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Positive parity states in 16O
Volume 29l3, number 5
POSITIVE
PHYSICS
LETTERS
PARITY
STATES
26 May 1969
IN 160
G. KLUGE and P. MANAKOS Institut
ftir Theoretische
Kemphysik
der
Technischen
Hochschule
Darmstadt,
Germany
Received 10 March 1969
The positive parity states in l60 are calculated in the shell model with 2ph and 4ph excitations free from spurious components. The energies, B(E2) values O-Otransition matrix elements agree fairly well with experiments.
In the past few years sucessful shell model calculations [l, 21 of the low lying positive parity levels of 160 have been made, essentially based on the picture [3] that these states are predominantly composed of 4ph and 2ph excitations from the lp into the 2sld shell. In the following, we report a calculation which we have performed using the supermultiplet and SU(3) classification in the harmonic oscillator shell model. We assumed for the particles in the sd-shell, as well as for the holes in the p-shell, maximum spatial symmetry. The SU(3) multiple& arising from these spatial symmetry classes, in Elliott’s (h, p) notation, are the following :
mine the spin-isospin wave function as (ST) = = (00), SU(4)-representation = [0], and thus the spatial symmetry of the 12 particles in the p and sd shell as [444]. In the 2ph case, we make the additional restriction (ST) = (00), thus allowing the spin-isospin couplings (01) X (01) (00) and (10) X (10) (00) which will be referred to as 13 and 31, respectively. The possible SU(4) representations for 15
-
0’
NO4 [41 + (80), (42), (04), (20)
-
c*
(sd)2 [2] + (40), (02)
-
0’
P8
-
2’ Y+
-
4’
-
2+
c441 -+ (04)
p10[442] - (02) The SU(3) multiplets for the 4ph and 2ph states are listed in table 1, together with the values of the Casimir operator, which give a rough measure of the intrinsic deformation. We retained the representations (84), (46), (08) and (42), (04) for the 4ph and 2ph case, respectively. They have maximum value of the Casimir operator if one disregards representations with odd X or ~1which contains no L = 0 state. Furthermore, the selected states are the only ones free from spurious center of mass components. For the 4ph states, the assumed spatial symmetry in the p and sd shell is sufficient to deter*This work made use of the computingfacilities of Deutsches Rechenzentrum, Darmstadt, supportedby Deutsche Forschungsgemeinsch.
10
2+
> r”
-
0’
=
5:
-
2,
=
$
-9 -
+ 2+
-
2,
o+
-
2.
-
0,
-
Exp.
0’
0. Theor.
5
0
Fig. 1. The positive parity spectrum of oxygen below 14 MeV.
277
Volume
29B, number
PHYSICS
5
A list of the irreducible
LETTERS
26 May 1969
Table 1 SU(3) representations occuring in the 4ph and 2ph spaces. tor are $@2+l.r2+3(X+p)+Xp).
The values of the Casimir
opera-
4ph states Representation
(84)
Multiplicity Casimir
(73)
(46)
(54)
(62)
(35)
11112
operator
Xi
148
(08)
(43)
(16)
(51)
(24)
11112
109
106
88
76
73
88
58
64
49
(32)
(40)
(13)
3
2
3
2
46
34
28
25
(21)
(02)
12 16
10
2ph states Representation Multiplicity Casimir
operator
Xf
the 2ph spin-isospin states are [0] and [22], corresponding to the spatial symmetry classes [444] and [4422]. We used an oscillator length parameter b = = 1.62 fm and the corresponding level distance &I = 16 MeV; therefore the unperturbed energies of the 2ph and 4ph states are 32 and 64 MeV, respectively. The two body interaction was taken to be Yukawa form with Rosenfeld exchange mixture (force range a = 1.37 fm, strength V, = = 49 MeV). Hartree-Fock contributions with respect to the closed p-shell have been excluded, since the level distance of the model corresponds to the average phenomenological one of 016. The calculated and experimental spectrum are shown in fig. 1; some of the calculated state vectors are given in table 2. The B(E2) values given in table 3 were cal-
Calculated
energies
and state vectors.
(42)
(31)
(04)
(12)
1
1
1
1
(20) 2
46
25
28
16
10
culated with effective charges 1.5 for protons and 0.5 for neutrons. The largest calculated B(E2) (from the first 2+ to the first excited O+ state) is essentially due to the components of those states belonging to the SU(3) representation (84). The second 2+ state also has a large component belonging to the same SU(3) representation, which is, however, nearly orthogonal to the previous one and gives a very small contribution to the B(E2). The importance of the (84) representation has been already emphasized by Brown and Green [3]. The matrix elements for the O-O transitions calculated (no effective charges) are by a factor 4 too small compared with the experimental values. Their relative magnitudes, however, which are not so sensitively dependent on the radial wave functions are in rough agreement with experiment (table 4).
Table 2 In case of multiple occurence of a L-value trary orthogonal base was chosen.
in a SU(3) representation,
L 0
-
2
4
Excitation Energ 0 6.12 11.28 13.85
‘OOph 7
312ph
132ph
(00) (04) 0.908 0.003 -0.175 0.072 0.199 -0.190 0.306 0.290
(42) 0.240 0.085 -0.231 -0.508
7.65 8.82 12.19
I II 0.060 -0.149 0.073 -0.061 -0.038 -0.122 -0.096 0.451 -0.207
9.21 11.16
I 0.008 -0.160 -0.055 -0.024
-
278
II -0.071 0.195
“4ph
I (08)
-0.0391 -0.333 0.418 -0.405
1
(46) -0.090 -0.422 0.461 -0.026
I
(84) -0.060 -0.803 -0.576 0.047
an arbi-
Volume
29B, number
Calculated
5
PHYSICS
Table 3 and experimental
z+(s.92)
+
0+(6.05)
2_(6.92)
+
g.s.
2+(9.84)
B(E2) values **
LETTERS
Calculated
Th,
Exp
83.92
99.73 * 18.8
26 May 1969 Table 4 and experimental O-O-transition matrix ments in fm2 ** (il r2 If)
Transition 2.23
7.25 f
-+ 0+(6.05)
0.121
2.88
2+(9.84)
+
0.043
0.08
2’.(11.52)
--) 0+(6.05)
2.97
6.41 *
1.35
2$(11.52)
+
3.05
3.34 f
0.4
4+(10.36) + **Transition oscillator
g.s.
g. s. 2+(6.92)
40.86
matrix elements length parameter
117
Tk
Exp.
g. 6.
0.80
3.8
s.
0.63
4.4
MeV - g. s.
0.82
?
0.72 6.05 MeV 12.05 MeV -g. 14.0
ele-
**Transition oscillator
matrix elements length parameter
were evalueted with an b = 1.73 fm.
10
were evaluated with an b = 1.73 fm.
References
There is recent experimental evidence [4] for the existence of a O+ state at 14 MeV having a ground state transition matrix element of the same order of magnitude as the first two excited O+ states, which is in agreement with our calculation.
1. L. S. Celenza, R. M. Dreizler, A. Klein and G. J. Dreiss, Phys. Letters 23 (1966) 241. 2. A. P. Zuker, B. Buck and J. B.McGrory, Phys. Letters 21 (1968) 39. 3. G. E. Brown and A.M. Green, Nuclear Phys. 75 (1966) 401. 4. M. Stroetzel and A. Goldmann, to be published. *****
279
POSITIVE
PHYSICS
LETTERS
PARITY
STATES
26 May 1969
IN 160
G. KLUGE and P. MANAKOS Institut
ftir Theoretische
Kemphysik
der
Technischen
Hochschule
Darmstadt,
Germany
Received 10 March 1969
The positive parity states in l60 are calculated in the shell model with 2ph and 4ph excitations free from spurious components. The energies, B(E2) values O-Otransition matrix elements agree fairly well with experiments.
In the past few years sucessful shell model calculations [l, 21 of the low lying positive parity levels of 160 have been made, essentially based on the picture [3] that these states are predominantly composed of 4ph and 2ph excitations from the lp into the 2sld shell. In the following, we report a calculation which we have performed using the supermultiplet and SU(3) classification in the harmonic oscillator shell model. We assumed for the particles in the sd-shell, as well as for the holes in the p-shell, maximum spatial symmetry. The SU(3) multiple& arising from these spatial symmetry classes, in Elliott’s (h, p) notation, are the following :
mine the spin-isospin wave function as (ST) = = (00), SU(4)-representation = [0], and thus the spatial symmetry of the 12 particles in the p and sd shell as [444]. In the 2ph case, we make the additional restriction (ST) = (00), thus allowing the spin-isospin couplings (01) X (01) (00) and (10) X (10) (00) which will be referred to as 13 and 31, respectively. The possible SU(4) representations for 15
-
0’
NO4 [41 + (80), (42), (04), (20)
-
c*
(sd)2 [2] + (40), (02)
-
0’
P8
-
2’ Y+
-
4’
-
2+
c441 -+ (04)
p10[442] - (02) The SU(3) multiplets for the 4ph and 2ph states are listed in table 1, together with the values of the Casimir operator, which give a rough measure of the intrinsic deformation. We retained the representations (84), (46), (08) and (42), (04) for the 4ph and 2ph case, respectively. They have maximum value of the Casimir operator if one disregards representations with odd X or ~1which contains no L = 0 state. Furthermore, the selected states are the only ones free from spurious center of mass components. For the 4ph states, the assumed spatial symmetry in the p and sd shell is sufficient to deter*This work made use of the computingfacilities of Deutsches Rechenzentrum, Darmstadt, supportedby Deutsche Forschungsgemeinsch.
10
2+
> r”
-
0’
=
5:
-
2,
=
$
-9 -
+ 2+
-
2,
o+
-
2.
-
0,
-
Exp.
0’
0. Theor.
5
0
Fig. 1. The positive parity spectrum of oxygen below 14 MeV.
277
Volume
29B, number
PHYSICS
5
A list of the irreducible
LETTERS
26 May 1969
Table 1 SU(3) representations occuring in the 4ph and 2ph spaces. tor are $@2+l.r2+3(X+p)+Xp).
The values of the Casimir
opera-
4ph states Representation
(84)
Multiplicity Casimir
(73)
(46)
(54)
(62)
(35)
11112
operator
Xi
148
(08)
(43)
(16)
(51)
(24)
11112
109
106
88
76
73
88
58
64
49
(32)
(40)
(13)
3
2
3
2
46
34
28
25
(21)
(02)
12 16
10
2ph states Representation Multiplicity Casimir
operator
Xf
the 2ph spin-isospin states are [0] and [22], corresponding to the spatial symmetry classes [444] and [4422]. We used an oscillator length parameter b = = 1.62 fm and the corresponding level distance &I = 16 MeV; therefore the unperturbed energies of the 2ph and 4ph states are 32 and 64 MeV, respectively. The two body interaction was taken to be Yukawa form with Rosenfeld exchange mixture (force range a = 1.37 fm, strength V, = = 49 MeV). Hartree-Fock contributions with respect to the closed p-shell have been excluded, since the level distance of the model corresponds to the average phenomenological one of 016. The calculated and experimental spectrum are shown in fig. 1; some of the calculated state vectors are given in table 2. The B(E2) values given in table 3 were cal-
Calculated
energies
and state vectors.
(42)
(31)
(04)
(12)
1
1
1
1
(20) 2
46
25
28
16
10
culated with effective charges 1.5 for protons and 0.5 for neutrons. The largest calculated B(E2) (from the first 2+ to the first excited O+ state) is essentially due to the components of those states belonging to the SU(3) representation (84). The second 2+ state also has a large component belonging to the same SU(3) representation, which is, however, nearly orthogonal to the previous one and gives a very small contribution to the B(E2). The importance of the (84) representation has been already emphasized by Brown and Green [3]. The matrix elements for the O-O transitions calculated (no effective charges) are by a factor 4 too small compared with the experimental values. Their relative magnitudes, however, which are not so sensitively dependent on the radial wave functions are in rough agreement with experiment (table 4).
Table 2 In case of multiple occurence of a L-value trary orthogonal base was chosen.
in a SU(3) representation,
L 0
-
2
4
Excitation Energ 0 6.12 11.28 13.85
‘OOph 7
312ph
132ph
(00) (04) 0.908 0.003 -0.175 0.072 0.199 -0.190 0.306 0.290
(42) 0.240 0.085 -0.231 -0.508
7.65 8.82 12.19
I II 0.060 -0.149 0.073 -0.061 -0.038 -0.122 -0.096 0.451 -0.207
9.21 11.16
I 0.008 -0.160 -0.055 -0.024
-
278
II -0.071 0.195
“4ph
I (08)
-0.0391 -0.333 0.418 -0.405
1
(46) -0.090 -0.422 0.461 -0.026
I
(84) -0.060 -0.803 -0.576 0.047
an arbi-
Volume
29B, number
Calculated
5
PHYSICS
Table 3 and experimental
z+(s.92)
+
0+(6.05)
2_(6.92)
+
g.s.
2+(9.84)
B(E2) values **
LETTERS
Calculated
Th,
Exp
83.92
99.73 * 18.8
26 May 1969 Table 4 and experimental O-O-transition matrix ments in fm2 ** (il r2 If)
Transition 2.23
7.25 f
-+ 0+(6.05)
0.121
2.88
2+(9.84)
+
0.043
0.08
2’.(11.52)
--) 0+(6.05)
2.97
6.41 *
1.35
2$(11.52)
+
3.05
3.34 f
0.4
4+(10.36) + **Transition oscillator
g.s.
g. s. 2+(6.92)
40.86
matrix elements length parameter
117
Tk
Exp.
g. 6.
0.80
3.8
s.
0.63
4.4
MeV - g. s.
0.82
?
0.72 6.05 MeV 12.05 MeV -g. 14.0
ele-
**Transition oscillator
matrix elements length parameter
were evalueted with an b = 1.73 fm.
10
were evaluated with an b = 1.73 fm.
References
There is recent experimental evidence [4] for the existence of a O+ state at 14 MeV having a ground state transition matrix element of the same order of magnitude as the first two excited O+ states, which is in agreement with our calculation.
1. L. S. Celenza, R. M. Dreizler, A. Klein and G. J. Dreiss, Phys. Letters 23 (1966) 241. 2. A. P. Zuker, B. Buck and J. B.McGrory, Phys. Letters 21 (1968) 39. 3. G. E. Brown and A.M. Green, Nuclear Phys. 75 (1966) 401. 4. M. Stroetzel and A. Goldmann, to be published. *****
279