Investigation of magnetic dipole 1*K = 1+0 states in 150Nd and 176Hf nuclei

INVESTIGATION OF MAGNETIC DIPOLE V K = VO STATES

IN 150Nd AND 176H f NUCLEI

^rtuğral F .,1,2 Guliyev E., *Akkaya R., lrTiitiincii H.M.

‘Sakarya University, Adapazari, Turkey 2Institute o f Physics, Baku, Azerbaijan

In this study, the properties of collective 1 ”K = TO states in doubly even deformed nuclei ,50Nd and

,76H f are investigated in the framework of the Random-Phase Approximation (RPA). Spin-spin interactions

have shown that the spectroscopic T states (up to energies on the order of 4 MeV) are weakly correlated and are characterized by small values of B(M1). The collective T states with the largest values of B(M1) are expected in the region of energies of 9-11 MeV (K=0). The dipole sum rule for Ml transitions are also investigated in this work. Our results show that B(M1) transition probabilities which have been calculated theoretically are in good agreement with the experimental data.

INTRODUCTION

The study of nuclear excitations with low multipolarity, in particular collective magnetic dipole ones, gives valuable information about nuclear structure and nuclear- nuclear forces at

low energy. At present magnetic dipole resonances

(T=T)

have been found experimentally in a

wide region beginning from light spherical nuclei up to actinides [1], For example, contrary to the well known El response of stable nuclei, the Ml response in a total of 12 open-shell nuclei could only be measured in recent years through recombination of differed high- resolution probes, i.e. inelastic electron, photon and proton scattering experiments. These experiments show that in heavy spherical and deformed nuclei there exists a very broad Ml resonance at energies between 7 and 11 MeV centred around 44A'1/3 MeV. Magnetic dipole excitations in deformed nuclei have been found to give Ml strength with major peaks corresponding to K=0 and K=1 mode of V excitations. In spherical nuclei, appearance of the collective V states is connected with transitions between levels of the spin-orbit doublet [2], Recent experimental and theoretical investigations show, as a rule, that the spin part of the residual interaction in odd nuclei leads to polarization effects that influence the magnetic moments and the probabilities of the allowed /? transitions greatly. These effects cause the gs factors of the nucleons in the nucleus to differ noticeably from the corresponding values for free nucleons. This is in fact equivalent to the renormalization of the single-particle matrix elements of the odd nuclei.

In spherical even-even nuclei

T

excitations are connected with particle- hole transitions

between the neutron-neutron (proton-proton) levels of the spin-orbit doublets. In spite of spherical nuclei, in deformed nuclei the picture is more complicated, owing to the mixing of states with different j. The selection rules with respect to the / and j can be strongly violated.

Besides, in axial symmetric deformed nuclei the

T

states split into excitations characterized by the angular momentum projection on the symmetry axis K=0 and K=1 with each of them being formed independently. Strong Ml transitions can occur in the case of AK=0 and AK=1.

In this paper, we investigate the spin-vibration I nK - l+0 excitations for I50Nd and n6H f nuclei at deformed region, using the QRPA, with the deformed Wood-Saxon potential, and single particle energies and wave functions.

HAMILTONIAN

The Random Phase Approximation is a successful method to describe many-body systems. In nuclear physics, RPA has been extensively exploited to calculate intensities of various nuclear reactions, decay probabilities of electromagnetic, beta and double beta decay transitions including ground state correlations. For the nuclei away from closed shells there appear static pairing correlations between nucleons giving rise to the energy gap in the excitation spectra. In

this case one uses the quasiparticle version of RPA (QRPA). In this work QRPA method is

responsible for the investigation of I 71 K = l +0 spin-vibration states in deformed nuclei

(150< A< 178).

In QRPA method, the model Hamiltonian of the system can be written as

H = H sqp+VaT

(1)

where Hsqp is the single- quasiparticle Hamiltonian with pairing interactions, V„T takes into account the isovector spin-spin interactions. These interactions generating the 1+ states in the deformed nuclei are written as

V = V'nt' + VcoIL _{G T G T G T} (4)_{\ /}

where V^'“ generates the collective I 71 = 1+ states in even- even deformed nuclei and V"'7 takes into account the collective state-quasiparticle interactions of odd nuclei.

Assuming the generation of high lying 1+ states due to the isovector spin-spin interactions, hamiltonian of the system can be written as

H = H ^ v : iL ■ (5)

In Quasi Particle Random Phase Approximation (QRPA) collective 1+ states are considered as one-phonon excitations described by

I'T') = Q* W = ^ -I[^ ;(r)C ;(r)-!';(r)C „ (r)]|> P 0) (6)

where Q l is the phonon creation operator, | is the phonon vacuum which corresponds to the

ground state of the even-even nucleus, Xp and Yp are the two-quasiparticle amplitudes. Employing the conventional procedure of RPA and solving the equations of motion

V s q P + V a T , Q , + i= a jlQ ; ₍₇₎

we obtain the dispersion equation

D(a>t ) = l + z [ F / > Q + F / > ; )] = 0 (8) for the excitation energy a>i of 1+ states. Where

M

E» < L\ _{i=n, p (9)}

Ep = Es + Es. are two-quasiparticle energies of nucleons, a>i equals to the energy of the phonon

The most characteristic quantity of the state is the reduced Ml transition probability, which can be written in the form;

B(M 1,0+ ^ 1 +) _{[ ' E j m s s L s s g ' s s ' + Y u m W L v v S \ A}

ss' v v '

( 11)

The value of B(M1) for individual excitations indicate, as a rule the degree of collectivity of the given state. On the other hand, only qualitative conclusions concerning the degree of collectivity of all the generated levels are possible from the sum rules.

We can write the energy-weighted sum rule for Ml transition operator as;

n

3 İ

( 12)

Behavior of this fuction enables us to assess the contribution of the different states to the sum rule and to determine the region of saturation. The sum rule

X n = Y a E ss'm l ^ ( I 3 )

ss' is independent from model.

RESULTS AND DISCUSSION

In numerical calculations, the experimental value of deformation parameter was taken from Ref.[3], The Nilson single-particle energies were obtained from Warsaw deformed Wood-Saxon potential [4], All energy levels from the bottom of the potential well to 6 MeV were considered for neutrons and protons. The pair-interaction constants A and A were chosen in accordance with Soloviev [5], For the strength parameter of the isovector spin- spin interactions we used %a=40/A

MeV, which has been obtained from the magnetic moments calculations. In the calculation we

used a scheme of single-particle levels including 294 neutron and 208 proton levels. The calculations have shown all the 1+ states with K=0, up to the order of 6-7 MeV, Table I shows an example of spectroscopically observed low-lying 1+ states of 150Nd and their interpretation. Table II lists the characteristics of a number of collective l +0 states in 176Hf. The group of strongly collective 1+ states with B(M1)>0.45 is found in the energy interval of 9-11 MeV. Besides, we have calculated the magnetic dipole sum rule for a number of rare-earth nuclei. The diagrams of the transition energies a>n, of the values of B(M 1,0 —> 1), and the sum rule for the

1+ states with K=0 in the nuclei 150Nd and 176Hf are shown Fig.l and Fig. 2. The dashed line (13) shows the single-quasiparticle estimate of the sum rule. The solid curves show the saturation function x n • The 1+ states in this region contribute about 70% to the sum rule. Here, the B(M1) value may be as large as a few single-particle units. The state of this region gives the main contribution to the sum rule. As seen from Fig.3, in heavy 176Hf isotopes there are several orbital collective levels in the energy internal of a> - 4 -1 2 MeV with a large probability B(M1)=(0.45-1.4)/^7 . The group of strongly collective 1+ states is found in the energy interval of 8-10 MeV. This figure only illustrates B(M1)>0.1 /j.2n .

ACKNOWLEDGEMENTS

Dr. E. Guliyev would like to thank The Scientific and Technical Research Council of Turkey (TUBITAK) for financial support.

Table I. The properties of 150Nd nuclei for

TK =l+0 excitations in spectroscopic region

CO , M e V d G e n li k V ss’ D u r u m la r ın Y a p ıs ı 3.028 0.027 -0.032 nn 521-510 0.011 nn 640-660 -0.011 nn 640-651 -0.011 nn 521-512 -0.706 nn 523-512 0.015 pp 550-550 -0.013 pp 420-411 0.011 pp 422-411 3.677 0.041 0.011 nn 530-521 0.703 nn 521-510 0.021 nn 640-660 -0.018 nn 640-651 -0.022 nn 521-512 -0.014 nn 651-651 0.011 nn 651-642 -0.029 nn 523-512 0.025 pp 550-550 -0.026 pp 420-411 -0.014 pp 541-532 0.032 pp 422-411 4.164 0.031 -0.027 nn 521-510 -0.018 nn 640-660 0.013 nn 640-651 0.019 nn 521-512 0.010 nn 651-651 -0.019 pp 550-550 0.028 pp 420-411 0.705 pp 422-411 4.784 0.072 -0.037 nn 530-521 -0.017 nn 521-510 -0.070 nn 640-660 0.025 nn 640-651 0.096 nn 521-512 0.021 nn 651-651 -0.013 nn 651-642 0.029 nn 514-503 -0.046 nn 550-550 -0.692 pp 420-411 0.018 pp 541-532 0.019 pp 422-411

Table II. The properties of 176Hf nuclei for

r K = l+0 excitations in high energy region CD , M e V B ( M 1 ) , £ G e n li k D u r u m la r ın Y a p ıs ı 5.751 0.261 0.138 nn 640-651 0.189 nn 651-651 0.642 pp 550-550 6.898 0.117 -0.254 nn 521-501 -0.476 nn 512-752 -0.361 pp 541-532 6.913 0.116 -0.175 nn 521-501 0.520 nn 512-752 -0.138 nn 642-622 -0.337 pp 541-532 7.719 0.213 -0.584 nn 512-501 -0.201 nn 651-642 -0.209 nn 523-514 -0.137 pp 422-413 7.744 0.449 0.396 nn 512-501 -0.275 nn 651-642 -0.127 nn 642-622 -0.341 nn 523-514 -0.118 pp 550-530 0.119 pp 541-532 -0.187 pp 422-413 8.146 0.177 0.586 nn 510-501 0.298 nn 523-514 0.103 pp 532-523 8.180 0.418 -0.390 nn 510-501 -0.141 nn 651-642 0.389 nn 523-514 0.187 pp 422-651 0.172 pp 532-523 0.161 pp 413-404 8.860 0.119 -0.361 nn 532-512 0.274 pp 541-521 9.060 1.215 0.258 nn 532-512 0.110 nn 523-514 0.257 pp 541-521 0.174 pp 431-411 -0.290 pp 532-523 -0.165 pp 413-404 9.503 0.987 0.233 nn 532-523 -0.359 nn 523-503 0.382 nn 514-505 -0.227 pp 431-640 -0.133 pp 532-523 9.719 0.671 -0.130 nn 530-501 -0.621 nn 532-523 0.158 nn 514-505 -0.125 pp 431-422 9.943 0.267 0.686 nn 431-422 10.813 0.126 -0.694 pp 532-512 11.016 0.212 0.117 nn 541-521 0.685 pp 550-521

transition of 150Nd nuclei.

oc,MeV

Fig. 3. Energy diagram for B(M1) values

In 176 Hf nuclei.

0 50 w „ 100 190 MeV

Fig. 2. Energy weight sum rule for Ml

o f 176 Hf nuclei.

REFERENCES

1. L.Nordheim, Phys.Rev., 78294 (1950)

2. O. Bohr and Mottelson Nuclear Structure, Vol.l, Benjamin,New York 1969

3. S. Raman et al. Atomic Data and Nuclear Data Tables,Vol 36,1(1987)

4. J. Dudek, et al J.Phys. G4( 1978) 1543