The role of unitarity in diffractional reactions

2.C

I I

Nuclear Physics All3 (1968) 367--375; (~) North-Holland Publishiny Co., Amsterdam N o t to be reproduced by photoprint or microfilm without written permission from the publisher

THE ROLE OF UNITARITY IN DIFFRACTIONAL REACTIONS GIULIANO SCHIFFRER Istituto di Fisica dell' Universitd, Catania t

Received 19 October 1967 Abstract: A diffractional reaction is defined by requiring that in its angular distribution the interference term between any pair of partial wave amplitudes of fixed modulus and given angular momentum is maximum. Some sufficient conditions for a binary reaction involving spinless particles to be diffractional are given, using only unitarity and symmetry of the S-matrix. One of these conditions is satisfied if a suitably defined strong absorption holds in the initial and in the final channel and if the partial reaction cross sections are equal in both channels for each angular momentum. The cases of diffractional reactions involving non vanishing spins and of a reaction between spinless heavy ions are briefly discussed and tests for diffraction are proposed.

1. Diffractional reactions Reactions exhibiting strongly oscillating angular distributions, similar to diffraction patterns in optics, have been observed both in nuclear and in elementary particle physics 1, 2). A l t h o u g h the main ideas o f the present paper can be used in both fields, the emphasis will be here on nuclear physics, where such reactions are k n o w n to take place at energies ranging f r o m a few MeV to several tens MeV [ref. 1)], at least. Terms like "nuclear diffraction" have been used by several authors, often in connection with special models, as it will be shortly reviewed in sect. 2. F o r the purpose o f the present work, we need a definition o f diffraction, or diffractional reaction, which should be as general as possible, f r o m the point o f view o f q u a n t u m theory. R e m e m b e r i n g a suggestion o f Blair 3), the following definition is proposed here for binary reactions involving spinless particles: a reaction will be said to be "diffractional" when in the angular distribution the interference term between any pair o f partial wave amplitudes o f fixed modulus and given angular m o m e n t u m attains its m a x i m u m absolute value. This is equivalent to put [ cos (~pl-~ov)[ = 1

(1)

for any l, l', where ~ot = arg St, St being the S-matrix element for the considered reaction in the angular m o m e n t u m representation. As a consequence of condition (1), the odd terms in cos 0 in the angular distribution are, in general, large; this is especially true for the coefficient o f the odd Legendre polynomial of the highest allowed order, where cancellation due to different signs in eq. (1) c a n n o t occur, because only one pair l, l ' is important. This follows in a straight* This work has been supported in part by CRRN, CSFN and E U R A T O M / C N E N - I N F N . 367

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G. SCHIFFRER

forward way from the general form of the angular distribution for reactions of the considered type, expressed in terms of Sg = p~ exp irpl by -dr2 - oc u2' . [ ( 2 1 + 1 ) ( 2 1 ' + 1 ) ] ~ ( 2 n + l )

0

0n

P~Pt, cos (qh-qh,)Pn(cos 0),

the notation being standard. Moreover, definition (I) can be better understood from the physical point of view also by remarking that it represents the limiting case opposite to compound nucleus reactions, for which, by averaging the cross section with respect to energy, the cosine turns, by definition, into 6,,. If the four particles have a non-vanishing spin, a condition analogous to eq. (1) can be written down in several coupling schemes for the angular momenta. To each scheme corresponds in general a different definition of diffractional reaction. A coupling scheme which is particularly meaningful physically seems to be the one labelled by the transferred angular momentum 4' s) L. Condition (1) in this coupling scheme, without the symbol of modulus, leads *' 6) to the well-known Butler rule 7) for the angular distributions of reactions, characterized by a single value of L. 2. A p p r o x i m a t i o n s and m o d e l s

If all partial wave amplitudes of a given reaction are slowly varying functions of the energy, so that condition (1) holds in a suitable energy interval, then a diffractional reaction is also direct. In such a case, the various methods available in the theory of direct reactions 8) can be applied to the study of diffraction. The diffractional character, expressed by condition (1), seems then to be a consequence of strong absorption of particles, taking place in the initial and in the final channel, in the sense that many approximate methods and models lead to condition (I), irrespectively of the nature of the basic assumptions they are founded on, provided strong absorption is taken into account in some way. As an example, consider the adiabatic approximation of Austern and Blair 9), where condition (1) derives, neglecting Coulomb effects, from some postulated properties of the Green function, which tend to be good approximations when strong absorption takes place. On the other hand, the distorted wave method 5, s) gives nearly the same results 10). For this reason, although here the relevant formulae are less transparent, it seems reasonable that the phases also obey approximately condition (1), due to strong absorption, introduced by means of optical potentials. Indeed, the work of Goldfarb and Hooper 1~) and of Dar 12) supports this view. In the case of the Butler theory v), condition (1) is satisfied because the amplitudes are real. Here, the radial cut-off simulates strong absorption and has the function of enhancing the contribution from angular momenta near the centrifugal barrier. The above discussion could help us to understand the surprising fact that so different assumptions, such as validity of Butler 7) or semiclassical ~3) approximation lead to results, which to a certain extent agree.

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I f a single value of L contributes to a reaction, condition (1), together with a specification of the sign of the cosines, is sufficient to determine roughly the main diffraction m a x i m u m of the angular distribution 4, 6). On the contrary, the secondary diffraction maxima are more sensitive to the choice of moduli of the S-matrix elements. According to all models taking into account in some way strong absorption, such moduli are important only in correspondence to angular momenta near the centrifugal barrier 3, 4, 10 - 1,). Other references on diffractional models are reported elsewhere 8, 12, 14, ~8).

3.

Unitarity

The discussion of sect. 2 suggests the conjecture that, since condition (1) seems to depend more on the assumption of strong absorption, than on other dynamical properties of the reaction, it should be possible to derive eq. (1) from more generally valid principles than the approximate methods and models just described and a suitable definition of strong absorption. According to this argument, it seems natural to investigate whether unitarity has any relation with condition (1). The role of unitarity in direct reactions has already been studied by other authors 15, 16). In particular, Schnitzer reports the work done using techniques of dispersion relations; Dar and Tobocman, among other things, discuss the connection between unitarity and the vanishing of the S-matrix elements for low partial waves. In this section, only binary reactions between spinless particles will be considered. In such a case, the unitarity condition for the S-matrix in the angular m o m e n t u m representation can be written as

S,)~(t). ± ¢(~)~o)* _~ ii ~aif T~"~if " f f ~

¢(t)¢(t).

L ~-'in '°fn n:~i, f

= 0,

(i #

f),

(2)

where i and f denote initial and final channel and time reversal invariance has been assumed. It is convenient to introduce the notation (t) P = ]Sif ],

~0 = arg c(t) ~'if,

c~ = arg S~li),

b

r] ~ - ]

Z ~'in~'(l)q'(1)*l~-'I, fn n~i,f

(t)

~---

[Sff l,

(t)

a = ISji l, o(t)

/~ = arg off,

0 = arg [ ~ ~"in ~(t)¢,)*l ~'fn _1" n4:i, f

Eq. (2) then becomes

p[a cos (a--q~)+ b cos (fi-q~)] + r / c o s O = 0, p[a sin (a-~o)-b sin (fl-q~)]+tl sin O = 0.

(3)

Let us consider first the case q > 0, p > 0, a > 0, b > 0. The angle 0 can be eliminated from eqs. (3), by solving with respect to cos 0 and sin 0, squaring and summing.

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The result can be put into the form cos (e + / / - 2go) = ~

_a2_b 2 .

(4)

The amplitudes for elastic scattering can be considered as k n o w n in both channels, so that q~ depends on only one u n k n o w n p a r a m e t e r ~ = rl/p; the latter must obey the two limitations la-b[ < ~ < a+b,

(5)

since the m o d u l u s of the cosine in eq. (4) cannot exceed one. A sufficient condition for the validity o f e q . (1) is that for all l either ~ = l a - b l

(6)

or

(7)

~ = a+b,

and ½(e+/~) is independent on l.

(8)

This is justified by observing that the right-hand side of eq. (4) is equal now either to 1, or to - 1, so that we can solve for go to get ~o = ½(c~+fl)+(k+½)n,

for

¢ = la-bl,

for

~ = a + b , (k = 0, _ 1 . . . . ),

(9) (10)

and if condition (8) holds, eq. (1) is satisfied. In the case t/ = 0, p > 0, the linear system (2) or (3), determining the u n k n o w n s Re S[[ ) and I m S}[ ), becomes homogeneous; therefore the existence o f p requires the vanishing of the determinant, i.e. a = b. The conditions t/ = 0 and a = b are thus completely equivalent, in the sense that each of them implies the other. If a > 0, eqs. (3) give the solution (9), as it could be obtained also f r o m eq. (4) in the limit ~ 0, taking into account the first inequality (5). The condition [ref. 16)] a = b, (or r/ = 0)

for all /,

(11)

although m o r e restrictive than the validity of eq. (6), has a simpler physical meaning: it implies the equality of each partial reaction cross sections in channel i and f. If it holds in a suitable energy range together with condition (8) and the requirement that q0t-~p r are continuous functions of the energy, it is a sufficient condition for a reaction to be direct, in the sense that it implies the validity o f e q . (1) in the same range. Of course, it is by no means necessary. Nevertheless, it is interesting to point out that Jl = 0 is also a necessary condition for a reaction to be direct, in the sense that its amplitude, not averaged over energy, is well described by D W B A in its standard form [ref. 17)].

DIFFRACTIONAL REACTIONS

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4. Strong absorption The purpose of this section is to investigate more carefully the physical meaning of condition (8). In particular, it will be shown that if a suitably defined strong absorption takes place in both channels, then condition (8) is satisfied. The definition of strong absorption related to the "Fraunhofer diffraction" theory [refs. 9, 1o)] is not very useful for the present purpose. I propose here a slightly less restrictive definition, which seems to translate fairly faithfully the physical idea involved: for each l, the corresponding reaction cross section a~t) is the largest, consistently with a given elastic scattering cross section a~°. This first part of the definition implies that -ii must be real, i.e. ~ = krc. The second part requires that a~° > cr~t) for each l; then S}() cannot be negative, i.e. ~ = 2kTt. As a consequence, we can deduce that in reactions with strong absorption in both channels, we need just one more condition, in order to have diffraction, in the sense (1); this condition could be either eq. (6), or (7), or (11). If the definition of strong absorption is changed, by keeping its first part and reformulating for a given partial wave the second part by requiring a~~) > a~~), it turns out that the corresponding elastic S-matrix element is negative. The physical meaning of this alternative definition can be easily understood. Suppose that in the considered channel a resonance of t h e / t h partial wave is superimposed to a strong absorption background a8), defined in the original way. If the resonance is well described by a Breit-Wigner formula, the elastic S-matrix element at the resonance energy has the form

S(~) = S(o') (1- 21-'S) ,

(12)

where S(ot) represents the background amplitude corresponding to strong absorption, F~t) the elastic partial width and Ftt) the total width; the channel indices have been omitted. We see from eq. (12) that if the resonant state decays most probably elastically, i.e. if F~t) > ½/-¢o, the left-hand side of eq. (12) has the opposite sign of So(t) and is therefore negative. It follows that the condition ~r~z) > a~z) must be satisfied. Thus, a reaction characterized in both channels and a wide energy range by ordinary strong absorption, except for a partial wave l of one channel, to which corresponds the alternative definition of strong absorption in the neighbourhood of a given energy, can be interpreted according to the previously quoted Frahn 18) picture. In this case, eq. (1) will be satisfied by all pairs of phases, except those containing l; for these pairs the right-hand side of eq. (1) is zero, if either eq. (9) or (10) holds. This means that the interference between the resonant partial wave and the remaining ones is destroyed; as a consequence, the angular distribution will tend to be more symmetrical with respect to ½re. This result, previously found 18) using a Breit-Wigner formula for the resonant amplitude, is seen here also from another point of view. The presence of Coulomb interaction in elastic scattering tends to invalidate the previously defined conditions of strong absorption. However, for our purpose we

372

G. SCHIFFRER

require only the validity of condition (8). Remembering that the difference between two Coulomb phase shifts is given in terms of the Coulomb parameter 7 by at+ 1 -o'~ = arctg[y/(/+ I)], our requirement is y / ( l + 1)<< 1. This condition is valid both for high l, corresponding to high energy, or heavy nuclei and for low y, corresponding to light nuclei or high energy. In the former case, usually there is no problem for the lowest values of l, because the corresponding reaction amplitude is negligible, according to the discussion of sect. 2. In the case of light nuclei, the condition can be satisfied down to energies of a few MeV. Another violation of condition (8) could derive from nuclear phases, which are never exactly constant, nor zero, as the definition of strong absorption requires. Anyway, an optical model analysis of elastic scattering of c~-particles on 58Ni at 43 MeV has shown t that the phase ~ is nearly constant with respect to l just in that interval of angular momenta, which is expected to be most important for the corresponding inelastic scattering. Therefore, we can conclude this section by noting that not only the given definition of strong absorption implies condition (8), but also that this condition seems to hold in real physical situations.

5. Non-vanishing spins In this case the development of sects. 3 and 4 does not hold. The very concept of diffractional reaction becomes, in general, ambiguous, as it has been pointed out in sect. 1. Nevertheless, it is worth while to spend some words about this topic, because in some particular cases it is possible to test rigorously the diffractional character. An essential point is that in reactions involving non-vanishing spins the S-matrix elements are labelled by other angular momenta, besides the total J; there are several ways of choosing the related coupling scheme; introducing the transferred angular momenta, even the total J can be eliminated. If we limit ourselves to the study of angular distributions with no polarization prepared in the initial state or detected in the final state, the measured quantity can be expressed by means of a sum, bilinear in the amplitudes, coherent with respect to a group of indices, incoherent with respect to another group. These two groups of quantum numbers will be referred to hereafter as coherent and incoherent indices. In general, the definition of diffraction will be analogous to eq. (1), except that now both phases will have the same incoherent indices and more, generally different, coherent indices. If all spins vanish, there is a single index, the total angular momentum, belonging to the first group. Suppose now that in a given coupling scheme a reaction is diffractional and there is more than one value of the incoherent indices. Each angular distribution, corresponding to a fixed incoherent index, shows in general a marked structure, which will tend to be "smoothed out" in the sum of such distributions. On the other hand, there are reactions where only one value of the incoherent indices is allowed: for instance, t See fig. 15 of ref. ~0), and ref. 2t) for other data.

DIFFRACTIONAL REACTIONS

373

binary reactions involving three spinless particles, or two spinless, two spin ½ particles and intrinsic parity change; in the latter case, parity conservation must be postulated. As far as the definition of diffraction is concerned, reactions of this type do not differ in any essential point from reactions involving only spinless particles: there are just more coherent indices, instead of one. They are distinguished, instead, from reactions involving arbitrary spins, because there is no more the "smoothing" effect, due to more values of incoherent indices. As a consequence, diffractional reactions with one value of incoherent indices are expected to have a distinctive feature: their angular distributions display a strongly oscillating pattern, with exceptionally sharp maxima. This seems to be, indeed, the general trend of observed angular distributions: the reactions J 2C(~, po)~SN, 13C(3He ' ~o)iZC, reported in refs. 8, ~9), are characterized by sharper oscillations than other reactions in comparable conditions. Among reactions involving three spinless particles, we mention inelastic scattering of alpha particles on even nuclei 9), while the next section will be devoted to reactions involving only spinless particles. Moreover, in the case of a single value of incoherent indices, the diffractional character can be tested experimentally very neatly, by measuring the polarization of emerging particles, when this makes sense. In fact, the definition of diffractional reaction analogous to eq. (1), Icos (~,~-~o,,,~,)1 = 1, implies that the vector polarization must vanish identically, since the latter has the form P(O) = Z Cu,zx,(O) sin (~o,z-~p,,~,), ll',~."

where 2, Z' are the additional coherent quantum numbers, specifying the S-matrix. If more values of incoherent indices were effective, we could not deduce the vanishing of polarization from the definition of diffraction. We can conclude this section by noting that it is not yet clear whether we can formulate some properly chosen conditions, in such a way that a definition of diffraction can be deduced for reactions involving non-vanishing spins. Even if this would be the case, it would be certainly more difficult to give a simple physical interpretation to such conditions.

6. The reaction 12c(lSo,

160)14C

One of the main fields of application to nuclear physics of the discussion of sects. 3 and 4 is the one of reactions involving spinless heavy ions, such as t2C(180, 160)14C, studied experimentally by Sachs and Goldring 2 0). AS Bromley 2 0) has pointed out, this reaction exhibits unusually sharp oscillations in the angular distribution, the maxima being two or more orders of magnitude larger than the neighbouring minima. Bromley argues that this behaviour is presumably due to the vanishing of all spins involved.

374

G. SCHIFFRER

If we succeed to prove that the above reaction is diffractional, the statement of Bromley could be understood, because we have just seen that the sharpest angular structure is expected to be found in diffractional reactions involving only one value of incoherent indices, in particular if all particles are spinless. For this purpose, it is reasonable to assume that the conditions defining strong absorption are approximately satisfied both in the initial and in the final channel, so that ½(~+/~) should be nearly independent on 1. Thus, we need only the condition (l 1), in order to be sure that the reaction is diffractional. Is this condition satisfied in our case? More experimental information on elastic scattering in both channels and an accurate phenomenological analysis on the basis of the optical or the strong absorption model will provide an answer to this important question. In the meantime, we can only rely on some features of the strong absorption model [ref. 18)], according to which the elastic amplitudes depend essentially, though not exclusively, on two parameters: the angular momenta l o and A, defining the location and the width of the region where the partial amplitudes change from 0 to 1. To the extent that these parameters depend only on kinematical quantities, such as energy and nuclear radii, one expects that in our case condition (11) will be approximately true if the energy E is much larger than the Q-value. It is interesting to point out that E >> Q is also a condition for the validity of the adiabatic approximation 8-10). In this sense, the adiabatic approximation seems to be related with diffraction. In the experiment by Sachs and Goldring 2o) we have E(c.m.) = 14 MeV, Q = - 6 . 1 0 MeV. This could cast some doubts on the validity of condition (11) at this energy. On the other hand, the pronounced structure and asymmetry of the angular distribution 20) might suggest that the reaction is diffractional. Clearly, these two statements are not conflicting, because eq. (11) is just a sufficient condition for the validity of eq. (1). It follows from all previous considerations that the role of a dynamical theory of a reaction, whose diffractional character has been established, is only to remove the ambiguity in sign in eq. (I) and to determine Is~/)l, as a function of l and E.

7. C o n c l u d i n g r e m a r k s

In conclusion, three sufficient conditions have been formulated for the validity of eq. (1), or in other words, for a reaction involving spinless particles to be diffractional. They are given by (8), together with either eq. (6) or (7), or (11). In particular, condition (8) holds if particles are strongly absorbed in both channels. It should be emphasized that only unitarity and symmetry of the S-matrix have been used in the derivation, which is therefore independent on any particular statement on nuclear structure, or reaction mechanism. Despite the generality of the arguments, only extremely elementary mathematical tools have been used. It should be added that eq. (10), and therefore condition (1), can be derived also

DIFFRACTIONAL

REACTIONS

375

from the formula S .i )f

~

~'~.)~(o~½~.) \~'~ii ~ " f f J ° i f ,

(13)

w h e r e S~/) is t h e B o r n a p p r o x i m a t i o n a m p l i t u d e , w h i c h is real. E q . (13) c a n be ded u c e d f r o m a s e m i c l a s s i c a l f o r m u l a t i o n o f d i s t o r t e d w a v e a p p r o x i m a t i o n 12, i s ) , o r f r o m v a r i o u s v e r s i o n s o f d i s p e r s i o n t h e o r y as). T h e p r e s e n t d e r i v a t i o n , h o w e v e r , s e e m s to be m o r e s a t i s f a c t o r y , b e c a u s e t h e c o n d i t i o n s for its v a l i d i t y are wider, or else c a n be b e t t e r u n d e r s t o o d p h y s i c a l l y ; m o r e o v e r , t h e u n d e r l y i n g m a t h e m a t i c s is far simpler. T h e d i f f r a c t i o n a l c h a r a c t e r o f a r e a c t i o n c a n be t e s t e d in a few p a r t i c u l a r cases: e i t h e r by a n a l y s i n g t h e elastic s c a t t e r i n g d a t a in the case o f spinless particles, o r by m e a s u r i n g the p o l a r i z a t i o n in t h e case d i s c u s s e d in sect. 5. I a m d e e p l y i n d e b t e d to Prof. A t t i l i o A g o d i for the s t i m u l a t i n g c o n v e r s a t i o n s w h i c h g a v e rise to this r e s e a r c h a n d for c o n s t r u c t i v e c r i t i c i s m d u r i n g the p r o g r e s s o f t h e w o r k .

References 1) E. Clementel and C. Villi (eds.), Direct interactions and nuclear reaction mechanisms (Gordon and Breach, New York, 1963) 2) B. Kozlowsky and A. Dar, Phys. Lett. 20 (1966) 311 3) J. S. Blair, reE 1), p. 1165 4) A. Agodi, Proc. of the int. summer meeting on problems of nuclear structure, Hercegnovi (1966), ed. by L. gips (Federal Nucl. Energy Commission of Yugoslavia); G. Schiffrer, seminar at Hercegnovi (1966), unpublished; A. Agodi and G. Schiffrer, Boll. Soc. ltal. di Fisica 50 (1966) 33 5) G. R. Satchler, Nucl. Phys. 55 (1964) 1 6) A. Agodi and G. Schiffrer, in preparation 7) S. T. Butler, Phys. Rev. 106 (1957) 272 8) W. Tobocman, Theory of direct nuclear reactions (Oxford Univ. Press, 1961) 9) N. Austern and J. S. Blair, Ann. of Phys. 33 (1965) 15 10) J. S. Blair, ref. 1), p. 669 11) L. J. B. Goldfarb and M. H. Hooper, Phys. Lett. 4 (1963) 148; M. B. Hooper, Nucl. Phys. 76 (1966) 449 12) A. Dar, Nucl. Phys. 82 (1966) 354 13) S. T. Butler, N. Austern and C. Pearson, Phys. Rev. 112 (1958) 1227 14) A. de-Shalit, H. Feshbach and L. van Hove (eds.), Preludes in theoretical physics (North-Holland Amsterdam, 1966) 15) H. J. Schnitzer, Revs. Mod. Phys. 37 (1965) 666 16 A. Dar and W. Tobocman, Phys. Rev. Lett. 12 (1964) 511 17 B. Buck and J. R. Rook, Nucl. Phys. A92 (1967) 513 18 W. E. Frahn, Lecture notes, Trieste (1966) 19 H. D. Holmgren, Proc. of the int. school of physics "E. Fermi", Varenna (1960), course 15, ed. by G. Racah (Academic Press, New York, 1962) 20 D . A . Bromley, Proc. of the int. school of physics ' E . Fermi", Varenna (1967), course 40, ed. by M. Jean, to be published by Academic Press, New York 21 L. McFadden and G. R. Satcbler, Nucl. Phys. 84 (1966) 177