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Spin and orbital contributions to collective M1 transitions in 46,48Ti
Volume 183, number 2
PHYSICS LETTERS B
8 January. 1987
SPIN AND ORBITAL C O N T R I B U T I O N S TO COLLECTIVE M I T R A N S I T I O N S IN 46"4STi~ R. NOJAROV ~.2, Amand FAESSLER and O. CIVITARESE 34 Institut fiir Theoretische Physik, Universitiit Tiibingen, D-7400 Tfibingen. l:ed. Rep. Germany Received 1 April 1986; revised manuscript received 7 October 1986
Low- and high-lying K " = 1 " states and M I transitions in 464gTi are studied. The model hamiltonian is treated in the quasiparticle random phase approximation (QRPA) with an exact restoration of its rotational invariance. A considerable spin contribution to the transition matrix elements is found for the low-energy (about 4 MeV ) strong M 1 transition (the orbital contribution being 30-70% of the spin one), although the microscopic structure of this state in ~"Ti is typical for an orbital isovector excitation. The calculated energies and B(MI ) values are in a good agreement with the experimental data. The results are compared to the estimates of the isovector scissor model.
Experimental evidences about the existence of strong magnetic dipole transitions in the nuclei 46Ti [ 1 ] and 48Ti [2] have been reported recently. The experimental results from inelastic proton and electron scattering show the presence of a low-energy (about 4 MeV) K ' = 1 + state in each one of these two nuclei, which has a strong M 1 transitions: B(M 1 )'~ = 0.5-1.0/t~ [ 1,2 ]. A dominance of the orbital term of the transition operator has been established [ 1 ] with a limiting value allowing an almost equal spin contribution in the case when both are in phase. Similar low-lying 1 '- states with 3 MeV ~
n=Ho+h, h = - ~ , t [ H o , _
l(oO]+[Ho, l(a)],
k=4
~ (E,+Ek)[12(ki,o~r,)+12(~i,o~t__)],
(1)
t: ,t.k > 0
where E,, Ek are the quasiparticle energies and l(ki, at;) are the quasiparticle matrix elements of the angular momentum operators: l(a= 1 ) =iZ~, l ( a = - l ) =Ix. The quasiparticle pairs (k, i) are defined by the selection Supported by the Bundesministerium f'fir Forschung und Technologie and the Alexander von Humboldt Foundation. Fellow of the Alexander yon Humboldt Foundation. 2 Permanent address: Institute of Nuclear Research and Nuclear Energy, Bulgarian Academy of Sciences, Sofia 1784, Bulgaria. Permanent address: Department of Physics, University of La Plata, 1900 La Plata, Argentina. * Fellow of the CONICET, Argentina.
122
0370-2693/87/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
Volume 183, number 2
PHYSICS LETTERS B
8 January 1987
Table 1 Deformation and pairing gap parameters. Nucleus
fl
.J~ [MeV l
d~ [MeV]
4~Ti 48Ti
0.27 0.22
1.5 1.2
1.2 1.2
rule Al2=S'2.k--£2i= 1 of the angular m o m e n t u m operators. Following ref. [ 13], the pairing force is taken into account by writing the hamiltonian (1) in a quasiparticle representation, the single-quasiparticle term being denoted by Ho. The mean field is given by a deformed, axially symmetric Woods-Saxon potential, which has been parametrized in the way shown in ref. [ 14]. The corresponding coefficients, following the notation ofref. [ 14], are V,= - 4 5 MeV, Va= - 5 5 MeV, a n = a p = 0 . 6 7 fin, r , = 1.30 fro, ro= 1.28 fro. 2~ =21.7, 2~= 17.2. The single-particle states, which are expanded in a cylindrical basis, include states up to No~, = 1 I. The deformation and gap parameters are given in table 1. The deformation of the nuclei considered was studied experimentally in ref. [ 15 ]. Near magic numbers the pairing gaps are substantially reduced [ 13]. This increases the M1 strength and decreases the excitation energy of the low-lying strong 1 ~ state. The formalism, based on the QRPA treatment of the hamiltonian ensures an exact orthogonality of the phonon wave functions to the spurious 1 ' state, which corresponds to isoscalar rotation of the nucleus as a whole. The main ingredients of the formalism are the following. (i) Structure of the 1 + states. In the laboratory frame the phonon wave functions have the form
II=I,M,K"=I+,v)=
~
~ [.~.,(oJ)+a~t.
,(oJ)]F+(a)l
),
(2)
O~=±I
where the intrinsic states are generated by the phonon creation operator: F , ' (c0 = 1 /
~
[~0~,"~(v, t_.)A+(ik, oa-) - ~ ' ~ ( v . tz)A(ik, oa=)
k,t.t~ = n,p
+2~7'(v, t:)A* ( ik, oetz) -la~, ~(v, t~)7t( ik, c~t:)].
(3)
In eq. (3), (~0, q~, 2, #) are the forward- and backward-going amplitudes, A + (A) are the pair creation (annihilation) operators and the terms with ,.~,~"~(v, t-) and a L ~ ( v , t~) correspond to configurations with IS'2kI = 1"(2,I = 1/2, .Q being the single-quasiparticle angular m o m e n t u m projection along the symmetry axis. (ii) Transition operator. The tensor components of the intrinsic M I operator read 37/'(M 1,/0
=
(3/4n)"2 rhu,
(4)
while in the laboratory frame we have h;/(M 1, U) = Z ~ t , , ( ° ) ) ~'/' ( M I , v),
(5)
v
where the intrinsic operators rh u are the tensor components of the magnetic dipole operator
~t= ~ gl(G)[(t~) +g,.(t-)g(t~). t: = n . p
The B ( M I )1' for a transition from the ground state to a 1 + state with an energy E,, is given by [ 1 1 ] B(M1, v ) ? = (3/4~r)~m=(E.),
(6)
with the intrinsic matrix element 123
Volume 183, number 2
PHYSICS LETTERS B
8 January, 1987
Table 2 Energies, B(M1 )" transition probabilities, orbital and spin contributions to the transition matrix elements, as given by the QRPA, for -'rTi and *STi. The B(MI )T values are obtained from: B(MI )T = (3/4n)(orbital+spin/2p~. The values given in the second and fourth lines correspond to the values given in table 2, case ~/--4 of ref. [9] with the orbital and spin contributions multiplied by the factor (4n/3) ,/2. in order to allow for the comparison with the present ones. Nucleus
Experiments
Theory
E [MeV]
B(M1)T [,u~]
E [MeV]
B(M1)T [,u~]
orbital contribution
spin contribution
4~Ti
4.32 ~'
1.0 +0.2 ~
4•515
1.29 4.41 c}
0.96 2.49 c,
1.36 1.81 ~'
48Ti
3.74 h~
0.50 z 0. I b~
3•546
0•47 3.25 ~
0•35 2.18 °
1.05 1.51 c~
~'From ref. [ I].
h'From ref. [21.
~'From ref. [9].
224•24 1.o
0.5
T. ~
ii I
o.o
224~26
0.5
:
ii
0.0
5
ENERGY 124
10
E [MeV]
Fig. 1. M1 strength distribution in '648Ti, Only states with B(M I ) T> 0.05 ,u~ are displayed• Theory: full lines, experiment [1,2]: dashed lines (certain assignment) and dotted lines (uncertain assignment).
Volume 183, number 2 m(E,,)=2
~
PHYSICS LETTERS B
8 January 1987
{m+(k,i,t~)[~o~ ~(v,t-)+~y~(v,t.)]+fft+(k,i,t-)[2~,'(v,t:)+U~7'(v,t~)]}.
(7)
t: = n.p
where rn+ (k, i, t.) and ET, (k, i, tz) are the quasiparticle matrix elements of the step-up operator fi ,. The RPA energy-weighted sum rule (EWSR) is S = ( [/t-, [H,p]] ) = ~ E , B ( M 1 , v)t.
(8)
v
The results are shown in table 2 and fig. 1. We have used the values &(n) =0, &(p) = I, g,(tz)=0.7&frc~(t,) for the orbital and spin gyromagnetic factors, respectively. The obtained 1 ~ states correspond to isovector vibrations since they are orthogonal to the isoscalar angular momentum, which generates the isoscalar vibrations. It is seen from table 2 that the experimental B(M1) values of this state are well reproduced, while the energy differs from the experimental value by about 0.2 MeV. The spin and orbital contributions to the transition matrix element are in phase. The same result was obtained from the rotational model [9], whose predictions are also listed in table 2. It is seen that the rotational model overestimates the transition probability, which is a typical feature of most collective models. The experiments [ 1,2] on the low-energy M1 transitions in the titanium nuclei considered, were stimulated, in fact, by previous estimates of Zamick [8], based on a single-j shell model. He predicted the energy and the transition probability of the magnetic dipole state in a fairly good agreement with the subsequent experimental data. In contrast to the rotational model [ 9 ], we found an appreciable spin contribution to the transition matrix element, the orbital contribution being 70% (46Ti) and 33% (48Ti) of the spin one. Although the present theoretical B(M1 )1 value for 46Ti is in good agreement with the experimental data, the ratio between the orbital and spin contributions is, in our case, about two times smaller than the lower limit 1.2 [ 1 ] of the experimental value. This limit is not obtained direclty from the experiment, but it depends substantially on the theoretical models used to derive the values of the spin and orbital matrix elements from the experimentally measured cross sections. In our case the main contribution to the transition matrix element in 48Ti results from a transition between proton states with asymptotic quantum numbers, [No,c, nz, A]O: [321] 3/2-+ [321] 1/2, where the spin-flip contribution is dominant. In contrast, the transition has a more pronounced orbital character in 46Ti, where it is shared almost equally between the neutron and proton 2qp pairs: nn[330] - 1/2, [321 ] 1/2 and pp[330] - 1/2, [321 ] 1/2. From the asymptotic quantum numbers one should expect a dominant orbital transition in this case but the realistic calculations show that the accumulated spin contribution due to spin-flip admixtures in the wave function is even larger than the orbital one. We have also investigated the dependence of transition strengths and excitation energies upon the singleparticle energy spacing. The results for 48Ti were obtained with proton Woods-Saxon energy g([ 330 ] - 1/2) = - 5 . 0 MeV, the original energy spacing with quite a low density being unable to produce enough M1 strength. The rest of the single-particle levels both for protons and neutrons correspond to the above parametrization of the Woods-Saxon potential. The M 1 strength distribution is shown in fig. 1, where all the 1 ~ states with B(M 1 ) T> 0.05/z~. are displayed. In contrast to the heavier (rare-earth) nuclei [ 11 ], very few strong states are present here. The energy interval, which is outside the figure, does not contain any state with a strength larger than the above minimum. The strong low-energy M 1 state in 46Ti (48Ti) exhausts 32% (7%) of the energy-weighted sum rule (8), which means that a large amount of strength is concentrated in a single 1 - state. The strong isovector state at 4.32 MeV in "STi with B(M1 )t = 0.61/t 2 has a microscopic structure, which is almost identical to that of the 4.51 MeV state in 46Ti and exhausts 12% of the EWSR. Therefore, one should expect that it will appear in the experiment in the energy region 4.5-4.7 MeV. together with the strong MI state at 4.46 MeV, which has mainly a spin-flip character. The main differences with respect to the case of quadrupole-quadrupole residual interaction [ 16] are (i) that we have now almost pure isovector vibrations with a more pronounced orbital character, (ii) the giant resonance at about 27 MeV, consisting of AN= 2 orbital 2qp excitations, disappears now since the M1 strength is concentrated in the low-energy region E < 8 MeV. 125
Volume 183, number 2
PHYSICS LETTERS B
8 January 1987
Finally, and in o r d e r to c o m p a r e the Q R P A results with lhe ones o f a phenomenoiogical model based on the microscopic evaluation o f the restoring force for the 1 " isovector m o d e [ 10], we have performed calculations by following the formalism o f ref. [ 10]. The results for 46Ti, obtained with 14 valence particles o f each kind, are E = 4 . 4 5 MeV, B(M1 )T = 0 . 6 8 / 1 ~ . The energy is closer to the experimental value in comparison with the Q R P A calculations, while the transition probability is u n d e r e s t i m a t e d by 30%, i.e., it is not far away from the error bar. F r o m our results we can draw the following conclusions: (i) the M 1 transitions in 46Ti and 48Ti have isovector character and the strength o f this m o d e is concentrated exclusively in the low-energy region E < 8 MeV; (ii) the wave function o f the strong M 1 state in 46Ti is equally distributed between the two-neutron and two-proton pair configurations [330] - 1/2, [321 ] 1/2, implying a p r e d o m i n a n t l y orbital transition; (iii) nevertheless, the transition matrix elements o f the low-lying ( a b o u t 4 M e V ) strong magnetic dipole states have larger contribution from the spin term o f the transition operator, the orbital contribution in 4648Ti being respectively 70% and 33% o f the spin one. In this respect, the situation concerning low-lying M I transitions in t i t a n i u m isotopes appears to be very similar to the case o f d e f o r m e d nuclei in the rare-earth region [ 1 I ], except for the presence o f a large spin contribution from the transition operator, a fact which could be attributed to differences between the W o o d s - S a x o n density o f single-particle states a r o u n d the Fermi surface in light and heavy nuclei. We would like to thank Professor N.I. Pyatov and Professor A. Richter for useful discussions. One o f us ( R . N . ) acknowledges a research grant from the Alexander von H u m b o l d t F o u n d a t i o n .
References [ 1] C. Djalali, N. Marry. M. Morlet, A. Willis. J.C. Jourdain, D. Bohle, U. Hartmann, G. Kiichler, A. Richter. G. Caskey, G.M. Crawley and A. Galonsky, Phys. Lett. B 164 (1985) 269. [2 ] D. Bohle et al., Contribution to the Annual Meeting of the German Phys. Soc. (Heidelberg, FRG, March 1986). Verhandlungen der DPG 4 (1986) 592; 1-. Guhr, D. Bohle and A. Richter, to be published. [3] D. Bohle, A. Richter, W, Steffen, A.E.L. Dieperink, N. Lo ludice. F. Palumbo and O. Scholten, Phys. l,ett. B 137 (1984) 27. [4] D. Bohle, G. Ktichler, A. Richter and W. Steffen. Phys. kerr. B 148 (1984) 260. [51 D.R. Bes and R. Broglia, Phys. Left. B 137 (1984) 141. [61 I. Hamamoto and S. Aberg, Phys. Lett. B 145 (1984) 163. [ 7 ] N. Lo ludice and F. Palumbo, Phys. Rev. Left. 41 (1978) 1532. [8] L. Zamick, Phys. Rev. C 31 (1985) 1955. [9] I.. Zamick, Phys. Lett. B 167 (1986) I. [ 101 R. Nojarov, Z. Bochnacki and A. Faessler, Z. Phys. A 324 (19861 289. [ 11 ] O. Civitarese, A. Faessler and R. Nojarov, to be published. [ 12] M.I. Baznat and N.I. Pyatov, Yad. Fiz. 21 (1975) 708. [ 13] V.G. Soloviev, Theory. of complex nuclei (Pergamon, Oxford, 1976). [ 14] R. Nojarov, J. Phys. G 10 (1984) 539. [ 15] H. Rebel, G. Hauser, G.W. Schweimer. G. Nowicki. W. Wiesner and D. Hartmann, Nucl. Phys. A 218 (1974) 13. [ 16 ] O. Civitarese, A. Faessler and R. Nojarov, Proc. Intern. Conf. on Nuclear structure, reactions and symmetries (Dubrovnik, Yugoslavia, June 1986) (World Scientific, Singapore), to be published.
126
PHYSICS LETTERS B
8 January. 1987
SPIN AND ORBITAL C O N T R I B U T I O N S TO COLLECTIVE M I T R A N S I T I O N S IN 46"4STi~ R. NOJAROV ~.2, Amand FAESSLER and O. CIVITARESE 34 Institut fiir Theoretische Physik, Universitiit Tiibingen, D-7400 Tfibingen. l:ed. Rep. Germany Received 1 April 1986; revised manuscript received 7 October 1986
Low- and high-lying K " = 1 " states and M I transitions in 464gTi are studied. The model hamiltonian is treated in the quasiparticle random phase approximation (QRPA) with an exact restoration of its rotational invariance. A considerable spin contribution to the transition matrix elements is found for the low-energy (about 4 MeV ) strong M 1 transition (the orbital contribution being 30-70% of the spin one), although the microscopic structure of this state in ~"Ti is typical for an orbital isovector excitation. The calculated energies and B(MI ) values are in a good agreement with the experimental data. The results are compared to the estimates of the isovector scissor model.
Experimental evidences about the existence of strong magnetic dipole transitions in the nuclei 46Ti [ 1 ] and 48Ti [2] have been reported recently. The experimental results from inelastic proton and electron scattering show the presence of a low-energy (about 4 MeV) K ' = 1 + state in each one of these two nuclei, which has a strong M 1 transitions: B(M 1 )'~ = 0.5-1.0/t~ [ 1,2 ]. A dominance of the orbital term of the transition operator has been established [ 1 ] with a limiting value allowing an almost equal spin contribution in the case when both are in phase. Similar low-lying 1 '- states with 3 MeV ~
n=Ho+h, h = - ~ , t [ H o , _
l(oO]+[Ho, l(a)],
k=4
~ (E,+Ek)[12(ki,o~r,)+12(~i,o~t__)],
(1)
t: ,t.k > 0
where E,, Ek are the quasiparticle energies and l(ki, at;) are the quasiparticle matrix elements of the angular momentum operators: l(a= 1 ) =iZ~, l ( a = - l ) =Ix. The quasiparticle pairs (k, i) are defined by the selection Supported by the Bundesministerium f'fir Forschung und Technologie and the Alexander von Humboldt Foundation. Fellow of the Alexander yon Humboldt Foundation. 2 Permanent address: Institute of Nuclear Research and Nuclear Energy, Bulgarian Academy of Sciences, Sofia 1784, Bulgaria. Permanent address: Department of Physics, University of La Plata, 1900 La Plata, Argentina. * Fellow of the CONICET, Argentina.
122
0370-2693/87/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
Volume 183, number 2
PHYSICS LETTERS B
8 January 1987
Table 1 Deformation and pairing gap parameters. Nucleus
fl
.J~ [MeV l
d~ [MeV]
4~Ti 48Ti
0.27 0.22
1.5 1.2
1.2 1.2
rule Al2=S'2.k--£2i= 1 of the angular m o m e n t u m operators. Following ref. [ 13], the pairing force is taken into account by writing the hamiltonian (1) in a quasiparticle representation, the single-quasiparticle term being denoted by Ho. The mean field is given by a deformed, axially symmetric Woods-Saxon potential, which has been parametrized in the way shown in ref. [ 14]. The corresponding coefficients, following the notation ofref. [ 14], are V,= - 4 5 MeV, Va= - 5 5 MeV, a n = a p = 0 . 6 7 fin, r , = 1.30 fro, ro= 1.28 fro. 2~ =21.7, 2~= 17.2. The single-particle states, which are expanded in a cylindrical basis, include states up to No~, = 1 I. The deformation and gap parameters are given in table 1. The deformation of the nuclei considered was studied experimentally in ref. [ 15 ]. Near magic numbers the pairing gaps are substantially reduced [ 13]. This increases the M1 strength and decreases the excitation energy of the low-lying strong 1 ~ state. The formalism, based on the QRPA treatment of the hamiltonian ensures an exact orthogonality of the phonon wave functions to the spurious 1 ' state, which corresponds to isoscalar rotation of the nucleus as a whole. The main ingredients of the formalism are the following. (i) Structure of the 1 + states. In the laboratory frame the phonon wave functions have the form
II=I,M,K"=I+,v)=
~
~ [.~.,(oJ)+a~t.
,(oJ)]F+(a)l
),
(2)
O~=±I
where the intrinsic states are generated by the phonon creation operator: F , ' (c0 = 1 /
~
[~0~,"~(v, t_.)A+(ik, oa-) - ~ ' ~ ( v . tz)A(ik, oa=)
k,t.t~ = n,p
+2~7'(v, t:)A* ( ik, oetz) -la~, ~(v, t~)7t( ik, c~t:)].
(3)
In eq. (3), (~0, q~, 2, #) are the forward- and backward-going amplitudes, A + (A) are the pair creation (annihilation) operators and the terms with ,.~,~"~(v, t-) and a L ~ ( v , t~) correspond to configurations with IS'2kI = 1"(2,I = 1/2, .Q being the single-quasiparticle angular m o m e n t u m projection along the symmetry axis. (ii) Transition operator. The tensor components of the intrinsic M I operator read 37/'(M 1,/0
=
(3/4n)"2 rhu,
(4)
while in the laboratory frame we have h;/(M 1, U) = Z ~ t , , ( ° ) ) ~'/' ( M I , v),
(5)
v
where the intrinsic operators rh u are the tensor components of the magnetic dipole operator
~t= ~ gl(G)[(t~) +g,.(t-)g(t~). t: = n . p
The B ( M I )1' for a transition from the ground state to a 1 + state with an energy E,, is given by [ 1 1 ] B(M1, v ) ? = (3/4~r)~m=(E.),
(6)
with the intrinsic matrix element 123
Volume 183, number 2
PHYSICS LETTERS B
8 January, 1987
Table 2 Energies, B(M1 )" transition probabilities, orbital and spin contributions to the transition matrix elements, as given by the QRPA, for -'rTi and *STi. The B(MI )T values are obtained from: B(MI )T = (3/4n)(orbital+spin/2p~. The values given in the second and fourth lines correspond to the values given in table 2, case ~/--4 of ref. [9] with the orbital and spin contributions multiplied by the factor (4n/3) ,/2. in order to allow for the comparison with the present ones. Nucleus
Experiments
Theory
E [MeV]
B(M1)T [,u~]
E [MeV]
B(M1)T [,u~]
orbital contribution
spin contribution
4~Ti
4.32 ~'
1.0 +0.2 ~
4•515
1.29 4.41 c}
0.96 2.49 c,
1.36 1.81 ~'
48Ti
3.74 h~
0.50 z 0. I b~
3•546
0•47 3.25 ~
0•35 2.18 °
1.05 1.51 c~
~'From ref. [ I].
h'From ref. [21.
~'From ref. [9].
224•24 1.o
0.5
T. ~
ii I
o.o
224~26
0.5
:
ii
0.0
5
ENERGY 124
10
E [MeV]
Fig. 1. M1 strength distribution in '648Ti, Only states with B(M I ) T> 0.05 ,u~ are displayed• Theory: full lines, experiment [1,2]: dashed lines (certain assignment) and dotted lines (uncertain assignment).
Volume 183, number 2 m(E,,)=2
~
PHYSICS LETTERS B
8 January 1987
{m+(k,i,t~)[~o~ ~(v,t-)+~y~(v,t.)]+fft+(k,i,t-)[2~,'(v,t:)+U~7'(v,t~)]}.
(7)
t: = n.p
where rn+ (k, i, t.) and ET, (k, i, tz) are the quasiparticle matrix elements of the step-up operator fi ,. The RPA energy-weighted sum rule (EWSR) is S = ( [/t-, [H,p]] ) = ~ E , B ( M 1 , v)t.
(8)
v
The results are shown in table 2 and fig. 1. We have used the values &(n) =0, &(p) = I, g,(tz)=0.7&frc~(t,) for the orbital and spin gyromagnetic factors, respectively. The obtained 1 ~ states correspond to isovector vibrations since they are orthogonal to the isoscalar angular momentum, which generates the isoscalar vibrations. It is seen from table 2 that the experimental B(M1) values of this state are well reproduced, while the energy differs from the experimental value by about 0.2 MeV. The spin and orbital contributions to the transition matrix element are in phase. The same result was obtained from the rotational model [9], whose predictions are also listed in table 2. It is seen that the rotational model overestimates the transition probability, which is a typical feature of most collective models. The experiments [ 1,2] on the low-energy M1 transitions in the titanium nuclei considered, were stimulated, in fact, by previous estimates of Zamick [8], based on a single-j shell model. He predicted the energy and the transition probability of the magnetic dipole state in a fairly good agreement with the subsequent experimental data. In contrast to the rotational model [ 9 ], we found an appreciable spin contribution to the transition matrix element, the orbital contribution being 70% (46Ti) and 33% (48Ti) of the spin one. Although the present theoretical B(M1 )1 value for 46Ti is in good agreement with the experimental data, the ratio between the orbital and spin contributions is, in our case, about two times smaller than the lower limit 1.2 [ 1 ] of the experimental value. This limit is not obtained direclty from the experiment, but it depends substantially on the theoretical models used to derive the values of the spin and orbital matrix elements from the experimentally measured cross sections. In our case the main contribution to the transition matrix element in 48Ti results from a transition between proton states with asymptotic quantum numbers, [No,c, nz, A]O: [321] 3/2-+ [321] 1/2, where the spin-flip contribution is dominant. In contrast, the transition has a more pronounced orbital character in 46Ti, where it is shared almost equally between the neutron and proton 2qp pairs: nn[330] - 1/2, [321 ] 1/2 and pp[330] - 1/2, [321 ] 1/2. From the asymptotic quantum numbers one should expect a dominant orbital transition in this case but the realistic calculations show that the accumulated spin contribution due to spin-flip admixtures in the wave function is even larger than the orbital one. We have also investigated the dependence of transition strengths and excitation energies upon the singleparticle energy spacing. The results for 48Ti were obtained with proton Woods-Saxon energy g([ 330 ] - 1/2) = - 5 . 0 MeV, the original energy spacing with quite a low density being unable to produce enough M1 strength. The rest of the single-particle levels both for protons and neutrons correspond to the above parametrization of the Woods-Saxon potential. The M 1 strength distribution is shown in fig. 1, where all the 1 ~ states with B(M 1 ) T> 0.05/z~. are displayed. In contrast to the heavier (rare-earth) nuclei [ 11 ], very few strong states are present here. The energy interval, which is outside the figure, does not contain any state with a strength larger than the above minimum. The strong low-energy M 1 state in 46Ti (48Ti) exhausts 32% (7%) of the energy-weighted sum rule (8), which means that a large amount of strength is concentrated in a single 1 - state. The strong isovector state at 4.32 MeV in "STi with B(M1 )t = 0.61/t 2 has a microscopic structure, which is almost identical to that of the 4.51 MeV state in 46Ti and exhausts 12% of the EWSR. Therefore, one should expect that it will appear in the experiment in the energy region 4.5-4.7 MeV. together with the strong MI state at 4.46 MeV, which has mainly a spin-flip character. The main differences with respect to the case of quadrupole-quadrupole residual interaction [ 16] are (i) that we have now almost pure isovector vibrations with a more pronounced orbital character, (ii) the giant resonance at about 27 MeV, consisting of AN= 2 orbital 2qp excitations, disappears now since the M1 strength is concentrated in the low-energy region E < 8 MeV. 125
Volume 183, number 2
PHYSICS LETTERS B
8 January 1987
Finally, and in o r d e r to c o m p a r e the Q R P A results with lhe ones o f a phenomenoiogical model based on the microscopic evaluation o f the restoring force for the 1 " isovector m o d e [ 10], we have performed calculations by following the formalism o f ref. [ 10]. The results for 46Ti, obtained with 14 valence particles o f each kind, are E = 4 . 4 5 MeV, B(M1 )T = 0 . 6 8 / 1 ~ . The energy is closer to the experimental value in comparison with the Q R P A calculations, while the transition probability is u n d e r e s t i m a t e d by 30%, i.e., it is not far away from the error bar. F r o m our results we can draw the following conclusions: (i) the M 1 transitions in 46Ti and 48Ti have isovector character and the strength o f this m o d e is concentrated exclusively in the low-energy region E < 8 MeV; (ii) the wave function o f the strong M 1 state in 46Ti is equally distributed between the two-neutron and two-proton pair configurations [330] - 1/2, [321 ] 1/2, implying a p r e d o m i n a n t l y orbital transition; (iii) nevertheless, the transition matrix elements o f the low-lying ( a b o u t 4 M e V ) strong magnetic dipole states have larger contribution from the spin term o f the transition operator, the orbital contribution in 4648Ti being respectively 70% and 33% o f the spin one. In this respect, the situation concerning low-lying M I transitions in t i t a n i u m isotopes appears to be very similar to the case o f d e f o r m e d nuclei in the rare-earth region [ 1 I ], except for the presence o f a large spin contribution from the transition operator, a fact which could be attributed to differences between the W o o d s - S a x o n density o f single-particle states a r o u n d the Fermi surface in light and heavy nuclei. We would like to thank Professor N.I. Pyatov and Professor A. Richter for useful discussions. One o f us ( R . N . ) acknowledges a research grant from the Alexander von H u m b o l d t F o u n d a t i o n .
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