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Production cross sections of Λ-hypernuclei in (K−,π−) reactions
Volume 99B, number 5
PHYSICS LETTF.RS
5 March 1981
PRODUCTION CROSS SECTIONS OF A-HYPERNUCLEI IN (K-, n - ) REACTIONS A. BOUYSSY Division de Physique Thdorique 1, Instimt de Physique Nucldaire, F-91406 Orsay Cedex, France Received 28 October 1980
Forward cross sections of A-hypernuclei produced in (K-, rr-) experiments are calculated in the distorted wave impulse ~-. 9~. approximation. Two kinds of approaches are discussed and a comparison with recent data ranging from 6ALl IO 2 0 ,\131 is made. The results support the assumption of a one-step process.
The observation of hypernuclear states produced in ( K - , rr-) reactions on nuclear targets has already attracted much interest [1]. Under suitable kinematical conditions the momentum transfer can be chosen very small compared to the Fermi m o m e n t u m of the nucleons in the nucleus. As a consequence strongly populated states are observed (at least in light nuclei) which have been interpreted as corresponding to recoilless or quasi-elastic transitions. In addition the main features of" the hypernuclear spectrum are reasonably well reproduced, although there aro still some open questions [2,3]. From the structure of hypernuclear excited states, mainly from the quasi-elastic ones, reliable information on the A-nucleus system have been obtained. This is, however, based on the assumption that the transitions to the hypernuclear states are one-step processes. In that case the reaction occurs on a single neutron, the rest o f the nucleus being unchanged, without any additional distortion of either the incoming K or the outgoing 7r-. If the high energies of the mesons and the forward rr- observation in the present experiments seem to favour such a process it is important to understand the nature o f the ( K - , 7r-) reaction mechanism. In this short letter we want to show an other piece of information supporting the assumption of a onestep process in the ( K - , r r - ) reaction, namely the for-
ward production cross section of A-liypernuclei. There has been already some calculations of the cross sections, either for some particular transitions or for the whole hypernuclear states produced in the case of light nuclei [ 2 - 8 ] , most o f them based on the distorted wave impulse approximation. Here a detailed comparison of two kinds of calculations [2,6] for the production cross sections in light and heavy nuclear targets will be made. A systematic study o f these two approaches will be presented and their results compared with recent data obtained for hypernuclei ranging from A6Li tO 209Bi, at three incident momenta [9]. If the one-step process dominates the ( K - , r r - ) reaction the indicated method of analysis is the DWIA. In that case the hypernuclear formation cross section at 0 ° for the strangeness exchange reaction K - + A z - + r r - + A,xz
(1)
is related to the elementary K - n -+ A r t - cross section by: do(0°)/dg2 = Nef f do(0°)/d~2 [K-n_,A~- .
(2)
The total effective number o f neutrons in the reaction is given by [2]: A Neff=~f
fx(-)*(r)(fl2 j=lU
(/)8(r-r/)li) (3)
X X(+)(r) drl 2 .
i Laboratoire associ6 au CNRS. 0 0 3 1 - 9 1 6 3 / 8 1 / 0 0 0 0 - 0 0 0 0 / $ 02.50 © North-Holland Publishing Company
373
Volume 99B, number 5
PHYSICS LETTERS
The operator U_ (/) is the U-spin lowering operator (_+) transforming a neutron into a A-particle, the Xn,K(r) being the appropriate distorted waves of the mesons. In a first approach, to calculate Nef f we choose the beam direction as axis of quantization and expand the product of distorted waves in partial waves
(4) = ~
L=O
[4rr(2L +
1)]l/2iLTL(qr)yO(f),
where the radial function ]'L(qr) reduces to the usual spherical Bessel function ]L(qr) in the absence of any distortion (q is the momentum transfer). We restrict here for simplicity the initial state to a doubly closedshell nucleus and describe the final state as a one particle-one hole excitation denoted by []AJn 1)J. Straightforward angular momentum algebra gives NeJff, the effective number of neutrons for a particular transition 10>-+ [5]:
[/AJnl>j
NJff = (2J + 1)(2/A + 1)(2in + 1) 1
F2(q) '
(5)
5 March 1981
actual number of neutrons N for no distortion. The total forward formation cross section is much easier to calculate if one uses the sum-rule approach. However, the advantage of using eq. (5) is that it allows the extraction of the amount of quasi-elastic A production which is believed to be the dominant process in forward ( K - , 7r-) reactions on light nuclei, and to give the magnitude of the cross section for a transition to a given state of the hypernucleus. Both approaches obviously need knowledge of the distorted waves of the mesons. The K - and ~r- wave functions can be generated by using a standard optical potential as was done in ref. [8] with or without a pand d-wave part. However, the number of partial waves needed in such a calculation and also the lack of very precise information on the K--nucleus interaction was our motivation to use a much simpler procedure, namely the eikonal approximation. The comparison made in ref. [8] between results obtained either from the fully distorted waves or from the eikonal waves shows that the eikonal approximation is good at these energies. With eikonal distortions N eff SR can be written;
NeSfl~= (N/A )(°K where
x F(q) = f r2dr RA(r)Rn(r)ij(qr ) ,
fd2b (exp[-onT(b)]
where
_
f pN(r)lx ~ )
(r)×(~)(r)12
dr
(7)
where PN(r) is the neutron density and SR means sum rule. The effective number of neutrons reduces to the 374
(8)
-exp[-OKT(b)]
) ,
(6)
with obvious notations. The total shell-model effective number of neutrons (called N sM) is therefore the sum over all the possible transitions corresponding to a given j,r and the sum over all the possible values of the angular momentum. This approach is the one used in ref. [6] including configuration mixing of different particle-hole states obtained after the diagonalization of the hamiltonian with a residual A - n u c l e o n interaction. An other approach has been used in refs. [2,8], where closure in the summation on final hypernuclear states in eq. (3) is done without any assumption on the hypernuclear structure. It includes coherent as well as incoherent excitation modes and in that case:
SRf Nef
0~)-1
T(b) is the
thickness function,
T(b) : f p(r) dz, normalized to the mass number of the target A. In eq. (8) 07r' K denotes the total isospin-averaged m e s o n nucleon cross section. In deriving eq. (8) the ratio of the real to the imaginary part of the isospin-averaged forward scattering amplitude for meson-nucleon scattering has been neglected since these quantities are not so well determined and since the result is not very sensitive to them. The quantities oTr,K are taken from experiment [10] and are typically, for the incident momenta we are interested in, 0K ~ 3 3 m b ,
0~r~28nab.
(9)
As concerns the quantity F(q) in eq. (6), the distorted radial function yj(qr) can be written, with the same assumptions as above, as
Volume 99B, number 5
/~(qr) =
PHYSICS LETTERS
(1/4n)(-i) J
(lo) x f d ~ es(cos 0) e x p ( i q ' r)G(b, z), with
-
--
c~
Z
0 where 1 a = ' ~ ( O K "t- Or/. ) ,
g = ~1 (o- K - a ~ ) .
(12)
The function G(b, z) is actually almost i n d e p e n d e n t o f z for the case we are discussing. The only point which remains to be settled is the nuclear radial part entering in eqs. (5) and (10) and eq. (8). The b o u n d state radial wave f u n c t i o n R (r) for the A-particle or the n e u t r o n were generated as indicated in ref. [6]. The effective n u m b e r o f n e u t r o n s given by eq. (5) can therefore be calculated for all particle- hole configurations needed. In the case o f very light nuclei the more appropriate intermediate coupling m o d e l was used instead o f t h e / f - c o u p l i n g m o d e l [11]. F o r N 4 = Z target nuclei an isos~in factor has to be added to formula,(5). The total NeSM is then
5 March 1981
obtained by summing over all configurations and angular m o m e n t a up to an excitation energy o f 60 MeV. In the case ofNStR different kinds o f nuclear target densities were used but we only present here two e x t r e m e choices o f shapes, as was done in ref. [2] : a u n i f o r m shape with a radius R = rOA1/3, a Fermi shape with a radius R = rlA 1/3 and a diffuseness a. In order to make a direct comparison the radii r 0 and r 1 have been chosen to reproduce in each case the experimeiltal root mean square radius. It is e x p e c t e d that NeSR depends quite strongly on the surface thickness (a = 0 for the u n i f o r m shape, a = 0.55 fm for the Fermi shape) because o f the strong absorption of the K - or the n - appearing in the damping factor exp [ - a T ( b ) ] . The results are shown in table 1 for the nuclei of interest and c o m p a r e d with the experimental value O f N e f f deduced from the recent data [9], assuming a constant value for the elementary forward reaction in the centre o f mass system, for all the three K - momenta: do(0°)CM/d~2 [K-n~aTr- = (0.95 -+ 0.13) m b / s r . The n u m b e r o f quasi-elastic transitions ( N sM (QE)) is ahnost constant over the whole mass range. If in light nuclei the quasi-elastic peaks are p r e p o n d e r a n t they have the t e n d e n c y to disappear with increasing mass target and in a large nucleus they are vanishing in a
Table 1 Comparison of the effective number of neutrons of different target nuclei and incident K- momenta. The experimentat value [9 ] is compared to the shell-model value calculated as explained in the text, and the number of quasi-elastic transitions (Ne~M(QF) f j is given. The effective number of neutrons calculated with closure approximation is given for two kinds of nuclear density; a . . . . zArSR(F)~ Fermi smtpc ti,ef f , and a uniform shape tArSR(U)~ ~'eff ~' Nucleus 6Li 7Li 9Be 12C 160 27A1 32S 4°Ca 51V
89y 2°9Bi
PK(MeV/c)
Neff (exp)
790 720 720 720 720 720 720 790 720 720 640
1.5 1.5 1.5 1.7 1.7 3.5 2.5 2.1 1.3 2.1 5.0
± 0.4 _+0.4 -+ 0.4 +_0.4 ± 0.6 ± 1.0 ± 0.7 ± 0.8 ± 0.6 _+ 1.7 _+1.8
SM Neff
NSM(QE) eff
, SR(F) eVeff
vSR(U) e eff
1.50 1.58 1.86 1.67 1.97 2.35 2.52 2.40 3.05 3.91 7.00
1.16 1.14 1.28 1.08 1.21 1.10 1.17 1.14 1.25 1.08 1.01
1.58 1.88 2.13 1.92 2.47 3.16 3.34 3.50 4.69 5.45 8.57
1.57 1,83 2.04 1.71 2.19 2.50 2.55 2.47 2.92 3.15 4.11
375
Volume 99B, number 5
PHYSICS LETTERS
broad structure coming from other transitions. These results are consistent with the observations made in ref. [12]. The shell-model effective number o f neutrons is similar but somewhat smaller than the one obtained using the sum-rule approach with a Fermi density NesR(F). This is not surprising since in the sum-rule approach A-hypernuclear states which lie at an excitation energy larger than 60 MeV are taken into account and may be also possible G-hypernuclear states. Moreover, in the sum-rule approach it is assumed that the momentum transfer is constant over the whole spectrum. Both values, N sM and NesR (F), are much larger than the effective number o f neutrons calculated using the uniform density, NesR(U), but that was expected as said previously. However, if a uniform density presents the advantage to give a completely analytical formula for NesR [2], such a shape is far from being realistic, especially in the case o f a strongly absorptive process. If one assumes the forward cross section to be given by a A c~ law, the predictions range between ~ ~ 0.50 with a Fermi density and a ~ 0.30 with a uniform density while the shell-model approach gives a ~ 0.45. These values are obtained when the calculation is done for the same incident m o m e n t u m (720 MeV/c) for every target nucleus in order to avoid deviations coming from different values of 0 K : r at different energies. It is, however, difficult to extract a value for from the present data, even if one tries to plot them versus the number of neutrons of the target nucleus, instead of the target mass number as is done in fig. 1. The overall agreement of NeSM and N SR with the experimental values is quite good although theoretical values are on the mean larger than the data, especially for A ~> 40. According to refs. [5,8] the scale factor necessary to get a better account of the data is believed to come mainly from a more refined description of nuclear and mesonic wave functions. Different effects which could reduce the magnitude o f the calculated values have been discussed recently, for either the total cross section [8] or partial cross sections obtained in the angular distributions of different hypernuclear states at small and large angles [ 13,14]. To conclude, we remark that the shell-model and the sum-rule approaches with a realistic density lead both to a rather successful account of the data. This seems to be a quite clear confirmation of the one-step 376
5 March 1981
/.// / /
~3 /
/
/
/
/
f /
/ /// o
/
:
/// /
ol o
: : x~
1 i[
+ I 50
100
150
2 0 'A Target moss number
Fig. 1. Effective number of neutrons plotted as a function of the target mass number. The shell-model values are indicated by crosses, while the sum-rule values are indicated by open circles (in the case of a Fermi density) and by open triangles (in the case of a uniform density). The dashed lines are only drawn to serve as a guide for the calculated values. The data are from ref. [9].
process. The great simplicity of the sum-rule approach allows to use it for the calculation, of the total formation cross section, especially for heavy targets where a broad structure is dominating the experimental spectrum. However, and this will be very useful for light nuclei, the main advantage o f the shell-model approach lies in the possibility to calculate the relative intensities to particular states, mainly the low-excited ones where quasi-elastic transitions can be easily identified. This will be very important in order to get reliable information on the structure o f these states when more realistic configuration mixed A - n e u t r o n states will be used. Par tof this work was done while the author was at the Institute der Sciences Nucl6aires, Grenoble. He would like to thank the warm hospitality extended to him, as well as many discussions with members of the nuclear theory group.
References [ 1] B. Povh, Nuclear physics with hyperons, preprint Heidelberg (1979), and references therein. [2] R.H. Dalitz and A. Gal, Phys. Lett. 64B (1976) 154.
Volume 99B, number 5
PHYSICS LETTERS
[3] A. Bouyssy, Phys. kett. 84B (1979) 41; tlyperons A et Z dans les noyaux, preprint Orsay (1980). [4] R.J. Esch, Can. J. Phys. 51 (1973) 1524. [5 ] J. Htifner, S.Y. Lee and H.A. Weidenmiiller, Nucl. Phys. A234 (1974) 429. [6] A. Bouyssy, Nucl. Phys. A290 (1977) 324. [7] L.S. Kisslinger and Nguyen Van Giai, Phys. Lett. 72B (1977) 19. [8] G.N. Epstein et al., Phys. Rev. C17 (1978) 1501. [9] R. Bartini et al., preprint CERN (1980).
5 March 1981
[10] Compilation of cross sections, zr- and n + induced reactions, CERN HERA 79-01; compilation of cross sections, K- and K+ induced reactions, CERN HERA 79-02. [11] W. Briickner et al., Proc. Kaon Factory Workshop (Vancouver, 1979), TRIUMF, TRI-79-1, p. 124. [12] B, Povh, Z. Phys. A279 (1976) 159. [13] C.B. Dover et al., Plays. Lett. 89B (1979) 26. [14] H.C. Chiang and J. Hfifner, Phys. Lett. 84B (1979) 393.
377
PHYSICS LETTF.RS
5 March 1981
PRODUCTION CROSS SECTIONS OF A-HYPERNUCLEI IN (K-, n - ) REACTIONS A. BOUYSSY Division de Physique Thdorique 1, Instimt de Physique Nucldaire, F-91406 Orsay Cedex, France Received 28 October 1980
Forward cross sections of A-hypernuclei produced in (K-, rr-) experiments are calculated in the distorted wave impulse ~-. 9~. approximation. Two kinds of approaches are discussed and a comparison with recent data ranging from 6ALl IO 2 0 ,\131 is made. The results support the assumption of a one-step process.
The observation of hypernuclear states produced in ( K - , rr-) reactions on nuclear targets has already attracted much interest [1]. Under suitable kinematical conditions the momentum transfer can be chosen very small compared to the Fermi m o m e n t u m of the nucleons in the nucleus. As a consequence strongly populated states are observed (at least in light nuclei) which have been interpreted as corresponding to recoilless or quasi-elastic transitions. In addition the main features of" the hypernuclear spectrum are reasonably well reproduced, although there aro still some open questions [2,3]. From the structure of hypernuclear excited states, mainly from the quasi-elastic ones, reliable information on the A-nucleus system have been obtained. This is, however, based on the assumption that the transitions to the hypernuclear states are one-step processes. In that case the reaction occurs on a single neutron, the rest o f the nucleus being unchanged, without any additional distortion of either the incoming K or the outgoing 7r-. If the high energies of the mesons and the forward rr- observation in the present experiments seem to favour such a process it is important to understand the nature o f the ( K - , 7r-) reaction mechanism. In this short letter we want to show an other piece of information supporting the assumption of a onestep process in the ( K - , r r - ) reaction, namely the for-
ward production cross section of A-liypernuclei. There has been already some calculations of the cross sections, either for some particular transitions or for the whole hypernuclear states produced in the case of light nuclei [ 2 - 8 ] , most o f them based on the distorted wave impulse approximation. Here a detailed comparison of two kinds of calculations [2,6] for the production cross sections in light and heavy nuclear targets will be made. A systematic study o f these two approaches will be presented and their results compared with recent data obtained for hypernuclei ranging from A6Li tO 209Bi, at three incident momenta [9]. If the one-step process dominates the ( K - , r r - ) reaction the indicated method of analysis is the DWIA. In that case the hypernuclear formation cross section at 0 ° for the strangeness exchange reaction K - + A z - + r r - + A,xz
(1)
is related to the elementary K - n -+ A r t - cross section by: do(0°)/dg2 = Nef f do(0°)/d~2 [K-n_,A~- .
(2)
The total effective number o f neutrons in the reaction is given by [2]: A Neff=~f
fx(-)*(r)(fl2 j=lU
(/)8(r-r/)li) (3)
X X(+)(r) drl 2 .
i Laboratoire associ6 au CNRS. 0 0 3 1 - 9 1 6 3 / 8 1 / 0 0 0 0 - 0 0 0 0 / $ 02.50 © North-Holland Publishing Company
373
Volume 99B, number 5
PHYSICS LETTERS
The operator U_ (/) is the U-spin lowering operator (_+) transforming a neutron into a A-particle, the Xn,K(r) being the appropriate distorted waves of the mesons. In a first approach, to calculate Nef f we choose the beam direction as axis of quantization and expand the product of distorted waves in partial waves
(4) = ~
L=O
[4rr(2L +
1)]l/2iLTL(qr)yO(f),
where the radial function ]'L(qr) reduces to the usual spherical Bessel function ]L(qr) in the absence of any distortion (q is the momentum transfer). We restrict here for simplicity the initial state to a doubly closedshell nucleus and describe the final state as a one particle-one hole excitation denoted by []AJn 1)J. Straightforward angular momentum algebra gives NeJff, the effective number of neutrons for a particular transition 10>-+ [5]:
[/AJnl>j
NJff = (2J + 1)(2/A + 1)(2in + 1) 1
F2(q) '
(5)
5 March 1981
actual number of neutrons N for no distortion. The total forward formation cross section is much easier to calculate if one uses the sum-rule approach. However, the advantage of using eq. (5) is that it allows the extraction of the amount of quasi-elastic A production which is believed to be the dominant process in forward ( K - , 7r-) reactions on light nuclei, and to give the magnitude of the cross section for a transition to a given state of the hypernucleus. Both approaches obviously need knowledge of the distorted waves of the mesons. The K - and ~r- wave functions can be generated by using a standard optical potential as was done in ref. [8] with or without a pand d-wave part. However, the number of partial waves needed in such a calculation and also the lack of very precise information on the K--nucleus interaction was our motivation to use a much simpler procedure, namely the eikonal approximation. The comparison made in ref. [8] between results obtained either from the fully distorted waves or from the eikonal waves shows that the eikonal approximation is good at these energies. With eikonal distortions N eff SR can be written;
NeSfl~= (N/A )(°K where
x F(q) = f r2dr RA(r)Rn(r)ij(qr ) ,
fd2b (exp[-onT(b)]
where
_
f pN(r)lx ~ )
(r)×(~)(r)12
dr
(7)
where PN(r) is the neutron density and SR means sum rule. The effective number of neutrons reduces to the 374
(8)
-exp[-OKT(b)]
) ,
(6)
with obvious notations. The total shell-model effective number of neutrons (called N sM) is therefore the sum over all the possible transitions corresponding to a given j,r and the sum over all the possible values of the angular momentum. This approach is the one used in ref. [6] including configuration mixing of different particle-hole states obtained after the diagonalization of the hamiltonian with a residual A - n u c l e o n interaction. An other approach has been used in refs. [2,8], where closure in the summation on final hypernuclear states in eq. (3) is done without any assumption on the hypernuclear structure. It includes coherent as well as incoherent excitation modes and in that case:
SRf Nef
0~)-1
T(b) is the
thickness function,
T(b) : f p(r) dz, normalized to the mass number of the target A. In eq. (8) 07r' K denotes the total isospin-averaged m e s o n nucleon cross section. In deriving eq. (8) the ratio of the real to the imaginary part of the isospin-averaged forward scattering amplitude for meson-nucleon scattering has been neglected since these quantities are not so well determined and since the result is not very sensitive to them. The quantities oTr,K are taken from experiment [10] and are typically, for the incident momenta we are interested in, 0K ~ 3 3 m b ,
0~r~28nab.
(9)
As concerns the quantity F(q) in eq. (6), the distorted radial function yj(qr) can be written, with the same assumptions as above, as
Volume 99B, number 5
/~(qr) =
PHYSICS LETTERS
(1/4n)(-i) J
(lo) x f d ~ es(cos 0) e x p ( i q ' r)G(b, z), with
-
--
c~
Z
0 where 1 a = ' ~ ( O K "t- Or/. ) ,
g = ~1 (o- K - a ~ ) .
(12)
The function G(b, z) is actually almost i n d e p e n d e n t o f z for the case we are discussing. The only point which remains to be settled is the nuclear radial part entering in eqs. (5) and (10) and eq. (8). The b o u n d state radial wave f u n c t i o n R (r) for the A-particle or the n e u t r o n were generated as indicated in ref. [6]. The effective n u m b e r o f n e u t r o n s given by eq. (5) can therefore be calculated for all particle- hole configurations needed. In the case o f very light nuclei the more appropriate intermediate coupling m o d e l was used instead o f t h e / f - c o u p l i n g m o d e l [11]. F o r N 4 = Z target nuclei an isos~in factor has to be added to formula,(5). The total NeSM is then
5 March 1981
obtained by summing over all configurations and angular m o m e n t a up to an excitation energy o f 60 MeV. In the case ofNStR different kinds o f nuclear target densities were used but we only present here two e x t r e m e choices o f shapes, as was done in ref. [2] : a u n i f o r m shape with a radius R = rOA1/3, a Fermi shape with a radius R = rlA 1/3 and a diffuseness a. In order to make a direct comparison the radii r 0 and r 1 have been chosen to reproduce in each case the experimeiltal root mean square radius. It is e x p e c t e d that NeSR depends quite strongly on the surface thickness (a = 0 for the u n i f o r m shape, a = 0.55 fm for the Fermi shape) because o f the strong absorption of the K - or the n - appearing in the damping factor exp [ - a T ( b ) ] . The results are shown in table 1 for the nuclei of interest and c o m p a r e d with the experimental value O f N e f f deduced from the recent data [9], assuming a constant value for the elementary forward reaction in the centre o f mass system, for all the three K - momenta: do(0°)CM/d~2 [K-n~aTr- = (0.95 -+ 0.13) m b / s r . The n u m b e r o f quasi-elastic transitions ( N sM (QE)) is ahnost constant over the whole mass range. If in light nuclei the quasi-elastic peaks are p r e p o n d e r a n t they have the t e n d e n c y to disappear with increasing mass target and in a large nucleus they are vanishing in a
Table 1 Comparison of the effective number of neutrons of different target nuclei and incident K- momenta. The experimentat value [9 ] is compared to the shell-model value calculated as explained in the text, and the number of quasi-elastic transitions (Ne~M(QF) f j is given. The effective number of neutrons calculated with closure approximation is given for two kinds of nuclear density; a . . . . zArSR(F)~ Fermi smtpc ti,ef f , and a uniform shape tArSR(U)~ ~'eff ~' Nucleus 6Li 7Li 9Be 12C 160 27A1 32S 4°Ca 51V
89y 2°9Bi
PK(MeV/c)
Neff (exp)
790 720 720 720 720 720 720 790 720 720 640
1.5 1.5 1.5 1.7 1.7 3.5 2.5 2.1 1.3 2.1 5.0
± 0.4 _+0.4 -+ 0.4 +_0.4 ± 0.6 ± 1.0 ± 0.7 ± 0.8 ± 0.6 _+ 1.7 _+1.8
SM Neff
NSM(QE) eff
, SR(F) eVeff
vSR(U) e eff
1.50 1.58 1.86 1.67 1.97 2.35 2.52 2.40 3.05 3.91 7.00
1.16 1.14 1.28 1.08 1.21 1.10 1.17 1.14 1.25 1.08 1.01
1.58 1.88 2.13 1.92 2.47 3.16 3.34 3.50 4.69 5.45 8.57
1.57 1,83 2.04 1.71 2.19 2.50 2.55 2.47 2.92 3.15 4.11
375
Volume 99B, number 5
PHYSICS LETTERS
broad structure coming from other transitions. These results are consistent with the observations made in ref. [12]. The shell-model effective number o f neutrons is similar but somewhat smaller than the one obtained using the sum-rule approach with a Fermi density NesR(F). This is not surprising since in the sum-rule approach A-hypernuclear states which lie at an excitation energy larger than 60 MeV are taken into account and may be also possible G-hypernuclear states. Moreover, in the sum-rule approach it is assumed that the momentum transfer is constant over the whole spectrum. Both values, N sM and NesR (F), are much larger than the effective number o f neutrons calculated using the uniform density, NesR(U), but that was expected as said previously. However, if a uniform density presents the advantage to give a completely analytical formula for NesR [2], such a shape is far from being realistic, especially in the case o f a strongly absorptive process. If one assumes the forward cross section to be given by a A c~ law, the predictions range between ~ ~ 0.50 with a Fermi density and a ~ 0.30 with a uniform density while the shell-model approach gives a ~ 0.45. These values are obtained when the calculation is done for the same incident m o m e n t u m (720 MeV/c) for every target nucleus in order to avoid deviations coming from different values of 0 K : r at different energies. It is, however, difficult to extract a value for from the present data, even if one tries to plot them versus the number of neutrons of the target nucleus, instead of the target mass number as is done in fig. 1. The overall agreement of NeSM and N SR with the experimental values is quite good although theoretical values are on the mean larger than the data, especially for A ~> 40. According to refs. [5,8] the scale factor necessary to get a better account of the data is believed to come mainly from a more refined description of nuclear and mesonic wave functions. Different effects which could reduce the magnitude o f the calculated values have been discussed recently, for either the total cross section [8] or partial cross sections obtained in the angular distributions of different hypernuclear states at small and large angles [ 13,14]. To conclude, we remark that the shell-model and the sum-rule approaches with a realistic density lead both to a rather successful account of the data. This seems to be a quite clear confirmation of the one-step 376
5 March 1981
/.// / /
~3 /
/
/
/
/
f /
/ /// o
/
:
/// /
ol o
: : x~
1 i[
+ I 50
100
150
2 0 'A Target moss number
Fig. 1. Effective number of neutrons plotted as a function of the target mass number. The shell-model values are indicated by crosses, while the sum-rule values are indicated by open circles (in the case of a Fermi density) and by open triangles (in the case of a uniform density). The dashed lines are only drawn to serve as a guide for the calculated values. The data are from ref. [9].
process. The great simplicity of the sum-rule approach allows to use it for the calculation, of the total formation cross section, especially for heavy targets where a broad structure is dominating the experimental spectrum. However, and this will be very useful for light nuclei, the main advantage o f the shell-model approach lies in the possibility to calculate the relative intensities to particular states, mainly the low-excited ones where quasi-elastic transitions can be easily identified. This will be very important in order to get reliable information on the structure o f these states when more realistic configuration mixed A - n e u t r o n states will be used. Par tof this work was done while the author was at the Institute der Sciences Nucl6aires, Grenoble. He would like to thank the warm hospitality extended to him, as well as many discussions with members of the nuclear theory group.
References [ 1] B. Povh, Nuclear physics with hyperons, preprint Heidelberg (1979), and references therein. [2] R.H. Dalitz and A. Gal, Phys. Lett. 64B (1976) 154.
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[3] A. Bouyssy, Phys. kett. 84B (1979) 41; tlyperons A et Z dans les noyaux, preprint Orsay (1980). [4] R.J. Esch, Can. J. Phys. 51 (1973) 1524. [5 ] J. Htifner, S.Y. Lee and H.A. Weidenmiiller, Nucl. Phys. A234 (1974) 429. [6] A. Bouyssy, Nucl. Phys. A290 (1977) 324. [7] L.S. Kisslinger and Nguyen Van Giai, Phys. Lett. 72B (1977) 19. [8] G.N. Epstein et al., Phys. Rev. C17 (1978) 1501. [9] R. Bartini et al., preprint CERN (1980).
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[10] Compilation of cross sections, zr- and n + induced reactions, CERN HERA 79-01; compilation of cross sections, K- and K+ induced reactions, CERN HERA 79-02. [11] W. Briickner et al., Proc. Kaon Factory Workshop (Vancouver, 1979), TRIUMF, TRI-79-1, p. 124. [12] B, Povh, Z. Phys. A279 (1976) 159. [13] C.B. Dover et al., Plays. Lett. 89B (1979) 26. [14] H.C. Chiang and J. Hfifner, Phys. Lett. 84B (1979) 393.
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