- Email: [email protected]
Magnetic dipole transitions near 208Pb
Volume
79B, number
1, 2
PHYSICS
MAGNETIC DlPOLE TRANSITIONS
LETTERS
6 November
1978
NEAR 208Pb a
Q. HO-KIM and N. QUANG-HOC D&artement
de Physique, C’niversiti Lavaf, QuPbec, Canada GIK 7P4
Received 4 May 1978 Revised manuscript received
8 August
1978
The amplitudes for the magnetic dipole transitions between low-lying states of simple nuclei around 208Pb are calculated with a large model space taking into account the core polarization and the meson exchange currents. The results are analyzed and compared with experiment.
It is well-known that the deviations of the observed magnetic moments and magnetic dipole transition rates from the shell model predictions arise in large parts from single particle excitations and collective core excitations, often referred to as the core polarization or configuration mixing effects [ 11. In many cases, however, these effects prove to be insufficient to explain the data. A case in point is that of the magnetic moment of 209Bi to which the core polarization contributes no more than half of what is needed. For some time the meson exchange current between interacting nucleons [2], and the excitations of nucleons into isobar states [3,4] have been suggested to play an important role in nuclear magnetic properties. The situation in the lead region is particularly interesting because magnetic properties of many neighbouring nuclei are known, making systematic analyses possible. In this report we focus our attention on some Ml transitions between low-lying states of single-particle or single-hole nuclei in the vicinity of 208Pb. These transitions have already been discussed in the past [5-81. In particular Arima and Huang-Lin [6] evaluated both core polarization effects (in the first order) and mesonic effects arising from the pionic current (in the Thomas-Fermi approximation). Since then more transitions have been observed. Our purpose is to obtain theoretical predictions of the transition rates including * Work supported in part by the National Research Council Canada, and the Minis&e de 1’Education du Quebec.
contributions from the core polarization, the meson exchange currents, and the isobar admixture to the same approximation. Our calculation was performed in the frarnework of the conventional shell-model under the assumption that 2osPb formed a good, spherical, closed-shell core. With two exceptions to be noted below, our model space included the neutron orbits 3s,,,, 2d5/2,3/2, 18712, lhI,/2, the proton orbits 4s1/2,3d,/, 3/2,287/2, lill,,, lj15j2, and the neutron-proton orbits 3p3/2,1/2, 3f7/2,5/2,
lh9/2
and li13/2.
The transition amplitudes induced by the one-body operator or the two-body exchange operators are expanded in powers of the residual nucleon-nucleon interaction. For the transition associated with the usual magnetic dipole operator Msp =g++g,s
t
(1)
there exist two first order diagrams corresponding to the excitations to the proton configuration [(hcII,2hg,2)1+] and the neutron configuration [(ici,2 i1112)1+]. They are the first terms of the well-known TDA series; if all backward going graphs are also included we have the RPA series (fig. 1a). The two-body exchange operator M,, includes translationally invariant and non-invariant terms, and can be expressed as follows [2,9,10]
of
19
Volume
79B, number
1, 2
PHYSICS
LETTERS
6 November
1978
[4.45 - 8.65 #,(x)] @i(x) ,
- T m
[I - 2.3 1 @,(x) + 8.77 @x)] #,(x) ,
F,,, = f2g [I - 0.6984 0 (x) 3 mll
+ (‘j ’ .Ck)zxjk ‘Yjk
LFI + Uj ’ QkFII +
SjkF*IIl)
where x = mnr = m,(ri - rk), yik = im,,(rj t r,+j, mn and mN are the mass of the pion and nucleon respectively. The spin tensor operators ‘Ti;c(e = k, X), and Sik are as defined in [2]. For the radial parts we have used the following expressions. (a) One-pion exchange: pion current plus pair excitation current plus nucleonic current
81 = [ $/$(2x-1)+& 7r
I
@o(x),
(3)
A, = ii = --A7?29+)(.7)> h,, =i11 =-f sy#y(x)
3
where f* = 0.08, n1 = 0.696, q2 = 0.619, $o(x) = eex/x and G2(x) = (1 + 3/x t ~/x*)@~(x). (b) pny and wny dissociation currents orI = 2j_,(mnlm,)3#, hl = iI = <,(m,lm,)3@,” hit =&I = 3.C,(mrr/mw)39y
qI
= 6C,(Qm,)3@~,
,
(4) ,
+
0.288@&)I4,(x> .
(6)
The lowest (zeroth) order contribution of the exchange operators to the transition amplitude represents the interaction of the valence nucleon with the core particle via the exchange operators (fig. lb). For this diagram our model space is enlarged to include all the occupied orbits from Is upwards. In the next order there are six diagrams (fig. lc-e), the first terms of six summable series. We calculated these six series including all the allowed particle-hole configurations of our model space in the intermediate states. The third mechanism we discuss is the conversion of nucleons into isobars. The simplest, and perhaps dominant, configurations are the isobar particle-nucleon hole pairs, with the created isobar occupying the same low spatial state left by the nucleon (fig. 1f). This TDA series was evaluated in an enlarged space to include all the occupied orbits from the 1s orbit upwards. The lowest isobars that can be excited by an MI operator are the NT,, 1,2 (1480) and the A,12 3,2 (1236). Since our calculation shows that the N* (1480) give insignificant effects, we shall limit our discussion to the second possibility. The AN-y magnetic dipole matrix moment can be deduced from photoproduction data yielding (AlpIN) = 4.21 pN [I I]. The transition potentials between the channels i, j are mediated by 77and p mesons, and are given by [12,13]
where m,, and m, are the mass of the p meson and w meson respectively, and cP = -0.69 and cw = -4.07, andwithx,=m,r(v=p,o): “v2 _ n1i ]&i(x) - (mv/m,)2@i(x,)] . (5) mvz (c) Multipyon exchange currents: their contributions are included by using the phenomenological Sachs moment associated with the Hamada-Johnston potential in the two and three pion range. The Sachs moment is given by $5; =
20
+ ‘ii’i,N,‘(T1 . ‘2)~~ I
(7)
where C and Tare spin and isospin transition operators, G and fare coupling constants (G = 680 MeV, fiN = 0.282, f& = 0.592,f;, = 1.78, f& = 3.73). The parameter CYis cy= m fPf.P/(n~~finfin), and the radial functions are givenPb; (x = 71,p; n = 0,2)
Volume
--_ I- ”
79B, number
1, 2
_4 OQ
PHYSICS
LETTERS
6 November
\lo
-_ !D4
-_P p
b
0
d
c
1978
f
e
I:ig. 1. Three mechanisms contributing to the magnetic dipole transitions in single-particle nuclei (dotted line: nucleon-nucleon interaction, wiggly line: electromagnetic interaction, horizontal line: exchange current, vertical line: nucleon, vertical heavy line: isobar: core polarization (a) ; meson current effects (b - e); isobar admixture (f). Time-reversed diagrams are implied.
;I;” W-) ’ n =@ n (mAr>- (A/mAn13@
(8)
with q+,(x) and @Q(X)as defined before, and the damping factor A chosen to be 7 fnr-l. Matrix elements of all operators have been calculated in a harmoiric oscillator basis with hw = 7.6 MeV. The radial integrals of the exchange operators have a lower cut-off at r = 0.4 fm. As a typical residual nucleon-nucleon interaction we have chosen the Gillet potential [ 141. The single particle energies are taken from experiment [ 151. The seven known Ml transitions in *n7T1, *OgBi, *07Pb and *ogPb are listed in the second column of table 1. Their amplitudes in the shell are given in the third column. In the model,~BMIsp, next four columns we show the successively accumulated sums of the contributions from the core polarization, the one-pion exchange currents, the po dissociation and multipion currents, and finally the isobar admixture. They are to be compared with entries in the last column which lists the differences between the measured amplitudes and the single-particle amplitudes, m - 4-i in pN [l&20]. Some features of these results are worth noting. Table 1 Magnetic ____~
207n
2o9Bi 207Pb
‘09Pb
dipole transition amplitude _ ~. ~~-~ _~~
3/2+ -t l/2+ 7/2- * 9/2S/2- * 7/23/2- + l/27/2---f 5/23/2- --* 5/21 l/2’ + 9/2+
a) The core polarization effects turn out to be large, almost sufficient to explain the data, especially in *07T1 and *OgBi [6,8]. This conclusion of course depends on the residual nucleon-nucleon interaction; for example another potential used in our calculation, the Perey potential [2 11, yielded slightly weaker effects. b) The meson exchange currents contribute predominantly in the zeroth order, although it would be a mistake to neglect higher-order terms especially in transitions between low-lying states. Because they are not constrained by the selection rules associated with the one-body operator (1) their effects do not vary dramatically from transition to transition making them relatively important in the I-forbidden transitions. As already noted by Arima et al. [6,10] the mesonic effects are small in the l-effective operator and thus give small contributions to the Al = 0 transitions. c) The corrections resulting from the A-admixture, being submitted to the selection rule Al = 0, are small in the forbidden transitions but large enough in the others to bring them in even closer agreement with the data. In the two cases where the differences between theory and experiment remain substantial, 5’ + f’
(in MN)
aq
RPA
+pion
+pw + nn
+A
expt _
0 0 -1.69 -1.08 -1.22 0 0
0.152 -0.108 0.720 0.348 0.484 0.140 -0.074
0.258 -0.158 0.599 0.375 0.429 0.186 -0.107
0.250 -0.150 0.676 0.415 0.412 0.186 -0.097
0.247 -0.152 0.750 0.459 0.524 0.193 -0.105
0.152 -0.064 0.82 0.44 0.52 0.221 -0.099
~ __ i * + k f + i
0.005 0.001 0.08 0.04 0.08 0.007 0.001
118)
[ 171 (161 [16] [16] 1201 [ 191
21
Volume
79B, number
1, 2
PHYSICS
(207Tl) and f- -+ z2 ( 2u9Bi) the residual nucleonnucleon interaction seems to’have been an important factor. Thus, with the Perey potential we have obtamed 0.184 and -0.112 for the above two transitions, respectively, in closer agreement with the data. We have calculated the effects of core polarization, meson currents and isobar admixture in essentially the same approximation in a large model space. We have limited our investigation to the seven observed Ml transitions in nuclei around 208Pb. It would be instructive to see if the above three mechanisms can systematically explain the other data available in this region. References
Ill A. Arima and H. Horie, Prog. Theor. Phys. 11 (1954) 509. 121 M. Chemtob and M. Rho, Nucl. Phys. Al63 (1971) 1; A212 (1973) 628. I31 L. Kisslinger, Phys. Lett. 29B (1969) 211. 141 H. Arenhovel, M. Danos and H.T. Williams, Nucl. Phys. Al62 (1971) 72. 151 J.D. Vergados, Phys. Lett. 36B (1971) 12. 161 A. Arima and L.J. Huang-Lin, Phys. Lett. 41B (1972) 429.
22
LETTERS [7] F:.C. Khanna
[ 101
[ 111 [ 121 [ 131 [14] [lS]
[16] [17] [18] [19] [20] [Zl]
1978
and 0. Hatisser, Phys. Lett. 45B (1973) 12. F.C. Khanna and 0. Hausser, Nucl. Phys. A277 (1977) 285. G. Konopka, M. Gari and J.G. Zabolitzky, Nucl. Phys. A290 (1977) 360. H. Hyuga and A. Arima, Exchange currents and effective g-factors, (to be published). M. Gourdin and Ph. Salin, Nuovo Cim. 27 (1963) 193. A.M. Green, Rep. Prog. Phys. 39 (1976) 1109. L. Kisslinger, Experimental tests of isobar components of nuclei (to be published). V. Gillet, A.M. Green and E.A. Sanderson, Phys. Lett. 11 (1964) 44. M.J. Martin, Nucl. Data Sheets for A = 2Oy9, BS (1971) 287; M.R. Schmorak and R.L. Auble, Nucl. Data Sheets for A = 207, B5 (1971) 207. 0. Haiisser, F.C. Khanna and D. Ward, Nucl. Phys. Al94 (1973) 113. W. Kratschmer, H.V. Klapdor and E. Grosse, Nucl. Phys. A201 (1973) 179. A.P. Komar, A.A. Vorobev, Yu. Zalite and GA. Korolev, Dokl. (Sov. Phys.) 15 (1970) 244. 0. Haiisser, A. Olin, D. Ward and W. Witthuhn, Phys. Lett. 43B (1973) 247. 0. Hafisser, D.B. Fossan, A. Olin, D. Ward, W. Witthuhn and R.E. Warner, Nucl. Phys. A225 (1974) 425. SM. Perez, Phys. Lett. 33B (1970) 317.
[S] IS. Towner, [9)
6 November
79B, number
1, 2
PHYSICS
MAGNETIC DlPOLE TRANSITIONS
LETTERS
6 November
1978
NEAR 208Pb a
Q. HO-KIM and N. QUANG-HOC D&artement
de Physique, C’niversiti Lavaf, QuPbec, Canada GIK 7P4
Received 4 May 1978 Revised manuscript received
8 August
1978
The amplitudes for the magnetic dipole transitions between low-lying states of simple nuclei around 208Pb are calculated with a large model space taking into account the core polarization and the meson exchange currents. The results are analyzed and compared with experiment.
It is well-known that the deviations of the observed magnetic moments and magnetic dipole transition rates from the shell model predictions arise in large parts from single particle excitations and collective core excitations, often referred to as the core polarization or configuration mixing effects [ 11. In many cases, however, these effects prove to be insufficient to explain the data. A case in point is that of the magnetic moment of 209Bi to which the core polarization contributes no more than half of what is needed. For some time the meson exchange current between interacting nucleons [2], and the excitations of nucleons into isobar states [3,4] have been suggested to play an important role in nuclear magnetic properties. The situation in the lead region is particularly interesting because magnetic properties of many neighbouring nuclei are known, making systematic analyses possible. In this report we focus our attention on some Ml transitions between low-lying states of single-particle or single-hole nuclei in the vicinity of 208Pb. These transitions have already been discussed in the past [5-81. In particular Arima and Huang-Lin [6] evaluated both core polarization effects (in the first order) and mesonic effects arising from the pionic current (in the Thomas-Fermi approximation). Since then more transitions have been observed. Our purpose is to obtain theoretical predictions of the transition rates including * Work supported in part by the National Research Council Canada, and the Minis&e de 1’Education du Quebec.
contributions from the core polarization, the meson exchange currents, and the isobar admixture to the same approximation. Our calculation was performed in the frarnework of the conventional shell-model under the assumption that 2osPb formed a good, spherical, closed-shell core. With two exceptions to be noted below, our model space included the neutron orbits 3s,,,, 2d5/2,3/2, 18712, lhI,/2, the proton orbits 4s1/2,3d,/, 3/2,287/2, lill,,, lj15j2, and the neutron-proton orbits 3p3/2,1/2, 3f7/2,5/2,
lh9/2
and li13/2.
The transition amplitudes induced by the one-body operator or the two-body exchange operators are expanded in powers of the residual nucleon-nucleon interaction. For the transition associated with the usual magnetic dipole operator Msp =g++g,s
t
(1)
there exist two first order diagrams corresponding to the excitations to the proton configuration [(hcII,2hg,2)1+] and the neutron configuration [(ici,2 i1112)1+]. They are the first terms of the well-known TDA series; if all backward going graphs are also included we have the RPA series (fig. 1a). The two-body exchange operator M,, includes translationally invariant and non-invariant terms, and can be expressed as follows [2,9,10]
of
19
Volume
79B, number
1, 2
PHYSICS
LETTERS
6 November
1978
[4.45 - 8.65 #,(x)] @i(x) ,
- T m
[I - 2.3 1 @,(x) + 8.77 @x)] #,(x) ,
F,,, = f2g [I - 0.6984 0 (x) 3 mll
+ (‘j ’ .Ck)zxjk ‘Yjk
LFI + Uj ’ QkFII +
SjkF*IIl)
where x = mnr = m,(ri - rk), yik = im,,(rj t r,+j, mn and mN are the mass of the pion and nucleon respectively. The spin tensor operators ‘Ti;c(e = k, X), and Sik are as defined in [2]. For the radial parts we have used the following expressions. (a) One-pion exchange: pion current plus pair excitation current plus nucleonic current
81 = [ $/$(2x-1)+& 7r
I
@o(x),
(3)
A, = ii = --A7?29+)(.7)> h,, =i11 =-f sy#y(x)
3
where f* = 0.08, n1 = 0.696, q2 = 0.619, $o(x) = eex/x and G2(x) = (1 + 3/x t ~/x*)@~(x). (b) pny and wny dissociation currents orI = 2j_,(mnlm,)3#, hl = iI = <,(m,lm,)3@,” hit =&I = 3.C,(mrr/mw)39y
qI
= 6C,(Qm,)3@~,
,
(4) ,
+
0.288@&)I4,(x> .
(6)
The lowest (zeroth) order contribution of the exchange operators to the transition amplitude represents the interaction of the valence nucleon with the core particle via the exchange operators (fig. lb). For this diagram our model space is enlarged to include all the occupied orbits from Is upwards. In the next order there are six diagrams (fig. lc-e), the first terms of six summable series. We calculated these six series including all the allowed particle-hole configurations of our model space in the intermediate states. The third mechanism we discuss is the conversion of nucleons into isobars. The simplest, and perhaps dominant, configurations are the isobar particle-nucleon hole pairs, with the created isobar occupying the same low spatial state left by the nucleon (fig. 1f). This TDA series was evaluated in an enlarged space to include all the occupied orbits from the 1s orbit upwards. The lowest isobars that can be excited by an MI operator are the NT,, 1,2 (1480) and the A,12 3,2 (1236). Since our calculation shows that the N* (1480) give insignificant effects, we shall limit our discussion to the second possibility. The AN-y magnetic dipole matrix moment can be deduced from photoproduction data yielding (AlpIN) = 4.21 pN [I I]. The transition potentials between the channels i, j are mediated by 77and p mesons, and are given by [12,13]
where m,, and m, are the mass of the p meson and w meson respectively, and cP = -0.69 and cw = -4.07, andwithx,=m,r(v=p,o): “v2 _ n1i ]&i(x) - (mv/m,)2@i(x,)] . (5) mvz (c) Multipyon exchange currents: their contributions are included by using the phenomenological Sachs moment associated with the Hamada-Johnston potential in the two and three pion range. The Sachs moment is given by $5; =
20
+ ‘ii’i,N,‘(T1 . ‘2)~~ I
(7)
where C and Tare spin and isospin transition operators, G and fare coupling constants (G = 680 MeV, fiN = 0.282, f& = 0.592,f;, = 1.78, f& = 3.73). The parameter CYis cy= m fPf.P/(n~~finfin), and the radial functions are givenPb; (x = 71,p; n = 0,2)
Volume
--_ I- ”
79B, number
1, 2
_4 OQ
PHYSICS
LETTERS
6 November
\lo
-_ !D4
-_P p
b
0
d
c
1978
f
e
I:ig. 1. Three mechanisms contributing to the magnetic dipole transitions in single-particle nuclei (dotted line: nucleon-nucleon interaction, wiggly line: electromagnetic interaction, horizontal line: exchange current, vertical line: nucleon, vertical heavy line: isobar: core polarization (a) ; meson current effects (b - e); isobar admixture (f). Time-reversed diagrams are implied.
;I;” W-) ’ n =@ n (mAr>- (A/mAn13@
(8)
with q+,(x) and @Q(X)as defined before, and the damping factor A chosen to be 7 fnr-l. Matrix elements of all operators have been calculated in a harmoiric oscillator basis with hw = 7.6 MeV. The radial integrals of the exchange operators have a lower cut-off at r = 0.4 fm. As a typical residual nucleon-nucleon interaction we have chosen the Gillet potential [ 141. The single particle energies are taken from experiment [ 151. The seven known Ml transitions in *n7T1, *OgBi, *07Pb and *ogPb are listed in the second column of table 1. Their amplitudes in the shell are given in the third column. In the model,~BMIsp, next four columns we show the successively accumulated sums of the contributions from the core polarization, the one-pion exchange currents, the po dissociation and multipion currents, and finally the isobar admixture. They are to be compared with entries in the last column which lists the differences between the measured amplitudes and the single-particle amplitudes, m - 4-i in pN [l&20]. Some features of these results are worth noting. Table 1 Magnetic ____~
207n
2o9Bi 207Pb
‘09Pb
dipole transition amplitude _ ~. ~~-~ _~~
3/2+ -t l/2+ 7/2- * 9/2S/2- * 7/23/2- + l/27/2---f 5/23/2- --* 5/21 l/2’ + 9/2+
a) The core polarization effects turn out to be large, almost sufficient to explain the data, especially in *07T1 and *OgBi [6,8]. This conclusion of course depends on the residual nucleon-nucleon interaction; for example another potential used in our calculation, the Perey potential [2 11, yielded slightly weaker effects. b) The meson exchange currents contribute predominantly in the zeroth order, although it would be a mistake to neglect higher-order terms especially in transitions between low-lying states. Because they are not constrained by the selection rules associated with the one-body operator (1) their effects do not vary dramatically from transition to transition making them relatively important in the I-forbidden transitions. As already noted by Arima et al. [6,10] the mesonic effects are small in the l-effective operator and thus give small contributions to the Al = 0 transitions. c) The corrections resulting from the A-admixture, being submitted to the selection rule Al = 0, are small in the forbidden transitions but large enough in the others to bring them in even closer agreement with the data. In the two cases where the differences between theory and experiment remain substantial, 5’ + f’
(in MN)
aq
RPA
+pion
+pw + nn
+A
expt _
0 0 -1.69 -1.08 -1.22 0 0
0.152 -0.108 0.720 0.348 0.484 0.140 -0.074
0.258 -0.158 0.599 0.375 0.429 0.186 -0.107
0.250 -0.150 0.676 0.415 0.412 0.186 -0.097
0.247 -0.152 0.750 0.459 0.524 0.193 -0.105
0.152 -0.064 0.82 0.44 0.52 0.221 -0.099
~ __ i * + k f + i
0.005 0.001 0.08 0.04 0.08 0.007 0.001
118)
[ 171 (161 [16] [16] 1201 [ 191
21
Volume
79B, number
1, 2
PHYSICS
(207Tl) and f- -+ z2 ( 2u9Bi) the residual nucleonnucleon interaction seems to’have been an important factor. Thus, with the Perey potential we have obtamed 0.184 and -0.112 for the above two transitions, respectively, in closer agreement with the data. We have calculated the effects of core polarization, meson currents and isobar admixture in essentially the same approximation in a large model space. We have limited our investigation to the seven observed Ml transitions in nuclei around 208Pb. It would be instructive to see if the above three mechanisms can systematically explain the other data available in this region. References
Ill A. Arima and H. Horie, Prog. Theor. Phys. 11 (1954) 509. 121 M. Chemtob and M. Rho, Nucl. Phys. Al63 (1971) 1; A212 (1973) 628. I31 L. Kisslinger, Phys. Lett. 29B (1969) 211. 141 H. Arenhovel, M. Danos and H.T. Williams, Nucl. Phys. Al62 (1971) 72. 151 J.D. Vergados, Phys. Lett. 36B (1971) 12. 161 A. Arima and L.J. Huang-Lin, Phys. Lett. 41B (1972) 429.
22
LETTERS [7] F:.C. Khanna
[ 101
[ 111 [ 121 [ 131 [14] [lS]
[16] [17] [18] [19] [20] [Zl]
1978
and 0. Hatisser, Phys. Lett. 45B (1973) 12. F.C. Khanna and 0. Hausser, Nucl. Phys. A277 (1977) 285. G. Konopka, M. Gari and J.G. Zabolitzky, Nucl. Phys. A290 (1977) 360. H. Hyuga and A. Arima, Exchange currents and effective g-factors, (to be published). M. Gourdin and Ph. Salin, Nuovo Cim. 27 (1963) 193. A.M. Green, Rep. Prog. Phys. 39 (1976) 1109. L. Kisslinger, Experimental tests of isobar components of nuclei (to be published). V. Gillet, A.M. Green and E.A. Sanderson, Phys. Lett. 11 (1964) 44. M.J. Martin, Nucl. Data Sheets for A = 2Oy9, BS (1971) 287; M.R. Schmorak and R.L. Auble, Nucl. Data Sheets for A = 207, B5 (1971) 207. 0. Haiisser, F.C. Khanna and D. Ward, Nucl. Phys. Al94 (1973) 113. W. Kratschmer, H.V. Klapdor and E. Grosse, Nucl. Phys. A201 (1973) 179. A.P. Komar, A.A. Vorobev, Yu. Zalite and GA. Korolev, Dokl. (Sov. Phys.) 15 (1970) 244. 0. Haiisser, A. Olin, D. Ward and W. Witthuhn, Phys. Lett. 43B (1973) 247. 0. Hafisser, D.B. Fossan, A. Olin, D. Ward, W. Witthuhn and R.E. Warner, Nucl. Phys. A225 (1974) 425. SM. Perez, Phys. Lett. 33B (1970) 317.
[S] IS. Towner, [9)
6 November