- Email: [email protected]
Meson exchange current effects in the electroexcitation of magnetic states in closed shell nuclei
PHYSICS LETTERS
Volume 146B, number 5
18 October 1984
MESON EXCHANGE CURRENT EFFECTS IN THE ELECTROEXCITATION
OF MAGNETIC STATES IN CLOSED SHELL NUCLEI
A.M. LALLENA, J.S. DEHESA Departamento
de Fisica Nuclear, Universidad de Granada, Granada, Spain
and S. KREWALD Institut ftir Kerphysik,
Kernforschungsanlage
Jiilich, D-51 70 Jiilich, West Germany
Received 12 June 1984
A microscopic approach to evaluate the contributions of one-pion exchange currents to the inelastic (e, e’) scattering form factor of closed shell nuclei is developed. Both seagull and pionic parts of the two-body term are treated on the same footing and their final expressions are simple, clear and physically transparent. Application to the magnetic stretched states 4-(18.98 MeV) of 160 and 14-(6.74 MeV) of 208Pb is done. Agreement with the available experimental data is discussed. It is found that the effect of the meson exchange currents is a smooth but non-negligible enhancement (215% at the first scattering maximum) at small momentum transfers and an increase by a factor bigger than 1.5 in the region of the second maximum.
A systematic theoretical investigation of meson exchange currents (MEC) effects on the inelastic electron scattering form factors of heavy nuclei (i.e. A 2 4) remains to be accomplished. However, except for two- and threebody systems [l-5], these effects are often difficult to disentangle from those associated with nuclear structure. The MEC effects cannot be ignored [6] if one wants to extract the full information of the accurate high-resolution data of the new generation of spectrometers and electron accelerators which are close to operate with a 100% duty cycle and beam intensities of 100 PA. Therefore reliable calculations of the electron-nucleus form factors with the inclusion of MEC’s are mandatory even in heavy nuclei. The purpose of the present letter is to develop a microscopic method to evaluate the one-pion exchange current contributions to the electroexcitation form factors of magnetic states in closed shell nuclei. Furthermore the applicability of the method is illustrated by calculating the form factors of the magnetic high-spin states 4-( 18.98 MeV) in 160 and 14- (6.74 MeV) in 208Pb, of which recent accurate data [7,8] are available. This method has three important advantages. Firstly: one can handle realistic mean fields such as Hartree-Fock or Woods-Saxon. Secondly: the seagull and pion-in-flight parts of the two-body contributions are treated on the same footing together with the one-body term, i.e. all of them are calculated by using the same realistic single-particle wavefunctions. In particular it is remarkable that contrary to other approaches [9-l 1] one can analytically evaluate within our method the pion-in-flight current term in any closed shell nucleus and for any multipolarity, e.g. we do not use any “pair approximation” to estimate it. Thirdly: simple, clear and physically transparent expressions, which resemble much to the one-body term, are found for the two-body contributions. The electromagnetic nuclear current operator J(r) is given in this work by the sum of the well-known one-body J’OB)(r), seagull J
294
= flB)k)
+ $G)(c!)
+ 8)(q)
9
(1) 0370-2693/84/$03.00 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
PHYSICS LETTERS
Volume 1468, number 5
18 October 1984
where i+‘(q)
= (fi/Z)(Jll
fik)(q)IIO)
for k = OB, SC, rr, and i’fk’(q) denotes the corresponding
a(q)
transverse magnetic multipole
operator, i.e.
Y%(a) * J@)(r) .
=Jd3rj&r)
Besides the nuclear wavefunction ].I) will be described within the RPA theory, so it will be characterized by the particle-hole amplitudes XJ(ph) and YJ(ph). Apart from the one-body term, which is already known a long time [ 12,131, the basic idea in our derivation of the two-body terms is to keep explicitly the dependence on the nucleon coordinates rl and rq. Given the nucleon wavefunction at the vertex ri, a virtual pion wavefunction may be created with the pion creation operator $(r -rj)
=ej .Vls3
un(q) exp[iq . (r -I$]
(2)
,
where u,(q) = 1/(q2 + rn$ denotes the bare pion propagator. tor can be written as
With this in mind, the pion exchange current opera-
dSG)(rl,r2;r)
-u$(r
= 47r(_&rrz~)(z1 XT&
{a16(r
-Q&r
-r2)
-rz)J(r
-rl)}
(3)
- rl)]]
(4)
for the seagull or contact part, and as
r2;4 = -47$&&~1
J(?rl,
Xt21z i&r
- r2)l - d
- rl)[VJ(r
for the pion-in-flight part. Here the nNN coupling constant fz = 0.079 and m, is the pion mass. After a long and appropriate use of angular momentum algebra [ 14,151 one can find simple and compact formulas for the two-body terms of the nuclear form factor which allow a direct and simple physical interpretation. We have found tiG’(q)
= -2i$
X
_:p/2 _s2
(
2
x
g
[XJ*(ph) + (-)J’l
s) ,&%@ph,.6h’,r
r$(/,+lh’+Ll+
l)if
[ +Eg([h’+lh+L2+
Ip
(’ l/2 ‘r :;2
l)ig
Y;(ph)](-)‘p4-(1,
+ lh +J t l)f&,
- 6ph,n6h’,v) jh’
Ll
-l/2 . _z2
2
0 11
d=2jdclx)Iph’(L~J)
x2 i&X)
;’
Rh’Rh
Ih’h(L2, X) Rp Rh’
1
(5)
2
for the seagull part, and @j(q)
= 4i g m
%$[X;(ph) + (-)J’l
YAph)l
E(lp + lh + J+
1) fpfh
nP X
c
(-l)ih’+ihj=?(6ph,vShf ,-6
ph,n6h’,u)~~~E(lp+lhl+Llf
2
_:;;
l)t;(lh’+lh+L2+
1)ifi;[L2(L2+1)11’2
h’
x (;
:h)
(f’
_:
;I)
( z2
tl)
(:;;
_:;2
Id2)Jbcx~~qx)zph’(L1,x)lh’h(L2,x)
t
(6)
295
a
18 October 1984
PHYSICS LETTERS
Volume 146B, number 5
(e,e')160 (L-J8 %MeV)
lO+
;fij&
b (e.e’) ollsody
P$-y5
E
I 1
2
3
L
12
3 q
[fm?
4
1
2
3
4
12
3
4
1234
1234
q Wm.‘1
Fig. 1. Absolute values of the transverse form factor, IF!$)(q)l, versus momentum transfer q in the cases 4-, 160 (a) and 14; “‘Pb (b). In each graph the one- and the twc+body contributions considered in this work with the harmonic oscillator (dashed lines) and Woods-Saxon (solid lines) mean fields are shown.
for the pion-in-flight part. Here li = (2a + 1) ‘i2 and g(a) = 1,0 according to a being even or odd respectively. The index h’ runs over all the hole states in the nucleus, and v, rr are the conventional notation for neutron and proton. Besides we have used
where K, = (Za - i,)(2i, + 1) and the R-symbols denote radial single-particle wavefunctions. It is remarkable that eqs. (5) and (6) seem to be the simplest way to describe the physical process under consideration, partially due to the fact that the angular momentum couplings appear in the easiest way and are reduced at maximum. Indeed, in eq. (5) there are three couplings: one is associated to the particle-hole nature of the nuclear wavefunctions and the other unavoidable two correspond to the scattering process of the virtual pion inherent to the seagull operator (3). Eq. (6) is slightly more intricate since the pionic operator (4) involves an additional angular momentum coupling which is coming from the interaction of the virtual photon of the electronnucleus process directly with the pion exchanged by the two nucleons; this is shown in the 6j-symbol and that is why one cannot further reduce this expression. In the actual calculations the inelastic form factor given by eq. (1) is improved to take into account the centerof-mass [ 161 and the nucleon finite-size [ 171 corrections. In fig. 1 the influence of the nuclear mean potential on the transverse form factors l@‘(q) I is analyzed. From left to right the one-body, the seagull and the pionic form factors of 160(4-) and 208Pb(14-) with both harmonic oscillator (dashed lines) and Woods-Saxon (solid lines) single-nucleon wavefunctions are shown. A partial similar analysis was done by Dubach et al. [ 181 in 6Li(l+) and this figure confirms and extends their statements about this point. One can make the following important remarks: (a) The existence of the first scattering minimum is obtained by using a mean potential more realistic than the harmonic oscillator model. Then, a nucleon potential of a Woods-Saxon type has to be used in any analysis of the MEC effects. (bj The exchange-current form factors show tha same qualitative aspect as the one-body form factor in the Woods-Saxon model although, as is well known, they are considerably smaller. The pionic value is everywhere much less than the seagull one, what is clearer at higher momentum transfer. (c) The exchange-current form factors have in the two potential models qualitatively similar but quantitatively very different values, especially at momentum transfer higher than -4 fm-l in I60 and -3 fm-’ in 208Pb. Fig. 2 shows the comparison between the, to the best of our information, most recent experimental data and the total squared form factor (FT.(~) I2 calculated in the Woods-Saxon model with solid lines) and without (dashed lines) including the MEC effects. The data of 160(4-; 18.98 MeV; I I d5/2p$$,) and 296
Volume
146B, number
5
PHYSICS
18 October
LETTERS
- b to+:
(e,e'l"'Pb
1984
(14~;6.74MeV)
serf[fm-‘I Fig. 2. Transverse form factor squared, lFf)(q)12, (a) and 14-, 2osPb (b). In addition the most recent
ith (solid lines) and without (dashed lines) MEC effects available experimental data [7,8] are shown.
for the cases 4-, 160
*08Pb(14-; 6.74 MeV; Ijl5/2iijl/$,) are from Bertozzi [7] and Lichtenstadt et al. [8], respectively. With respect to the theoretical calculations one observes that the pion exchange currents produce in the form factors of both states an enhancement of more than 15% at the first scattering maximum and even bigger at higher momentum transfer. The increase at the second maximum is 75% in the 4- case and 52% in the 14-, thus producing a value of the total form factor within the reach of the available electron accelerators. An overall qualitative agreement of our calculations with the available experimental data in the states under consideration is found. The inclusion of the core polarization [ 10,11,19] or the 2p-2h components of the nuclear wavefunctions [20] cause our theoretical results to be closer to the experimental data, so we make the agreement almost quantitatively around the first maximum and we slightly vary the previous statements made in the region of the second scattering maximum. Calculations of this type have been done and will be published in a more extensive paper [ 141. We are aware that for a proper comparison with experiment we should take into account the contributions of short-range correlations and the p-meson exchange together with the A-isobar currents. However the role of the short-range correlations is not at all clear in spite of the efforts of Mathiot and Desplanques [21], and besides there are arguments to believe the existence of an almost exact cancellation between the last two contributions mentioned [ 11,221. In summary, we have found the simplest analytical and compact expressions for the MEC contributions to the inelastic electron scattering form factor. Application to stretched high spin states in light and heavy doubly closed nuclei produces an overall agreement with the available experimental data and indicates a possible way to detect the MEC effects in nuclei with A S 4. References [l] M. Bernheim et al., Phys. Rev. Lett 46 (1982) 402. [2] J.M. Cavedon et al., Phys. Rev. Lett. 49 (1982) 986; P.C. Dunn et al., Phys Rev. C27 (1983) 71. [3] D.O. Riska and GE. Brown, Phys. Lett. 38B (1972) 193. [4] C. Hadjimichael et al., Phys Lett. 39B (1972) 59.
297
Volume 146B, number 5 [5] [6] [7] [8] [9] [ 101 [ll] 121
PHYSICS LETTERS
J. Hockert et al., Nucl. Phys A217 (1973) 14. J. Dubach and W.C. Haxton, Phys Rev. Lett 41 (1978) 1453. W. Bertozzi, Nucl. Phys. A374 (1982) 109~. J. Lichtenstadt et al., Phys. Rev. C20 (1979) 497. J. Dubach, Nucl. Phys. A340 (1980) 271. T. Suzuki et al., Phys. Rev. C26 (1982) 750. T. Suzuki and H. Hyuga, Nucl. Phys. A402 (1983) 491. J.L.
[ 141 J.S. Dehesa, S. Krewald, A.M. Lallena and T.W. Donnelly, preprint (1984), to be published. [ 151 A.M. Lallena, Ph. D. thesis, Universidad de Granada (1984). [16] L.J. Tassie and F.C. Barker, Phys. Rev. 111 (1958) 940. Janssens et al., Rev. 142 922. Dubach, J.H. Koch and T.W. Donnelly, Nucl. Phys A271 (1976) 279. [19] I. Hamamoto et al., Phys. Lett. 93B (1980) 213. [20] S. Krewald and J. Speth, Phys. Rev. Lett. 45 (1980) 417. [21] J.F. Mathiot and B. Desplanques, Phys Lett. 101B (1981) 141. [22] B. Sommer, Nucl. Phys. A308 (1978) 263.
[ 171 [ 181
298
18 October 1984
Volume 146B, number 5
18 October 1984
MESON EXCHANGE CURRENT EFFECTS IN THE ELECTROEXCITATION
OF MAGNETIC STATES IN CLOSED SHELL NUCLEI
A.M. LALLENA, J.S. DEHESA Departamento
de Fisica Nuclear, Universidad de Granada, Granada, Spain
and S. KREWALD Institut ftir Kerphysik,
Kernforschungsanlage
Jiilich, D-51 70 Jiilich, West Germany
Received 12 June 1984
A microscopic approach to evaluate the contributions of one-pion exchange currents to the inelastic (e, e’) scattering form factor of closed shell nuclei is developed. Both seagull and pionic parts of the two-body term are treated on the same footing and their final expressions are simple, clear and physically transparent. Application to the magnetic stretched states 4-(18.98 MeV) of 160 and 14-(6.74 MeV) of 208Pb is done. Agreement with the available experimental data is discussed. It is found that the effect of the meson exchange currents is a smooth but non-negligible enhancement (215% at the first scattering maximum) at small momentum transfers and an increase by a factor bigger than 1.5 in the region of the second maximum.
A systematic theoretical investigation of meson exchange currents (MEC) effects on the inelastic electron scattering form factors of heavy nuclei (i.e. A 2 4) remains to be accomplished. However, except for two- and threebody systems [l-5], these effects are often difficult to disentangle from those associated with nuclear structure. The MEC effects cannot be ignored [6] if one wants to extract the full information of the accurate high-resolution data of the new generation of spectrometers and electron accelerators which are close to operate with a 100% duty cycle and beam intensities of 100 PA. Therefore reliable calculations of the electron-nucleus form factors with the inclusion of MEC’s are mandatory even in heavy nuclei. The purpose of the present letter is to develop a microscopic method to evaluate the one-pion exchange current contributions to the electroexcitation form factors of magnetic states in closed shell nuclei. Furthermore the applicability of the method is illustrated by calculating the form factors of the magnetic high-spin states 4-( 18.98 MeV) in 160 and 14- (6.74 MeV) in 208Pb, of which recent accurate data [7,8] are available. This method has three important advantages. Firstly: one can handle realistic mean fields such as Hartree-Fock or Woods-Saxon. Secondly: the seagull and pion-in-flight parts of the two-body contributions are treated on the same footing together with the one-body term, i.e. all of them are calculated by using the same realistic single-particle wavefunctions. In particular it is remarkable that contrary to other approaches [9-l 1] one can analytically evaluate within our method the pion-in-flight current term in any closed shell nucleus and for any multipolarity, e.g. we do not use any “pair approximation” to estimate it. Thirdly: simple, clear and physically transparent expressions, which resemble much to the one-body term, are found for the two-body contributions. The electromagnetic nuclear current operator J(r) is given in this work by the sum of the well-known one-body J’OB)(r), seagull J
294
= flB)k)
+ $G)(c!)
+ 8)(q)
9
(1) 0370-2693/84/$03.00 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
PHYSICS LETTERS
Volume 1468, number 5
18 October 1984
where i+‘(q)
= (fi/Z)(Jll
fik)(q)IIO)
for k = OB, SC, rr, and i’fk’(q) denotes the corresponding
a(q)
transverse magnetic multipole
operator, i.e.
Y%(a) * J@)(r) .
=Jd3rj&r)
Besides the nuclear wavefunction ].I) will be described within the RPA theory, so it will be characterized by the particle-hole amplitudes XJ(ph) and YJ(ph). Apart from the one-body term, which is already known a long time [ 12,131, the basic idea in our derivation of the two-body terms is to keep explicitly the dependence on the nucleon coordinates rl and rq. Given the nucleon wavefunction at the vertex ri, a virtual pion wavefunction may be created with the pion creation operator $(r -rj)
=ej .Vls3
un(q) exp[iq . (r -I$]
(2)
,
where u,(q) = 1/(q2 + rn$ denotes the bare pion propagator. tor can be written as
With this in mind, the pion exchange current opera-
dSG)(rl,r2;r)
-u$(r
= 47r(_&rrz~)(z1 XT&
{a16(r
-Q&r
-r2)
-rz)J(r
-rl)}
(3)
- rl)]]
(4)
for the seagull or contact part, and as
r2;4 = -47$&&~1
J(?rl,
Xt21z i&r
- r2)l - d
- rl)[VJ(r
for the pion-in-flight part. Here the nNN coupling constant fz = 0.079 and m, is the pion mass. After a long and appropriate use of angular momentum algebra [ 14,151 one can find simple and compact formulas for the two-body terms of the nuclear form factor which allow a direct and simple physical interpretation. We have found tiG’(q)
= -2i$
X
_:p/2 _s2
(
2
x
g
[XJ*(ph) + (-)J’l
s) ,&%@ph,.6h’,r
r$(/,+lh’+Ll+
l)if
[ +Eg([h’+lh+L2+
Ip
(’ l/2 ‘r :;2
l)ig
Y;(ph)](-)‘p4-(1,
+ lh +J t l)f&,
- 6ph,n6h’,v) jh’
Ll
-l/2 . _z2
2
0 11
d=2jdclx)Iph’(L~J)
x2 i&X)
;’
Rh’Rh
Ih’h(L2, X) Rp Rh’
1
(5)
2
for the seagull part, and @j(q)
= 4i g m
%$[X;(ph) + (-)J’l
YAph)l
E(lp + lh + J+
1) fpfh
nP X
c
(-l)ih’+ihj=?(6ph,vShf ,-6
ph,n6h’,u)~~~E(lp+lhl+Llf
2
_:;;
l)t;(lh’+lh+L2+
1)ifi;[L2(L2+1)11’2
h’
x (;
:h)
(f’
_:
;I)
( z2
tl)
(:;;
_:;2
Id2)Jbcx~~qx)zph’(L1,x)lh’h(L2,x)
t
(6)
295
a
18 October 1984
PHYSICS LETTERS
Volume 146B, number 5
(e,e')160 (L-J8 %MeV)
lO+
;fij&
b (e.e’) ollsody
P$-y5
E
I 1
2
3
L
12
3 q
[fm?
4
1
2
3
4
12
3
4
1234
1234
q Wm.‘1
Fig. 1. Absolute values of the transverse form factor, IF!$)(q)l, versus momentum transfer q in the cases 4-, 160 (a) and 14; “‘Pb (b). In each graph the one- and the twc+body contributions considered in this work with the harmonic oscillator (dashed lines) and Woods-Saxon (solid lines) mean fields are shown.
for the pion-in-flight part. Here li = (2a + 1) ‘i2 and g(a) = 1,0 according to a being even or odd respectively. The index h’ runs over all the hole states in the nucleus, and v, rr are the conventional notation for neutron and proton. Besides we have used
where K, = (Za - i,)(2i, + 1) and the R-symbols denote radial single-particle wavefunctions. It is remarkable that eqs. (5) and (6) seem to be the simplest way to describe the physical process under consideration, partially due to the fact that the angular momentum couplings appear in the easiest way and are reduced at maximum. Indeed, in eq. (5) there are three couplings: one is associated to the particle-hole nature of the nuclear wavefunctions and the other unavoidable two correspond to the scattering process of the virtual pion inherent to the seagull operator (3). Eq. (6) is slightly more intricate since the pionic operator (4) involves an additional angular momentum coupling which is coming from the interaction of the virtual photon of the electronnucleus process directly with the pion exchanged by the two nucleons; this is shown in the 6j-symbol and that is why one cannot further reduce this expression. In the actual calculations the inelastic form factor given by eq. (1) is improved to take into account the centerof-mass [ 161 and the nucleon finite-size [ 171 corrections. In fig. 1 the influence of the nuclear mean potential on the transverse form factors l@‘(q) I is analyzed. From left to right the one-body, the seagull and the pionic form factors of 160(4-) and 208Pb(14-) with both harmonic oscillator (dashed lines) and Woods-Saxon (solid lines) single-nucleon wavefunctions are shown. A partial similar analysis was done by Dubach et al. [ 181 in 6Li(l+) and this figure confirms and extends their statements about this point. One can make the following important remarks: (a) The existence of the first scattering minimum is obtained by using a mean potential more realistic than the harmonic oscillator model. Then, a nucleon potential of a Woods-Saxon type has to be used in any analysis of the MEC effects. (bj The exchange-current form factors show tha same qualitative aspect as the one-body form factor in the Woods-Saxon model although, as is well known, they are considerably smaller. The pionic value is everywhere much less than the seagull one, what is clearer at higher momentum transfer. (c) The exchange-current form factors have in the two potential models qualitatively similar but quantitatively very different values, especially at momentum transfer higher than -4 fm-l in I60 and -3 fm-’ in 208Pb. Fig. 2 shows the comparison between the, to the best of our information, most recent experimental data and the total squared form factor (FT.(~) I2 calculated in the Woods-Saxon model with solid lines) and without (dashed lines) including the MEC effects. The data of 160(4-; 18.98 MeV; I I d5/2p$$,) and 296
Volume
146B, number
5
PHYSICS
18 October
LETTERS
- b to+:
(e,e'l"'Pb
1984
(14~;6.74MeV)
serf[fm-‘I Fig. 2. Transverse form factor squared, lFf)(q)12, (a) and 14-, 2osPb (b). In addition the most recent
ith (solid lines) and without (dashed lines) MEC effects available experimental data [7,8] are shown.
for the cases 4-, 160
*08Pb(14-; 6.74 MeV; Ijl5/2iijl/$,) are from Bertozzi [7] and Lichtenstadt et al. [8], respectively. With respect to the theoretical calculations one observes that the pion exchange currents produce in the form factors of both states an enhancement of more than 15% at the first scattering maximum and even bigger at higher momentum transfer. The increase at the second maximum is 75% in the 4- case and 52% in the 14-, thus producing a value of the total form factor within the reach of the available electron accelerators. An overall qualitative agreement of our calculations with the available experimental data in the states under consideration is found. The inclusion of the core polarization [ 10,11,19] or the 2p-2h components of the nuclear wavefunctions [20] cause our theoretical results to be closer to the experimental data, so we make the agreement almost quantitatively around the first maximum and we slightly vary the previous statements made in the region of the second scattering maximum. Calculations of this type have been done and will be published in a more extensive paper [ 141. We are aware that for a proper comparison with experiment we should take into account the contributions of short-range correlations and the p-meson exchange together with the A-isobar currents. However the role of the short-range correlations is not at all clear in spite of the efforts of Mathiot and Desplanques [21], and besides there are arguments to believe the existence of an almost exact cancellation between the last two contributions mentioned [ 11,221. In summary, we have found the simplest analytical and compact expressions for the MEC contributions to the inelastic electron scattering form factor. Application to stretched high spin states in light and heavy doubly closed nuclei produces an overall agreement with the available experimental data and indicates a possible way to detect the MEC effects in nuclei with A S 4. References [l] M. Bernheim et al., Phys. Rev. Lett 46 (1982) 402. [2] J.M. Cavedon et al., Phys. Rev. Lett. 49 (1982) 986; P.C. Dunn et al., Phys Rev. C27 (1983) 71. [3] D.O. Riska and GE. Brown, Phys. Lett. 38B (1972) 193. [4] C. Hadjimichael et al., Phys Lett. 39B (1972) 59.
297
Volume 146B, number 5 [5] [6] [7] [8] [9] [ 101 [ll] 121
PHYSICS LETTERS
J. Hockert et al., Nucl. Phys A217 (1973) 14. J. Dubach and W.C. Haxton, Phys Rev. Lett 41 (1978) 1453. W. Bertozzi, Nucl. Phys. A374 (1982) 109~. J. Lichtenstadt et al., Phys. Rev. C20 (1979) 497. J. Dubach, Nucl. Phys. A340 (1980) 271. T. Suzuki et al., Phys. Rev. C26 (1982) 750. T. Suzuki and H. Hyuga, Nucl. Phys. A402 (1983) 491. J.L.
[ 141 J.S. Dehesa, S. Krewald, A.M. Lallena and T.W. Donnelly, preprint (1984), to be published. [ 151 A.M. Lallena, Ph. D. thesis, Universidad de Granada (1984). [16] L.J. Tassie and F.C. Barker, Phys. Rev. 111 (1958) 940. Janssens et al., Rev. 142 922. Dubach, J.H. Koch and T.W. Donnelly, Nucl. Phys A271 (1976) 279. [19] I. Hamamoto et al., Phys. Lett. 93B (1980) 213. [20] S. Krewald and J. Speth, Phys. Rev. Lett. 45 (1980) 417. [21] J.F. Mathiot and B. Desplanques, Phys Lett. 101B (1981) 141. [22] B. Sommer, Nucl. Phys. A308 (1978) 263.
[ 171 [ 181
298
18 October 1984