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Multiple scattering analysis of K−-d interactions
I~
NuclearPhysics 66 8.B
[
(1965) 673--687; ( ~
North-Holland Publishing Co., Amsterdam
Not to be reproduced by photoprint or microfilm without written permission from the publisher
M U L T I P L E SCATTERING A N A L Y S I S OF K - - d INTERACTIONS N. M. QUEEN
Department of Mathematical Physics, University of Birmingham, England Received 21 September 1964 Abstract: Corrections to the impulse approximation for K - meson scattering by the deuteron are studied. Certain multiple scattering contributions are summed exactly to all orders and are shown to give unitarity corrections to the single scattering approximation. The formalism is applied to calculations of the elastic and total cross sections for K - - d interactions at low energies, using six sets of ~ N scattering parameters given by Chand. The accuracy of the results and their relation to those of previous models are discussed. It is found that the K - - d cross sections considered cannot serve to distinguish between the various sets of K N parameters which fit the available K - - p data.
1. Introduction
Interactions of K mesons with nucleons have been extensively studied in recent years. The analysis of K N processes is complicated by the existence of many competing channels, even at low energies. Ross and Humphrey 1) constructed two sets of zero-range S wave K.N scattering parameters, generally known as solutions I and II, which give an adequate description of all available experimental data on K - - p scattering and absorption processes for K - laboratory momenta below 275 MeV/c. Akiba and Capps 2) have shown, however, that the low-energy behaviour of solution I is inconsistent with data on 2~rc production at higher energies obtained by Tripp et al. z), if the observed smooth variation of the Z - / Z + ratio as a function of energy is taken into account. Thus solution II appears to hold at low energies. Watson 4) has carried out a least-squares analysis of all the K - - p data in the range 350 to 450 MeV/c, including the P wave amplitudes, and found that these data can be fitted quite well in terms of two sets of zero-range S wave scattering parameters, called solutions A and B. These two sets are similar in character, but differ strongly in some respects from those of Ross and Humphrey. More recently, Chand 5) has pointed out that certain properties of Watson's solution B are inconsistent with the existence of poles in the amplitudes for K N processes arising from the 1385 MeV Y* resonance. Chand concluded that the K N amplitudes should be described by an interpolation between the Ross-Humphrey solution II at low energies and the Watson solution A at higher energies, and thus constructed six new sets 6)t of S wave K N interaction parameters, designated as solutions X1, X2, Y1, Y2, Z1 and Z2, which fit the K - - p data in the energy range of interest. These solutions also satisfy ? These six sets are members of a continuous family of solutions. 673
674
N.M.
Qt~EN
the condition that the 1405 M e V Y* resonance bc dcscribcd as an S wave bound state of the K N system. The scattcring amplitudes of these solutions are given by effective-range theory 7), in which p cotg 5x = A[ 1+½RIp2,
(1)
where p is the c.m. wave number, At and RI are thc scattering length and effective range for isospin I (I = 0, 1), and 51 is the corresponding S wave phase shift. Becausc of the absorption channels leading to hyperon production, the phase shifts 5I, and hence thc parameters AI and RI, are complex-valued. Comparison of existing K - - p scattering data with predictions 6) based on Chand's parameters provides little discrimination bctwcen the various sets, although solutions X1 and X2 are somewhat favoured on the basis of the absorption reactions. Since K - - p and K - - n processes depend differently on the two KN isospin states, information on the K - - n interaction would constitute an independent condition which may make it possible to distinguish between yarious sets of scattering parameters which fit the K - - p data equally well. Such information would also provide a direct test of the hypothesis of charge independence for KN interactions.The general philosophy of the impulse approximation s) suggests that the deuteron, because of its simple structure and loose binding, is particularly advantageous as a target nucleus for studying K - - n interactions. Chand 6) has carried out detailed calculations of K - - d scattering and absorption processes on the basis of his six sets of KN scattering parameters, using a static model 9). Although this model has the advantage of taking into account multiple scattering, it treats the target nucleons as infinitely heavy scattering centres and therefore neglects the recoil of the struck nucleons, their initial momentum distribution and the role of two-body transition matrix elements off the energy shell. These neglected effects may bc particularly important for collisions involving large momentum transfers. Various calculations of N-d and K - - d processes t0-1a) have included some of the effects associated with the finite nucleon mass, but have neglected all multiple scattering contributions beyond single or single and double scattering. Since scattering by the deuteron at low energies may bc sensitive to both types of effect lo-14), it is desirable to have a formalism which is capable of including all the effects mentioned above. As reported in ref. 14), such a formalism was used with some success in interpreting N-d elastic scattering in terms of two-body interactions. It was shown there that certain multiple scattering contributions of all orders can easily be evaluated in terms of the single scattering contributions which, in turn, can be related to two-body scattering amplitudes by means of the impulse approximation. The "complete impulse approximation", in contrast with the "simple impulse approximation" 1¢), includes all three effects of the finite target mass referred to above. In sect. 2 of the present paper we adapt the formalism of ref. 14) to the K - - d problem. A simple procedure for summing the multiple scattering contributions
K--d
INTERACTIONS
675
more systematically than in the previous work is developed in sect. 3, where it is shown that the leading multiple scattering terms give unitarity corrections to the single scattering approximation. Results of numerical calculations for elastic and total K - - d cross sections are presented in sect. 4. In sect. 5 we discuss the accuracy of our results and compare them with other approximations. It is shown that it is not possible to distinguish between the various sets of K N scattering parameters on the basis of the elastic or total K - - d cross sections, because of the present uncertainty in the deuteron wave function. One of the main purposes of this paper is to examine the nature and magnitude of various effects in K---d scattering, in the hope of shedding light on some aspects of the three-body problem which are not yet well understood. 2. Application of the Multiple Scattering Formalism to K--d Processes In calculating K - - d scattering amplitudes by means of the multiple scattering formalism t in the present work, we neglect the contributions from all intermediate states other than K.NN states, although absorption from the Y-~NN channels due to hyperon production is taken into account by the imaginary parts of the K.N interaction parameters. We are concerned here only with the non-absorptive processes, viz. K-+d
--* K - + d ,
(2a)
K- +n+p,
(2b)
K°+n+n.
(2c)
We label the incident meson 1, the deuteron d and the individual nucleons 2 and 3. The mass of particle i is denoted by rni, the reduced mass of i a n d j by mij, and the interaction between i and j by V~j. In order to simplify our calculations, we neglect the K - - p Coulomb force and the mass differences between particles belonging to the same isospin multiplet. Our calculations are therefore not applicable at very low energies, where deviations from charge independence arising from these effects are important is). The amplitudes for the processes (2) are given by the matrix elements F(f, i) = (--4n2mld/h2)(f[tli), (3) where t is the transition operator satisfying the equation
t = Via + VIalGt;
(4)
here G = ( E - H o + i e ) -1, E is the total energy of the three-body system, and [i) and If) are the initial and final eigenstates of the unperturbed Hamiltonian Ho. The operator G has the spectral representation G = ~ ( E - E a + is)- l la>
* A more detailed discussion of the formalism is given in ref. 14).
(5)
676
N.M. QUF.~N
where the sum is over a complete set of states satisfying Ho[a) = state is o f the form
la> - Ik, ~ , Zj>,
E, la>. Each
such
(6)
i.e. a product of a meson plane-wave state k, a nucleon-nucleon state ~b~in coordinate or momentum space, and a K N N isospin state Zj; the corresponding energy is given by (7)
E,, = E~,(k) - ( h 2 / 2 m l d ) k 2 + E . .
We consider two-body interactions in S states only, so that there is no possibility o f nucleon spin-flip and we can neglect the spin coordinates. The initial K - - d state is given by li) = [k0, ~o, Zo),
(8)
where q~o is the ground state of the deuteron and Zo = 2-~[IKi - P2na>-[K~- n2p3)']
(9)
is an isospin eigenstate with I = ½, I s = -½. The only other isospin state to which transitions can occur is Zt = 6-½['21K°n2na)-[K~- p 2 n a ) - I K ~ - n2pa>],
(10)
also with I = ½, I 3 = -½. In Zo the two nucleons have total isospin I = 0, whereas in Z1 they have I = 1. Therefore only Xo is associated with the ground state of the deuteron. When it is necessary to indicate the continuum states explicitly, we denote by ~b~ the two-nucleon state with c.m. wave vector q and isospin I. In calculating amplitudes for the processes (2b, c) involving deuteron disintegration, the appropriate final states If) in the matrix elements (3) contain two-nucleon continuum states with ingoiny spherical waves at infinity l~); the ~b~ here always refer to states o f this type. The multiple scattering expansion can be written 17) t = T2+T3-I-T3GT2+T2GTa+T2GT3GT2+
....
(11)
where T~ (i = 2, 3) is the transition operator satisfying the equation T, = V l , + V I , G T , .
(12)
Substituting eq. (5) into the expansion (11) and taking matrix elements, we obtain an expansion for the required matrix elements of t in terms of those of T2 and T3. In the multiple scattering terms of this expansion, corresponding to each intermediate state there is a contribution from the ground state of the deuteron with associated three-body isospin function Zo, as well as integrals over all the continuum states of the two-nucleon system for both Zo and Zx. As seen from eq. (10), transitions between the states Xo and Z1 include contributions from the charge-exchange process (2c).
K--d
677
INTERACTIONS
In order to evaluate matrix elements between definite isospin states •j, we expand the Xj in terms of I(N isospin states r/,(1, i) for particles 1 and i (i = 2, 3) in the form Xj = ~ , 2j, r/,(1, i), where the r/~(1, i)which occur and their corresponding eigenvalues are given by r/o(1, i) = 2-~EIK°n,) -IK~- p,)]
(I = 0, I a = 0),
(13a)
rh(1, i) - 2-~ElK°n,> + IK~"P,)-I
(I = 1, 13 = 0),
(13b)
r/2(1, i) = IK~- n,)
(I = 1, I 3 = - 1 ) ,
(13c)
and the coefficients 2jr involve isospin states of the third particle. We now apply the impulse approximation in the form used in ref. 14) to express the T matrix element for scattering of the meson by each of the bound nucleons in terms of matrix elements for free meson-nucleon scattering. This approximation is obtained by assuming that the meson is scattered by the bound nucleon as though it were a free nucleon with the momentum distribution appropriate to the ground state of the deuteron. We obtain
(k', (a~, xjITiIk, ~bp, Xk) ~ f f dqdq'tS[q'-q +½( k' - k )]g~(q * ')gp(q) x ~. (2j, lAks)(Ak'--Bq', ~lr[ti(e)[Ak-Bq, r/s>, (14) where the g,(q) - (ql~b~) are the N N wave functions in momentum space; the twobody transition operator ti(e) is defined by ti(e ) = Vii+ Vli(e-ho+ie)-lti(e),
(15)
where h o is the unperturbed Hamiltonian of the two particles; the energy e in eq. (14) is taken as e = (h2/2mli)],,lk-Bq[ 2, and A and B are constants given by B = m l / ( m l + m i ) and A = 1-½B. The matrix elements of T i and t i are taken between Y-, momenta measured in the K - d and K - N c.m. systems, respectively, and the kinematical factors A and B arise from the transformation between these two systems. Because of conservation of I and I a and charge independence, all the matrix elements (~/,It~lr/,) vanish for r # s and the remaining isospin sum in eq. (14) reduces to a linear combination of the amplitudes t[ for the channels I = O, I defined by t ° = (r/olt#/o) and t~ = Q/little1) = (r/2ltllr/2). E.g., the impulse approximation for the matrix elements (ZolZ~lXo)depends on the combination ¼t°+¼t~. The Y-,N matrix elements in the integrand o f eq. (14) are in general off the energy shell. Unless some approximation is introduced to relate them to corresponding on-shell matrix elements, it is necessary to evaluate them in a specific model of the K N interactions.
3. The Multiple Scattering Corrections We turn now to the systematic evaluation and summation of the multiple scattering corrections. Although we are especially concerned here with K - - d scattering,
678
N.M. QUEEN
in order to derive certain results which are applicable to a more general class of three-body problems we assume initially in this section that the two nucleons of the deuteron are distinguishable and have different interactions with the incident meson. It is convenient to decompose the operator G into the sum G = G' + G", where 14) G'
= --inf dk6[E-Eo(k)'llk, ~Po,Zo)
(16)
is the contribution to the sum (5) arising from states which are on the energy shell and contain the deuteron in its ground state (we refer to such states as "on-shell elastic" states), while G" is the remainder, arising from states which are off-shell or have the nucleons in an excited state. The multiple scattering expansion (11) is equivalent to the set of coupled equations t = 1412+Wa,
(17)
W 2 = R2+R23+R2G'Wa+R23G'W2,
(18a)
W 3 = Ra+Ra2+R3G'W2+R32G'W3,
(18b)
where the quantities Ri and R o are the solutions to the equations (19a)
R 2 = T2+T2G"Ra2, R 3 = Ta+T3G"R23
,
(19b)
R23 -- T 2 G ' ' R 3 ,
(19c)
R32 -----T3 G"R2.
(19d)
A formal iterative solution of eqs. (17)-(19) leads back to the multiple scattering expansion, although these equations may be verified more rigorously by standard operator methods and their validity is independent of the convergence of the expansion. The advantages of using these equations instead of the original expansion are that they avoid possible convergence problems, provide a neat separation of effects from different types of intermediate states, make it possible to sum certain multiple scattering contributions exactly to all orders, and give rise to approximate solutions which satisfy unitarity conditions on the S matrix. The role of the quantities Ri and R 0 here is somewhat similar to that of the usual reaction matrix 18) in two-body scattering theory, and eqs. (18) are analogous to the Heitler "radiation damping" equation. Assuming that we have an approximate solution of eqs. (19), we now solve eqs. (18) for W2 and Wa. Let us make a partial-wave expansion for the on-energy-shell elastic scattering matrix elements of each of the operators t, W2, W3, R2, R3, R23, R 3 2 , T 2 and T3; thus, cO
(-4n2mlak/h2)(k ', (So, Zoltlk, q~o, Zo)
-- ~ (2/+ 1)t~t~(cos Okk,), |=0
(20)
K--d INTERACTIONS
679
with a similar notation for the other cases. In the case of each of the transition operators, the partial-wave amplitudes can be expressed in terms of phase shifts in the form t t = (S t - 1)/2i,
St = exp(2i8t),
(21)
and similarly for T2t and Tat. Using eq. (16) and the identity 19) (2l+ 1)fPt(cos Ok~,,)Pr(cos Ot,,t,,,)df2k, = 4nfwPt(cos Okk"),
(22)
we obtain the partial-wave decomposition of eqs. (18) in the form W2t -~ R2t q- R23t-t" iR2t W31+iR231W21 ,
(23a)
W31 ~- R3tq-R32tq-iR31W21q-iR32 t W3t.
(23b)
Solving for W2t and W3t and substituting in eq. (17), we have
h = R2t + R3t + R23t + R32t + 2iR2t R3t-- 2iR23t Ra2z (1 -- iR2at)(1 - iRa2t) + Rat Rat
(24)
The complete K - - d elastic scattering amplitude is then given by eq. (20). We now require an approximation for the R, and R, i. The simplest possible approximation, namely -R2 ~, T2,
R3 ~,~ T3,
R23 ~ 0,
R32 ~ 0,
(25)
is obtained from eqs. (19) by neglecting the contributions from all intermediate off-shell and inelastic states. We now show that our lowest-order multiple scattering approximation t ~°), defined by eqs. (24) and (25), gives unitarity corrections to the single scattering approximation t ..~ 7"2+T3, which violates elastic unitarity badly at low energies 14). Using the definition of the quantities St, S2t and Sat by means of relations of the form (21), eqs. (24) and (25) yield St = S2t "°r.$3! --t-.3S2t S3t - 1
(26)
S21 + Sat- S21S3t + 3 Since T 2 and 7"3 are the exact transition operators for scattering of the incident meson by the bound nucleons, the unitarity condition for these processes requires that IS2tl _-< 1 and [S3t I _< 1. Combining these inequalities with eq. (26), some tedious but straightforward algebra leads to the result [S~I < 1, which shows that our lowestorder multiple scattering approximation automatically preserves elastic unitarity if the approximations used for T 2 and T3 individually satisfy this condition. In the special case in which no inelastic channels are open, our initial unitarity restriction reduces to IS2tl -- ISazl = 1 and eq. (26) then leads to the expected result IS11 = 1. Multiple scattering approximations of higher order are obtained in a similar manner by iterating eqs. (19) and substituting the result into eq. (24), The first
680
N.M. QUEEN
iteration gives R2 ~ T2,
R3 ~' T3,
R23 ~
T2G"T3,
R32 ~
T3G"T2.
(27)
In general, we define the nth iteration of eqs. (19) as that which includes all terms containing up to n occurrences of the factor G", and we denote the corresponding approximation for t by t ~"). By substituting this nth iteration into the formal iterative solution of eqs. (17) and (18) and comparing the result with the original multiple scattering expansion (I1), it is readily verified that the approximation t c") contains all contributions to the complete expansion (11) except those which correspond to more than n consecutive occurrences of the operator G" in any multiple scattering term. If all the multiple scattering terms with up to n intermediate states are known, it is possible to calculate t t") exactly by the methods outlined above. We cannot prove any unitary properties of the approximations t t") with n _~ 1 similar to those of t t°), since these higher approximations involve off-energy-shell matrix elements of/'2 a n d / 3 , which are not bounded by unitarity. However, this limitation may be of little practical importance because the higher approximations t ~") appear to represent small corrections to t ~°) (see sect. 5). Although we have so far considered in detail only the K---d elastic scattering, similar approximations can be formulated for the inelastic processes, in which the final state If) involves a two-nucleon continuum state. In lowest order, the amplitudes for inelastic scattering are obtained by taking the appropriate matrix elements of the approximation t (°) = 7'2 + Ta + 7"2 G'W(°)+ 7"3 G'W(2°),
(28)
where the required (on-shell elastic) matrix elements of W2t°) and W3t°), defined as the solutions of eqs. (18) under the approximation (25), can be evaluated by means of the preceding analysis of the elastic scattering. Approximation (28) includes contributions to all orders of multiple scattering arising from intermediate on-shell elastic states only. The specialization of the preceding results to the K - - d problem is trivial. The correct isospin algebra is taken into account as described in sect. 2, and the Pauli principle for the two nucleons is automatically satisfied by using completely antisymmetric NN states. Inelastic scattering processes lead to final states in the two isospin channels (9) and (10), so that the differential cross sections for deuteron disintegration (without charge exchange) and for charge exchange, i.e. for reactions (2b) and (2c) respectively, are given by dO'dis = do- 0 .k-~do- 1,
do-,. = ~do "1,
(29a) (29b)
with
do-x = (k'/k)lF(k', d~, X~; k, dpo, Zo)12dt2vdq,
(30)
K - - d INTERACTIONS
681
where the amplitude F is defined by the matrix element (3). The main simplification in the treatment of elastic K---d scattering results from the fact that corresponding elastic scattering matrix elements o f / ' 2 and T3 are equal. Thus, our lowest-order multiple scattering approximation reduces to t~o)_
2Tt ,
(31)
1-iT, where Tl -- T2~ = Tat. Several interesting observations follow from eq. (31). Firstly, it is clear that the multiple scattering contributions which are effectively summed in each partial-wave amplitude h(°) form a simple geometric series. Furthermore, each such series converges if [T~[ < 1, which always holds unless the corresponding phase shift has a resonance value of an odd multiple of ½n. These remarks generalize the conclusions about convergence which were derived heuristically in ref. 1~) for the special case when S wave scattering dominates. Thus the original multiple scattering expansion diverges (by oscillation) at any resonance, although eq. (31) remains valid. This illustrates one of the advantages of the present approach, in which various multiple scattering contributions are summed in closed form. Secondly, eq. (31) supports the frequentlymade conjecture 11, 14) that the total effect of the multiple scattering corrections is always to reduce the cross sections of the single scattering approximation. Since unitarity requires that Im T t => 0, eq. (31) leads immediately to the inequality ITS°)[ __<2[Tll, i.e. the multiple scattering corrections of this type reduce each partialwave amplitude. 4. Calculations
Our detailed numerical calculations in the present work are confined to the K - - d elastic scattering amplitude, from which we obtain both the cross section for elastic scattering and the total cross section, where the latter is given by the optical theorem. These calculations are based on the multiple scattering approximation t ~°), defined in terms of the transition operator T by eq. (31). The required matrix elements of T are evaluated in the impulse approximation. For the ground state of the deuteron in momentum space we use the Hulth6n function 1 1 1 go(q)-- [ab(a+b)]~[ 7r(b-a) q2+a2 q2-bb2' normalized so that SgE(q)dq= 1, and we take the parameter values 20) a =
(32)
0.232 fm-1 and b = 5.2 a. Effects of both the D state and the hard core in the deuteron wave function are expected to be small for scattering at low energies 21) and are neglected here. Very few calculations have been attempted using the impulse approximation in a form such as eq. (14) because of the difficulty of evaluating the off-energy-shell two-body matrix elements in any realistic model and then performing the integration
682
N . M . QUEEN
over them. It is customary to evaluate these matrix elements instead at some nearby point of the energy shell. If this point is fixed as q and q' vary ("simple impulse approximation"), the matrix element becomes a constant and can be removed from the integral in eq. (14). However, it is well known lo, 12.14) that, although this procedure is very accurate for forward scattering, it may introduce serious errors for large momentum transfers. We therefore adopt a compromise between the complete and the simple impulse approximations by replacing all matrix elements of the form (where e = hZp2/2ml~, and p' ~ p in general) by , where c is chosen so that cp' = p. Thus, these matrix elements are all evaluated on the energy shell and can be expressed directly in terms of phase shifts, but the integral (14) still receives contributions from K.N scattering over a range of energies. Some justification can be given for this procedure. The K N interaction is generally believed to be of short range compared, for example, with the N N interaction. Writing eq. (15) in the form (p'lt,(e)lp> = (P'I VI,lp> + (P'I Vl,lp">
(33)
it is readily verified that in the limit of zero-range forces, in which VIi is proportional to a tS-function in coordinate space, the matrix elements of t~(e) depend only on e TABLE 1 Elastic a n d total K - - d cross sections calculated for the six sets o f i ( N scattering p a r a m e t e r s o f C h a n d K- laboratory m o m e n t u m (MeV/c)
Cross sections (mb) X1
X2
Y1
Y2
Z1
Z2
~el
150 200 250 300 350 400 ~)
114.4 65.7 38.2 22.7 13.9 8.9
119.0 68.5 39.5 23.1 14.0 8.9
114.1 64.5 37.4 22.3 13.8 8.9
119.4 67.4 38.7 22.8 13.9 8.9
115.5 64.3 37.1 22.1 13.8 8.9
121.4 67.3 38.3 22.6 13.9 8.9
O'tot
150 200 250 300 350 400 a)
312.8 215.6 153.5 112.2 84.2 64.8
316.3 220.6 156.7 113.7 84.7 64.8
312.5 213.4 151.7 111.1 83.8 64.8
316.3 218.5 154.9 112.6 84.3 64.8
314.7 213.1 150.9 110.5 83.5 64.8
318.6 218.2 154.1 112.0 84.0 64.8
s) T h e six solutions X1 to Z2 give identical results at 400 M e V / c because they all agree with the W a t s o n solution A at this m o m e n t u m .
and not on the initial and final momenta, since (p'[ VI~Ip) is then independent of p and p'. In general, the shorter the range of the forces, the smaller the dependence is expected to be. In addition, it should be pointed out that the K---d elastic scattering
K - - d INTERACTIONS
683
in the energy range of interest is strongly peaked in the forward direction and that the forward scattering amplitude depends primarily on the single scattering term T. Since matrix elements of T in the forward direction are given accurately by the simple impulse approximation, we conclude that the neglect of K N scattering off the energy shell is a good approximation in evaluating the K---d total cross section given by the optical theorem and, to a lesser extent, the integrated elastic cross section. If at a later stage it is desired to investigate the differential cross section in greater detail, the off-shell ~.N matrix elements can be computed in a potential model. In table 1 we present values of the elastic and total K - - d cross sections calculated for the six sets of K N scattering parameters of Chand, at several K - laboratory momenta between 150 and 400 MeV/c. Our neglect o f the K ° - K - mass difference may be unjustified at the lower end of this momentum range, which is close to the charge-exchange threshold at 97 MeV/c. The calculation would probably also be inapplicable above 400 MeV/c, because of the inadequacy of the effective-range representation of the K N amplitudes. The shape of the angular dependence of the differential elastic cross section in our calculations is found to be in good agreement with that obtained by Chand 6) in the static model, and is therefore not given here. Unfortunately, experimental data are not yet available for comparison with the results listed in table 1. Such data would be extremely useful as a test of the multiple scattering formalism and the correctness of the phenomenological K N scattering parameters. However, it is shown in the following section that, even if accurate data were available and if there were no uncertainties in our method of calculation, it would not be possible to distinguish between the six sets of K N scattering parameters on the basis of the K - - d cross sections under consideration because of the present uncertainty in the deuteron wave function.
5. Comparison and Discussion of Various Approximations Since our formalism takes into account a number of effects which have been neglected in many previous models of the three-body scattering process, the present results serve as a means of studying the importance of these effects. Various approximations for the elastic and total K - - d cross sections for solution X1 are compared in table 2 for several energies. We now explain these approximations and discuss their accuracy. The "impulse approximation" here refers to the procedure of retaining only the single scattering terms and evaluating them as described in the preceding section. The "double scattering approximation" is obtained by adding to the single scattering terms the double scattering contributions with intermediate on-shell elastic states, with the required matrix elements evaluated in the same way as in the single scattering terms; the result is equivalent to the approximation t t ~ 2 T l ( l + i T ~ ) , although the actual calculation was made by a different method 14). As can be seen in table 2, the multiple scattering corrections of the type considered arise almost entirely from
684
N.M. QUEEN
double scattering processes, particularly at the higher end of the energy range. The multiple scattering corrections are somewhat more important for the elastic than for the total cross section; the reason for this appears to be that the single elastic scattering, unlike the multiple scattering, is strongly peaked in the forward direction, so that the correction to the single scattering is of greatest importance at large scattering angles. There is little advantage in using the double scattering approximation as a first-order correction to the single scattering, even when the higher-order effects are small (as they are here), because the multiple scattering approximation t (°) is no m o r e difficult to evaluate and has the additional advantage of giving unitarity corrections to the single scattering approximation. TABLE 2
Comparison of various approximations for the elastic and total K---d cross sections for solution X1 150 ~rel(mb)
O'tot(mb)
K- laboratory momentum (MeV/c) 200 300 400
Impulse approximation Double scattering
209.7 114.7
114.1 66.1
33.0 22.7
11.3 8.9
Static model e) Present work
117.8 114.4
74.8 65.7
27.3 22.7
8.9
crnq-ap Glauber approximation
399.4 267.3
263.2 205.5
126.5 113.2
69.8 65.8
Impulse approximation Double scattering Static model 6) Present work
389.7 330.5 321.6 312.8
258.7 224.3 225.2 215.6
125.7 113.9 118.0 112.2
69.6 65.1 64.8
The "static model" in table 2 refers to the results obtained by Chand 6) in a formalism which includes multiple scattering effects completely but treats the nucleons as fixed, infinitely heavy scattering centres. Thus the difference between the results of the static model and those of the present work gives a measure of the effects associated with the finite mass of the nucleons. The static model, like the simple impulse approximation, neglects the internal m o m e n t u m distribution of the deuteron and the associated two-body scattering off the energy shell. The corrections due to these effects have generally been found 1o, 12, 14) to reduce the magnitude of the single scattering terms, although only slightly in the forward direction. For reasons explained above, the effect of these corrections is therefore small in the case of the total cross section. The results given in table 2 exhibit the expected behaviour. The simplest possible approximation for the total K - - d cross section, namely trtot ~ t r , + % , where trn and trp are the K - - n and K - - p total cross sections at the same K - laboratory energy, is obtained by direct application of the optical theorem to the simple impulse approximation. As expected, this approximation predicts slightly larger values of atot than the complete impulse approximation.
K - - d INTERACTIONS
685
22) expresses atot in the form ,~ a n + o'p + (4rc/k2)Re(fnfp)(r-2)d,
The Glauber approximation o'to t
(34)
w h e r e f , a n d f p are the K - - n and K - - p forward scattering amplitudes, and ( r - 2 ) a is the expectation value of the inverse square of the n - p separation in the ground state of the deuteron. This simple relation is particularly useful for extracting information on interactions with neutrons from experimental data on scattering by the deuteron. It can be adequately justified at very high energies, but the precise conditions for its validity are not well understood. Since the Glauber approximation is derived by making several further simplifications in the double scattering approximation, it seems worthwhile to use the results of our more exact calculations to study its validity. The Hulth6n function (32) yields (r-2)d =
4ab(a+b)(b-a)-2[aln(2a)+bln(2b)-(a+b)ln(a+b)].
(35)
Treating the nucleons as identical particles, the isospin algebra for the K - - d system is taken into account in eq. (34) by setting an = % = ¼a ° +¼a 1 andfn = fp = ¼fo +¼ f l , where a t a n d f ~ are the corresponding cross section and amplitude for isospin L The numerical results in table 2 indicate that the Glauber approximation is reliable only at the higher energies considered, although even at low energies it represents an improvement over approximations which neglect multiple scattering. TABLe 3 Comparison of the calculated K---d cross sections for solution X1, using deuteron wave functions of the Hulth~n and Gartenhaus type K- laboratory
Hulth6n (b = 5.2a)
Hulth6n (b = 7a)
momentum (MeV/c)
ael(mb)
atot(mb)
ael(mb)
150 200 300 400
114.4 65.7 22.7 8.9
312.8 215.6 112.2 64.8
118.8 69.6 24.9 10.0
atot(mb) 306.8 211.2 110.3 64.0
Gartenhaus ael(mb)
atot(mb)
114.2 65.3 22.2 8.5
313.9 216.4 112.6 65.0
Any analysis of scattering by the deuteron necessarily depends to some extent on the assumed form of the N N interaction. This interaction enters our detailed calculations only through the deuteron wave function. Since one of the main reasons for studying low-energy K - - d processes at the present time is the hope of narrowing the choice a m o n g a continuous family of K N interaction parameters, it is essential to determine how sensitive the calculations are to the assumed deuteron wave function. A function of the Hulth6n form (32) is usually adopted for the S state of the deuteron. The parameter a is fixed by the binding energy, although the best value for the parameter b is uncertain and estimates 20, 23) vary f r o m b = 5.2a to b = 7a.
686
N.M.
QUEEN
Typical K---d cross sections calculated using these two values of b are compared in table 3. The range of variation in these cross sections due to the uncertainty in the parameter b is in all cases at least as great as the variation (table 1) corresponding to the possible K N parameters. Thus, the uncertainty in the deuteron wave function prevents the possibility of distinguishing between the various sets of g.N parameters on the basis of the K - - d cross sections given. It should be mentioned that earlier calculations 24) based on the original Ross-Humphrey solutions I and II gave quite distinct results for these two solutions, although these calculations are no longer relevant. Some justification is required for the direct comparison (table 2) of the results of Chand in the static model with those of the present work, since Chand assumed an analytic form of the deuteron wave function which very accurately approximates the Gartenhaus 25) (hard-core) wave function, whereas we assume here the much simpler Hulth6n function. Numerical results (table 3) using these two wave functions differ only slightly, provided we assume the Hulth6n parameters b = 5.2a. We next consider the error in our lowest-order multiple scattering approximation t ~o) due to the neglect of off-shell and inelastic intermediate states. In recent calculations x4, z4) of N - d and K - - d processes, we have made a crude estimate of the leading corrections to t c°). It was found that, although the double scattering contribution from the intermediate states neglected in t ~°) may be rather large, this correction tends to be cancelled almost completely by similar terms of higher order. This cancellation arises from phase differences between terms of various orders; it tends to be stronger than the cancellation among the terms effectively summed in t ~°), since in any multiple scattering term with n intermediate states there are in general many possible ways to be considered in which exactly m ( m < n) of these states are on-shell elastic, whereas there is only a single possibility in which all n states are of this type. The nature of this cancellation can be better understood, however, by examining successively higher approximations after t ~°) in our formalism. E.g., the next approximation, t ~1), is derived from eqs. (24) and (27). After some algebraic manipulation the result can be expressed as t~l) = t~0)q_
2D, (1 - i T , ) ( 1 - i T , - iD,) '
(36)
where D~ is the partial-wave amplitude corresponding to each of the double scattering contributions T 2 G"T3 and T 3 G"T2. The effect of the cancellation is contained in the denominator of the second term in eq. C36). Assuming the very plausible result Im D l > --Im Ts, the inequality Im Tl >= 0 (see sect. 3) implies that each factor in the denominator is of modulus not less than unity, so that the denominator leads to a reduction in the modulus of the second term which is in fact stronger than the corresponding reduction in the case of t[ °). Calculations of the inelastic K - - d processes (2b, c) are particularly difficult and have many features in common with those of the multiple scattering contributions
K - - d INTERACTIONS
687
involving inelastic i n t e r m e d i a t e states. A n estimate o f the o r d e r o f m a g n i t u d e o f the inelastic cross sections in the earlier calculations 24) i n d i c a t e d that, as in the case o f the elastic scattering, o u r f o r m a l i s m gives slightly smaller inelastic cross sections t h a n those o b t a i n e d b y C h a n d 6). T h e only a p p r o x i m a t i o n in o u r f o r m a l i s m which has n o t been a d e q u a t e l y tested or justified is the use o f the impulse a p p r o x i m a t i o n for the m a t r i x elements o f / ' 2 a n d 7"3. Little is k n o w n a b o u t this a p p r o x i m a t i o n , a l t h o u g h it is believed to be sufficiently a c c u r a t e at m o d e r a t e l y high energies. The b i n d i n g effects which are neglected in the i m p u l s e a p p r o x i m a t i o n are p a r t i c u l a r l y difficult to study, since they involve those effects m o s t closely a s s o c i a t e d with the s i m u l t a n e o u s i n t e r a c t i o n o f all three particles. I n conclusion, we c o m p a r e a n d c o n t r a s t the K - - d with the N - d p r o b l e m , which has been s t u d i e d earlier x4) b y closely r e l a t e d m e t h o d s . The m a i n c o m p l i c a t i o n in the N - d p r o b l e m is the P a u l i principle; a l t h o u g h the identity o f the t w o target nucleons can be r i g o r o u s l y t a k e n into account, difficulties arise when the incident particle is also a n u c l e o n because the incident a n d the target particles are n o t treated on an equal f o o t i n g in this f o r m a l i s m . The a c c u r a c y o f the N--d calculations is also m o r e u n c e r t a i n because the v a r i o u s c o r r e c t i o n s to the simple impulse a p p r o x i m a t i o n are o f greater i m p o r t a n c e t h a n for K - - d scattering at the s a m e energies. I n addition, the d e s c r i p t i o n o f the t w o - b o d y interactions a n d the spin a n d isospin a l g e b r a are m o r e c o m p l e x for the N - d t h a n for the K - - d system. F o r these reasons, o u r calculations o f K - - d scattering a r e believed to be m u c h m o r e reliable t h a n the p r e v i o u s l y p u b l i s h e d analysis o f N - d scattering.
References 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14) 15) 16) 17) 18) 19) 20) 21) 22) 23) 24) 25)
R. R. Ross and W. E. Humphrey, Phys. Rev. 127 (1962) 1305 T. Akiba and R. H. Ca1313s, Phys. Rev. Lett. 8 (1962) 457 R. D. Tri1313, M. B. Watson and M. Ferro-Luzzi, Phys. Rev. Lett. 8 (1962) 175 M. B. Watson, UCRL-10175 (1962) R. Chand, Nuovo Cim. 30 (1963) 1445 R. Chand, Nuovo Cim. 31 (1964) 827, 33 (1964) 148 J. D. Jackson, D. G. Ravenhall and H. W. Wyld, Nuovo Cim. 9 (1958) 834 G. F. Chew, Phys. Rev. 80 (1950) 196; G. F. Chew and G. C. Wick, Phys. Rev. 85 (1952) 636 R. Chand, Ann. of Phys. 22 (1963) 438 T. Fulton and P. Schwed, Phys. Rev. 115 (1959) 973 A. Everett, Phys. Rev. 126 (1962) 831 K. L. Kowalski and D. Feldman, Phys. Rev. 130 (1963) 276 A. K. Bhatia and J. Sucher, Phys. Rev. 132 (1963) 855 N. M. Queen, Nuclear Physics 55 (1964) 177 J. D. Jackson and H. W. Wyld, Nuovo Cim. 13 (1959) 85 K. M. Watson, Phys. Rev. 88 (1952) 1163 K. M. Watson, Phys. Rev. 89 (1953) 575 M. L. Goldberger, Phys. Rev. 84 (1951) 929 D. M. Brink and G. R. Satchler, Angular momentum (Oxford University Press, 1962) 13. 121 M. J. Moravcsik, Nuclear Physics 7 (1958) 113 Y. Sakamoto and T. Sasakawa, Prog. Theor. Phys. 21 (1959) 879 R. J. Glauber, Proc. Conf. on Nuclear Forces and the Few-Nucleon Problem, London, 1959, ed. by T. C. Gritiith and E. A. Power (Pergamon Press, London, 1960) p. 233 R. L. Gluckstern and H. A. Bethe, Phys. Rev. 81 (1951) 761 N. M. Queen, Ph.D. thesis, University of Bristol (1963) S. Gartenhaus, Phys. Rev. 100 (1955) 900
NuclearPhysics 66 8.B
[
(1965) 673--687; ( ~
North-Holland Publishing Co., Amsterdam
Not to be reproduced by photoprint or microfilm without written permission from the publisher
M U L T I P L E SCATTERING A N A L Y S I S OF K - - d INTERACTIONS N. M. QUEEN
Department of Mathematical Physics, University of Birmingham, England Received 21 September 1964 Abstract: Corrections to the impulse approximation for K - meson scattering by the deuteron are studied. Certain multiple scattering contributions are summed exactly to all orders and are shown to give unitarity corrections to the single scattering approximation. The formalism is applied to calculations of the elastic and total cross sections for K - - d interactions at low energies, using six sets of ~ N scattering parameters given by Chand. The accuracy of the results and their relation to those of previous models are discussed. It is found that the K - - d cross sections considered cannot serve to distinguish between the various sets of K N parameters which fit the available K - - p data.
1. Introduction
Interactions of K mesons with nucleons have been extensively studied in recent years. The analysis of K N processes is complicated by the existence of many competing channels, even at low energies. Ross and Humphrey 1) constructed two sets of zero-range S wave K.N scattering parameters, generally known as solutions I and II, which give an adequate description of all available experimental data on K - - p scattering and absorption processes for K - laboratory momenta below 275 MeV/c. Akiba and Capps 2) have shown, however, that the low-energy behaviour of solution I is inconsistent with data on 2~rc production at higher energies obtained by Tripp et al. z), if the observed smooth variation of the Z - / Z + ratio as a function of energy is taken into account. Thus solution II appears to hold at low energies. Watson 4) has carried out a least-squares analysis of all the K - - p data in the range 350 to 450 MeV/c, including the P wave amplitudes, and found that these data can be fitted quite well in terms of two sets of zero-range S wave scattering parameters, called solutions A and B. These two sets are similar in character, but differ strongly in some respects from those of Ross and Humphrey. More recently, Chand 5) has pointed out that certain properties of Watson's solution B are inconsistent with the existence of poles in the amplitudes for K N processes arising from the 1385 MeV Y* resonance. Chand concluded that the K N amplitudes should be described by an interpolation between the Ross-Humphrey solution II at low energies and the Watson solution A at higher energies, and thus constructed six new sets 6)t of S wave K N interaction parameters, designated as solutions X1, X2, Y1, Y2, Z1 and Z2, which fit the K - - p data in the energy range of interest. These solutions also satisfy ? These six sets are members of a continuous family of solutions. 673
674
N.M.
Qt~EN
the condition that the 1405 M e V Y* resonance bc dcscribcd as an S wave bound state of the K N system. The scattcring amplitudes of these solutions are given by effective-range theory 7), in which p cotg 5x = A[ 1+½RIp2,
(1)
where p is the c.m. wave number, At and RI are thc scattering length and effective range for isospin I (I = 0, 1), and 51 is the corresponding S wave phase shift. Becausc of the absorption channels leading to hyperon production, the phase shifts 5I, and hence thc parameters AI and RI, are complex-valued. Comparison of existing K - - p scattering data with predictions 6) based on Chand's parameters provides little discrimination bctwcen the various sets, although solutions X1 and X2 are somewhat favoured on the basis of the absorption reactions. Since K - - p and K - - n processes depend differently on the two KN isospin states, information on the K - - n interaction would constitute an independent condition which may make it possible to distinguish between yarious sets of scattering parameters which fit the K - - p data equally well. Such information would also provide a direct test of the hypothesis of charge independence for KN interactions.The general philosophy of the impulse approximation s) suggests that the deuteron, because of its simple structure and loose binding, is particularly advantageous as a target nucleus for studying K - - n interactions. Chand 6) has carried out detailed calculations of K - - d scattering and absorption processes on the basis of his six sets of KN scattering parameters, using a static model 9). Although this model has the advantage of taking into account multiple scattering, it treats the target nucleons as infinitely heavy scattering centres and therefore neglects the recoil of the struck nucleons, their initial momentum distribution and the role of two-body transition matrix elements off the energy shell. These neglected effects may bc particularly important for collisions involving large momentum transfers. Various calculations of N-d and K - - d processes t0-1a) have included some of the effects associated with the finite nucleon mass, but have neglected all multiple scattering contributions beyond single or single and double scattering. Since scattering by the deuteron at low energies may bc sensitive to both types of effect lo-14), it is desirable to have a formalism which is capable of including all the effects mentioned above. As reported in ref. 14), such a formalism was used with some success in interpreting N-d elastic scattering in terms of two-body interactions. It was shown there that certain multiple scattering contributions of all orders can easily be evaluated in terms of the single scattering contributions which, in turn, can be related to two-body scattering amplitudes by means of the impulse approximation. The "complete impulse approximation", in contrast with the "simple impulse approximation" 1¢), includes all three effects of the finite target mass referred to above. In sect. 2 of the present paper we adapt the formalism of ref. 14) to the K - - d problem. A simple procedure for summing the multiple scattering contributions
K--d
INTERACTIONS
675
more systematically than in the previous work is developed in sect. 3, where it is shown that the leading multiple scattering terms give unitarity corrections to the single scattering approximation. Results of numerical calculations for elastic and total K - - d cross sections are presented in sect. 4. In sect. 5 we discuss the accuracy of our results and compare them with other approximations. It is shown that it is not possible to distinguish between the various sets of K N scattering parameters on the basis of the elastic or total K - - d cross sections, because of the present uncertainty in the deuteron wave function. One of the main purposes of this paper is to examine the nature and magnitude of various effects in K---d scattering, in the hope of shedding light on some aspects of the three-body problem which are not yet well understood. 2. Application of the Multiple Scattering Formalism to K--d Processes In calculating K - - d scattering amplitudes by means of the multiple scattering formalism t in the present work, we neglect the contributions from all intermediate states other than K.NN states, although absorption from the Y-~NN channels due to hyperon production is taken into account by the imaginary parts of the K.N interaction parameters. We are concerned here only with the non-absorptive processes, viz. K-+d
--* K - + d ,
(2a)
K- +n+p,
(2b)
K°+n+n.
(2c)
We label the incident meson 1, the deuteron d and the individual nucleons 2 and 3. The mass of particle i is denoted by rni, the reduced mass of i a n d j by mij, and the interaction between i and j by V~j. In order to simplify our calculations, we neglect the K - - p Coulomb force and the mass differences between particles belonging to the same isospin multiplet. Our calculations are therefore not applicable at very low energies, where deviations from charge independence arising from these effects are important is). The amplitudes for the processes (2) are given by the matrix elements F(f, i) = (--4n2mld/h2)(f[tli), (3) where t is the transition operator satisfying the equation
t = Via + VIalGt;
(4)
here G = ( E - H o + i e ) -1, E is the total energy of the three-body system, and [i) and If) are the initial and final eigenstates of the unperturbed Hamiltonian Ho. The operator G has the spectral representation G = ~ ( E - E a + is)- l la>
* A more detailed discussion of the formalism is given in ref. 14).
(5)
676
N.M. QUF.~N
where the sum is over a complete set of states satisfying Ho[a) = state is o f the form
la> - Ik, ~ , Zj>,
E, la>. Each
such
(6)
i.e. a product of a meson plane-wave state k, a nucleon-nucleon state ~b~in coordinate or momentum space, and a K N N isospin state Zj; the corresponding energy is given by (7)
E,, = E~,(k) - ( h 2 / 2 m l d ) k 2 + E . .
We consider two-body interactions in S states only, so that there is no possibility o f nucleon spin-flip and we can neglect the spin coordinates. The initial K - - d state is given by li) = [k0, ~o, Zo),
(8)
where q~o is the ground state of the deuteron and Zo = 2-~[IKi - P2na>-[K~- n2p3)']
(9)
is an isospin eigenstate with I = ½, I s = -½. The only other isospin state to which transitions can occur is Zt = 6-½['21K°n2na)-[K~- p 2 n a ) - I K ~ - n2pa>],
(10)
also with I = ½, I 3 = -½. In Zo the two nucleons have total isospin I = 0, whereas in Z1 they have I = 1. Therefore only Xo is associated with the ground state of the deuteron. When it is necessary to indicate the continuum states explicitly, we denote by ~b~ the two-nucleon state with c.m. wave vector q and isospin I. In calculating amplitudes for the processes (2b, c) involving deuteron disintegration, the appropriate final states If) in the matrix elements (3) contain two-nucleon continuum states with ingoiny spherical waves at infinity l~); the ~b~ here always refer to states o f this type. The multiple scattering expansion can be written 17) t = T2+T3-I-T3GT2+T2GTa+T2GT3GT2+
....
(11)
where T~ (i = 2, 3) is the transition operator satisfying the equation T, = V l , + V I , G T , .
(12)
Substituting eq. (5) into the expansion (11) and taking matrix elements, we obtain an expansion for the required matrix elements of t in terms of those of T2 and T3. In the multiple scattering terms of this expansion, corresponding to each intermediate state there is a contribution from the ground state of the deuteron with associated three-body isospin function Zo, as well as integrals over all the continuum states of the two-nucleon system for both Zo and Zx. As seen from eq. (10), transitions between the states Xo and Z1 include contributions from the charge-exchange process (2c).
K--d
677
INTERACTIONS
In order to evaluate matrix elements between definite isospin states •j, we expand the Xj in terms of I(N isospin states r/,(1, i) for particles 1 and i (i = 2, 3) in the form Xj = ~ , 2j, r/,(1, i), where the r/~(1, i)which occur and their corresponding eigenvalues are given by r/o(1, i) = 2-~EIK°n,) -IK~- p,)]
(I = 0, I a = 0),
(13a)
rh(1, i) - 2-~ElK°n,> + IK~"P,)-I
(I = 1, 13 = 0),
(13b)
r/2(1, i) = IK~- n,)
(I = 1, I 3 = - 1 ) ,
(13c)
and the coefficients 2jr involve isospin states of the third particle. We now apply the impulse approximation in the form used in ref. 14) to express the T matrix element for scattering of the meson by each of the bound nucleons in terms of matrix elements for free meson-nucleon scattering. This approximation is obtained by assuming that the meson is scattered by the bound nucleon as though it were a free nucleon with the momentum distribution appropriate to the ground state of the deuteron. We obtain
(k', (a~, xjITiIk, ~bp, Xk) ~ f f dqdq'tS[q'-q +½( k' - k )]g~(q * ')gp(q) x ~. (2j, lAks)(Ak'--Bq', ~lr[ti(e)[Ak-Bq, r/s>, (14) where the g,(q) - (ql~b~) are the N N wave functions in momentum space; the twobody transition operator ti(e) is defined by ti(e ) = Vii+ Vli(e-ho+ie)-lti(e),
(15)
where h o is the unperturbed Hamiltonian of the two particles; the energy e in eq. (14) is taken as e = (h2/2mli)],,lk-Bq[ 2, and A and B are constants given by B = m l / ( m l + m i ) and A = 1-½B. The matrix elements of T i and t i are taken between Y-, momenta measured in the K - d and K - N c.m. systems, respectively, and the kinematical factors A and B arise from the transformation between these two systems. Because of conservation of I and I a and charge independence, all the matrix elements (~/,It~lr/,) vanish for r # s and the remaining isospin sum in eq. (14) reduces to a linear combination of the amplitudes t[ for the channels I = O, I defined by t ° = (r/olt#/o) and t~ = Q/little1) = (r/2ltllr/2). E.g., the impulse approximation for the matrix elements (ZolZ~lXo)depends on the combination ¼t°+¼t~. The Y-,N matrix elements in the integrand o f eq. (14) are in general off the energy shell. Unless some approximation is introduced to relate them to corresponding on-shell matrix elements, it is necessary to evaluate them in a specific model of the K N interactions.
3. The Multiple Scattering Corrections We turn now to the systematic evaluation and summation of the multiple scattering corrections. Although we are especially concerned here with K - - d scattering,
678
N.M. QUEEN
in order to derive certain results which are applicable to a more general class of three-body problems we assume initially in this section that the two nucleons of the deuteron are distinguishable and have different interactions with the incident meson. It is convenient to decompose the operator G into the sum G = G' + G", where 14) G'
= --inf dk6[E-Eo(k)'llk, ~Po,Zo)
(16)
is the contribution to the sum (5) arising from states which are on the energy shell and contain the deuteron in its ground state (we refer to such states as "on-shell elastic" states), while G" is the remainder, arising from states which are off-shell or have the nucleons in an excited state. The multiple scattering expansion (11) is equivalent to the set of coupled equations t = 1412+Wa,
(17)
W 2 = R2+R23+R2G'Wa+R23G'W2,
(18a)
W 3 = Ra+Ra2+R3G'W2+R32G'W3,
(18b)
where the quantities Ri and R o are the solutions to the equations (19a)
R 2 = T2+T2G"Ra2, R 3 = Ta+T3G"R23
,
(19b)
R23 -- T 2 G ' ' R 3 ,
(19c)
R32 -----T3 G"R2.
(19d)
A formal iterative solution of eqs. (17)-(19) leads back to the multiple scattering expansion, although these equations may be verified more rigorously by standard operator methods and their validity is independent of the convergence of the expansion. The advantages of using these equations instead of the original expansion are that they avoid possible convergence problems, provide a neat separation of effects from different types of intermediate states, make it possible to sum certain multiple scattering contributions exactly to all orders, and give rise to approximate solutions which satisfy unitarity conditions on the S matrix. The role of the quantities Ri and R 0 here is somewhat similar to that of the usual reaction matrix 18) in two-body scattering theory, and eqs. (18) are analogous to the Heitler "radiation damping" equation. Assuming that we have an approximate solution of eqs. (19), we now solve eqs. (18) for W2 and Wa. Let us make a partial-wave expansion for the on-energy-shell elastic scattering matrix elements of each of the operators t, W2, W3, R2, R3, R23, R 3 2 , T 2 and T3; thus, cO
(-4n2mlak/h2)(k ', (So, Zoltlk, q~o, Zo)
-- ~ (2/+ 1)t~t~(cos Okk,), |=0
(20)
K--d INTERACTIONS
679
with a similar notation for the other cases. In the case of each of the transition operators, the partial-wave amplitudes can be expressed in terms of phase shifts in the form t t = (S t - 1)/2i,
St = exp(2i8t),
(21)
and similarly for T2t and Tat. Using eq. (16) and the identity 19) (2l+ 1)fPt(cos Ok~,,)Pr(cos Ot,,t,,,)df2k, = 4nfwPt(cos Okk"),
(22)
we obtain the partial-wave decomposition of eqs. (18) in the form W2t -~ R2t q- R23t-t" iR2t W31+iR231W21 ,
(23a)
W31 ~- R3tq-R32tq-iR31W21q-iR32 t W3t.
(23b)
Solving for W2t and W3t and substituting in eq. (17), we have
h = R2t + R3t + R23t + R32t + 2iR2t R3t-- 2iR23t Ra2z (1 -- iR2at)(1 - iRa2t) + Rat Rat
(24)
The complete K - - d elastic scattering amplitude is then given by eq. (20). We now require an approximation for the R, and R, i. The simplest possible approximation, namely -R2 ~, T2,
R3 ~,~ T3,
R23 ~ 0,
R32 ~ 0,
(25)
is obtained from eqs. (19) by neglecting the contributions from all intermediate off-shell and inelastic states. We now show that our lowest-order multiple scattering approximation t ~°), defined by eqs. (24) and (25), gives unitarity corrections to the single scattering approximation t ..~ 7"2+T3, which violates elastic unitarity badly at low energies 14). Using the definition of the quantities St, S2t and Sat by means of relations of the form (21), eqs. (24) and (25) yield St = S2t "°r.$3! --t-.3S2t S3t - 1
(26)
S21 + Sat- S21S3t + 3 Since T 2 and 7"3 are the exact transition operators for scattering of the incident meson by the bound nucleons, the unitarity condition for these processes requires that IS2tl _-< 1 and [S3t I _< 1. Combining these inequalities with eq. (26), some tedious but straightforward algebra leads to the result [S~I < 1, which shows that our lowestorder multiple scattering approximation automatically preserves elastic unitarity if the approximations used for T 2 and T3 individually satisfy this condition. In the special case in which no inelastic channels are open, our initial unitarity restriction reduces to IS2tl -- ISazl = 1 and eq. (26) then leads to the expected result IS11 = 1. Multiple scattering approximations of higher order are obtained in a similar manner by iterating eqs. (19) and substituting the result into eq. (24), The first
680
N.M. QUEEN
iteration gives R2 ~ T2,
R3 ~' T3,
R23 ~
T2G"T3,
R32 ~
T3G"T2.
(27)
In general, we define the nth iteration of eqs. (19) as that which includes all terms containing up to n occurrences of the factor G", and we denote the corresponding approximation for t by t ~"). By substituting this nth iteration into the formal iterative solution of eqs. (17) and (18) and comparing the result with the original multiple scattering expansion (I1), it is readily verified that the approximation t c") contains all contributions to the complete expansion (11) except those which correspond to more than n consecutive occurrences of the operator G" in any multiple scattering term. If all the multiple scattering terms with up to n intermediate states are known, it is possible to calculate t t") exactly by the methods outlined above. We cannot prove any unitary properties of the approximations t t") with n _~ 1 similar to those of t t°), since these higher approximations involve off-energy-shell matrix elements of/'2 a n d / 3 , which are not bounded by unitarity. However, this limitation may be of little practical importance because the higher approximations t ~") appear to represent small corrections to t ~°) (see sect. 5). Although we have so far considered in detail only the K---d elastic scattering, similar approximations can be formulated for the inelastic processes, in which the final state If) involves a two-nucleon continuum state. In lowest order, the amplitudes for inelastic scattering are obtained by taking the appropriate matrix elements of the approximation t (°) = 7'2 + Ta + 7"2 G'W(°)+ 7"3 G'W(2°),
(28)
where the required (on-shell elastic) matrix elements of W2t°) and W3t°), defined as the solutions of eqs. (18) under the approximation (25), can be evaluated by means of the preceding analysis of the elastic scattering. Approximation (28) includes contributions to all orders of multiple scattering arising from intermediate on-shell elastic states only. The specialization of the preceding results to the K - - d problem is trivial. The correct isospin algebra is taken into account as described in sect. 2, and the Pauli principle for the two nucleons is automatically satisfied by using completely antisymmetric NN states. Inelastic scattering processes lead to final states in the two isospin channels (9) and (10), so that the differential cross sections for deuteron disintegration (without charge exchange) and for charge exchange, i.e. for reactions (2b) and (2c) respectively, are given by dO'dis = do- 0 .k-~do- 1,
do-,. = ~do "1,
(29a) (29b)
with
do-x = (k'/k)lF(k', d~, X~; k, dpo, Zo)12dt2vdq,
(30)
K - - d INTERACTIONS
681
where the amplitude F is defined by the matrix element (3). The main simplification in the treatment of elastic K---d scattering results from the fact that corresponding elastic scattering matrix elements o f / ' 2 and T3 are equal. Thus, our lowest-order multiple scattering approximation reduces to t~o)_
2Tt ,
(31)
1-iT, where Tl -- T2~ = Tat. Several interesting observations follow from eq. (31). Firstly, it is clear that the multiple scattering contributions which are effectively summed in each partial-wave amplitude h(°) form a simple geometric series. Furthermore, each such series converges if [T~[ < 1, which always holds unless the corresponding phase shift has a resonance value of an odd multiple of ½n. These remarks generalize the conclusions about convergence which were derived heuristically in ref. 1~) for the special case when S wave scattering dominates. Thus the original multiple scattering expansion diverges (by oscillation) at any resonance, although eq. (31) remains valid. This illustrates one of the advantages of the present approach, in which various multiple scattering contributions are summed in closed form. Secondly, eq. (31) supports the frequentlymade conjecture 11, 14) that the total effect of the multiple scattering corrections is always to reduce the cross sections of the single scattering approximation. Since unitarity requires that Im T t => 0, eq. (31) leads immediately to the inequality ITS°)[ __<2[Tll, i.e. the multiple scattering corrections of this type reduce each partialwave amplitude. 4. Calculations
Our detailed numerical calculations in the present work are confined to the K - - d elastic scattering amplitude, from which we obtain both the cross section for elastic scattering and the total cross section, where the latter is given by the optical theorem. These calculations are based on the multiple scattering approximation t ~°), defined in terms of the transition operator T by eq. (31). The required matrix elements of T are evaluated in the impulse approximation. For the ground state of the deuteron in momentum space we use the Hulth6n function 1 1 1 go(q)-- [ab(a+b)]~[ 7r(b-a) q2+a2 q2-bb2' normalized so that SgE(q)dq= 1, and we take the parameter values 20) a =
(32)
0.232 fm-1 and b = 5.2 a. Effects of both the D state and the hard core in the deuteron wave function are expected to be small for scattering at low energies 21) and are neglected here. Very few calculations have been attempted using the impulse approximation in a form such as eq. (14) because of the difficulty of evaluating the off-energy-shell two-body matrix elements in any realistic model and then performing the integration
682
N . M . QUEEN
over them. It is customary to evaluate these matrix elements instead at some nearby point of the energy shell. If this point is fixed as q and q' vary ("simple impulse approximation"), the matrix element becomes a constant and can be removed from the integral in eq. (14). However, it is well known lo, 12.14) that, although this procedure is very accurate for forward scattering, it may introduce serious errors for large momentum transfers. We therefore adopt a compromise between the complete and the simple impulse approximations by replacing all matrix elements of the form
(33)
it is readily verified that in the limit of zero-range forces, in which VIi is proportional to a tS-function in coordinate space, the matrix elements of t~(e) depend only on e TABLE 1 Elastic a n d total K - - d cross sections calculated for the six sets o f i ( N scattering p a r a m e t e r s o f C h a n d K- laboratory m o m e n t u m (MeV/c)
Cross sections (mb) X1
X2
Y1
Y2
Z1
Z2
~el
150 200 250 300 350 400 ~)
114.4 65.7 38.2 22.7 13.9 8.9
119.0 68.5 39.5 23.1 14.0 8.9
114.1 64.5 37.4 22.3 13.8 8.9
119.4 67.4 38.7 22.8 13.9 8.9
115.5 64.3 37.1 22.1 13.8 8.9
121.4 67.3 38.3 22.6 13.9 8.9
O'tot
150 200 250 300 350 400 a)
312.8 215.6 153.5 112.2 84.2 64.8
316.3 220.6 156.7 113.7 84.7 64.8
312.5 213.4 151.7 111.1 83.8 64.8
316.3 218.5 154.9 112.6 84.3 64.8
314.7 213.1 150.9 110.5 83.5 64.8
318.6 218.2 154.1 112.0 84.0 64.8
s) T h e six solutions X1 to Z2 give identical results at 400 M e V / c because they all agree with the W a t s o n solution A at this m o m e n t u m .
and not on the initial and final momenta, since (p'[ VI~Ip) is then independent of p and p'. In general, the shorter the range of the forces, the smaller the dependence is expected to be. In addition, it should be pointed out that the K---d elastic scattering
K - - d INTERACTIONS
683
in the energy range of interest is strongly peaked in the forward direction and that the forward scattering amplitude depends primarily on the single scattering term T. Since matrix elements of T in the forward direction are given accurately by the simple impulse approximation, we conclude that the neglect of K N scattering off the energy shell is a good approximation in evaluating the K---d total cross section given by the optical theorem and, to a lesser extent, the integrated elastic cross section. If at a later stage it is desired to investigate the differential cross section in greater detail, the off-shell ~.N matrix elements can be computed in a potential model. In table 1 we present values of the elastic and total K - - d cross sections calculated for the six sets of K N scattering parameters of Chand, at several K - laboratory momenta between 150 and 400 MeV/c. Our neglect o f the K ° - K - mass difference may be unjustified at the lower end of this momentum range, which is close to the charge-exchange threshold at 97 MeV/c. The calculation would probably also be inapplicable above 400 MeV/c, because of the inadequacy of the effective-range representation of the K N amplitudes. The shape of the angular dependence of the differential elastic cross section in our calculations is found to be in good agreement with that obtained by Chand 6) in the static model, and is therefore not given here. Unfortunately, experimental data are not yet available for comparison with the results listed in table 1. Such data would be extremely useful as a test of the multiple scattering formalism and the correctness of the phenomenological K N scattering parameters. However, it is shown in the following section that, even if accurate data were available and if there were no uncertainties in our method of calculation, it would not be possible to distinguish between the six sets of K N scattering parameters on the basis of the K - - d cross sections under consideration because of the present uncertainty in the deuteron wave function.
5. Comparison and Discussion of Various Approximations Since our formalism takes into account a number of effects which have been neglected in many previous models of the three-body scattering process, the present results serve as a means of studying the importance of these effects. Various approximations for the elastic and total K - - d cross sections for solution X1 are compared in table 2 for several energies. We now explain these approximations and discuss their accuracy. The "impulse approximation" here refers to the procedure of retaining only the single scattering terms and evaluating them as described in the preceding section. The "double scattering approximation" is obtained by adding to the single scattering terms the double scattering contributions with intermediate on-shell elastic states, with the required matrix elements evaluated in the same way as in the single scattering terms; the result is equivalent to the approximation t t ~ 2 T l ( l + i T ~ ) , although the actual calculation was made by a different method 14). As can be seen in table 2, the multiple scattering corrections of the type considered arise almost entirely from
684
N.M. QUEEN
double scattering processes, particularly at the higher end of the energy range. The multiple scattering corrections are somewhat more important for the elastic than for the total cross section; the reason for this appears to be that the single elastic scattering, unlike the multiple scattering, is strongly peaked in the forward direction, so that the correction to the single scattering is of greatest importance at large scattering angles. There is little advantage in using the double scattering approximation as a first-order correction to the single scattering, even when the higher-order effects are small (as they are here), because the multiple scattering approximation t (°) is no m o r e difficult to evaluate and has the additional advantage of giving unitarity corrections to the single scattering approximation. TABLE 2
Comparison of various approximations for the elastic and total K---d cross sections for solution X1 150 ~rel(mb)
O'tot(mb)
K- laboratory momentum (MeV/c) 200 300 400
Impulse approximation Double scattering
209.7 114.7
114.1 66.1
33.0 22.7
11.3 8.9
Static model e) Present work
117.8 114.4
74.8 65.7
27.3 22.7
8.9
crnq-ap Glauber approximation
399.4 267.3
263.2 205.5
126.5 113.2
69.8 65.8
Impulse approximation Double scattering Static model 6) Present work
389.7 330.5 321.6 312.8
258.7 224.3 225.2 215.6
125.7 113.9 118.0 112.2
69.6 65.1 64.8
The "static model" in table 2 refers to the results obtained by Chand 6) in a formalism which includes multiple scattering effects completely but treats the nucleons as fixed, infinitely heavy scattering centres. Thus the difference between the results of the static model and those of the present work gives a measure of the effects associated with the finite mass of the nucleons. The static model, like the simple impulse approximation, neglects the internal m o m e n t u m distribution of the deuteron and the associated two-body scattering off the energy shell. The corrections due to these effects have generally been found 1o, 12, 14) to reduce the magnitude of the single scattering terms, although only slightly in the forward direction. For reasons explained above, the effect of these corrections is therefore small in the case of the total cross section. The results given in table 2 exhibit the expected behaviour. The simplest possible approximation for the total K - - d cross section, namely trtot ~ t r , + % , where trn and trp are the K - - n and K - - p total cross sections at the same K - laboratory energy, is obtained by direct application of the optical theorem to the simple impulse approximation. As expected, this approximation predicts slightly larger values of atot than the complete impulse approximation.
K - - d INTERACTIONS
685
22) expresses atot in the form ,~ a n + o'p + (4rc/k2)Re(fnfp)(r-2)d,
The Glauber approximation o'to t
(34)
w h e r e f , a n d f p are the K - - n and K - - p forward scattering amplitudes, and ( r - 2 ) a is the expectation value of the inverse square of the n - p separation in the ground state of the deuteron. This simple relation is particularly useful for extracting information on interactions with neutrons from experimental data on scattering by the deuteron. It can be adequately justified at very high energies, but the precise conditions for its validity are not well understood. Since the Glauber approximation is derived by making several further simplifications in the double scattering approximation, it seems worthwhile to use the results of our more exact calculations to study its validity. The Hulth6n function (32) yields (r-2)d =
4ab(a+b)(b-a)-2[aln(2a)+bln(2b)-(a+b)ln(a+b)].
(35)
Treating the nucleons as identical particles, the isospin algebra for the K - - d system is taken into account in eq. (34) by setting an = % = ¼a ° +¼a 1 andfn = fp = ¼fo +¼ f l , where a t a n d f ~ are the corresponding cross section and amplitude for isospin L The numerical results in table 2 indicate that the Glauber approximation is reliable only at the higher energies considered, although even at low energies it represents an improvement over approximations which neglect multiple scattering. TABLe 3 Comparison of the calculated K---d cross sections for solution X1, using deuteron wave functions of the Hulth~n and Gartenhaus type K- laboratory
Hulth6n (b = 5.2a)
Hulth6n (b = 7a)
momentum (MeV/c)
ael(mb)
atot(mb)
ael(mb)
150 200 300 400
114.4 65.7 22.7 8.9
312.8 215.6 112.2 64.8
118.8 69.6 24.9 10.0
atot(mb) 306.8 211.2 110.3 64.0
Gartenhaus ael(mb)
atot(mb)
114.2 65.3 22.2 8.5
313.9 216.4 112.6 65.0
Any analysis of scattering by the deuteron necessarily depends to some extent on the assumed form of the N N interaction. This interaction enters our detailed calculations only through the deuteron wave function. Since one of the main reasons for studying low-energy K - - d processes at the present time is the hope of narrowing the choice a m o n g a continuous family of K N interaction parameters, it is essential to determine how sensitive the calculations are to the assumed deuteron wave function. A function of the Hulth6n form (32) is usually adopted for the S state of the deuteron. The parameter a is fixed by the binding energy, although the best value for the parameter b is uncertain and estimates 20, 23) vary f r o m b = 5.2a to b = 7a.
686
N.M.
QUEEN
Typical K---d cross sections calculated using these two values of b are compared in table 3. The range of variation in these cross sections due to the uncertainty in the parameter b is in all cases at least as great as the variation (table 1) corresponding to the possible K N parameters. Thus, the uncertainty in the deuteron wave function prevents the possibility of distinguishing between the various sets of g.N parameters on the basis of the K - - d cross sections given. It should be mentioned that earlier calculations 24) based on the original Ross-Humphrey solutions I and II gave quite distinct results for these two solutions, although these calculations are no longer relevant. Some justification is required for the direct comparison (table 2) of the results of Chand in the static model with those of the present work, since Chand assumed an analytic form of the deuteron wave function which very accurately approximates the Gartenhaus 25) (hard-core) wave function, whereas we assume here the much simpler Hulth6n function. Numerical results (table 3) using these two wave functions differ only slightly, provided we assume the Hulth6n parameters b = 5.2a. We next consider the error in our lowest-order multiple scattering approximation t ~o) due to the neglect of off-shell and inelastic intermediate states. In recent calculations x4, z4) of N - d and K - - d processes, we have made a crude estimate of the leading corrections to t c°). It was found that, although the double scattering contribution from the intermediate states neglected in t ~°) may be rather large, this correction tends to be cancelled almost completely by similar terms of higher order. This cancellation arises from phase differences between terms of various orders; it tends to be stronger than the cancellation among the terms effectively summed in t ~°), since in any multiple scattering term with n intermediate states there are in general many possible ways to be considered in which exactly m ( m < n) of these states are on-shell elastic, whereas there is only a single possibility in which all n states are of this type. The nature of this cancellation can be better understood, however, by examining successively higher approximations after t ~°) in our formalism. E.g., the next approximation, t ~1), is derived from eqs. (24) and (27). After some algebraic manipulation the result can be expressed as t~l) = t~0)q_
2D, (1 - i T , ) ( 1 - i T , - iD,) '
(36)
where D~ is the partial-wave amplitude corresponding to each of the double scattering contributions T 2 G"T3 and T 3 G"T2. The effect of the cancellation is contained in the denominator of the second term in eq. C36). Assuming the very plausible result Im D l > --Im Ts, the inequality Im Tl >= 0 (see sect. 3) implies that each factor in the denominator is of modulus not less than unity, so that the denominator leads to a reduction in the modulus of the second term which is in fact stronger than the corresponding reduction in the case of t[ °). Calculations of the inelastic K - - d processes (2b, c) are particularly difficult and have many features in common with those of the multiple scattering contributions
K - - d INTERACTIONS
687
involving inelastic i n t e r m e d i a t e states. A n estimate o f the o r d e r o f m a g n i t u d e o f the inelastic cross sections in the earlier calculations 24) i n d i c a t e d that, as in the case o f the elastic scattering, o u r f o r m a l i s m gives slightly smaller inelastic cross sections t h a n those o b t a i n e d b y C h a n d 6). T h e only a p p r o x i m a t i o n in o u r f o r m a l i s m which has n o t been a d e q u a t e l y tested or justified is the use o f the impulse a p p r o x i m a t i o n for the m a t r i x elements o f / ' 2 a n d 7"3. Little is k n o w n a b o u t this a p p r o x i m a t i o n , a l t h o u g h it is believed to be sufficiently a c c u r a t e at m o d e r a t e l y high energies. The b i n d i n g effects which are neglected in the i m p u l s e a p p r o x i m a t i o n are p a r t i c u l a r l y difficult to study, since they involve those effects m o s t closely a s s o c i a t e d with the s i m u l t a n e o u s i n t e r a c t i o n o f all three particles. I n conclusion, we c o m p a r e a n d c o n t r a s t the K - - d with the N - d p r o b l e m , which has been s t u d i e d earlier x4) b y closely r e l a t e d m e t h o d s . The m a i n c o m p l i c a t i o n in the N - d p r o b l e m is the P a u l i principle; a l t h o u g h the identity o f the t w o target nucleons can be r i g o r o u s l y t a k e n into account, difficulties arise when the incident particle is also a n u c l e o n because the incident a n d the target particles are n o t treated on an equal f o o t i n g in this f o r m a l i s m . The a c c u r a c y o f the N--d calculations is also m o r e u n c e r t a i n because the v a r i o u s c o r r e c t i o n s to the simple impulse a p p r o x i m a t i o n are o f greater i m p o r t a n c e t h a n for K - - d scattering at the s a m e energies. I n addition, the d e s c r i p t i o n o f the t w o - b o d y interactions a n d the spin a n d isospin a l g e b r a are m o r e c o m p l e x for the N - d t h a n for the K - - d system. F o r these reasons, o u r calculations o f K - - d scattering a r e believed to be m u c h m o r e reliable t h a n the p r e v i o u s l y p u b l i s h e d analysis o f N - d scattering.
References 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14) 15) 16) 17) 18) 19) 20) 21) 22) 23) 24) 25)
R. R. Ross and W. E. Humphrey, Phys. Rev. 127 (1962) 1305 T. Akiba and R. H. Ca1313s, Phys. Rev. Lett. 8 (1962) 457 R. D. Tri1313, M. B. Watson and M. Ferro-Luzzi, Phys. Rev. Lett. 8 (1962) 175 M. B. Watson, UCRL-10175 (1962) R. Chand, Nuovo Cim. 30 (1963) 1445 R. Chand, Nuovo Cim. 31 (1964) 827, 33 (1964) 148 J. D. Jackson, D. G. Ravenhall and H. W. Wyld, Nuovo Cim. 9 (1958) 834 G. F. Chew, Phys. Rev. 80 (1950) 196; G. F. Chew and G. C. Wick, Phys. Rev. 85 (1952) 636 R. Chand, Ann. of Phys. 22 (1963) 438 T. Fulton and P. Schwed, Phys. Rev. 115 (1959) 973 A. Everett, Phys. Rev. 126 (1962) 831 K. L. Kowalski and D. Feldman, Phys. Rev. 130 (1963) 276 A. K. Bhatia and J. Sucher, Phys. Rev. 132 (1963) 855 N. M. Queen, Nuclear Physics 55 (1964) 177 J. D. Jackson and H. W. Wyld, Nuovo Cim. 13 (1959) 85 K. M. Watson, Phys. Rev. 88 (1952) 1163 K. M. Watson, Phys. Rev. 89 (1953) 575 M. L. Goldberger, Phys. Rev. 84 (1951) 929 D. M. Brink and G. R. Satchler, Angular momentum (Oxford University Press, 1962) 13. 121 M. J. Moravcsik, Nuclear Physics 7 (1958) 113 Y. Sakamoto and T. Sasakawa, Prog. Theor. Phys. 21 (1959) 879 R. J. Glauber, Proc. Conf. on Nuclear Forces and the Few-Nucleon Problem, London, 1959, ed. by T. C. Gritiith and E. A. Power (Pergamon Press, London, 1960) p. 233 R. L. Gluckstern and H. A. Bethe, Phys. Rev. 81 (1951) 761 N. M. Queen, Ph.D. thesis, University of Bristol (1963) S. Gartenhaus, Phys. Rev. 100 (1955) 900