Absolute cross sections for stripping reactions

Nuclear Physics 81 (1966) 289--295; (~) North-Holland Publishing Co., Amsterdam Not to be reproduced by photoprint or microfilm without written permission from the.-publisher

ABSOLUTE CROSS SECTIONS FOR STRIPPING REACTIONS R. N. G L O V E R

A. W. R. E. Aldermaston, Berks

and A. D. W. JONES and J. R. ROOK Nuclear Physics Laboratory, Oxford Received 16 November 1965 Abstract: The absolute cross sections for (t, d) and (t, p) reactions are calculated using a simple wave function to describe the motion of the nucleons in the triton. In both cases we obtain reasonable agreement with the experimental data. 1. Introduction Considerable attention has been given to the problem of obtaining agreement between the experimental and theoretical absolute cross sections for (d, p) stripping reactions. According to Satchler 1) agreement to within 20 ~o can be obtained provided the Hulthdn wave function is used to describe the internal structure of the deuteron. Recently however Smith z) has shown that the absolute cross section is uncertain by a factor of two indicating that the (d, p) reaction cannot be used to obtain reliable absolute fractional parentage coefficients. Nevertheless the agreement obtained gives us a certain confidence that the stripping assumptions for (d, p) reactions are basically correct. For more complicated reactions the stripping assumptions are expected to be less valid since competing processes are a priori more likely 3). It is rather difficult to decide whether this is in fact the case solely from angular distribution measurements since these are not particularly sensitive to the detailed reaction mechanism 4). Thus the added information that can be obtained from the absolute cross section becomes especially useful. In this paper we restrict ourselves to (t, d) and (t, p) reactions but we should note that our results may be applied directly to other reactions. For instance isobaric spin invariance, which is usually satisfactory for nuclear reactions, immediately extends our results to (3He, d) and (3He, n) reactions while many (3He, p) reactions may be considered in a similar way. We consider first (t, d) reactions. Relatively little data are at present available but for (d, t) and (d, 3He) reactions 5,6) analyses indicate that the distorted wave theory with zero-range interaction predicts cross sections too small by about a factor of two. We shall show that this factor can be obtained by a more careful treatment of the internal structure of the triton or 3He. After outlining in sect. 2 sufficient of the usual direct reaction theory we describe in sect. 3 the triton wave function we use. 289

290

It. N. GLOVERe t

al.

In sect. 4 we apply our results to (t, d) reactions. In sect. 5 we show that the triton wave function which accounts for the (t, d) cross section also reproduces roughly the correct cross sections for (t, p) reactions. 2. Direct Reaction Theory

We consider a stripping reaction A(x, y)B. The distorted wave theory gives for the reaction amplitude M = (~be~byUylVxyJt~AtkxUx),

(1)

where the ~b represent the internal wave functions of the various particles, U the distorted waves describing the relative motion of the various projectiles and V,y the interaction between the nucleons in y and the nucleons in x which are captured into the nuclear state B. For a derivation of this formula the reader is referred to Tobocman 7). Since V~y does not affect the nuclear state we may write ---- ~. (SAO"A j¢lsBtrs)(sal2lj~)
(2)

where sa,/2 and j~ are the spin, orbital angular momentum and total angular momentum numbers of the transferred particles in the reaction considered while 0 is a fractional parentage coefficient. Thus eq. (1) may be written M = ~ (sAa Aj¢lsB ~B)(salX[j¢)Os,j< Uy:/AI OI Ux>,

(3)

where o = <~,: salV,,lff,>.

(4)

Now the magnetic quantum number dependence of eq. (4) can be extracted to give O = <~br: sllV~yll~b~>(sy~rysals~cr~),

(5)

where the double bar matrix element is defined by eqs. (4) and (5). We assume throughout our work that O has zero range, that is (~by: sllV~yll~b~ = #6(rx-ry),

(6)

where r~ and ry are the coordinates of the centres of mass of particles x and y. Thus g can be obtained by integrating eq. (6) over rx for a fixed value of ry. An equation for M can be obtained by inserting eqs. (4)-(6) into eq. (3) and the result is the usual distorted wave expression multiplied by g. It is conventional in computer programmes to use a value for g obtained by assuming that Vxy itself has zero range but this underestimates the cross section in general. We now write down explicit expressions for (t, d) and (t, p) reactions. We let suffices 1 and 2 denote neutrons and 3 the proton, use X and ~kto denote spin and space wave functions and assume that tk = Z~k. Thus using C to denote the ClebschGordan coefficient appearing in eq. (5) we obtain for (t, d) reactions and for (t, p)

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291

CO = (X(1)Za(23)~ka(23)[ V12 + V131Xt(123)~bt(123)),

(7)

Co' = ()C(12)X(3)1V13 + l/'231X,(123)~k,(123)).

(8)

reactions, respectively,

In eqs. (7) and (8) we perform the integration over r~ mentioned in coonection with eq. (6). In these formulae suttees d and t refer to deuteron and triton wave functions, X (1) is a spin wave function for a neutron while, in view of the usual selection rules for (t, p) reactions )C (12) is a spin zero state of the two neutrons. We assume that Xd is a spin one state while )fi consists of the two neutrons in a spin zero state coupled to the proton to give a spin ½ state. Finally in eqs. (7) and (8) Vu represents the potential between nucleons i and j. We assume throughout that ~kd has the usual HultMn form, that is =

N(e-"--e-P')lr,

(9)

where 0c = 0.23 fm-1, /~ = 1.58 fm-1 and N is a normalizing constant. The numerator of this wave function peaks at 1.4 fm, that is at the nuclear force range, as would be expected. 3. Triton Wave Function

The simplest triton wave function has the form I//t =

N' exp (--½y E r~),

(10)

where rlj = r i - r j . If we put X = rx--½(r2+r3),

y = r2--r3,

R = ~(rl+r2+r3),

we obtain Ct = N' exp [ - ~ ( x 2 +¼y2)].

(11)

We see that the wave function separates into a part describing the motion of a proton-neutron system and a part describing the motion of the remaining neutron about this system. This separation greatly simplifies calculations and we shall assume that such a separation is possible. The asymptotic form of this wave function is not correct and a correct asymptotic form for the x part of the wave function is required for our calculation. Thus our method is to retain the y part of eq. (11) and modify the x part to have the correct asymptotic form and a peak at 4 fm as indicated in re£ s). The simplest possibility would be to take the Hulth~n form but this cannot be made to satisfy these conditions. The next simplest possibility is to use a wave function derived from a square well and this is acceptable for our purpose. Thus our triton wave function is

~O, = NV(x)e - t ' ~ ,

(12)

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GLOVER et al.

where U(x) = e -k*

=AsinKx

x > R x < R.

The value of k is given by the appropriate separation energy while K and A are obtained by matching values and derivatives at R which is an adjustable parameter. We thus have a theory depending on two parameters R and V. The value of V which fits the root mean square radius of the triton is ~, = 0.14 fm -2 and we adopt this value throughout our calculation. There are two ways by which we can obtain an estimate of R. Firstly according to ref. 8) we require a wave function which peaks around 4 fm and this gives R = 4.5 fm. Secondly we may use the fact that eq. (10) reproduces the root mean square radius of the triton. We thus compare the root mean square radii of the x part of the wave functions of eqs. (10) and (12). The respective values, assuming R = 4.5 fm in eq. (12) are 2.3 fm and 2.6 fm which are sufficiently close for us to accept a value of R between 4 and 5 fm for our calculations.

4. The (t, d) Reaction We need to evaluate eq. (7). The object is to use the Schr6dinger equations for ~bt and ~d to reduce the matrix element to a simple form. Denoting the kinetic energy operator for particle i by T i, the triton binding energy by Et and the deuteron binding energy by Ed we write

(T~+T2+T3+V~+V~3+V23--EM,, =

0,

giving from eq. (5) c g = (Z(1)Xd(23)~d(23)IE,-- T~ -- T 2 - T3-- V~31X,(123)~,(123)>.

Using the Schr6dinger equation for q~d in a similar way we obtain CO = (g(1)Xd(23)@a(23)lEr-- TrlZ,(123)g,t(123)),

(13)

where T~ is the kinetic energy operator for the relative motion of nucleon 1 with respect to the nucleons 2 and 3. Similarly E~ is the separation energy of the triton into deuteron and a neutron. This formula is exact and is evaluated with the approximate wave function of sect. 3. We obtain

=

(14) k R + (1 + k 2 / K 2) sin E K R

In this formula V is the depth of the square potential which binds the neutron by

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293

E r while m r is the reduced mass of the neutron. The expression for F is

f 2 -f e - ~t~Y2y2 dy where 0d is the normalized Hulth6n wave function for the deuteron. Finally we note that eq. (14) contains a factor of ¼, since we need only that part of the triton wave function for which the neutron proton system is in a triplet state, and a factor of two since there are two neutrons. Together these account for the factor of ½. The value of F 2 we obtain is 0.3 while the value of V required to give the correct triton binding energy is 16 MeV, assuming R = 4 fm. As stated above it is conventional to include the factor 2k(h2/2mr) 2 corresponding to zero-range interaction, into distorted wave calculations so the remaining factors multiply the actual results obtained from computer programmes. Collecting the results together we find that the zero-range results should be multiplied by 2.3 for R = 4 fm and 3.1 for R = 5 fm. These numbers compare with the figure of about two obtained in refs. s,6) for (d, 3He) and (d, t) reactions. We rather expect that the results of refs. 5.6) are influenced by uncertainties in the nuclear matrix elements and optical potentials involved. However in the case of (t, d) reactions a particularly favourable set of results is available 9). These compare (t, d) and (d, p) reactions on 26Mg with measured triton and deuteron optical potentials 10). The merit of comparing (t, d) and (d, p) reactions is that the nuclear matrix elements cancel out providing we assume both reactions proceed by pure stripping. The authors of ref. 9) obtain for the parameter

t

A = \a(d, p)/experiment

×

\ ~ ( ~ d~/distorted ....

an average value of 1.8. Remembering that the use of the Hulth~n wave function for the deuteron increases 11) the (d, p) cross sections by 1.5, we obtain for A the theoretical values 1.5 and 2.1 for R = 4 and 5 fm. Thus we see that a value of R between 4 and 5 fm gives agreement with experiment. This value is in accord with our estimate of R in sect. 3.

5. The (t, p) Reaction We now need to evaluate eq. (8). The triton wave function used in sect. 4 is not obviously applicable to this case since in sect. (4) we considered the motion of a neutron about a deuteron whereas for (t, p) reactions we need to know the motion of the proton about the di-neutron. The true triton wave function is however approximately spatially symmetric so we expect that the error introduced by using eq. (12) to describe the triton as a separable proton-dineutron system will not be large.

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R.N. GLOWER et al.

With this approximation and using methods analogous to those used in the case of (t, d) reactions we obtain

Cg' =

(Z(3)X(12)IEr - TrlXt(123)~kt(123)).

(15)

Hence

9'2

=(8n~8nk(h212 ( V i 2 (l+kR)Usin2KR \371 \~m¢/ \V------EJ kR+(l+k2/K2)sin 2 KR'

(16)

where the notation is that of eq. (14). We should stress that the manipulations leading to eq. 06) are by no means as rigorous as those leading to eq. (13) since the presence of ~bd in eq. (13) projects out the motion of the odd nucleon while in eq. (15) we need to make the separable assumption for ~bt. Consequently the answers of this section are not expected to be as reliable as those of sect. 4. TABLE 1 Comparison of experimental and theoretical maximum cross sections at lO MeV bombarding energy Reaction

1600, p)IsO (0) leO(t, p)IsO (I)

Experiment (mb/sr)

Theory (mb/sr)

25 8

28 7.4

Parameter set A A

4°Ca(t, p)~2Ca (0)

3.3

2.3

B

4°Ca(t, p)42Ca (1) 4eCa(t, p)~2Ca (4)

0.48 0.18

0.46 0.19

B B

TABLE 2 Optical model parameters used in calculations of table 1 Parameter

Triton

Proton

set

A B

V (MeV)

W (MeV)

re (fm)

a (fm)

V (MeV)

W (MeV)

re (fm)

a (fro)

147 129

18 53

1.4 1.4

0.55 0.64

50 52

20 8.5

1.25 1.3

0.55 0.5

We show in table 1 a comparison between the theoretical and experimental t2) results for 160(t, p)~SO and 4°Ca(t, p)42Ca taking the extreme single particle model to describe the states of 1SO and 42Ca. The computer programme was that of Yates based on the work of Rook and Mitra 4). The optical potentials used had entirely Saxon-Woods wells with the same radial shape parameters for the real and imaginary parts, the details being given in table 2. The triton parameters were obtained from analyses of triton scattering data 10) while the proton parameters were extracted from the tables of Hodgson 13). These rather old results were used in view of the limitation to Saxon-Woods wells imposed by the computer programme.

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295

We see from table 1 that the theoretical calculations are in good agreement with experiment for the transitions to the states shown. The agreement for the excited states is perhaps surprising since we expect some mixing of the core states. A detailed quantitative investigation of the (t, p) reaction is in progress and these results will be published separately.

6. Conclusions The agreement we have obtained for both (t, d) and (t, p) reactions leads us to believe that the stripping mechanism is predominant for these processes. Similar results may be expected for reactions such as (3He, d) (3He, n), (3He, p) and their inverses. We can thus feel added confidence in using these reactions to extract spectroscopic information although the uncertainties remaining in a calculation of an absolute cross section from the distorted wave theory must limit the accuracy obtained. We thank Mr. M. Yates for making his programme available to us. One of us (A.D.W.J.) is indebted to the Commissioners of the Exhibition of 1851 for a Senior Studentship while another (J.R.R.) was supported by the Science Research Council.

References 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12)

G. R. Satchler, Prec. Conf. on nuclear structure with direct reactions, Chicago (1964) ANL 6878 W. R. Smith, Phys. Rev. 137 (1965) B913 L S. Shapiro, Selected topics in nuclear theory (IAEA Vienna, 1963) p. 85 J. R. Rook and D. Mitra, Nuclear Physics 51 (1964) 96 J. H. Bjerregard et aL, Phys. Rev. 136 (1964) B1348 J. L. Yntema and G. R. Satchler, Phys. gev. 134 (1964) B976 W. Tobocman, Theory of direct nuclear reactions (Oxford University Press, London, 1961) A. Werner, Nuclear Physics 1 (1956) 9 R. N. Glover, A. D. W. Jones and J. R. Rook, Phys. Lett., 19 (1965) 493 R. N. Glover and A. D. W. Jones, Nuclear Physics 81 (1966) 268 R. H. Bassel, R. M. Drisko and G. R. Satchler, O R N L 3240 (1962) R. Middleton and D. J. Pullen, Nuclear Physics 51 (1964) 63; S. Hinds and R. Middleton, private communication (1965) 13) P. E. Hodgson, The optical model of elastic scattering (Oxford University Press, London, 1963)