Theory of nuclear reactions

Theory of nuclear reactions

Nuclear Physics 58 (1964) 1--8; (~ North-Holland Publishing Co., Amsterdam Not to be reproduced by photoprint or microfilm without written permissi...

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Nuclear Physics 58 (1964) 1--8;


North-Holland Publishing Co., Amsterdam

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

THEORY OF NUCLEAR REACTIONS V. Penetration Factors for Charged Particles in Low Energy Nuclear Reactions J. P. J E U K E N N E t

Theoretical Nuclear Physics, University of Liege tt Received 6 March 1964 The choice of a definition for a penetration factor is discussed, with emphasis on the condition of channel radius independence. As illustrative examples, the experimental data for the reactions HeS(d, p) and HS(d, n) are analysed and it is shown that the best fit is most readily obtained for the penetration factors independent o f the channel radii.


1. Introduction: On the Separation of the Penetration Effects Numerous experimental data on isolated resonances in nuclear reactions have been fitted to cross sections derived from phenomenological or more elaborate theories. A common feature of these cross sections is that they may be expressed as a product of two factors which are interpreted as corresponding respectively to the penetration of the Coulomb and centrifugal barriers and to the effect of nuclear forces. The former is exponentially decreasing when the energy becomes smaller and smaller; the latter has exactly or approximately the form of a Lorentz factor. On the one hand, such a separation in penetration and nuclear effects is well justified qualitatively because it corresponds to the behaviour of experimental cross sections. On the other hand, however, it is also evident that an exact and complete separation of the penetration and nuclear effects is intrinsically impossible from the point of view of wave mechanics: the penetration of a barrier depends on the behaviour of the interaction on the two sides o f the barrier; while in the external region it is possible to separate the Coulomb and centrifugal energy terms, such a separation is obviously impossible in the interior region of the configuration space. Under such conditions, there exists no exact and unique way of defining a penetration factor in a phenomenological theory of nuclear reactions. Just this arbitrariness, however, opens the possibility of choosing a penetration factor which exhibits, as the cross section itself, the essential property of being independent of the channel radii. Such a penetration factor independent of the channel radii has in fact been recently suggested by Humbler 1). Since his choice was primarily guided by analytical rather than physical arguments, it is still necessary to discuss whether it is in agreement with the physical idea of a penetration factor: this is done in sect. 2. Two examples t Chercheur A g r ~ , Institut Interuniversitaire des Sciences Nucl6aires. tt c/o Institut de Math6matique, 15, avenue des Tilleuls, Liege, Belgium. 1 September





are then discussed in sect. 3, to show how theoretical cross sections can be fitted to the low energy experimental data for the reactions Hea(d, p)He 4 and Ha(d, n)He 4 which have also been discussed extensively by Breit 2). Our notations are those of refs. 1, 7, s). 2. Penetration Factors and Cross Sections

Let us first briefly recall how the conventional penetration factor has been introduced in phenomenological theories, because it is too often erroneously considered as a quantity proportional to the transmission coefficient describing accurately the process of formation and disintegration of the compound nucleus. The cross section of a nuclear reaction at low energy is usually taken proportional to the well known transmission factor s, 4)

4KaP T = S2+(e+ra)2,


which applies to the following approximate picture of the first stage of the reaction leading to the formation of the compound nucleus a): a one-dimensional beam of charged particles comes from r = + oo with k as wave number, partly penetrates the region r < a, and then proceeds undisturbed towards r = - o o with a wave number

K = (k2+2MUo/h2) ~ = (k 2 + K g ) ~,


corresponding to a well of constant depth Uo = h2K~/2M. In eq. (1), S arid P, for a beam of angular momentum l, are defined in terms of the Coulomb wave-functions Ft, Ot according to


ka =




St =

A-} [F,(a, k)F;(a, k)+ Gt(a, k)Gi(a, k)],





CrY(a, k)+ G (a,


when the energy is very small, namely when k << K, the factor Ka is much larger than Pt and St and, instead o f e q . (1), we may write 4Pt_

Tl -- Ka




This is the relation that justifies the interpretation of Pt as a penetration factor. Obviously, Pt is not independent of the channel radius a.



The same quantity P~ also appears in the R-matrix theory, in which the one-level approximation of the cross section for the reaction ~--* ~' reads, with the usual notations 5, 6), cru, = ~ Os L


i r



In the last expression, the total width F. and the shift A, are functions of the total energy; the level n has a spin J and a speeified parity. In the fitting of experimental data to the cross section (7), one is immediately faced with the a priori choice of the channel radii a,, ao,. On the one hand, the absolute value of Pz P r varies strongly with a, and a~, ; while, on the other hand, the exact cross section is "independent" 7, s) of such radii. The importance of such an anomaly seems better realized now and in the framework of the R-matrix theory itself, recent computations have been made 6, 9), which tend to justify an a posteriori modification of P~ and Pv in eq. (7). The argument given is that the penetration factor (3) corresponds to a nucleus with a sharply defined surface (the potential well), rather than to an actual nucleus with a diffuse surface. To find a completely satisfactory solution to such objections was precisely one of the main purposes pursued by Humblet and Rosenfeld 7) in their complex eigenvalue theory. In the last paper of the series, Humblet 1) has given an expansion of the collision matrix that implicitly introduces penetration factors which are completely independent of the channel radii a¢, a~,. The one-level approximation without background of the integrated cross section is k ck¢, /-/r/-/l k.

...,. xo.xo..In,,.n,.I

F¢~ F~,n



in this formula, the background term of the collision matrix is assumed to be negligible and the penetration factor Hc is defined by 1) Hc = e?k~'


1 I/2 r/2 8tZ = ~ (12 + k--~¢2)" " " (1 + ~ ) e2~/k¢2ntl/kc_



One is led to the separation of the factor e~e~, in the cross section (8) by purely analytical considerations 1), namely the necessity of applying the expansion theorem to a quantity that is continuous on the integration contour involved in the proof of this theorem. Nevertheless, Humblet a) has also noticed that the conventional penetration factor P~ could be introduced in the complex eigenvalue theory as well. Then, however,




the corresponding one-level approximation lacks the a-independence property; it reads, instead of eq. (8),

IO¢.O ,.I


tru" = ~2 yjq2 E PIP,, (E°_e°.)2 +¼r k~ s, s'r re. ac re,, a c,


It is interesting to point out the relation existing between Pz and e~, and according to which the cross section (8) is nothing else than the limit of cross section (11) for vanishing channel radii a~ and a e . It is indeed well known that, when k¢a¢ is sufficiently small, one has

G2(a~, k~) >> F~(a©, kc) and, in eq. (11), I Gl(a¢, k..) lim IO~"12Pl(a¢, k¢) = lim .o-.o kca~ .o-.O Gl(a~, kc)


--21812 K©



Ik~.e+.121 2 '

in agreement with the behaviour 14) of G~ near a¢ = 0. This formula (12) suggests that the best way to take the penetration effect into account is to introduce a penetration down to the "centre" of the target nuclei, rather than to its surface. But one should certainly not be justified to take such an interpretation too literally. If, on the one hand, it is true that the cross section (8) is exactly the limit of the cross section (11) for vanishing radii, one should also, on the other hand, keep in mind the restrictions attached to the interpretation of Pt as a penetration factor; moreover, considered alone, the quantity Pz tends t o zero like a 21+1 for vanishing a¢. Another way of looking at the problem which is discussed here is the following one. Let us definef~(a¢, k¢) by

Pz(ao,k~) =


2 (ko a~) 2t+1 e,~fl(ac, k¢).


In agreement with eq. (12), fl(0, kc) = 1 for any ko, but for a c # 0, it is a well defined function of kc and a~, in which the ao and ko dependences are not separable. By its occurrence in the expression (11), the factor f l introduces into this expression an a¢-dependence which is a priori bad, since the exact expression for the cross section should in fact be at-independent s). In view of these circumstances, we cannot reasonably expect this factor fz to improve the energy dependence of the one-level approximation in the low energy region. In fact, the best choice suggested by this argument is just f| = 1. In order to study the situation more concretely, it is not superfluous to treat examples. We have chosen two reactions discussed by Breit in the R-matrix theory and already analysed by Mahaux lo) by eq. (8). We shall assume that eq. (11) is



an adequate basis for a practical discussion of the best choice of the penetration factor when varying channel radii ac, ao, are adopted.

3. The Reactions He3(d, p)He 4 and HS(d, n)He 4 Between 100 and 800 keV, the experimental cross section of the reaction He3(d, p) has a pronounced maximum at about 630 keV. It corresponds to a J = ½+ broad resonance state t. In the cross sections (7), (8) and (11), the only important term corresponds to the smallest possible values of l and I', namely l=0,


1' = 2, s' = ½.


If a cross section of type (11) is adapted to the fitting of the experimental data, the values of the quantity

kd A~(kd, ad)A2(k,, ap)%,,

Xdp ~ kp


corresponding to the experimental cross section %p should lie on a Lorerttz curve, since according to eq. (12), const Zdp = (Ea_Ed.) 2 +¼F.2 •


In eq. (15), the coefficients A are defined according to eq. (5) and computed according to the tables of Bloch et al. H); the indices d and p are used respectively for the incoming and outgoing channels. The width and the position of the maximum of the Lorentz curve (16) should be exactly the total width and the energy of the involved resonance state. The value of Zap has been computed from the experimental data of Kunz et al. 12) for different entrance channel radii %. The excitation energy is so high ( ~ 17 MeV) that the variation with kp of A~/kp is negligible in the energy range considered. For each chosen ad, we have drawn a Lorentz curve fitting as exactly as possible the experimental data below and just above the resonance. It is clear from fig. 1 that the best fit is obtained for the smallest radius, i.e. for aa = 0. One also sees that to different radii should correspond different total widths and resonances energies, they are given in table 1 together with the height of the Coulomb barrier B = Z 1Z 2


which is added for reference. t For references, see Breit's 2) and Mahaux's 10) papers.



J . P . JEUKEIqN]~

3He(d,p)~;He Ed.p


°/keV =274


!: i~



Pn c


", ac:Zfm




,r ....... i!

~Pn=322.5keV i




• Lorentz curve o Experimentol points











Fig. 1. Lorentz curve Y'ap and the corresponding experimental points with different channel radii for the reaction HeS(d, p). The comparison o f the quality o f the different fittings has been made easier by normalizing the factor Y'ap to 10 at the maximum.

TABLE 1 Position and total width o f the low energy resonance o f the reaction Hen(d, p) for different channel radii ad.

a e (fm) B












E . (keY)






/-'n (keV)







Similar computations have been made for the mirror reaction HS(d, n). The experimental data of Conner et al. 13) have been used in the range of 15 to 160 keV. The cross section has a maximum at 110 keV corresponding to a J = ½+ resonance 3H(d,n)~ He $'d,n


10 O O


,)/ I ~n=39 key ) " Lo entz curve . . . . . ~ o Experimentat poir~ts


4i, 0

, iO

:/0 3'0 4'o

5to ~o

Ed(CM)keV 7~) 8'o 9o


Fig. 2. Lorentz curve ]~ap and the corresponding experimental points with different channel radii for the reaction HS(d, n). The comparison of the quality of the different fittings has been made easier by normalizing the factor T.ep to I0 at the maximum. TABLE 2 Position and total width of the low energy resonance of the reaction HS(d, n ) H e ' for different channel radii ad ad (fm)











E~ (keV)





-P. (keV)








level. The only important term in the sums (7), (8) and (11) again corresponds to the quantum numbers (14). Again, also, according to fig. 2 the best fitting corresponds to a a = 0; the widths and resonance energies corresponding to the different radii are given in table 2. To avoid any misunderstanding on the interpretation procedure adopted here, we must emphasize that it does not require the "background" term -~c,c of the collision matrix ~'c,c to be always negligible; other fittings just achieved by Lejeune 15) and Mahanx 16) introduce along with the penetration factor (9) a non-vanishing background. The results obtained here in the framework of the complex eigenvalue theory may help to understand better the origin of some of the difficulties encountered in the fitting of experimental data, such as those discussed in this section, to the R-matrix theory cross section (7) 2). Our results suggest that the energy variations of the shift and total width should compensate that o f the function f~. We certainly agree, however, that more studies of broad and isolated resonances as that occurring in the above (d, p) reaction are necessary to support such an argument. Since the position of an isolated resonance is determined accurately and unambiguously by formulae such as (8) and (11), let us also hope that more experimental data will permit cross determinations of such resonance levels. This should certainly be an excellent test for any theory. It requires the possibility of forming the same compound nucleus by at least two different entrance channels, the inverse reaction not being considered. We are very grateful to Professor J. Humblet for having suggested this work and for stimulating discussions. We are indebted to Professor Rosenfeld for several improvements of earlier versions of our manuscript and to Mr. C. Mahaux for communication of the details of his computations.

References 1) J. Humblet, Nuclear Physics 50 (1964) 1

2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14) 15) 16)

G. Breit, Hdb. der Physik, 41/1, p. 231 (Springer, 1959) J. M. Blatt and V. F. Weisskopf, Theoretical nuclear physics (Wiley, 1952) M. A. Preston, Physics of the nucleus (Addison-Wesley, 1962) A. M. Lane and R. G. Thomas, Revs. Mod. Phys. 30 (1958) 257 E. Vogt, Revs. Mod. Phys. 34 (1962) 723 J. Humblet and L. Rosenfeld, Nuclear Physics 26 (1960) 529 J. Humblet, Nuclear Physics 31 (1962) 544 J. P. Schiffer, Nuclear Physics 46 (1963) 246 C. Mahaux, Bull. Soc. Sc. Li6ge 32 (1963) 70 I. Bloch, M. H. Hull, A. A. Broyles, W.G. Bouricius, B. E. Freeman and G. Breit, Revs. Mod. Phys. 23 (1951) 147 W. E. Kunz, Phys. Rev. 97 (1955) 456 J. P. Conner, T. W. Bonner and J. R. Smith, Phys. Rev. 88 (1952) 473 M. H. Hull and G. Breit, Hdb der Physik, 41/1 p. 408 (Springer, 1959) A. Lejeune, private communication (1963) C. Mahaux, private communication (1963)