Backbending Phenomena for Some Even Nuclei in Medium Light Region

Transcription

1 nternational Journal of Sciences: Basic and Applied Research (JSBAR) SSN (Print & Online) Backbending Phenomena for Some Even Nuclei in Medium Light Region Alaseri S. M. a*, Bakeer J. H. b, Nafie H. O. c a,b Physics Department, Faculty Of Science, University of Tabuk ( KSA) c Physics Department, Faculty Of Science, University of Benha (Egypt) a sh_7ma@hotmail.com b jhasan@ut.edu.sa c hnafei@yahoo.com Abstract The backbending phenomena for the nuclei Se 7, S 7,Kr 7,Kr 76 Zn 6,Ge 6 and Ge 66 is studied in the frame work of Broken polynomial model (BM) and Exponential model (EXPM).The predicted results of the two models are compared with the experimental data. The two models describe the backbending phenomena in good manner. But the BM model describes the phenomena in finest one comparing with EXPM model. Keywords: rotational bands; backbending; angular momentum; softness parameter. 1. ntroduction The are many attempts are carried to explain the backbending phenomena in deformed nuclei [1,,3,] to describe the rapid change of the moment of inertia (JM) versus the square of angular momentum ( ω). There are many reasons for this phenomena like as rotation alignment [5], pairing collapse [1], centrifugal stretching and etc., the change of the JM against ( ω ). Reaches its maximum where the backbending occurs. On the reduction of pairing correlation Sood P. C and Jain A * Corresponding author. 19

2 Authors in reference [1] proposed rotational energy levels of the ground band the exponential form as: 1 ( ) ( 1)exp E = + 1 o ϕ o c Whereϕ, o and o are free parameters. This formula fitted well the experimental in deformed region c ( 15 A 19 ) []. The authors in [5] improved the previous equation to be as: ( ) ( 1)exp E = + 1 ϕ o o c 1 ν Where introduce the new parameter ν. R. K. Gupta [,3] proposed that moment of inertia as broken polynomial n this article we calculated, the moment JMP(), energy levels EP () and the square angular frequency ( ω),the predicting results JMP(),Ep(),and ( ω) are tabulated in Table (1-b) and Table (- a),the fitted parameters are tabulated in Table (1-a) and Table (-a). Also we plotted the moment of inertia JM()/JMP() versus ( ω) in figures 1 and. Methods of calculations We carried out the calculations on the chosen nuclei which are Se 7, S 7,Kr 7,Kr 76 Zn 6,Ge 6 and Ge 66 by using the following two methods: 1. Broken Polynomial Model: For ground-state bands of even-even nuclei, the γ-transition energy is characterized as E transition, where: ( ) = E( ) E( ) E (1) The angular frequency ω and the moment of inertia are defined by the following equations [,3]: ω = ( ) ( + 1) E, θ = E( ) [ ( + 1) ] () ( ω) = + 1 ( ) E( ) E 1 ω θ = J, J = 1 [ + ( + 1) ( )( 1)] (3) One can reproduce both the known features of the spectra and the expected asymptotic behavior by broken expressions in terms of the angular momentum, 15

3 ( ) = P[ ( + 1) ] Q[ ( + 1) ] E or ω = J φ = P J q J () Where P, Q, p, q are assumed to be polynomials with few terms. n systematic fits to the data with Equation () it is suitable to choose the coefficients in the numerator polynomial as varied parameters, while the denominator polynomial is used to achieve a prescribed asymptotic behavior. The following expression is considered: J φ a + bj + cj (5) = = ω kj 1+ Where k is a positive constant and l an even integer. Equation (5) is the simplest possible polynomial ratio which has three linear parameters (a, b, and c) and fulfils the requirements for both low and high spins. We choose to extent the application of broken polynomial expression in terms of the angular momentum for describing the energy levels up to spin values >3 +, including the crossing in the yrast band.. Exponential Model We also are calculated the ground state of rotational band of chosen by using the exponential model EXPOM [1,.5] Which is written as: E( ) 1 ν = ( + 1) exp. 1 (6) ϕ c Where, ϕ. ν and c are fitting parameters. 3. Results and Discussion The experimental moment of inertia JM () and square angular frequency ( ω) Are calculate as Equation () in the previous section. The predicted energy levels EP(),and predicted moment of inertia JMP() ' for the chosen nuclei (subject of this article) are calculated as predicted upon the bases of the two methods, BM and EXPM as explained in the last section. Broken polynomial model equation (5) as mentioned in the methods of calculations section was used to calculated the energy levels/moment of inertia (EP()/JMP()) and was compared with the experimental data (E()- value / JM() value) as mentioned in the experimental references [1,11]. The results are tabulated in Table (1).The moment of inertia versus ( ω) are plotted in Figure (1). 151

4 By similar manner the energy levels and moment of inertia, square angular frequency are calculated using EXPOM exponential model Equation (6).The results are tabulated in Table, and also, The moment of inertia versus ( ω) are plotted in Figure (). The mean deviation as noted in the last row, column three in tables table (1-b) and table (-b) is calculated by the formula deviation = ( E ( ) EP( ) )/ N Similarly; the mean deviation as noted in the last row, column fifth in tables table(1-b) and table (-b) is calculated by the formula deviation ( JM ( ) JMP( ) )/ N taken data values and takes,,6,--,--.n. =, where N is the number of the The values of the parameters and a,b,c L and K as given in Equation (5) are obtained by fitting experimental energies (E) using best fit method, and the obtained values of the parameters are listed in table (1-a). Also, the parameters A=, B=, C and v for exponential model Equation (6), the parameters are given by the ϕ same method and tabulated in table (-a). The chosen nuclei are around mass number A 6, for Se 7 the energy levels EP() and moment of inertia JMP() are calculated up to + =8 for Se 7 up to + =, For Kr 7 up to + =3, for Kr 76 up to + =, for Zn 6 up to + =8 for Ge 6 up to + =8 and for Gn 66 up to + =16. We may note that the calculation include the backbending /up bending region for the chosen nuclei in this article. Table 1-a: The parameters of Broken polynomial model as in Equation (5) Nucleus a b c L k Se Se Kr E- 5E-9 Kr Zn Ge Ge E-3 E-3 15

5 Table (1-b): the experimental energy E(),the predicted energy EP() as Equation (5) in Mev,experimental moment of inertia JM() and predicted moment of inertia JMP() in (Mev -1 ) and ( angular frequency (Mev ) ) () the square of + (ħω()) E() EP() JM() JMP() Se Deviation E E-7 Kr E E Deviation E-6 Kr

6 Deviation E() EP() JM() JMP() (ħω()) E E- 3.93E-6 + (ħω()) E() EP() JM() JMP() Zn Deviation E-7 Se

7 Deviation E-6 Ge 6 + E() EP() JM() JMP() () Deviation E E-6 Ge 66 + E() EP() JM() JMP() () Deviation 7.19E E-6 155

8 Figure 1: The moment of inertia " J" versus square angular frequency" (hω) " for chosen nuclei dashed line for predicted values Equation 5 and "O" for experimental values 156

9 Table -a: parameters for exponential Model Equation (6) Nucleus A= B= ϕ S E C ν S E Kr E Kr E Zn E Ge E Ge E *The parameters C and ν are taken from ref.5 Table (-b): the experimental energy E(),the predicted energy as.equation (6), EP() in Mev,experimental moment of inertia JM() and predicted moment of inertia JMP) in (Mev -1) and square of angular frequency(ħω()) (Mev ) Se

10 Deviation Se Deviation Kr E E

11 Deviation, Kr E E Deviation E Zn

12 Deviation E E- Ge Deviation.89659E Ge Deviation E E- 16

13 Figure (): The moment of inertia " J" versus square angular frequency" (hω) " for chosen nuclei dashed line for predicted values Equation 6 and "O" for experimental values From figures 1 and, it is clear that the two models Equation (5) and Equation (6) described well the behavior of the moment of inertia JMP () versus the square angular frequency (h ) including the back bending/up bending phenomena for the nuclei under consideration. 161

14 Also by comparing the tabulated data E(), EP(),JM() and JMP() in table (1-b) and table(-b) and the deviations in the last raw in the same tables we noted that the broken polynomial Equation (5) is better comparing with experimental data than the exponential model Equation (6).. Conclusion The present models broken polynomial Equation (5),and exponential Equation (6) are predict the yrast state rotational bands of the deformed chosen nuclei, and can also describe well the behaviors of moment of inertia versus square angular (h ),but the pre compared with the results of exponential model Equation (6). Acknowledgment The authors would like to acknowledge financial support for this work, from the Deanship of Scientific Research (DSR), University of Tabuk (Tabuk, Saudi Arabia, under grant no S ) References [1] P.C.Sood and A.K.Jain " Expoential Model with pairing attenuation amd backbending phynomenon " Phys.Eev.C18,196-(1978) [] R. K. Gupta (1971). Nuclear-softness model of Ground state Bands in even-even nuclei Phys Rev. Lett. Vol. 36B, No. 3 pp [3] J. S. Batra and R. K. Gupta (1991). Determination of the variable. [] S.U.El-Kamesy,H.H.Alharabi, and H.A.Alhndi"Backbending phenomena in light nucleiat A 6 mass region" arxiv;nucl-th/5915v1 7 Sep 5. [5] H.H.Alharbi,H.A.Alhend,and S.U.El-Kamessy. "Nuclear Structure Of Some Actinide Nuclei" arxiv;nuclth/517v1 6 Feb 5. [6] D. Bonatsos and A. Klein. (198). Generalized Phenomenological models of yrast band Phys.Rev.C Vol. 9 pp [7] D. Bonatsos and A. Klein (198). Energies of Ground-state bands of even-even Nuclei from generalized variable moment of inertia moment of inertia model in terms of nuclear softness Phys. Rev.C Vol.3 pp. 175 [8] J.H.Bakeer and S.M.Alaseri (1) Description of Rotational Bands for Some Even-Even\Nuclei in Actinide Region " JSBAR,P [9] Klein. (198). Perspective in the theory of nuclear collective motion Nucl. Phys. A Vol. 37, pp [1] Klein (198). Rotation of variable moment of inertia (VM) Concept with the interacting model Phys. Litt.B. Vol. 93No. 1, pp 1 Edition.) Plenum press, New York. models Nucl. Data Tables Vol. 3, pp