In the frame work of the concept of softness parameter ? of nucle

Gado, K. A.

Madani, J. H.

Nafie, H. O.

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International Journal of Sciences: Basic and Applied Research (IJSBAR)ISSN 2307-4531 http://gssrr.org/index.php?journal=JournalOfBasicAndAppliedYrast Band Description of 150Sm, 152Sm, 154Gd and 192Os Nuclei Using VAVMNS Model H. O. Nafiea, J. H. Madanib, and K. A. Gadoc*a,b,c Department of physics, Faculty of science, Tabuk University, KSA Abstract In the frame work of the concept of softness parameter  σ of nuclei  a new formulation of the variable a harmonicvibrator model (VAVM)  is  formulated , which is denoted by ( VAVMNS)  The model  is used  in calculating  the energies of rotational ground bands of 150Sm,152Sm, 154Gd and 192Os  nuclei . The predicted results of the (VAVMNS) are in close agreements with experimental data and other theoretical ones.  Key words: rotational bands; variable a harmonic variable moment of inertia (VAVM); angular momentum, softness parameter (σ).  1. Introduction:Many models have achieved a significant success for the representations of the energies of ground bands of well deformed nuclei, like the variable moment of inertia model (VMI) [1-3] and variable anharmonic variable moment of inertia model (VAVM) [4]. Generalized variable of inertia model (GVMI) [4, 5]. Cranking model [6] and its phenomenological equivalent [7, 8, 9]. The nuclear softness (NS) is well accepted concept in nuclear study especially in even- even nuclei. This concept is used through the softness parameter 𝜎𝜎  which is the relative increase of moment of inertia with angular momentum J. Batra and Gupta [10, 11] have extended the NS model concept to the VAVMNS model The purpose of this work is to extend the concept of NS to the VAVM model (referred as VAVMNS). The VAVMNS model is tested by applying it on the chosen nuclei which gives good results compared with experimental data and other theoretical ones. 2. Theory:The ground state rotational band of deformed nuclei was described by the semi classical formula [1]: 𝐸𝐸(𝐽𝐽) = ℏ22𝜃𝜃(𝐽𝐽) 𝐽𝐽(𝐽𝐽 + 1) … … (1) ------------------------------------- * Corresponding author. Tel.: +966-(0)535646329E-mail address: jado76@yahoo.com 11  International Journal of Sciences: Basic and Applied Research (IJSBAR) (2014) Volume 13, No  1, pp 11-17Where 𝜃𝜃(𝐽𝐽)  is the nuclear  moment of inertia, and   𝐽𝐽  is the angular momentum, which takes the sequence 𝐽𝐽𝜋𝜋 =2+, 4+, 6+, …  For higher spin states, Eq. (1) deviates from experimental data, this deviation may be attributed to the change in moment of inertia 𝜃𝜃(𝐽𝐽) So, the energy is written as: 𝐸𝐸(𝐽𝐽) = ℏ22𝜃𝜃(𝐽𝐽) 𝐽𝐽(𝐽𝐽 + 1) + 12𝐶𝐶(𝜃𝜃(𝐽𝐽) − 𝜃𝜃0)2 … … (2) Where the first term represents the rotational energy and the second represents the vibrational energy. Eq.(2)  is well known variable moment of inertia model (VMI)Batra and Gupta [11] have   reformulated the VMI   in terms of nuclear softness parameter which is called VMINS model. Klein et al [4]; developed the (VMI) by proposed VAVM 𝐸𝐸(𝐽𝐽) = 𝐴𝐴𝐽𝐽 + 𝐽𝐽(𝐽𝐽 − 2)𝜃𝜃(𝐽𝐽) + 12𝐶𝐶(𝜃𝜃(𝐽𝐽) − 𝜃𝜃0)2 … … (3) where A, C and  θο  are fitting parameters. Also. Klein et al; [2] proposed the generalized variable moment ofinertia model (GVMI) which written as: 𝐸𝐸(𝐽𝐽) = 𝐽𝐽 + 𝑥𝑥𝐽𝐽(𝐽𝐽 − 2)𝜃𝜃(𝐽𝐽) + 12𝐾𝐾(𝜃𝜃(𝐽𝐽) − 𝜃𝜃0)2 … …     (4) Where 𝑥𝑥  , 𝜃𝜃0 and 𝐾𝐾 are parameters. For 𝑥𝑥 = 1/3 GVMI model Eq.(4) reduces to VMI  model  Eq.(2). Since the formulation of VMI model in terms of nuclear softness (VMINS) model, which is deduced by Batra and Gupta [11] gives good agreement with experimental data and since VAVM model is special case of generalized model (GVMI), this encourages one to formulate VAVM model in terms of softness parameter. By using Taylor expansion the moment of inertia as: 𝜃𝜃(𝐽𝐽) = 𝜃𝜃0(1 + 𝜎𝜎1𝐽𝐽 + 𝜎𝜎2𝐽𝐽2 + 𝜎𝜎3𝐽𝐽3 + ⋯ ) … …  (5) Where 𝜎𝜎𝑛𝑛  is the softness parameter  𝜎𝜎𝑛𝑛 = 1𝑛𝑛! 𝛿𝛿𝑛𝑛𝜃𝜃(𝐽𝐽)𝛿𝛿𝐽𝐽𝑛𝑛 �𝐽𝐽=0 … …  (6) 𝑛𝑛 = 1, 2,3, … Takes only 𝑛𝑛 = 1, the first approximation, and substitute in Eq (3), one gets 𝐸𝐸(𝐽𝐽) = 𝐴𝐴𝐽𝐽 + 𝐽𝐽(𝐽𝐽 − 2)𝜃𝜃0(1 + 𝜎𝜎1𝐽𝐽) + 12𝐶𝐶(𝜃𝜃0𝜎𝜎1𝐽𝐽)2 … … (7) 12  International Journal of Sciences: Basic and Applied Research (IJSBAR) (2014) Volume 13, No  1, pp 11-17Using the experimental excitations energies 𝐸𝐸(2), 𝐸𝐸(4),𝐸𝐸(6) and 𝐸𝐸(8) and the Equation (7) one can find the parameters 𝜎𝜎1,𝜃𝜃0,𝐴𝐴 and 𝐶𝐶 as follows: 𝜎𝜎1 = � 9𝐸𝐸(2) − 9𝐸𝐸(4) + 3𝐸𝐸(6)3𝐸𝐸(4) − 𝐸𝐸(2) + 𝐸𝐸(8) − 3𝐸𝐸(6) − 3�24 … …  (8) 𝜃𝜃0 = −48𝜎𝜎1[3𝐸𝐸(2) − 3𝐸𝐸(4) + 𝐸𝐸(6)][1 + 4𝜎𝜎1][1 + 6𝜎𝜎1] …  … (9)𝐴𝐴 = 124 � 48𝜃𝜃0(1 + 8𝜎𝜎1) + 16𝐸𝐸(2) − 𝐸𝐸(8)�… …  (10) 𝐶𝐶 = 12 �𝐸𝐸(2) − 2𝐴𝐴(𝜃𝜃0𝜎𝜎1)2 �…  … (11) 3. Results and Discussion:Using the experimental excitation energies E(2), E(4), E(6) & E(8) and Equations (8,9,10 &11), the parameters  𝜎𝜎1,𝜃𝜃0,𝐴𝐴 and 𝐶𝐶   are calculated.    Using Eq (7) “VAVMNS model “ and  the given parameters  𝜎𝜎1,𝜃𝜃0,𝐴𝐴 &𝐶𝐶 ,we predicted the energies for chosen nuclei  150Sm ,152Sm ,154Gd and 192Os .which is listed in table (2). The deviation from experimental data 𝜏𝜏 = 1𝑁𝑁∑ �𝐸𝐸𝐶𝐶𝐶𝐶𝐶𝐶 − 𝐸𝐸𝑒𝑒𝑥𝑥𝑒𝑒 �2𝑁𝑁𝑖𝑖=1 .  Table (1): The fitting parameters of VAVMNS model as in Eq. (7) Nucleus Parameters  σ1 θ0 A C × 10−4 150Sm 0.261 60.024 0.157 0.412 152Sm 0.068 67.091 0.054 3.490 154Gd 0.049 58.304 0.059 3.087 192Os 0.068 52.231 0.091 9.564 The calculated results for the ground state rotational bands are given systematically in table 2. From this table we notice that the calculations are carried out for 150Sm,152Sm, 154Gd and 192Os  nuclei whose yrast bands are observed experimentally up to 𝐽𝐽𝜋𝜋 = 16+for 150Sm, up to 𝐽𝐽𝜋𝜋 = 14+ for 152Sm, up to 𝐽𝐽𝜋𝜋 = 18+   for  154Gd and 𝐽𝐽𝜋𝜋 = 12+ up to for 192Os nuclei. And the energies calculated according to (VAVMNS) model in comparison with experimental data [14] and the energies calculated by VAVM model for the chosen nuclei. As can be seen, the results are excellent for all nuclei, in the vast majority nuclei, results of VAVMNS better than those predicted by the VAVM model.  So from this table we notice that the agreements between the calculated and observed data are excellent.   The present study can also be useful in study the third term of equation (7) i.e. potential energy term with spin of the nucleons and with ground state of moment of inertia. The variation of potential energy term based on the valence particle and hole pairs consideration [8] with spin has been shown in fig. (1), it is clear that the value of potential energy term is increasing almost linear with increasing spin for all nuclei. It is apparent from fig. (2), that the potential energy term is decreases almost exponentially with ground state of moment of inertia. 13  International Journal of Sciences: Basic and Applied Research (IJSBAR) (2014) Volume 13, No  1, pp 11-17Table (2):  Experimental and Theoretical Energies in (Mev) of the Yrast band s of 150Sm, 152Sm, 154Gd and 192Os nuclei. 150Sm Nucleus Spin 𝐽𝐽𝜋𝜋  𝐸𝐸𝑒𝑒𝑥𝑥𝑒𝑒𝑒𝑒𝑒𝑒𝑖𝑖𝑒𝑒𝑒𝑒𝑛𝑛𝑒𝑒𝐶𝐶𝐶𝐶   (Mev) 𝐸𝐸𝑉𝑉𝐴𝐴𝑉𝑉𝑉𝑉  (Mev) 𝐸𝐸𝑉𝑉𝐴𝐴𝑉𝑉𝑉𝑉𝑁𝑁𝑉𝑉 (Mev) 2+ 4+ 6+ 8+ 10+ 12+ 14+ 16+ 0.340 0.774 1.279 1.837 2.432 3.043 3.646 4.305 0.340 0.774 1.279 1.837 2.417 3.034 4.676 4.340 0.340 0.774 1.279 1.837 2.443 3.093 3.787 4.523 Mean Deviation 0.019 0.052 152Sm Nucleus Spin 𝐽𝐽𝜋𝜋  𝐸𝐸𝑒𝑒𝑥𝑥𝑒𝑒𝑒𝑒𝑒𝑒𝑖𝑖𝑒𝑒𝑒𝑒𝑛𝑛𝑒𝑒𝐶𝐶𝐶𝐶 (Mev) 𝐸𝐸𝑉𝑉𝐴𝐴𝑉𝑉𝑉𝑉 (Mev) 𝐸𝐸𝑉𝑉𝐴𝐴𝑉𝑉𝑉𝑉𝑁𝑁𝑉𝑉  (Mev) 2+ 4+ 6+ 8+ 10+ 12+ 14+ 0.122 0.367 0.707 1.125 1.609 2.149 2.736 0.122 0.366 0.698 1.120 1.591 2.111 2.671 0.122 0.367 0.707 1.125 1.610 2.152 2.747 Mean Deviation 0.056 0.001 14  International Journal of Sciences: Basic and Applied Research (IJSBAR) (2014) Volume 13, No  1, pp 11-17154Gd Nucleus Spin 𝐽𝐽𝜋𝜋  𝐸𝐸𝑒𝑒𝑥𝑥𝑒𝑒𝑒𝑒𝑒𝑒𝑖𝑖𝑒𝑒𝑒𝑒𝑛𝑛𝑒𝑒𝐶𝐶𝐶𝐶  (Mev) 𝐸𝐸𝑉𝑉𝐴𝐴𝑉𝑉𝑉𝑉  (Mev) 𝐸𝐸𝑉𝑉𝐴𝐴𝑉𝑉𝑉𝑉𝑁𝑁𝑉𝑉  (Mev) 2+ 4+ 6+ 8+ 10+ 12+ 14+ 16+ 18+ 0.123 0.371 0.718 1.145 1.637 2.185 2.778 3.405 4.017 0.123 0.371 0.718 1. 1401.622 2.154 2.730 3.343 3.992 0.123 0.371 0.718 1.145 1.638 2.187 2.784 3.422 4.098 Mean Deviation 0.026 0.008 192Os Nucleus Spin 𝐽𝐽𝜋𝜋  𝐸𝐸𝑒𝑒𝑥𝑥𝑒𝑒𝑒𝑒𝑒𝑒𝑖𝑖𝑒𝑒𝑒𝑒𝑛𝑛𝑒𝑒𝐶𝐶𝐶𝐶 (Mev) 𝐸𝐸𝑉𝑉𝐴𝐴𝑉𝑉𝑉𝑉  (Mev) 𝐸𝐸𝑉𝑉𝐴𝐴𝑉𝑉𝑉𝑉𝑁𝑁𝑉𝑉  (Mev) 2+ 4+ 6+ 8+ 10+ 12+ 0.206 0.580 1.089 1.708 2.411 3.212 0.206 0.580 1.089 1.700 2.393 3.154 0.206 0.580 1.089 1.708 2.424 3.224 Mean Deviation 0.031 0.002 15  International Journal of Sciences: Basic and Applied Research (IJSBAR) (2014) Volume 13, No  1, pp 11-17Fig. (1): The variation of potential energy term with spin.  Fig. (2): The variation of potential energy term with ground state moment of    inertia. 00.020.040.060.080.10.120.140.160.180.20 2 4 6 8 10 12 14 16 18 20Poten. Energy (Mev)Spin150Sm152Sm154Gd192Os00.0020.0040.00625 30 35 40 45 50 55 60 65 70Poten. Energy (Mev)Ground state moment of inertia16  International Journal of Sciences: Basic and Applied Research (IJSBAR) (2014) Volume 13, No  1, pp 11-174. ConclusionThe present VAVMNS model Eq. (7) is practically fit to predict the ground state rotational bands of deformed even-even nuclei, and can also be applied to other nuclei. It includes three parameters which are determined straight forward using Equations 8, 9, 10 and 11. On the other hand, we observe that correlation between potential energy and spin, moment of inertia. The results of our calculation show good agreement with experimental data. References [1] M. A. J. Mariscotti, G. Scharfrf-Goldhaber and B. Buck. (1969).”Phenomenological Analysis of Ground state Bands in Even-Even Nuclei” Phys. Rev. Lett.Vol. 178, No 4 pp1864-1868. [2] A. Klein. (1980).” Perspective in the theory of nuclear collective motion” Nucl. Phys. A Vol. 347, pp. 3-30.  [3] G. Scharff-Goldhaber,C. Dover and A. L. Goodman. (1976).” Model and Theories of nuclear collective motion” Annu Rev.Nucl. Soci Vol. 26 pp 239. [4] D. Bonatsos and A. Klein. (1984). “Generalized Phenomenological models of yrast band”  Phys.Rev.C Vol. 29 pp 1879.  [5] S. M. Harris. (1965).” Higher Order Corrections to the Cranking Model “ Phys. .Rev., 138B, pp. 509-513. [6] M. J. A. De Voigt, J. Dudck, and Z. Szymanski. (1983) “ High-spin phenomena in atomic nuclei”  Rev. Mod.Phys. Vol. 55, No. 4 pp. 949.  [7] M. I. El Zaiki, H.O Nafie. and K. E. Abd El Maged. (1992) “Phenomenological descriptions of the yrast bands in 160, 162, 164, 166 Yb nuclei band crossing and moment of inrtia” I. J. of Pure and applied Phys. Vol. 30, pp.113.  [8] H.O Nafie. (1994) “ Yrast band description for some Even-Even nuclei by new formulation of GVMI model” Egypt J. Phys. Vol. 25 No 1-2 ,pp. 65-72.    [9] H. Moringa (1966).” Rotational bands of well deformed nuclei studied form gamma rays following (𝛼𝛼, 𝑥𝑥𝑛𝑛)reactions” Nucl. Phys.Vol. 75, pp.385.  [10] R. K. Gupta (1971). “Nuclear-softness model of Ground state Bands in even-e ven nuclei “ Phys Rev. Lett. Vol. 36B, No. 3 pp. 173.  [11] J. S. Batra and R. K.  Gupta (1991). “Determination of the variable moment of inertia model in terms of nuclear softness” Phys. Rev.C Vol.43 pp. 1725. [12] A. Klein (1980). “Rotation of variable moment of inertia (VMI) concept with the interacting model” Phys. Lett. B Vol.  93No. 1, pp 1-6.  [13] F. Iachello (1979, June).” Interacting Bosons in Nuclear Physics” (1st Edition.) Plenum press, New York.  [14] D. Bonatsos and A. Klein (1984).”Energies of Ground-state bands of even-even Nuclei from generalized variable moment of inertia models” Nucl. Data Tables Vol.  30, pp. 27. 17 

Yrast Band Description of 150Sm, 152Sm, 154Gd and 192Os Nuclei  Using VAVMNS Model

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Yrast Band Description of 150Sm, 152Sm, 154Gd and 192Os Nuclei Using VAVMNS Model

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