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The absorption of K− mesons in deuterium
8.A.3: [
Nuclear Physics 77 (1966) 689--695; ~ ) North-Holland Publishin# Co., Amsterdam
8.A.8
Not to be reproduced py photoprint or microfilm without written permission from the publisher
[
THE A B S O R P T I O N OF K - M E S O N S IN D E U T E R I U M G. N. F O W L E R and P. N. POULOPOULOS
Physics Department, University of Newcastle Upon Tyne Received 31 May 1965 The two-nucleon capture rate of K - mesons in deuterium is evaluated using the impulse approximation and taking account of Y* production. The results are compared in detail with previous calculations which neglect the Y* processes considered here.
Abstract:
1. Introduction
The mechanism of absorption of K - mesons in complex nuclei is a problem which is still not satisfactorily resolved in spite of the efforts of a number of different groups. The difficulty lies in accounting for the high proportion of multinucleon capture processes, 20 % in heavy nuclei, if, as most workers agree, the capture process occurs on the nuclear surface. The various models which have been proposed have recently been summarized by one of the authors 1) and the conclusion remains that reasonable objections can be raised to all the models which have so far been proposed. In the present paper we present a variant of a model already put forward by Eisenberg z) according to which hyperon resonance processes play a vital role. According to this model the surface capture takes place on a single nucleon in the nuclear potential, with the formation in many cases of a Y* or a Y*. This is then supposed to live long enough to encounter another nucleon and undergo non-mesonic decay in much the same way as a A hyperon does in hyperfragment decay 3). However, the experimental evidence 4) does not appear to show the proportionality to A ~r which should be a characteristic of this model. Nevertheless it is true that at least 10 % of all captures in deuterium s), for example, proceed through the formation of a Y* and in helium 65 % of A producing events proceed 6) through a Y*. When we recall the enhancement of the photodisintegration of the deuteron through the intervention of the N~, ~ at high photon energies it is plausible that the corresponding K - process, namely twonucleon capture, ought similarly to be enhanced. The principal difference lies in the fact that in the photon case the resonance involved plays the same role in single nucleon pion photo-production whereas in K - meson absorption another nucleon must be present because the Y* lies below the K - N threshold. It is therefore to be expected that in the K - D case the two-nucleon capture process if it proceeds through a Y* must be very sensitive indeed to the high momentum components in the deuteron wave functioja. Thus the results of rather crude calculations, for example those 689
690
O. N. FOWLER AND P. N. POULOPOULOS
of Rook 7) and Biswas 3) are probably unreliable even as to order of magnitude. Furthermore if the two-nucleon correlations in the nuclear surface of complex nuclei are only slightly stronger than they are in the deuteron it may well be the case that they are sufficient to enhance the two-nucleon capture rate from the 1 ~ observed in the deuteron to the 20 ~ observed in complex nuclei. In other words we regard the Y* production and subsequent non-mesonic decay as a two-nucleon process in which the same pair of nucleons is involved in both stages and so we do not regard it as analogous to hyperfragment formation and decay. Indeed the great difference in lifetime between the Y* and the A hyperon makes such an analogy implausible. A better analogy is rather with the high-energy photodisintegration of the deuteron. We are therefore led to consider in more detail the two-nucleon capture process of K - in deuterium and we first examine the only detailed calculations of this process which have been made so far, namely those of Burhop et al. 8). The next section will therefore be concerned with a detailed discussion of their calculation. We then go on to consider in more detail the Y* capture model and to show that it could play an important role in the two-nucleon capture process in deuterium and also therefore in more complex nuclei. For this purpose we consider the Y* process. The Y~ contribution has been discussed elsewhere lo) and will be discussed further below.
2. The Burhop Model The Burhop model is based on the Chew-Low approximation corresponding to the diagrams of fig. 1. This is essentially an impulse approximation in which a pion or kaon at the capturing vertex is captured by the spectator nucleon. The authors ~___ N
N
(a) Fig. 1. C h e w - L o w d i a g r a m s for K - + D
Y
(b) --* Y + N
reactions.
claim that their resuks has a stronger foundation than would appear to be the case by virtue of the fact that their approximation ought to be good near the one pion pole in the capture amplitude. This means that the two vertices in the diagram may be replaced by the physical single K - capture or elastic scattering amplitudes and the K - N or nN coupling constant where appropriate. Since these are all known, or may be inferred, from the experimental data on single K - meson processes one may hope to make a meaningful calculation of the two-nucleon capture amplitude.
K - MESONS
691
However disregarding the fact that there may be other singularities of the amplitude nearer to the physical region it is true that the four m o m e n t u m carried by the pion is such that the process is not in fact close to the one pion pole. This presumably means that off-shell effects may be significant at the two vertices and that higher order processes involving several pions may be important. It is however arguable that at least at the K - vertex the corrections ought not to be too large since the main effect occurs in the dependence of the production or scattering angle on the off shell momenta and the single K - process is mostly s-wave. Furthermore multipion processes ought to correspond to rescattering corrections and we might hope to take account of these in a phenomenological way. Nevertheless this use of on-shell amplitudes at the K - N vertex precludes the occurrence of real Y* capture processes of the type discussed above. The Y* will only play a role indirectly in determining the dynamics of the K - N capture. Thus if we wish to include the real Y* process we must modify the K - N vertex so that the nucleon is allowed off the mass shell and this is the main point of the present paper. Before proceeding to the details some further comments should be made concerning the numerical results of the Burhop model since their result does in fact agree with experiment and therefore appears to leave very little r o o m for a Y* contribution. Their result may be questioned on three grounds apart from the neglect of the Y* process. These are, first of all, that in the K - D process for K - mesons at rest it is clear from angular m o m e n t u m and parity conservation that the final state must be a p-state of total angular m o m e n t u m 1. (An s-wave final state was used in the calculations of Burhop et al. since at that time the hyperon nucleon relative parity was not known.) Secondly this approximation to the (nN) or ( K N ) vertex also leads to an incorrect form for the pion or kaon propagator which effects the results considerably, particularly when a hard core is introduced into the final state wave functions. The third criticism refers to the fact that no final state interaction is included by Burhop et aL We have included a final state core and find that this makes a significant difference to the result. The approximation to the n N vertex made by Burhop et al. is
Iql g~5 ~ g 2M Here q is the four-momentum carried by the propagating meson. For present purposes it is sufficient certainly for the pion exchange case, to use the static approximation for the nucleons. This means that a'q
g~s ~ g - - , 2M
where q is the three-momentum transfer and a the nucleon spin operator. The pion
692
G. N. F O W L E R A N D P. N. P O U L O P O U L O S
propagator including the vertex factor is now
AFF ~
g
(1)
~ •q ,
q 2 .-F-#2 2M
and this is to be compared with
AFF ~ g
lql 2M q2+#2 '
(2)
where # is the pion mass. This means that apart from irrelevant factors, the amplitude, which should be
(3) becomes
T ~: g f ~(p'-k) ( ~ ~K) (p,_Ip'-Pl p)2 + 2 q~D(p)dpdp, '
(4)
assuming a plane wave final state. Here qbk(p') is the final state wave function, ~o(P) the deuteron wave function, trK the K - N capture cross section, and k, k' are the initial and final c.m. momenta of th~ K - N system. We have evaluated the Fourier transform of the product (kar./k') ~ ~D(P) numerically using a deuteron wave function with repulsive core, the resulting function being denoted by Go(r) and we have taken average values for certain q dependent kinematic factors which also does not affect the result significantly. This enables us to work in coordinate space directly. Going over to coordinate space in (3) we find
T oc g dp*(r)a" V
GD(r)dr.
(5)
The form (4) becomes
T oc g fe'k"F(Irl)GD(r)dr,
(6)
where the function F(Irl) will be discussed further below. Evidently the form (6) leads to an s-wave final state. With regard to the function F(Ir[) we have F(Irl)
= 1 ('e_i~. , [q{ dq (2n)a3 q2+~t2 1
(2n) a 1
r e _ , . . , I[l_qI
1
27~2 r 2
/a e - ~ '
4n
r
#2 ,q[(q~+#2)l d q
(7)
K-
693
MESONS
which is to be c o m p a r e d with l (~) ---a'V 4n
1 e -~r " [~--+ 4-n er ~
=
p c ; u"]
(8)
.
TABLE 1 Ratio o f n o n - m e s o n i c to m e s o n i c m a t r i x elements
Present calculation n exch.
K exch.
Without final state core
1.33
0.63
With final state core
1.0
0.254
TABLE 2 Ratio of non-mesonic to mesonic capture rates (in 7oo) Burhop et al.
Present calculation (no resonance)
With resonance
Exp.
max
vain
max
min
max
min
1.01
0.23
0.87
0.29
1.64
0.82
0.9!0.22
We have evaluated the matrix element (5) and the results for both n and K exchange are displayed in table 1, normalized as shown for the A channel. In deriving these results we have used the K i m 9) solution I parameters and a final state core o f 0.5 fm radius. It will be seen that the theoretical result is n o w somewhat smaller than ,hat given by B u r h o p e t a L but is still in agreement with experiment. We n o w turn to the Y* production process.
3. The Y* Production Model In the Burhop model the K - N vertex o f fig. 1 is proportional to (O'K(¢-O))½, where aK(co) is the total cross section for the appropriate K - N process and co is the total c.m. energy o f the system. K-
\ D N2
Fig. 2. Triangle diagram representing K - + D --~ Y-t-N reaction.
694
G. N. FOWLER AND P. N. POULOPOULOS
In fact the diagram which is actually being approximated is that of fig. 2 and Burhop's approximation consists in putting N2 on the mass shell and evaluating A as though N t were on the mass shell. We shall modify this by allowing NI off the mass shell so that the vertex A may describe real Y* production. This means that we replace a~(co) according to aK(09) ~
C (o92- to2)2 + c02 F2
(9)
We have 032 =
I(py..l..pn)2
=
(.Ey_l..gt)2
_
(pe+p~_p,,)2 = ( g v + E ~ ) 2 _ p ~ = (MD+MK--EN)2--p 2,
when PN and p~ are the initial and final three momenta of the outgoing nucleon. We have inserted (9) for (,ic(09))~ in (3) and integrated the result numerically. The mesonic decay case has been treated by making a similar replacement in the Burhop formula for this case. Since 10 ~ of the one-nucleon capture processes are assumed to proceed through the Y* we may evaluate the constant C. Then from (5) we can find the enhancement of the two nucleon capture process. The result is shown in column 5 of table 2. It will be seen that our result is large enough to account for the experimental observations even if there were significant corrections arising from our use of the impulse approximation. 4. Discussion
We have seen that by including Y* processes we can achieve an increase by a factor of two in the two-nucleon capture process. The reason for this increase lies partly in the resonance denominator of (9) which enhances the amplitude for processes involving high relative momentum in the final state and partly in the constant C which has been chosen large enough to ensure that 10 ~ of the mesic decay events occur through the Y*. The fact that the ratio of non-mesonic to mesonic decay of the Y* is as large as 1 0 ~ is a result of the fact that to produce a Y* at all requires that the two nucleons should be well correlated so that the extra degree of correlation required to give a non-mesonic decay of the Y* does not drastically affect the result. The contribution from a Y~ resonance has been discussed by the present authors and A. D. Crossland 1o) using the analytic continuation of the T matrix given by Kim solution I. This predicts a yield of 8~o for the mesonic and 15~o for the nonmesonic cases respectively when the final state core is included. (A numerical error in the results of ref. lo) has also been corrected). The contribution of the Y~ to the non-mesonic process must therefore be small since the corresponding mesonic process must occur in << 8 ~ of cases.
K - MESONS
695
The a u t h o r s are grateful to the staff of the C o m p u t i n g L a b o r a t o r y , University of Newcastle u p o n T y n e for their assistance with the n u m e r i c a l work. One of us (P.N.P.) is indebted to the G r e e k State Scholarship F o u n d a t i o n for a scholarship d u r i n g the tenure of which this work was carried out. T h a n k s are also due to Professor E. H. S. B u n n h o p for c o m m e n t s o n a n earlier draft.
References 1) 2) 3) 4) 5) 6) 7) 8) 9) 10)
G. N. Fowler, Nuclear Physics 57 (1964) 100 Y. Eisenberg, M. Friedrnann, G. Alexander and D. Kessler, Nuovo Cim. 22 (1961) 1 N. N. Biswas, Nuovo Cim. 22 (1961) 654 G.T. Condo and R. D. Hill, Phys. Rev. 129 (1963) 388 O. Dahl et al., Phys. Rev. 6 (1961) 142 M. Block et al., Nuovo Cim. 20 (1961) 724 J. R. Rook, Nuclear Physics 43 (1963) 363 E. H. S. Burhop, A. K. Common and K. Higgins, Nuclear Physics 39 (1962) 644 J. K. Kim, Phys. Rev. Lett. 14 (1965) 29 A. D. Crossland, G. N. Fowler and P. N. Poulolaoulos, Proceedings of the Brussels conference on elementary particles and nuclear structure, 1965, to be published
Nuclear Physics 77 (1966) 689--695; ~ ) North-Holland Publishin# Co., Amsterdam
8.A.8
Not to be reproduced py photoprint or microfilm without written permission from the publisher
[
THE A B S O R P T I O N OF K - M E S O N S IN D E U T E R I U M G. N. F O W L E R and P. N. POULOPOULOS
Physics Department, University of Newcastle Upon Tyne Received 31 May 1965 The two-nucleon capture rate of K - mesons in deuterium is evaluated using the impulse approximation and taking account of Y* production. The results are compared in detail with previous calculations which neglect the Y* processes considered here.
Abstract:
1. Introduction
The mechanism of absorption of K - mesons in complex nuclei is a problem which is still not satisfactorily resolved in spite of the efforts of a number of different groups. The difficulty lies in accounting for the high proportion of multinucleon capture processes, 20 % in heavy nuclei, if, as most workers agree, the capture process occurs on the nuclear surface. The various models which have been proposed have recently been summarized by one of the authors 1) and the conclusion remains that reasonable objections can be raised to all the models which have so far been proposed. In the present paper we present a variant of a model already put forward by Eisenberg z) according to which hyperon resonance processes play a vital role. According to this model the surface capture takes place on a single nucleon in the nuclear potential, with the formation in many cases of a Y* or a Y*. This is then supposed to live long enough to encounter another nucleon and undergo non-mesonic decay in much the same way as a A hyperon does in hyperfragment decay 3). However, the experimental evidence 4) does not appear to show the proportionality to A ~r which should be a characteristic of this model. Nevertheless it is true that at least 10 % of all captures in deuterium s), for example, proceed through the formation of a Y* and in helium 65 % of A producing events proceed 6) through a Y*. When we recall the enhancement of the photodisintegration of the deuteron through the intervention of the N~, ~ at high photon energies it is plausible that the corresponding K - process, namely twonucleon capture, ought similarly to be enhanced. The principal difference lies in the fact that in the photon case the resonance involved plays the same role in single nucleon pion photo-production whereas in K - meson absorption another nucleon must be present because the Y* lies below the K - N threshold. It is therefore to be expected that in the K - D case the two-nucleon capture process if it proceeds through a Y* must be very sensitive indeed to the high momentum components in the deuteron wave functioja. Thus the results of rather crude calculations, for example those 689
690
O. N. FOWLER AND P. N. POULOPOULOS
of Rook 7) and Biswas 3) are probably unreliable even as to order of magnitude. Furthermore if the two-nucleon correlations in the nuclear surface of complex nuclei are only slightly stronger than they are in the deuteron it may well be the case that they are sufficient to enhance the two-nucleon capture rate from the 1 ~ observed in the deuteron to the 20 ~ observed in complex nuclei. In other words we regard the Y* production and subsequent non-mesonic decay as a two-nucleon process in which the same pair of nucleons is involved in both stages and so we do not regard it as analogous to hyperfragment formation and decay. Indeed the great difference in lifetime between the Y* and the A hyperon makes such an analogy implausible. A better analogy is rather with the high-energy photodisintegration of the deuteron. We are therefore led to consider in more detail the two-nucleon capture process of K - in deuterium and we first examine the only detailed calculations of this process which have been made so far, namely those of Burhop et al. 8). The next section will therefore be concerned with a detailed discussion of their calculation. We then go on to consider in more detail the Y* capture model and to show that it could play an important role in the two-nucleon capture process in deuterium and also therefore in more complex nuclei. For this purpose we consider the Y* process. The Y~ contribution has been discussed elsewhere lo) and will be discussed further below.
2. The Burhop Model The Burhop model is based on the Chew-Low approximation corresponding to the diagrams of fig. 1. This is essentially an impulse approximation in which a pion or kaon at the capturing vertex is captured by the spectator nucleon. The authors ~___ N
N
(a) Fig. 1. C h e w - L o w d i a g r a m s for K - + D
Y
(b) --* Y + N
reactions.
claim that their resuks has a stronger foundation than would appear to be the case by virtue of the fact that their approximation ought to be good near the one pion pole in the capture amplitude. This means that the two vertices in the diagram may be replaced by the physical single K - capture or elastic scattering amplitudes and the K - N or nN coupling constant where appropriate. Since these are all known, or may be inferred, from the experimental data on single K - meson processes one may hope to make a meaningful calculation of the two-nucleon capture amplitude.
K - MESONS
691
However disregarding the fact that there may be other singularities of the amplitude nearer to the physical region it is true that the four m o m e n t u m carried by the pion is such that the process is not in fact close to the one pion pole. This presumably means that off-shell effects may be significant at the two vertices and that higher order processes involving several pions may be important. It is however arguable that at least at the K - vertex the corrections ought not to be too large since the main effect occurs in the dependence of the production or scattering angle on the off shell momenta and the single K - process is mostly s-wave. Furthermore multipion processes ought to correspond to rescattering corrections and we might hope to take account of these in a phenomenological way. Nevertheless this use of on-shell amplitudes at the K - N vertex precludes the occurrence of real Y* capture processes of the type discussed above. The Y* will only play a role indirectly in determining the dynamics of the K - N capture. Thus if we wish to include the real Y* process we must modify the K - N vertex so that the nucleon is allowed off the mass shell and this is the main point of the present paper. Before proceeding to the details some further comments should be made concerning the numerical results of the Burhop model since their result does in fact agree with experiment and therefore appears to leave very little r o o m for a Y* contribution. Their result may be questioned on three grounds apart from the neglect of the Y* process. These are, first of all, that in the K - D process for K - mesons at rest it is clear from angular m o m e n t u m and parity conservation that the final state must be a p-state of total angular m o m e n t u m 1. (An s-wave final state was used in the calculations of Burhop et al. since at that time the hyperon nucleon relative parity was not known.) Secondly this approximation to the (nN) or ( K N ) vertex also leads to an incorrect form for the pion or kaon propagator which effects the results considerably, particularly when a hard core is introduced into the final state wave functions. The third criticism refers to the fact that no final state interaction is included by Burhop et aL We have included a final state core and find that this makes a significant difference to the result. The approximation to the n N vertex made by Burhop et al. is
Iql g~5 ~ g 2M Here q is the four-momentum carried by the propagating meson. For present purposes it is sufficient certainly for the pion exchange case, to use the static approximation for the nucleons. This means that a'q
g~s ~ g - - , 2M
where q is the three-momentum transfer and a the nucleon spin operator. The pion
692
G. N. F O W L E R A N D P. N. P O U L O P O U L O S
propagator including the vertex factor is now
AFF ~
g
(1)
~ •q ,
q 2 .-F-#2 2M
and this is to be compared with
AFF ~ g
lql 2M q2+#2 '
(2)
where # is the pion mass. This means that apart from irrelevant factors, the amplitude, which should be
(3) becomes
T ~: g f ~(p'-k) ( ~ ~K) (p,_Ip'-Pl p)2 + 2 q~D(p)dpdp, '
(4)
assuming a plane wave final state. Here qbk(p') is the final state wave function, ~o(P) the deuteron wave function, trK the K - N capture cross section, and k, k' are the initial and final c.m. momenta of th~ K - N system. We have evaluated the Fourier transform of the product (kar./k') ~ ~D(P) numerically using a deuteron wave function with repulsive core, the resulting function being denoted by Go(r) and we have taken average values for certain q dependent kinematic factors which also does not affect the result significantly. This enables us to work in coordinate space directly. Going over to coordinate space in (3) we find
T oc g dp*(r)a" V
GD(r)dr.
(5)
The form (4) becomes
T oc g fe'k"F(Irl)GD(r)dr,
(6)
where the function F(Irl) will be discussed further below. Evidently the form (6) leads to an s-wave final state. With regard to the function F(Ir[) we have F(Irl)
= 1 ('e_i~. , [q{ dq (2n)a3 q2+~t2 1
(2n) a 1
r e _ , . . , I[l_qI
1
27~2 r 2
/a e - ~ '
4n
r
#2 ,q[(q~+#2)l d q
(7)
K-
693
MESONS
which is to be c o m p a r e d with l (~) ---a'V 4n
1 e -~r " [~--+ 4-n er ~
=
p c ; u"]
(8)
.
TABLE 1 Ratio o f n o n - m e s o n i c to m e s o n i c m a t r i x elements
Present calculation n exch.
K exch.
Without final state core
1.33
0.63
With final state core
1.0
0.254
TABLE 2 Ratio of non-mesonic to mesonic capture rates (in 7oo) Burhop et al.
Present calculation (no resonance)
With resonance
Exp.
max
vain
max
min
max
min
1.01
0.23
0.87
0.29
1.64
0.82
0.9!0.22
We have evaluated the matrix element (5) and the results for both n and K exchange are displayed in table 1, normalized as shown for the A channel. In deriving these results we have used the K i m 9) solution I parameters and a final state core o f 0.5 fm radius. It will be seen that the theoretical result is n o w somewhat smaller than ,hat given by B u r h o p e t a L but is still in agreement with experiment. We n o w turn to the Y* production process.
3. The Y* Production Model In the Burhop model the K - N vertex o f fig. 1 is proportional to (O'K(¢-O))½, where aK(co) is the total cross section for the appropriate K - N process and co is the total c.m. energy o f the system. K-
\ D N2
Fig. 2. Triangle diagram representing K - + D --~ Y-t-N reaction.
694
G. N. FOWLER AND P. N. POULOPOULOS
In fact the diagram which is actually being approximated is that of fig. 2 and Burhop's approximation consists in putting N2 on the mass shell and evaluating A as though N t were on the mass shell. We shall modify this by allowing NI off the mass shell so that the vertex A may describe real Y* production. This means that we replace a~(co) according to aK(09) ~
C (o92- to2)2 + c02 F2
(9)
We have 032 =
I(py..l..pn)2
=
(.Ey_l..gt)2
_
(pe+p~_p,,)2 = ( g v + E ~ ) 2 _ p ~ = (MD+MK--EN)2--p 2,
when PN and p~ are the initial and final three momenta of the outgoing nucleon. We have inserted (9) for (,ic(09))~ in (3) and integrated the result numerically. The mesonic decay case has been treated by making a similar replacement in the Burhop formula for this case. Since 10 ~ of the one-nucleon capture processes are assumed to proceed through the Y* we may evaluate the constant C. Then from (5) we can find the enhancement of the two nucleon capture process. The result is shown in column 5 of table 2. It will be seen that our result is large enough to account for the experimental observations even if there were significant corrections arising from our use of the impulse approximation. 4. Discussion
We have seen that by including Y* processes we can achieve an increase by a factor of two in the two-nucleon capture process. The reason for this increase lies partly in the resonance denominator of (9) which enhances the amplitude for processes involving high relative momentum in the final state and partly in the constant C which has been chosen large enough to ensure that 10 ~ of the mesic decay events occur through the Y*. The fact that the ratio of non-mesonic to mesonic decay of the Y* is as large as 1 0 ~ is a result of the fact that to produce a Y* at all requires that the two nucleons should be well correlated so that the extra degree of correlation required to give a non-mesonic decay of the Y* does not drastically affect the result. The contribution from a Y~ resonance has been discussed by the present authors and A. D. Crossland 1o) using the analytic continuation of the T matrix given by Kim solution I. This predicts a yield of 8~o for the mesonic and 15~o for the nonmesonic cases respectively when the final state core is included. (A numerical error in the results of ref. lo) has also been corrected). The contribution of the Y~ to the non-mesonic process must therefore be small since the corresponding mesonic process must occur in << 8 ~ of cases.
K - MESONS
695
The a u t h o r s are grateful to the staff of the C o m p u t i n g L a b o r a t o r y , University of Newcastle u p o n T y n e for their assistance with the n u m e r i c a l work. One of us (P.N.P.) is indebted to the G r e e k State Scholarship F o u n d a t i o n for a scholarship d u r i n g the tenure of which this work was carried out. T h a n k s are also due to Professor E. H. S. B u n n h o p for c o m m e n t s o n a n earlier draft.
References 1) 2) 3) 4) 5) 6) 7) 8) 9) 10)
G. N. Fowler, Nuclear Physics 57 (1964) 100 Y. Eisenberg, M. Friedrnann, G. Alexander and D. Kessler, Nuovo Cim. 22 (1961) 1 N. N. Biswas, Nuovo Cim. 22 (1961) 654 G.T. Condo and R. D. Hill, Phys. Rev. 129 (1963) 388 O. Dahl et al., Phys. Rev. 6 (1961) 142 M. Block et al., Nuovo Cim. 20 (1961) 724 J. R. Rook, Nuclear Physics 43 (1963) 363 E. H. S. Burhop, A. K. Common and K. Higgins, Nuclear Physics 39 (1962) 644 J. K. Kim, Phys. Rev. Lett. 14 (1965) 29 A. D. Crossland, G. N. Fowler and P. N. Poulolaoulos, Proceedings of the Brussels conference on elementary particles and nuclear structure, 1965, to be published