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Threshold pion photoproduction on 6Li
Volume 69B, number 4
PHYSICS LETTERS
THRESHOLD
29 August 1971
PION PHOTOPRODUCTION
ON 6Li
S.K. SINGH and I. AHMAD Physics Department,
A.M. U., Aligarh, U.P., India
Received 20 February 1977 Revised manuscript received 9 May 1977 The a-d cluster model which satisfactorily explains the electron scattering data is applied to study the threshold pion photoproduction on 6Li. The results demonstrate that the existing theoretical situation is not improved merely by having a better description of the target.
Pion photoproduction and radiative pion capture processes provide an excellent tool to test the validity of soft pion theorems in nuclei [ 11. In order to avoid the complications arising in isolating the S-wave contribution to the radiative pion capture processes for which the soft pion theorems apply [2] it has been suggested to study the pion photoproduction processes in nuclear targets sufficiently close to threshold to eliminate L = 0 contributions as much as possible [3]. Such an experiment performed on 6 Li target [4] finds that the experimental total cross section results for the process y + 6 Li -+ II+. + 6Heg s are considerably lower than the corresponding thebretical calculations using harmonic oscillator wavefunctions for 6Li [3, 51. This has led to a reexamination of the soft pion theorems as applied to pion photoproduction processes in nuclei [6]. Several processes involving pion and weak or electromagnetic interactions of nuclei studied recently seem to emphasize the inadequacy of soft pion results when applied to nuclei and have proposed ways to improve them [6,7]. Cammarata and Donnelly [8] have on the other hand studied this process in various nuclear models and the elementary particle model used to describe the weak and electromagnetic reactions on 6Li. They have however not considered the (a-d) cluster model of 6Li which is quite successful in explaining various electron scattering data on 6~i [9-l 11. We have in this letter considered this model to calculate the total cross section for the threshold pion photoproduction on 6Li with a hope to reduce the above discrepancy but find that the results are higher than the experimental results in this case tco as is the case with all other models considered by Cammarata and Donnelly [8]. The matrix element for the threshold photoproduc422
tion of pions in soft pion limit neglecting possible contributions from meson exchange currents [ 121 is given by tl,31 @eF
m =m
(fldr ~(k, X)*jT(r)e’k’rli),
n
(1)
where e = &cy, F = J4?7(1 tm,/rn~)-‘f,,~ = 0.09 and c(k, A) is the polarisation vector of the incident photon, k is the momentum of the photon given to the nuclear system and i?(r)
=7
a (i)<_(i)S(r
-ri)
(2)
where i runs over all the nucleons in the target nucleus. The nuclear wavefunctions Ii) and If> are written in terms of .$JMgiven by [ 111
(3)
where A stands for antisymmetrisation. 4: o and $“i are the wavefunctions for the initial and dinucleon clusters in the initial and final nuclei and $1”’ describes their relative motion. Assuming these wave functions for the nuclear states and neglecting effects of antisymmetrisation * and D state of deuteron the matrix * It has been shown by Kudeyarov et al. [ 111 in case of inelastic magnetic electron scattering from 6Li (l+-tO+ transitions) that the intercluster nucleon exchange effects are small at low momentum transfers (q < 1.3 fm-I) if we take $8(r) to have no modes. Since the transition involved here is similar to that involved in the inelastic magnetic electron scattering we assume these effects are small in our case too.
Volume 69B, number 4
PHYSICS LETTERS
29 August 1917
element can be written as m =$
@;*(r + $R)(a(S)*c
- a(6)*&)
A cs X ~~~(ti(r+aR)liL~(R)12eiq’rd,
1
(4)
,
where R = d ~~=l’i -4 Z$+ri is the relative coordinate between and dinucleon clusters and Q is the momentum transfer given to pion. In writing down the above matrix element we have assumed plane waves for the outgoing pions thus neglecting the final state interactions. These can be easily included by putting the outgoing pion in an optical potential and solving for its wavefunction [3], Using the matrix element given in eq. (4) the differential cross section is calculated to be da -= da
4afz(l+
m,,/m,)-’
3mi
k,, x lF,(4)t21F,_,(24/3)12, (5)
where
&Jr) and &(r) being the radial parts of 4”,(r) and 4,,,,(r) respectively and
We have taken the same Gaussian function for describing the radial part of the wavefunction for the triplet deuteron and singlet dinucleon clusters. The relative motion of the internal clusters is also described by a Gaussian +i(r) = Nr2e-2r2/3
(6)
where r is measured in units of b. The expressions we have used for the form factor are then derived to be be [lo] F
d
(q) =
e-b:S2/8
Fold(2q/3) = & edX2 [4x4 - 20x2 + 151
(7) \ .
where x2 = h q2b2 . Results for the total cross section as calculated from eq. (5) are shown in fig. 1(b) along with the corre-
2.0
1.0 (E,-E,)
Fig. 1. Total cross section q(pb) - the photon energy (ET-Eo) above threshold in various theoretical models. (a) Harmonic oscillator model with no final state interactions. (b) 0t-d cluster model with no final state interactions. (c) Harmonic oscillator model with final state interactions. (d) SASK-A model with final state interactions. Figs. l(a) and l(c) are from Koch and Donnelly [3] and fig. l(d) is from Commarata and Donnelly [8].
sponding results of a shell model calculation of Koch and Donnelly [3] in fig. l(a). Fig. 1(b) corresponds to the parameters b2 = 5 - 36 fm2; bi/b2 * 1, which are determined from electron lithium scattering [lo]. One should however use parameter values as determined from inelastic magnetic scattering (1+-O+ excitation) on 6Li. Such an analysis, though not in a simple model as ours, favours a smaller value of bi to fit the data if one keeps b2 fixed to the elastic scattering value [ 111. The numerical calculations performed with b2 = 5.36 fm2 and bs/b2 < 1 increases the total cross section towards fig. i(a). We have also calculated the total cross section u using Eckart function for 4:(r) where parameters are fixed by electron scattering data [9]. The results obtained with this wavefunction are very close to fig. l(b) and therefore are not shown in the figure. If we use the distorted wavefunctions for outgoing pions instead of plane waves the total cross sections will decrease as is the case with the shell model calculation for which the results are shown in fig. l(c). Various shell model wavefunctions whose parameters are determined by fitting the recent electron scattering and muon capture data are used by Cammarata and Donnelly [8] to calculate u which all lie between 423
Volume 69B, number 4
PHYSICS LETTERS
figs. l(c) and l(d) and are 50% higher than the experimental results. Including the final state interaction by putting the pions in an optical potential will expectedly bring down the curve l(b) to lie somewhere between figs. l(c) and l(d). This reduction obviously offers no improvement over the existing models analysed so far. Our analysis can be further improved by including the intercluster nucleon exchanges and using the parameters determined from analysing the Bergstrom’s data [5] but it seems unlikely that the results will change drastically to bring about an agreement with the experiment. In this light, we may conclude that all the nuclear and elementary particle models studied so far including the cluster model overestimate the threshold pion photoproduction cross section on 6Li and are unable to explain the experimental results. Theoretical emphasis should therefore be put in studying the various corrections to the interaction Hamiltonian rather than the nuclear models. However the recent experimental results on the threshold pion photoproduction on 12C [ 131 show that the disagreement is not as large as found in the case of 6Li. This experimental result [ 131 combined with the theoretically unsucessful attempts to explain the large discrepancy in the case of 6 Li strongly indicates the need for having a second look at the experimental analysis performed by Deutsch et al. [4] *. * A recent experimental analysisperformed
with improved techniques seems to have reduced this discrepancy to 20% (see C. Tzara quoted by A.M. Bernstein et al. [ 131). Our .results become more significant in the light of this development as they might bring an agreement with the experimental results if pion absorption is taken into account (compare figs. l(a) and l(b)). Such an analysis is presently under study.
424
29 August 1917
References [1] M. Ericson and M. Rho, Phys. Rep. SC (1972) 59. (21 M. Ericson and A. Figureau, Nucl. Phys. B3 (1967) 609; 511 (1969) 621; Phys. Rev. C8 (1973) 2029; P. Trio1 et al., Phys. Rev. Lett. 32 (1974) 1238. 131 J.H. Koch and T.W. Donnelly, Nucl. Phys. B64 (1973) 478. 141 J. Deutsch, Proc. High energy physics and nuclear structure Conf. (A.I.P., New York 1975) p. 445; J. Deutsch et al., Phys. Rev. Lett. 33 (1974) 316. 151 J.H. Koch and T.W. Donnelly, Phys. Rev. Cl0 (1974) 2618; F. Cannata et al., Phys. Rev. Lett. 33 (1974) 1316; J.C. Bergstrom et al., Nucl. Phys. A251 (1975) 401. M. Moreno et al., Phys. Rev. Cl2 (1975) 514. M. Ericson, Institut de Physique Nucleaire, Preprint, CEN/7552, May 1975; C.W. Kim and J. Kin, Phys. Rev. Dll (1975) 3347; C.W. Kim and J.S. Tounsend, Phys. Rev. Dll (1975) 656. 3.B. Cammarata and T.W. Donnelly, Nucl. Phys. A267 (1976) 365. [9] J.V. Noble, Phys. Rev. C9 (1974) 1209. [lo] R.B. Raphach, Nucl. Phys. A201 (1973) 62. [ll] A. Kudeyarov et al., Nucl. Phys. Al63 (1971) 316. [12] J. Delrome et al., Nucl. Phys. A240 (1975) 453; M. Chemtob and M. Rho, Nucl. Phys. Al63 (1971) 1. [13] A.M. Bernstein et al., Phys. Rev. Lett. 37 (1976) 819.
PHYSICS LETTERS
THRESHOLD
29 August 1971
PION PHOTOPRODUCTION
ON 6Li
S.K. SINGH and I. AHMAD Physics Department,
A.M. U., Aligarh, U.P., India
Received 20 February 1977 Revised manuscript received 9 May 1977 The a-d cluster model which satisfactorily explains the electron scattering data is applied to study the threshold pion photoproduction on 6Li. The results demonstrate that the existing theoretical situation is not improved merely by having a better description of the target.
Pion photoproduction and radiative pion capture processes provide an excellent tool to test the validity of soft pion theorems in nuclei [ 11. In order to avoid the complications arising in isolating the S-wave contribution to the radiative pion capture processes for which the soft pion theorems apply [2] it has been suggested to study the pion photoproduction processes in nuclear targets sufficiently close to threshold to eliminate L = 0 contributions as much as possible [3]. Such an experiment performed on 6 Li target [4] finds that the experimental total cross section results for the process y + 6 Li -+ II+. + 6Heg s are considerably lower than the corresponding thebretical calculations using harmonic oscillator wavefunctions for 6Li [3, 51. This has led to a reexamination of the soft pion theorems as applied to pion photoproduction processes in nuclei [6]. Several processes involving pion and weak or electromagnetic interactions of nuclei studied recently seem to emphasize the inadequacy of soft pion results when applied to nuclei and have proposed ways to improve them [6,7]. Cammarata and Donnelly [8] have on the other hand studied this process in various nuclear models and the elementary particle model used to describe the weak and electromagnetic reactions on 6Li. They have however not considered the (a-d) cluster model of 6Li which is quite successful in explaining various electron scattering data on 6~i [9-l 11. We have in this letter considered this model to calculate the total cross section for the threshold pion photoproduction on 6Li with a hope to reduce the above discrepancy but find that the results are higher than the experimental results in this case tco as is the case with all other models considered by Cammarata and Donnelly [8]. The matrix element for the threshold photoproduc422
tion of pions in soft pion limit neglecting possible contributions from meson exchange currents [ 121 is given by tl,31 @eF
m =m
(fldr ~(k, X)*jT(r)e’k’rli),
n
(1)
where e = &cy, F = J4?7(1 tm,/rn~)-‘f,,~ = 0.09 and c(k, A) is the polarisation vector of the incident photon, k is the momentum of the photon given to the nuclear system and i?(r)
=7
a (i)<_(i)S(r
-ri)
(2)
where i runs over all the nucleons in the target nucleus. The nuclear wavefunctions Ii) and If> are written in terms of .$JMgiven by [ 111
(3)
where A stands for antisymmetrisation. 4: o and $“i are the wavefunctions for the initial and dinucleon clusters in the initial and final nuclei and $1”’ describes their relative motion. Assuming these wave functions for the nuclear states and neglecting effects of antisymmetrisation * and D state of deuteron the matrix * It has been shown by Kudeyarov et al. [ 111 in case of inelastic magnetic electron scattering from 6Li (l+-tO+ transitions) that the intercluster nucleon exchange effects are small at low momentum transfers (q < 1.3 fm-I) if we take $8(r) to have no modes. Since the transition involved here is similar to that involved in the inelastic magnetic electron scattering we assume these effects are small in our case too.
Volume 69B, number 4
PHYSICS LETTERS
29 August 1917
element can be written as m =$
@;*(r + $R)(a(S)*c
- a(6)*&)
A cs X ~~~(ti(r+aR)liL~(R)12eiq’rd,
1
(4)
,
where R = d ~~=l’i -4 Z$+ri is the relative coordinate between and dinucleon clusters and Q is the momentum transfer given to pion. In writing down the above matrix element we have assumed plane waves for the outgoing pions thus neglecting the final state interactions. These can be easily included by putting the outgoing pion in an optical potential and solving for its wavefunction [3], Using the matrix element given in eq. (4) the differential cross section is calculated to be da -= da
4afz(l+
m,,/m,)-’
3mi
k,, x lF,(4)t21F,_,(24/3)12, (5)
where
&Jr) and &(r) being the radial parts of 4”,(r) and 4,,,,(r) respectively and
We have taken the same Gaussian function for describing the radial part of the wavefunction for the triplet deuteron and singlet dinucleon clusters. The relative motion of the internal clusters is also described by a Gaussian +i(r) = Nr2e-2r2/3
(6)
where r is measured in units of b. The expressions we have used for the form factor are then derived to be be [lo] F
d
(q) =
e-b:S2/8
Fold(2q/3) = & edX2 [4x4 - 20x2 + 151
(7) \ .
where x2 = h q2b2 . Results for the total cross section as calculated from eq. (5) are shown in fig. 1(b) along with the corre-
2.0
1.0 (E,-E,)
Fig. 1. Total cross section q(pb) - the photon energy (ET-Eo) above threshold in various theoretical models. (a) Harmonic oscillator model with no final state interactions. (b) 0t-d cluster model with no final state interactions. (c) Harmonic oscillator model with final state interactions. (d) SASK-A model with final state interactions. Figs. l(a) and l(c) are from Koch and Donnelly [3] and fig. l(d) is from Commarata and Donnelly [8].
sponding results of a shell model calculation of Koch and Donnelly [3] in fig. l(a). Fig. 1(b) corresponds to the parameters b2 = 5 - 36 fm2; bi/b2 * 1, which are determined from electron lithium scattering [lo]. One should however use parameter values as determined from inelastic magnetic scattering (1+-O+ excitation) on 6Li. Such an analysis, though not in a simple model as ours, favours a smaller value of bi to fit the data if one keeps b2 fixed to the elastic scattering value [ 111. The numerical calculations performed with b2 = 5.36 fm2 and bs/b2 < 1 increases the total cross section towards fig. i(a). We have also calculated the total cross section u using Eckart function for 4:(r) where parameters are fixed by electron scattering data [9]. The results obtained with this wavefunction are very close to fig. l(b) and therefore are not shown in the figure. If we use the distorted wavefunctions for outgoing pions instead of plane waves the total cross sections will decrease as is the case with the shell model calculation for which the results are shown in fig. l(c). Various shell model wavefunctions whose parameters are determined by fitting the recent electron scattering and muon capture data are used by Cammarata and Donnelly [8] to calculate u which all lie between 423
Volume 69B, number 4
PHYSICS LETTERS
figs. l(c) and l(d) and are 50% higher than the experimental results. Including the final state interaction by putting the pions in an optical potential will expectedly bring down the curve l(b) to lie somewhere between figs. l(c) and l(d). This reduction obviously offers no improvement over the existing models analysed so far. Our analysis can be further improved by including the intercluster nucleon exchanges and using the parameters determined from analysing the Bergstrom’s data [5] but it seems unlikely that the results will change drastically to bring about an agreement with the experiment. In this light, we may conclude that all the nuclear and elementary particle models studied so far including the cluster model overestimate the threshold pion photoproduction cross section on 6Li and are unable to explain the experimental results. Theoretical emphasis should therefore be put in studying the various corrections to the interaction Hamiltonian rather than the nuclear models. However the recent experimental results on the threshold pion photoproduction on 12C [ 131 show that the disagreement is not as large as found in the case of 6Li. This experimental result [ 131 combined with the theoretically unsucessful attempts to explain the large discrepancy in the case of 6 Li strongly indicates the need for having a second look at the experimental analysis performed by Deutsch et al. [4] *. * A recent experimental analysisperformed
with improved techniques seems to have reduced this discrepancy to 20% (see C. Tzara quoted by A.M. Bernstein et al. [ 131). Our .results become more significant in the light of this development as they might bring an agreement with the experimental results if pion absorption is taken into account (compare figs. l(a) and l(b)). Such an analysis is presently under study.
424
29 August 1917
References [1] M. Ericson and M. Rho, Phys. Rep. SC (1972) 59. (21 M. Ericson and A. Figureau, Nucl. Phys. B3 (1967) 609; 511 (1969) 621; Phys. Rev. C8 (1973) 2029; P. Trio1 et al., Phys. Rev. Lett. 32 (1974) 1238. 131 J.H. Koch and T.W. Donnelly, Nucl. Phys. B64 (1973) 478. 141 J. Deutsch, Proc. High energy physics and nuclear structure Conf. (A.I.P., New York 1975) p. 445; J. Deutsch et al., Phys. Rev. Lett. 33 (1974) 316. 151 J.H. Koch and T.W. Donnelly, Phys. Rev. Cl0 (1974) 2618; F. Cannata et al., Phys. Rev. Lett. 33 (1974) 1316; J.C. Bergstrom et al., Nucl. Phys. A251 (1975) 401. M. Moreno et al., Phys. Rev. Cl2 (1975) 514. M. Ericson, Institut de Physique Nucleaire, Preprint, CEN/7552, May 1975; C.W. Kim and J. Kin, Phys. Rev. Dll (1975) 3347; C.W. Kim and J.S. Tounsend, Phys. Rev. Dll (1975) 656. 3.B. Cammarata and T.W. Donnelly, Nucl. Phys. A267 (1976) 365. [9] J.V. Noble, Phys. Rev. C9 (1974) 1209. [lo] R.B. Raphach, Nucl. Phys. A201 (1973) 62. [ll] A. Kudeyarov et al., Nucl. Phys. Al63 (1971) 316. [12] J. Delrome et al., Nucl. Phys. A240 (1975) 453; M. Chemtob and M. Rho, Nucl. Phys. Al63 (1971) 1. [13] A.M. Bernstein et al., Phys. Rev. Lett. 37 (1976) 819.