rj arctan

x 2k

= 8=7T), 1 (Ui Uf

3

’

(2.17)

308

P. J. A. BUTTLE

AND

L. J. B. GOLDFARB

where q and E are the arithmetic means of ri, qr and ki , k;, respectively. If Ui # ai, the last term usually acts so as to reduce the cross section exponentially. It is also of interest to compare the situation with that of Coulomb excitation, where it is found that the parameter 5 = 1~~- qrj should be small for a large yield. In Coulomb excitation a massless photon is exchanged instead of a nucleon. This is equivalent to the restrictions 2 = 0 and qikj = qtkE in our eq. (2.16), which then reduces to the form:

1’ = exp

i

-4e arctan (5 cot $3) -e(n+B)]

.

(2.18)

This may be compared with a classical Coulomb excitation expression [eq. (2.41) of ref. 16)] if we use the classical identification &I-’ = cot 30. In addition, if 8 = 180”, we obtain the factor exp (-2x5) which again is familiar in CouIomb excitation. The approximate expressions (2.15) and (2.16) are particularly useful for reactions with very heavy targets such as ‘*‘Pb. A conventional DWBA program using a partial-wave analysis would tend to break down here unless allowance is made for the very large number of partial waves that are needed. Consider for example the reaction 208Pb(170, 160)209Pb with a c.m. energy of 60 MeV. Fig. 5 shows how the total cross section or and the differential cross section at 180” vary with the maximum number of partial waves L,,, for the 1 = 0 part of the calculation. The prediction of the Ter-Martirosyan calculation is also shown. The total cross section appears to have reached a reasonably steady value for Lmax x 70. The Ter-Martirosyan value is about 5 y0 higher, which could either arise from the approximations in the TerMartirosyan treatment or from systematic errors in the DWBA program. The differential cross section at 180” has clearly not settled down even for 76 partial waves. This can be understood for I = 0 transfers where the cross section is expressible simply in terms of the square modulus of a sum of Legendre polynomials which satisfy P,(cos 180°) = (-)“. The discussion of the early part of this section concerning the need to match the initial and final distances of closest approach also applies to reactions involving very heavy targets. In fig. 6 the Coulomb wave functions and integrands are shown for the 2 = 0 part of the reaction 208Pb(“70, 160)209Pb. Two cases are shown corresponding to Q = 0 when Ui ,Y a; and Q = 5 MeV when ai --a; z 1 fm. In the latter case it is clear that oscillations in the integral must reduce the cross section severely and that the transfer is no longer localized. This is evident from the plot of the integrand shown in fig. 6. The plot of yield as a function of Q-values shown in fig. 7 confirms that the transfer is more likely the smaller the magnitude of Aa = ai-a;. Proton transfers are especially interesting for these heavy targets. The transfer of a charged particle necessitates a large change in outgoing energy in order to balance the incoming and outgoing distances of closest approach. The maximum yield is still expected when cli 3 a;, but this requirement will no longer be associated with a zero Q-value. Fig. 8 shows the yield for the reaction 208Pb(‘60, 15N)209Bi and con-

NUCLEON

TRANSFER

309

firms this view. The sharpness of the peak seems to be less than that of the neutron transfer and this could possibly be because the Coulomb wave functions do not overlap so closely even when Iii = a;.

Fig. 6. The radial part of the Coulomb wave functions and the carrespondig integrand for the reaction soaPb(170, 160)20gPb, The two top curves are for Q = 0 MeV and the two lower curves are for Q = 5 MeV. The c.m. energy is 65 MeV.

Althaugh the basic post and prior amplitudes (2.2) are equal, different results in actual calculations reflect the approximations made in the evaluation of the amplitudes. Previous arguments “> as to the choice of representation have been made on the basis of the relative binding energy of the nucleon to the two cores. In the post

310

P. J. A. BUTTLE AND L. J. B. GOLDFARB Aa

in fm

-0.5 I

0 I

0.5 I

10 I

208Pb(‘70,160~

163 -4

' -3

1 -2

1 -1

1 0

1 1

‘ 2

Pt? (Ln)

0 4

3

' 5

*

'A 6

W&e in MeV Fig. 7. Variation of the 180” differential cross section with Q-value for the reaction z0SPb(‘70, 160) *09Pb. The binding energy in I’0 is fixed at 4.14 MeV while the binding energy in *OsPb is varied. acl -10 I

-05 /

in fm 0 I

05 1

10 I

‘mpb( ‘“0, ‘%J)203~~w1LPI

l-

69 IMeV ai = 14.92fm

10" -12

' -11

8 -10

' -9

' -8

' -7

' -6

I -5

' -4

-3

Q-value in MeV Fig. 8. Variation of the 180” differential cross section with Q-value for the reaction 208Pb(‘60, “N) *09Bio.90. The binding energy in IsO is fixed at 12.13 MeV and the bindingenergyin *09Bi is varied.

NUCLEON

311

TRANSFER

representation, the approximation (2.3) is suggested as being the more reasonable the higher the binding to core 1. In contrast, the prior representation approximates ul,(rl) with a Hankel function and would seem to be more reasonable tor strong binding to core 2. A post treatment therefore seems proper for negative Q-values and a prior treatment for positions Q-values. Either representation could be used for very low Q-values.

.Q-value

in MeV

Fig. 9. Comparison of post and prior calculations for the reaction 14N(r4N, 13N)“N. The solid curves are the results of calculations without recoil effects. The dashed curve is a post calculation and the dotted curve is a prior calculation, both including recoil effects. The variation in Q-value is produced by varying the binding energy in 15N and the physical value is shown for the ground state transition.

The r4N(14N, 13N)15N reaction with Q = 0.28 MeV could clearly be analysed with either treatment. Fig. 9 shows the results of the two calculations for varying Q-values. The prescription would in this case be to use the lower curves, post for Q < 0 and prior for Q > 0. The difference is surprisingly great for Q-values of &2 MeV. It seems unlikely that deviations of this magnitude should be ascribed to the Hankel-function approximation [eq. (2.3)]. We are forced therefore to reconsider the first approximation mentioned after eq. (2.2) the neglect of the so-called recoil or A- r effects. The vectors ri, rf can be written: ri

=

r-

mt

6,

rl,

where m, is the mass of the transferred

yf = mc”r+

ma, particle.

m,

rl,

ma2

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P. J. A. BUTTLE

AND

L. J. B. GOLDFARB

The approximation appropriate to the post representation is to ignore the part depending on Ye. In the prior treatment, we express ri and r, as follows:

and neglect the parts involving r2. A study of the six-dimensional integral in the amplitude (2.2) suggests that it would be more reasonabfe to replace rr andr, in eqs. (3.1) and (3.2) by physicafiy acceptable non-zero values. For simplicity, we advocate replacing ri and rt by some multiple of the vector r so that the integral may be done in a convenient numerical way. We consider first the contributions for fixed values of r, i.e., for fixed positions of the cores. In the post representation, the main contribution to the rr integral must eome from inside or very close to the core c tr since the main part of the interaction VJr,) falls off very quickly outside this core. On the other hand the bound state wave function uf,{rZ) favours contributions from the region nearest to c2. The largest contributian should therefore come where the line of centres c1 c2 crosses the nuclear surface of cl, i.e., when r1 M (RJrl)r, where R, is the radius of cr. Turning now to the I’ integral, the main contribution arises when Y is approximately equal to the distance of closest approach IPI M a = Zqjk. Since this may differ in the two channels we will assume that a denotes the average value. Thus rl x (R, ./a)~. We are therefore led to the approximation:

~irnj~arl~, in the prior representation

we suggest the following approximation:

where RZ is the radius of nueteus 2. We refer to these improved approximations as recoil corrections, as opposed to the previous basic treatment which assumed m, = 0 in eqs. (3.1) and (3.2). Since the distorted waves are Coulomb wave functions, the wave numbers kr and kf always appear in the wave functions in the form kiri or k,r,. The recoil correction is therefore equivalent to replacing these quantities by kfr or k;r, where in the post treatment:

P-5) The wave numbers ki and ki, appearing in the approximate Ter-Martirosyan expressions (2.15) and (2=16), should therefore be replaced by the expressions (3.5) in the post representation. A similar modification should occur in the prior formalism with

-4

-3

-2

_,

0

1

2

3

4

Q-value in MeV Fig. 10. Comparison of post and prior calculations for the reaction 40Ca(14N, 13N)%a. Recoil corrections are omitted for the solid curves and included for the broken curves. The dashed curve is a post calculation and the dotted curve a prior calculation. The variation in Q-value is produced by varying the binding energy in ‘%a and the physical value for the ground state transition is indicated.

Fig. 11. Comparison of post and prior calculations for Recoil corrections are omitted for the solid curves and variation of Q-value is produced by varying the binding the ground state transition

the I = 3 reaction z0sPb(‘60, 170)207Pb. are included for the broken curves. The energy in 208Pb and the physical value for is indicated.

314

suggested

P. J. A. BUTTLE

AND

L. J. B. GOLDFARB

values ki’ and 1~;’ given by: k;’

The broken curves r4N(14N, 13N)=N.

=

‘72,,+ !!k R? ki, ma, ma, a >

k;’ =

(I- ,z:)

k f'

(3.6)

in fig. 9 show the resultant improvement for the reaction Th e agreement between the post and prior treatments after including the recoil corrections is now quite acceptable. Fig. 10 shows similar results for the reaction 40Ca(‘4N, 13N)41Ca. In this case the recoil corrections have actually widened the gap between the post and prior treatments at the experimental ground state energy; but the overall agreement at other Q-values (unphysical though these may be) is clearly much better. In fig. 11 post and prior results are shown for the reaction 2osPb(‘60, 170)207Pb and we see that these can differ by factors of several hundred in places. It is worth asking why the A-’ corrections are more important for very heavy targets than for light ones. The distances ignored in the usual approximation are of the type m,ma~‘rl or m,maJ’rl [see eq. (3.1)], w h ere rl is of the order of R, . Thus, we ignore distances of the order of the radius of one nucleus R, , divided by the mass number of the other (A,). For I60 on 208Pb, RI/A2 could be as large as 0.5 fm. Since the wavelength of a distorted wave may be as low as 1 fm (for a 70 MeV oxygen beam), a shift of this magnitude may cause considerable errors. The problem is less acute for light targets because R, /A2 z 0.25 fm and since the Coulomb barrier is much lower, larger wavelengths (Y” 4 fm) are likely. The above arguments show that for reactions involving very heavy targets, the choice between post and prior representations should be made according as the heavy target is receiving or donating the transferred particle, i.e., whether the target is a, or c2 in the notation of sect. 2. Suppose first that the heavy target is a, and that it donates a nucleon. In the post treatment we ignore distances Rl/Al and RI/A2 in eq. (3.1). Clearly, RI/A2 is quite large. The prior treatment involves ignoring the corrections terms R2/A1 and RJA2 that appear in eq. (3.2) and although the latter distance is uncomfortably large, the error is far less. Thus, the prior treatment is better used if a very heavy target serves as a particle donor. Conversely, the post treatment is to be preferred for a light-donor nucleus and a very heavy receiver. If target and projectile are of similar size, the Q-value criterion should be used to choose between post and prior representation. Recoil corrections improve both calculations considerably, but the same general conclusion concerning the choice of representation is valid. Returning to fig. 11, we note the improvement owing to recoil corrections. At the experimental ground state energy however a difference of a factor of 1.5 still remains and gives some cause for concern. Still, we believe the result based on the prior treatment as being more reasonable here since a heavy donor is involved. The factor of 1.5 does not represent a discrepancy between two equally acceptable results.

NUCLEON

315

TRANSFER

We show in fig. 12 similar curves for the proton transfer reactions 208Pb(160, “N) “‘Bi. The basic treatments differ by an order of magnitude whereas the inclusion of recoil effects reduces the discrepancy to a factor of 2 at the ends of the range. We prefer the result based on the post treatment in this case since a light donor is involved. An attempt was made to test the sensitivity of our calculations to these recoil approximations as follows: we note that the six-dimensional integral in the amplitude

-12

-11

-10

-9

-8

-7

-6

-5

-4

-3

Q-value in MeV Fig. 12. Comparison of post and prior calculations for the reaction 208Pb(‘60, 1sN)20qBi. Recoil corrections are omitted for the solid curves and are included for the broken curves. The variation of Q-value is produced by varying the binding energy in zoqBi and the physical value corresponding to the first excited state in z0q3i is indicated.

normally involves integration variables Y and either r1 or r2 according to the choice of representation. We propose to use instead of rI the variabIe vector p which is a linear combination of Yi and rr: P =

Prff(i-P)ri

(3.7)

and to let p vary from 0 to I. The cross sections should be independent of p, but only if the calculation is an exact one. Any p-dependence reflects on the reliability of the recoil approximations where yi and ur are expressed in terms of p alone in analogous fashion to eq. (3.3). Calculations for the proton and neutron transfers associated with 69 MeV 160 ions incident on “*Pb show deviations which are never greater than

316

P. J. A. BUTTLE

15 “/Lin either representation, tion of integration variables to have been much calculations.

more

AND

L. J. B. GOLDFARB

in spite of the fact that the Jacobian for the transformais in certain cases quite sizeable. If the sensitivity were than

this, it would

4. Systematics

cast doubt

on the reliability

of our

of nuclear transfer

The strong Q-dependence of reaction cross sections, which reflects the need to balance the distance of closest approach before and after the nucleon transfer, severely limits the field of fruitful experimentation. The large variety of choices for targets and projectiles, on the other hand, compensates for this deficiency. Since the kinematical constraint is simple to describe, we can easily present optimum conditions for yields for arbitrary choices of targets and projectiles. In terms of the c.m. energies and nuclear charges, the requirement Ui = a; can be written: Z,, Z,, E; ’ = Z,, Z,, E; I. (4.1) This implies whereas:

that

Ei x Ef for optimum

conditions

Q 1 __w--_Ei

zc,

in the transfer

1 za,

of a neutron

(4.2) ’

for proton transfer if we ignore quantities of second order in Z-‘. In the cases of light targets and projectiles, the charges are similar and Ei is restricted to lie below the Coulomb barrier, so that the optimum Q-value is within about 1 MeV of zero for proton transfers as well as neutron transfers. In considering very heavy targets it is convenient to specify transfers according to the size of the donor nucleus. Thus, Ln or Lp refer to reactions in which the light projectile donates a neutron or proton, whereas Hn or Hp refer to reactions in which the heavy target donates a neutron or a proton. The Coulomb barrier is much higher and the energy E, is likely to be 50 MeV or more. Also, the charges Z,, and Z,, are quite different. Thus the optimum proton transfer region will lie some way from Q =O. Considering first the Lp reactions, the quantity Z,,-’ is fairly negligible and then the optimum

Q-value

is:

Q z -EiZ,‘,

(4.3)

(LP),

where a, refers to the light nucleus donor. Similarly for Hp reactions, a, is the heavy nucleus and c2 the light nucleus and the largest yields are to be expected for reactions in which: Q ~ +EiZ,‘, On the other hand, light targets.

low Q-values

are favoured

(4.4)

(HP). for neutron

transfer

as in the case of

NUCLEON

317

TRANSFER

The situation is summarized in fig. 13. In general the proton transfer regions seem to be broader than the neutron region, but this is offset by the larger peak height for neutron transfer. This, no doubt, reflects the fact that proton transfer is inhibited by the Coulomb barrier which the proton must cross. A particular example of the two types of proton transfer is provided by the reactions 17F)207T1. The experimental Q-values for zo8Pb(160, 15N)‘09Bi and z”*Pb(‘60, both

these

reactions

are around

-8

MeV for a ground

state

transition.

Barnett

Fig. 13. Expected dependence of yield on Q-value for very heavy targets. Z is the charge of the projectile and Ei the incident energy.

et al. ‘*) have observed the first reaction but not the second using a 160 beam with c.m. energy Ei = 64 MeV. Thus, EilZ = 8 MeV. The first reaction is of the Lp type, so that the experimental Q-values lie on the peak of the Lp curve. The second reaction is however of the Hp type and the experimental Q-value is clearly in the low tail of the Hp curve. In fact, a calculation of the cross section for the second reaction gave a cross section of 10e20 mb. The distances of closest approach for the second reaction differ by over 4 fm. A survey of possible reactions with relatively high yields shows that for beams of light-ions incident on heavy targets, reactions of the Hp type are particularly attractive since they favour positive Q-transitions, which in turn provides some favourable experimental conditions. The final binding of the proton to the projectile must be greater than its initial binding within the target. Possible projectile choices extend to “N, 19F or 23Na with final binding energies of 12.1, 12.8 and 11.7 MeV, respectively. Assuming a much heavier target, optimum conditions obtain when Q z 0.1 Ei. Thus, for example, we are interested in those target nuclei in the 90Zr to “‘Sn region having protons bound with energies of around 8 MeV. The optimum region has a

318

P. J. A. BU-ITLE

AND L. J. B. GOLDFARB

width of several MeV however, as seen from fig. 8. Many such examples present themselves, but the possibility of probing deep proton hole states of the target is limited. Reactions of the Lp type are favoured if the binding to the projectile donor exceeds that to the final heavy nucleus. Suitable projectile beams are izC, 13C, I60 and isO with binding energies of 15.9, 17.5, 12.1 and 15.9 MeV, respectively, and optimum conditions are associated with Q-values of about -0.15 Ei. This would lead to ground states and other low-lying states in the final heavy nucleus. For Hn reactions, choices of beam include 14N, 21Ne, “Mg and 33S with binding energies of about 10 MeV for the final configuration of neutron plus projectile. These are useful for probing neutron hole states of the target with similar binding energies. We are interested particularly in odd nucIei such as “0, “Ne, 2‘Mg, 29Si, 33S, 41Ca, 43Ca, 47Ti, 4gTi, 53Cr etc. which have characteristically much less binding energy and we are in a position to study a large variety of hole states. The Ln reactions favour final nuclear states with binding similar to that of the projectile. Thus, ’ 7O favours final states with a very loose binding, such as obtained with targets having 50 or 82 neutrons. It is instructive to consider the feasibility of studies below the Couiomb barrier using an ‘$0 beam. Since neutrons are tightly bound to 160 with B, = 15.7 MeV it is difficult to find low negative Q-transfers of the Ln type. On the other hand, Lp transfers are quite feasible. We note that in this case B, = 12.1 MeV and optimum conditions are associated with fairly large negative Q-values as would be the case with many heavy targets. Transfers of the Hp type lead to 17F ions and these are ruled out as the ground state is bound by only 0.6 MeV and Iarge positive Q-values are needed to provide good yields. The Hn reactions leading to I’0 are more likely to occur; however we note that the binding energy B, is only 4.1 MeV. Most processes lead to negative Q-values but reasonable yields should be obtained if this value is not more than a few MeV from zero. Thus, Lp transfers provide the most interest if we use I60 beams. One suggested study could be of the tin isotopes ranging from l”Sn with B2 = 2.5 MeV to 124Sn with B, = 7.3 MeV. The Q-values for the ground state transitions range from -9.4 to -4.8 MeV. The optimum Q-value is at the higher end of this range, favouring the heavier isotopes. 5. Conclusion Throughout our analysis higher-order effects such as virtual Coulomb excitation before or after the transition have been ignored. Some estimates by Breit 17) suggest the effects to be very small; however they may be more important for very heavy targets. More detailed consideration must await more intensive experimental study. Recoil effects appear to be of considerable importance for beams incident on heavy targets. This is contrary to what one might expect, since recoil effects are of order

NUCLEON

TRANSFER

319

A-r; however, sub-Coulomb energies are much higher for very heavy targets, the wavelength of relative motion is fairly small and therefore the recoil of the lighter core has a greater effect. Related to this from the viewpoint of calculations is the question of the choice of post or prior representation. If the target and projectile are of similar size, choice of the post or prior form depends on whether the Q-value is negative or positive. If projectiles are incident on a much heavier target, use of the post or prior form is dictated by whether the transferred particle originates in the projectile or in the target, respectively. In either case, the recoil corrections should be included and this helps to minimise differences between the post and prior formalisms. The attraction of heavy-ion transfer reactions below the Coulomb barrier is that nuclear spectroscopic information is obtainable without complications arising from a full nuclear collision; therefore few unknown parameters need be introduced. One of the disadvantages is that yields are likely to be small at these energies. It is therefore necessary to search for optimum conditions when planning experiments. One such condition is that the initial and final relative wave functions should exhibit their first maxima in the same region of space. More briefly, one requires that the distances of closest approach should be similar before and after the reaction. For all neutron transfers and for proton transfers involving projectiles and targets of similar size, this implies dealing with reactions having a low Q-value. For proton transfers involving light projectiles on very heavy targets, Q-values of + Ei/Z are favoured, where Ei is the beam energy and Z is the projectile charge. The sign is positive for reactions in which the proton is transferred from the heavy to the light nucleus (Hp reactions) and negative for the reverse situation (Lp reactions). The optimum region may extend over several MeV. These constraints considerably limit one’s choice of reaction. The wide range of available targets and projectiles on the other hand presents us a wide field open for future study.

References 1) Proc. Int. Conf. on nucl. reactions induced by heavy ions, ed. R. Bock and W. R. Hering (North Holland, Amsterdam, 1970) 2) G. Breit and M. E. Ebel, Phys. Rev. 103 (1956) 679 3) See for example G. Breit, J. A. Polak and D. A. Torchia, Phys. Rev. 161 (1967) 993 4) T. Kammuri and R. Nakasima, Proc. Third Conf. on reactions between complex nuclei, Ghiorso, Diamond and Conzett (Univ. of California, 1963); T. L. Abelishvili, JETP (Sov. Phys.) 13 (1963) 1010 5) P. J. A. Buttle and L. J. B. Goldfarb, Nucl. Phys. 78 (1966) 409 6) P. 3. A. Buttle and L. J. B. Goldfarb, Nucl. Phys. All5 (1968) 461 7) L. J. B. Goldfarb and 5. W. Steed, Nucl. Phys. All6 (1968) 321; T. Sawaguri and W. Tobocman, J. Math. Phys. 8 (1967) 2223; F. Schmittroth, W. Tobocman and A. A. Golestaneh, Phys. Rev. Cl (1970) 377; J. C. Jacmart et al. ref. r) p. 128; W. Von Oertzen et al. ref. ‘) p. 156 8) T. Kammuri and H. Yoshida, Nucl. Phys. A129 (1969) 625 9) G. Breit, Phys. Rev. 135 (1964) B1323

ed.

320

P. J. A. BUl-I’LE AND L. J. B. GOLDFARB

10) L. R. Dodd and K. R. Greider, Phys. Rev. 180 (1969) 1187 II) K. A. Ter-Martirosyan, JETP (Sov. Phys.) 2 (1956) 620; F. B. Morinigo, Phys. Rev. 133 (1964) B65; L. C. Biedenham, K. Boyer and M. Goldstein, Phys. Rev. 104 (1956) 383 12) G. Breit, Handbuch der Phys., vol. XL1 (Springer-Verlag, Berlin, 1959) part 1 sect. 48 13) F. B. Morinigo, Phys. Rev. 134B (1964) 1243; J. Cejpek, Czech. J. Phys. B16 (1966) 186; Phys. Lett. 21 (1966) 433 14) L. J. B. Goldfarb and K. Wong, Nucl. Phys. A90 (1967) 361 15) A. Sommerfeld, Atombau und Spektrallinen, vol. 2 (F. Vieweg, Braunschweig, 1953) 16) L. C. Biedenham and P. J. Brussaard, Coulomb excitation (Oxford Univ. Press, Oxford, 1965) 17) G. Breit, Proc. Nat. Acad. Sci. US 57 (1967) 849 18) A. R. Barnett, W. R. Phillips, P. J. A. Buttle and L. J. B. Goldfarb, Nucl. Phys. 176 (1971) 321