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Magnetic properties of 166Er and the collective model
Volume 57B, number 5
PHYSICS LETTERS
MAGNETIC PROPERTIES
4 August 1975
O F 166Er A N D T H E C O L L E C T I V E M O D E L
K.R. BAKER 1 , J.H. HAMILTON, J. LANGE 2 , A.V. RAMAYYA and L. V A R N E L L 3 Physics Department 4, Vanderbilt University, Nashville, Tennessee 37235, USA V. MARUHN-REZWANI, J.J. PINAJIAN and J.A. MARUHN Oak Ridge National Laboratory s , Oak Ridge, Tennessee 37830, USA Received 18 June 1975 The E2/M1 admixtures for nine ~-ground band transitions in 166Er were measured. Mixing ratios, tifE2/M1), were calculated based on a general treatment of magnetic properties in the framework of the collective model. The agreement with the experimental values is generally good.
If one assumes irrotational flow o f nuclear matter and an equal distribution of neutrons and protons, M1 transitions are forbidden in the collective model [1 ]. Generally quite small M1 admixtures are seen in transitions between K 'r = 0 + and 2 +,/~- and 7-type vibrational and ground-state bands in deformed nuclei [2], but sizable M1 contributions are seen in some cases at higher spins. A decrease in 6(E2/M1) with spin in/~-ground band transitions has been observed with a sharp drop above spin 4, (in 174Hf, 6(69 ~ 6g) ~ 1 [3]). For ")'-bands a decrease in 6 with spin was first observed [4] in 154Gd(6 = 10.1,2. r ~ 2g to 6 = 4.4, 47 ~ 4g). The properties of 166Er have long provided classic tests of the collective model [5]. It is the best o f only two or three examples [2,5] where B(E2) ratios yield a consistent 7-ground band mixing parameter Z 2. Five transitions had to be assumed to be pure E2, however, in the Z 2 calculations in 166Er since M1 admixtures were unknown. Systematic measurements of transitions from 1= 3 - 8 states in 166Er were made to test the spin dependence of 6(7 ~ g) to high spin and to verify the assumed E2 character made in extracting Z 2 values. While successes have been achieved in specific regions, more general theoretical treatments have not yielded detailed agreement with experimental 6-values. No detailed calculations have been done in well deformed nuclei. Recently a general treatment of magnetic properties of even-even nuclei in the framework of the collective model has been proposed [6] as described here. Because 166Er has been a classic test case o f the collective model, and because data are available to higher spin, it offers an important test of this new approach to the magnetic properties of even-even nuclei. We have calculated for 166Er the first detailed predictions of 6-values for a well deformed nucleus. Our results generally compare very favorably with our experimental data. Measurements were made with a chemically purified 15/aCi source in 0.1M HCI of 1200 y 166Ho. Correlations between the 184 keV, 4 + --* 2 +, ground band transition and those in coincidence with it were measured with a NaI-Ge(Li) system [2]. The correlations of transitions in coincidence with the (2 + ~ 0 +) 80 keV and (6 + ~ 4 +) 280 keV 7-rays were measured with a Ge(Li)-Ge(Li) system. A digital gate unit routed spectra from the 80 and 280 keV photopeaks and the Compton backgrounds above them into different 1024 channels of the analyzer. Data were recorded at five angles. Corrections for decentering of the source, measured chance coincidences and the measured Compton backgrounds in the gate and under the photopeaks were made. t s . The data were fitted to W(0) = 1 +A'2P2(cosO ) +A4P4(cosO ) where A K = A K Q K G K. Corrections were made 4for the detector finite solid angle QK and the 1.8 ns lifetime of the 2g state. Hyperfine interactions attenuate i Present address: ERDA, Washington, D.C. 2 Present address: Universitiit Bochum, 463 Bochum Universit/itsstr. 150, Gebaude NT, W. Germany. a Present address: Physics Department, Yale University, New Haven, Conn. 4 Supported in part by "agrant from the National Science Foundation. s Research sponsored by ERDA under contract with Union Carbide Corporation. 441
Volume 57B, number 5
PHYSICS LETTERS
4 August 1975
prescription of Gneuss and Greiner [14]. The strong coupling between ap and a n allows one to use the harmonic oscillator approximation for H(~) with a level spacing of the order of the energy of the giant dipole resonance. The interaction potential V(a, ~) is to lowest order given by V(a, ~) = C[a × ~] [01 + D ( [ a X a] X ~)[01 with C a n d D as yet unknown constants to be determined from experiment. If spin effects are neglected, the magnetic moment operator in this model is simply the orbital angular momentum of the protons, which does not commute with the total Hamiltonian. However, if the interaction V(a, ~) is treated in perturbation theory, the M1 matrix elements may be expressed as matrix elements of an effective operator Mleff=iVr~
(__ff BnBp Bp + C 1, -BnBp . , 5 2_~BT(/,:,I[[a x a]2x rr2]lll~., ) - I (/al[aXTr]llj',,)+lC2~
(1)
2B 2 I
between the unperturbed eigenstates of H(a) alone. Clearly this effective M1 operator contains a deformation dependent g-factor and is not in general parallel to the angular momentum operator. The E2 operator is also modified by the presence of the ~-degree of freedom in an effective E2 operator
, Bn I0 -#,]
10
2
]
., • X a 2] 2 I/~>
(2)
From eqs. (1) and (2), one finds
8-
~
Er
(/illE2effll/f)
10 ~c (/illMleff[I/f) "
8°° IMv •
.,
. . . .
i
. . . . . . . .
i
. . . .
i,.
.t
03
0'I 0
c~
0~
' '
MeV
0.'3 '
'
O:q' '
0)5 '
'
0.'6 B [E2)
O; [XPT. +
6+
__6
5+
2;
EXPT.
5.78
01 -- 22
O. 17
0.21
2[
2;
0.10
0.08
Lt[ -- 6[
2.80
2.27
,
-
5+
4 +
.4 +
3+ 2 +
6+
+
THEORY
5.86
-
*
THEORY _ _ o
¢
'
A o : B
-
_ _ 6
4+
+
-
3+ 2*
166E r
4+
2 +
_
0÷
- -
_
2 +
0 ÷
Fig. 1. The upper part of the figure shows the potential energy surface and potential obtained by comparison with experimental B(E2) values [15] and excitation energies as descried in the text. The fits to the experimental values are shown in the lower part of the figure. 442
Volume 57B, number 5
PHYSICS LETTERS
4 August 1975
Table 1 Error-weighted averaged results of "r-'r directional correlation measurements of transitions from the 3"-band to the ground-state band and comparison with theoretical results. 1.~ I;
5 exp
5 th
2+ ~ 2+
_(16+~ 2) a
-21.3 -29.8
3+ ~ 2+
-(18+_9 )
3+ --. 4 +
_ (9+_531 9)
4 + -~ 4 +
-(10+_427)
--
9.6
- 11.4
+3.0
-(3"3-1.2) 5+ ~ 4 +
+4 -(20_4)
-22.9
_(37+_1°) a 5+ ~ 6+
-(25+_33)
-5.40
6 + --, 6 +
-(20+_9°)
-8.0
7+ ~ 6 ÷
-(22+_~)
*)
7+~8 +
-80>5 >30
*)
8+ --~ 8+
151>2
*)
a Ref. [9]. * Program limitations restricted the theoretical calculations to I < 6. angular correlations when a cascade involves this level as intermediate state. Averaging results o f five cascades with the 80 keV transition gave the integral attenuation G 2 = 0.78 + 0.03. From G 2 and the measured ratio [7] X4/~. 2 = 2.84 -+ 0.20 for 166Er in diluted HC1, we get G 4 = 0.56 + 0.10. For the same perturbation mechanism for the 4 + state (0.17 + 0.07 ns lifetime), the ~ 2% theoretical integral attenuation is negligible within our accuracy of ~ 10%. Table 1 gives our 8(E2/M1) values (phase convention o f ref. [8] ), along with previous values from Coulomb excitation [9]. The errors reflect the insensitivity o f A 2 and A 4 for large f-values but in a sense are deceiving since for +0 34 E2. The data do not show the sharp drop to ~ ~ 1 above the 4 + state = - ( 2 0 _+90 9 ) the transition is 99.35_0:17% seen in/3-bands. The transitions are all essentially E2 so that previous extractions o f Z 2 are unchanged. For irrotational flow o f nuclear matter, and an equal distribution of neutrons and protons, the gyromagnetic ratio g (the ratio of the m o m e n t o f inertia o f the charged particles to that o f the total nucleus), is independent o f deformation and the M1 operator is parallel to the angular m o m e n t u m operator. So the M1 transition matrix elements vanish between angular m o m e n t u m eigenstates. Weak M1 transitions could arise for example from the different behavior o f protons and neutrons with respect to the collective coordinates either statistically by different ground state deformations [10], or dynamically [11 ], or from K = 1 quasiparticle admixtures in the collective states [12]. Microscopic calculations indicate that the contribution o f spin effects to the expectation values o f the total magnetic moment is very small [12]. The collective operators have non-vanishing matrix elements only between different magnetic substates o f a given nuclear state, unless the g-factors depend on intrinsic variables such as the quadrupole deformation. Such a dependence o f the g-factors on nuclear deformation results quite naturally in a recently d e v e l o ~ d generalization o f the collective model [6], which allows for independent quadrupole deformations tensors o~[21 and tx[2] for protons and neutrons. It is advantageous to introduce an average deformation ~[21 = (Bp42]' ~_BntXn[21)'/B and a deviation tensor ~[21 = Ot[n21 _ 4 2 ] where Bp and B n are the mass parameters for protons and neutrons and B = Bp + B n. The Hamiltonian may be set up as H = H(ct) + H(~) + H(et, ~), where H(a) is the hamiltonian obtained in the 443
Volume 57B, number 5
PHYSICS LETTERS
4 August 1975
The b-values may now be evaluated and the unknown quantities Ci and Ci can be determined from a least-squares fti.to the experimental data. Fig. 1 shows the potential energy surface and the theoretical low-energy spectrum and B(E2) values obtained by fitting the Gneuss-Greiner Hamiltonian to the experimental data. The collective eigenstates obtained from this fit were then used to fit the theoretical b-values as discussed above to the experimental data. Table 1 gives the results. Expect possibly for the transitions 6; -+ 6: and 5; + 66 the agreement seems quite satisfactory. Possibly one of these states may not be described well by the model. The potential energy surface shows that 166Er is a good rotator with a minimum at -10 MeV on the prolate axis and a ground-state deformation of PO = 0.37. The relatively steep ascent of the surface around the minimum indicates that the nucleus is stiff with respect to P-7 vibrations, which causes comparatively smaller dynamical deformation differences between protons and neutrons. Thus this nucleus should have weaker Ml transitions than “softer” nuclides like 152Sm and 152$154Gd. Indeed, Ml matrix elements extracted from experimental data are twice as large in these nuclei as in 166Er.
References [l] P.O. Lipas, Phys. Lett. 8 (1964) 279. [2] J.H. Hamilton, Jzv. Akad. Nauk Ser. Fiz. 36 (1972) 17. [3] H. Ejiri and G.B. Hagemann, Nucl. Phys. 161 (1971) 449. [4] J.H. Hamilton, A.V. Ramayya and L.C. Whitlock, F’hys.Rev. Letters 23 (1969) 1178. [5] B.R. Mottelson, J. Phys. Japan Supp. 24 (1968) 87. [6] V. Maruhn-Rezwani, J.A. Maruhn and W. Greiner, Univ. of Frankfort, to be published. [7] E. Gerdau et al., Z. Physik 174 (1963) 389. [ 81 K.S. Krane and R.M. Steffen, Phys. Rev. C2 (1970) 724. [9] J.M. Domingos, G.D. Symons and A.C. Douglas, Nucl. Phys. A180 (1972) 600. [ 101 W. Greiner, Nucl. Phys. 80 (1966) 417. [ 111 K. Kumar, The Electromagnetic Interaction, ed. W.D. Hamilton, North-Holland Publ. Co. (in press). [12] K. Kumar and M. Baranger, Nucl. Phys. A92 (1967) 608. [ 131 D.R. Bes, P. Federman, E. Magqueda and A. Zuker, Nucl. Phys. 65 (1965) 1. [ 141 G. Gneuss and W. Greiner, Nucl. Phys. Al 71 (1971) 449. [15] P.H. Stelson and L. Grodzins, Nucl. Data Al (1965) 21; Y. Yoshizawa, B. Elbek, B. Herskind and M.C. Olesen, Nucl. Phys. 73 (1965) 273; R.O. Sayer et al., Phys. Rev. Cl (1970) 1525.
444
PHYSICS LETTERS
MAGNETIC PROPERTIES
4 August 1975
O F 166Er A N D T H E C O L L E C T I V E M O D E L
K.R. BAKER 1 , J.H. HAMILTON, J. LANGE 2 , A.V. RAMAYYA and L. V A R N E L L 3 Physics Department 4, Vanderbilt University, Nashville, Tennessee 37235, USA V. MARUHN-REZWANI, J.J. PINAJIAN and J.A. MARUHN Oak Ridge National Laboratory s , Oak Ridge, Tennessee 37830, USA Received 18 June 1975 The E2/M1 admixtures for nine ~-ground band transitions in 166Er were measured. Mixing ratios, tifE2/M1), were calculated based on a general treatment of magnetic properties in the framework of the collective model. The agreement with the experimental values is generally good.
If one assumes irrotational flow o f nuclear matter and an equal distribution of neutrons and protons, M1 transitions are forbidden in the collective model [1 ]. Generally quite small M1 admixtures are seen in transitions between K 'r = 0 + and 2 +,/~- and 7-type vibrational and ground-state bands in deformed nuclei [2], but sizable M1 contributions are seen in some cases at higher spins. A decrease in 6(E2/M1) with spin in/~-ground band transitions has been observed with a sharp drop above spin 4, (in 174Hf, 6(69 ~ 6g) ~ 1 [3]). For ")'-bands a decrease in 6 with spin was first observed [4] in 154Gd(6 = 10.1,2. r ~ 2g to 6 = 4.4, 47 ~ 4g). The properties of 166Er have long provided classic tests of the collective model [5]. It is the best o f only two or three examples [2,5] where B(E2) ratios yield a consistent 7-ground band mixing parameter Z 2. Five transitions had to be assumed to be pure E2, however, in the Z 2 calculations in 166Er since M1 admixtures were unknown. Systematic measurements of transitions from 1= 3 - 8 states in 166Er were made to test the spin dependence of 6(7 ~ g) to high spin and to verify the assumed E2 character made in extracting Z 2 values. While successes have been achieved in specific regions, more general theoretical treatments have not yielded detailed agreement with experimental 6-values. No detailed calculations have been done in well deformed nuclei. Recently a general treatment of magnetic properties of even-even nuclei in the framework of the collective model has been proposed [6] as described here. Because 166Er has been a classic test case o f the collective model, and because data are available to higher spin, it offers an important test of this new approach to the magnetic properties of even-even nuclei. We have calculated for 166Er the first detailed predictions of 6-values for a well deformed nucleus. Our results generally compare very favorably with our experimental data. Measurements were made with a chemically purified 15/aCi source in 0.1M HCI of 1200 y 166Ho. Correlations between the 184 keV, 4 + --* 2 +, ground band transition and those in coincidence with it were measured with a NaI-Ge(Li) system [2]. The correlations of transitions in coincidence with the (2 + ~ 0 +) 80 keV and (6 + ~ 4 +) 280 keV 7-rays were measured with a Ge(Li)-Ge(Li) system. A digital gate unit routed spectra from the 80 and 280 keV photopeaks and the Compton backgrounds above them into different 1024 channels of the analyzer. Data were recorded at five angles. Corrections for decentering of the source, measured chance coincidences and the measured Compton backgrounds in the gate and under the photopeaks were made. t s . The data were fitted to W(0) = 1 +A'2P2(cosO ) +A4P4(cosO ) where A K = A K Q K G K. Corrections were made 4for the detector finite solid angle QK and the 1.8 ns lifetime of the 2g state. Hyperfine interactions attenuate i Present address: ERDA, Washington, D.C. 2 Present address: Universitiit Bochum, 463 Bochum Universit/itsstr. 150, Gebaude NT, W. Germany. a Present address: Physics Department, Yale University, New Haven, Conn. 4 Supported in part by "agrant from the National Science Foundation. s Research sponsored by ERDA under contract with Union Carbide Corporation. 441
Volume 57B, number 5
PHYSICS LETTERS
4 August 1975
prescription of Gneuss and Greiner [14]. The strong coupling between ap and a n allows one to use the harmonic oscillator approximation for H(~) with a level spacing of the order of the energy of the giant dipole resonance. The interaction potential V(a, ~) is to lowest order given by V(a, ~) = C[a × ~] [01 + D ( [ a X a] X ~)[01 with C a n d D as yet unknown constants to be determined from experiment. If spin effects are neglected, the magnetic moment operator in this model is simply the orbital angular momentum of the protons, which does not commute with the total Hamiltonian. However, if the interaction V(a, ~) is treated in perturbation theory, the M1 matrix elements may be expressed as matrix elements of an effective operator Mleff=iVr~
(__ff BnBp Bp + C 1, -BnBp . , 5 2_~BT(/,:,I[[a x a]2x rr2]lll~., ) - I (/al[aXTr]llj',,)+lC2~
(1)
2B 2 I
between the unperturbed eigenstates of H(a) alone. Clearly this effective M1 operator contains a deformation dependent g-factor and is not in general parallel to the angular momentum operator. The E2 operator is also modified by the presence of the ~-degree of freedom in an effective E2 operator
, Bn I0 -#,]
10
2
]
., • X a 2] 2 I/~>
(2)
From eqs. (1) and (2), one finds
8-
~
Er
(/illE2effll/f)
10 ~c (/illMleff[I/f) "
8°° IMv •
.,
. . . .
i
. . . . . . . .
i
. . . .
i,.
.t
03
0'I 0
c~
0~
' '
MeV
0.'3 '
'
O:q' '
0)5 '
'
0.'6 B [E2)
O; [XPT. +
6+
__6
5+
2;
EXPT.
5.78
01 -- 22
O. 17
0.21
2[
2;
0.10
0.08
Lt[ -- 6[
2.80
2.27
,
-
5+
4 +
.4 +
3+ 2 +
6+
+
THEORY
5.86
-
*
THEORY _ _ o
¢
'
A o : B
-
_ _ 6
4+
+
-
3+ 2*
166E r
4+
2 +
_
0÷
- -
_
2 +
0 ÷
Fig. 1. The upper part of the figure shows the potential energy surface and potential obtained by comparison with experimental B(E2) values [15] and excitation energies as descried in the text. The fits to the experimental values are shown in the lower part of the figure. 442
Volume 57B, number 5
PHYSICS LETTERS
4 August 1975
Table 1 Error-weighted averaged results of "r-'r directional correlation measurements of transitions from the 3"-band to the ground-state band and comparison with theoretical results. 1.~ I;
5 exp
5 th
2+ ~ 2+
_(16+~ 2) a
-21.3 -29.8
3+ ~ 2+
-(18+_9 )
3+ --. 4 +
_ (9+_531 9)
4 + -~ 4 +
-(10+_427)
--
9.6
- 11.4
+3.0
-(3"3-1.2) 5+ ~ 4 +
+4 -(20_4)
-22.9
_(37+_1°) a 5+ ~ 6+
-(25+_33)
-5.40
6 + --, 6 +
-(20+_9°)
-8.0
7+ ~ 6 ÷
-(22+_~)
*)
7+~8 +
-80>5 >30
*)
8+ --~ 8+
151>2
*)
a Ref. [9]. * Program limitations restricted the theoretical calculations to I < 6. angular correlations when a cascade involves this level as intermediate state. Averaging results o f five cascades with the 80 keV transition gave the integral attenuation G 2 = 0.78 + 0.03. From G 2 and the measured ratio [7] X4/~. 2 = 2.84 -+ 0.20 for 166Er in diluted HC1, we get G 4 = 0.56 + 0.10. For the same perturbation mechanism for the 4 + state (0.17 + 0.07 ns lifetime), the ~ 2% theoretical integral attenuation is negligible within our accuracy of ~ 10%. Table 1 gives our 8(E2/M1) values (phase convention o f ref. [8] ), along with previous values from Coulomb excitation [9]. The errors reflect the insensitivity o f A 2 and A 4 for large f-values but in a sense are deceiving since for +0 34 E2. The data do not show the sharp drop to ~ ~ 1 above the 4 + state = - ( 2 0 _+90 9 ) the transition is 99.35_0:17% seen in/3-bands. The transitions are all essentially E2 so that previous extractions o f Z 2 are unchanged. For irrotational flow o f nuclear matter, and an equal distribution of neutrons and protons, the gyromagnetic ratio g (the ratio of the m o m e n t o f inertia o f the charged particles to that o f the total nucleus), is independent o f deformation and the M1 operator is parallel to the angular m o m e n t u m operator. So the M1 transition matrix elements vanish between angular m o m e n t u m eigenstates. Weak M1 transitions could arise for example from the different behavior o f protons and neutrons with respect to the collective coordinates either statistically by different ground state deformations [10], or dynamically [11 ], or from K = 1 quasiparticle admixtures in the collective states [12]. Microscopic calculations indicate that the contribution o f spin effects to the expectation values o f the total magnetic moment is very small [12]. The collective operators have non-vanishing matrix elements only between different magnetic substates o f a given nuclear state, unless the g-factors depend on intrinsic variables such as the quadrupole deformation. Such a dependence o f the g-factors on nuclear deformation results quite naturally in a recently d e v e l o ~ d generalization o f the collective model [6], which allows for independent quadrupole deformations tensors o~[21 and tx[2] for protons and neutrons. It is advantageous to introduce an average deformation ~[21 = (Bp42]' ~_BntXn[21)'/B and a deviation tensor ~[21 = Ot[n21 _ 4 2 ] where Bp and B n are the mass parameters for protons and neutrons and B = Bp + B n. The Hamiltonian may be set up as H = H(ct) + H(~) + H(et, ~), where H(a) is the hamiltonian obtained in the 443
Volume 57B, number 5
PHYSICS LETTERS
4 August 1975
The b-values may now be evaluated and the unknown quantities Ci and Ci can be determined from a least-squares fti.to the experimental data. Fig. 1 shows the potential energy surface and the theoretical low-energy spectrum and B(E2) values obtained by fitting the Gneuss-Greiner Hamiltonian to the experimental data. The collective eigenstates obtained from this fit were then used to fit the theoretical b-values as discussed above to the experimental data. Table 1 gives the results. Expect possibly for the transitions 6; -+ 6: and 5; + 66 the agreement seems quite satisfactory. Possibly one of these states may not be described well by the model. The potential energy surface shows that 166Er is a good rotator with a minimum at -10 MeV on the prolate axis and a ground-state deformation of PO = 0.37. The relatively steep ascent of the surface around the minimum indicates that the nucleus is stiff with respect to P-7 vibrations, which causes comparatively smaller dynamical deformation differences between protons and neutrons. Thus this nucleus should have weaker Ml transitions than “softer” nuclides like 152Sm and 152$154Gd. Indeed, Ml matrix elements extracted from experimental data are twice as large in these nuclei as in 166Er.
References [l] P.O. Lipas, Phys. Lett. 8 (1964) 279. [2] J.H. Hamilton, Jzv. Akad. Nauk Ser. Fiz. 36 (1972) 17. [3] H. Ejiri and G.B. Hagemann, Nucl. Phys. 161 (1971) 449. [4] J.H. Hamilton, A.V. Ramayya and L.C. Whitlock, F’hys.Rev. Letters 23 (1969) 1178. [5] B.R. Mottelson, J. Phys. Japan Supp. 24 (1968) 87. [6] V. Maruhn-Rezwani, J.A. Maruhn and W. Greiner, Univ. of Frankfort, to be published. [7] E. Gerdau et al., Z. Physik 174 (1963) 389. [ 81 K.S. Krane and R.M. Steffen, Phys. Rev. C2 (1970) 724. [9] J.M. Domingos, G.D. Symons and A.C. Douglas, Nucl. Phys. A180 (1972) 600. [ 101 W. Greiner, Nucl. Phys. 80 (1966) 417. [ 111 K. Kumar, The Electromagnetic Interaction, ed. W.D. Hamilton, North-Holland Publ. Co. (in press). [12] K. Kumar and M. Baranger, Nucl. Phys. A92 (1967) 608. [ 131 D.R. Bes, P. Federman, E. Magqueda and A. Zuker, Nucl. Phys. 65 (1965) 1. [ 141 G. Gneuss and W. Greiner, Nucl. Phys. Al 71 (1971) 449. [15] P.H. Stelson and L. Grodzins, Nucl. Data Al (1965) 21; Y. Yoshizawa, B. Elbek, B. Herskind and M.C. Olesen, Nucl. Phys. 73 (1965) 273; R.O. Sayer et al., Phys. Rev. Cl (1970) 1525.
444