B(E2) values of Te isotopes with even N (68-74) by means of interacting boson model-1

Abstract

The electric reduced transition probabilities B(E2)[arrow down] of even-neutron-rich ^sup 120-126^Te nuclei by means of interacting boson model-1 (IBM-1) are calculated. The R^sub 4/2^ values of the first 4^sup +^ and 2^sup +^ levels are also calculated and the U(5) symmetry for these nuclei is indicated. The transition rate R = B(E2: L^sup +^[arrow right](L-2)^sup +^)/ B(E2: 2^sup +^[arrow right] 0^sup +^) of a few low-lying quadrupole collective states is studied systematically and compared with the experimental data. The electric reduced transition probabilities of even ^sup 120-126^Te nuclei of gamma transition states of 8^sup +^[arrow right]6^sup +^, 6^sup +^[arrow right]4^sup +^, 4^sup +^[arrow right]2^sup +^ and 2^sup +^[arrow right]0^sup +^ are studied and compared with the experimental values. In addition, the systematic B(E2) values, intrinsic quadrupole moments and deformation parameters of even neutrons of N= 68-74 in Te isotopes are also studied. All the computed values from the present study are in good agreement with the experimental data. The IBM-1 formula for the reduced transition probabilities B(E2) is analytically deduced in U(5) limit for a few ground-state transitions in even ^sup 120-126^Te isotopes.

Headnote

Abstract: The electric reduced transition probabilities B(E2)[arrow down] of even-neutron-rich ^sup 120-126^Te nuclei by means of interacting boson model-1 (IBM-1) are calculated. The R^sub 4/2^ values of the first 4^sup +^ and 2^sup +^ levels are also calculated and the U(5) symmetry for these nuclei is indicated. The transition rate R = B(E2: L^sup +^[arrow right](L-2)^sup +^)/ B(E2: 2^sup +^[arrow right] 0^sup +^) of a few low-lying quadrupole collective states is studied systematically and compared with the experimental data. The electric reduced transition probabilities of even ^sup 120-126^Te nuclei of gamma transition states of 8^sup +^[arrow right]6^sup +^, 6^sup +^[arrow right]4^sup +^, 4^sup +^[arrow right]2^sup +^ and 2^sup +^[arrow right]0^sup +^ are studied and compared with the experimental values. In addition, the systematic B(E2) values, intrinsic quadrupole moments and deformation parameters of even neutrons of N= 68-74 in Te isotopes are also studied. All the computed values from the present study are in good agreement with the experimental data. The IBM-1 formula for the reduced transition probabilities B(E2) is analytically deduced in U(5) limit for a few ground-state transitions in even ^sup 120-126^Te isotopes.

Keywords: reduced transition probabilities, Te isotopes, quadrupole moments, deformation parameter, yrast state band

(ProQuest: ... denotes formulae omitted.)

INTRODUCTION

The interacting boson model-1 (IBM-1) was developed by Iachello and Arima [1, 2]. This model is essential for explaining the collective nuclear structure and successfully predicts the low-lying states. The IBM-1 effectively describes the electromagnetic transition rates in medium-mass nuclei. Only the s-boson pair and d-boson pair with angular momentum L =0 and L= 2 respectively are taken into account in the first approximation. The limiting symmetries U(5), SU(3) and O(6) with an inherent group structure are associated with this model [1, 3].

In recent years extensive experimental as well as theoretical studies were carried out to explore the even-even tellurium isotopes Te (Z=52) with special focus on the experimental data via collective models. It is an attractive challenge to search for collective properties because these exist near the magic number 50, which is found in single closed-shell Sn nuclei. Yrast states up to Ip= 8+ in Z= 52 isotones have been found as ... configurations for Z=50 closed shells. There are many experimental and theoretical studies concerning the low-lying collective quadrupole E2 excitations that occur in even-even nuclei Z=52 [4-7]. In the framework of semi-microscopic model, the B(E2) values and electric quadrupole moments of Te isotopes have been studied [8]. The two-proton core coupling model [9], dynamic deformation model [10] and interacting boson model-2 have also been used for the same purpose [11-13].

The IBM-1 model has been employed theoretically to study the intruder configuration and configuration mixing around the shell closure Z=50. The empirical spectroscopic study within the configuration mixing calculation was conducted in IBM [14, 15] and IBM-associated models such as the configuration mixing model in strong connection with shell model [16, 17], the conventional collective Hamiltonian approach [18, 19] and the microscopic energy-density function [20]. The evolution properties of even-even 100-110Pd [21] and electromagnetic reduced transition probabilities of even-even 104-112Cd [22], 102-106Pd [23], 108-112Pd [24] and 100-102Ru [25] have been studied very recently.

The reduced E2 probability B(E2[arrow up]) merely depends on the magnitude of intrinsic quadrupole moment of the nucleus, which depends on the deformation [26]. The aim of the present work is to do a microscopic study of the even-even Te isotopes within the IBM in order to obtain a comprehensive view of these isotopes in a rather simple way. The transition rates of yrast state band up to 8+[arrow right]6+ level are calculated by using the E2 transition strengths, deformation parameter and intrinsic quadrupole moment. With more data accumulated over the past 12 years, it is interesting to re-examine the situation. For this purpose, we do an extensive analysis of the low-lying structure of even 120-126Te isotopes by IBM-1. We particularly focus our attention to the analytically deduced U(5) symmetry for a yrast state transition in even 120-126Te isotopes.

METHODS

Reduced Transition Probabilities B(E2)

In the simplest version of the IBM, it is assumed that the low-lying collective states in medium and heavy even-even nuclei away from closed shells are dominated by excitation of the valence protons and the valence neutrons only (i.e. particles outside the major closed shells at 2, 8, 20, 28, 50, 82 and 126), while the closed shell core is inert. Furthermore, it is assumed that the particle configurations are coupled together, forming pairs of angular momentum 0 and 2. These proton (neutron) pairs are treated as bosons. Proton (neutron) bosons with angular momentum L = 0 are denoted by sp (sn) and are called s-bosons, while proton (neutron) bosons with angular momentum L = 2 are denoted by dp(dn) and are called d-bosons. The underlying structure of the six dimensional unitary groups U(6) of the model leads to a simple Hamiltonian, capable of describing the three specific types of collective structure with classical geometrical analogs, namely vibrational U(5), rotational SU(3) and γ-unstable O(6). Hamiltonian H can be written explicitly in terms of boson creation (St, dt) and annihilation (s, d) operators [2], such that

... (1)

where it can be written in general form as [2]:

... (2)

where ..., the total number of dboson operator; ..., the pairing operator; ..., the angular momentum operator; ... the octupole and hexadecapole operator; ..., the boson energy; and ..., the quadrupole operator. The parameters a0, a1, a2, a3 and a4 designate the strength of the pairing, angular momentum, quadrupole, octupole and hexadecapole interactions respectively between the bosons.

Even-even nuclei of low-lying levels (Li =2, 4, 6, 8.......) usually decay by E2 transition to lower-lying yrast level with Lf = Li-2. The reduced transition probabilities in IBM-1 are given for anharmonic vibration limit U(5) [27] by:

... (3)

where L is the angular momentum and N is the boson number. The boson number is half of the valence nucleons (proton and neutrons). For every isotope, the value of parameter ... (square of effective charge) can be calculated by using the experimental value B(E2) of transition 2+[arrow right]0+, and then the ... value is employed to compute the transition 8+[arrow right]6+, 6+[arrow right]4+, 4+[arrow right]2+ and 2+[arrow right]0+.

Quadrupole Moments and Deformation Parameters

The intrinsic quadrupole moments (Q0) of the nuclei can be derived as [28]:

... (4)

The upward electromagnetic quadrupole transition probability B(E2)[arrow up] is related by [29]:

... (5)

... (6)

The relationship between the value of B(E2) in unit of e2b2 and B(E2) in Weisskopf unit (W.u.) is [30]:

... (7)

where e denotes the charge of electron, b (barn) is a unit of area and A is the mass number of nucleus.

The probability B(E2)[arrow up] of transition 01+[arrow right]21+ is related to the quadrupole deformation parameter β [31] of nucleus shape in equilibrium as:

... (8)

and β can be calculated as [31]:

... (9)

where Z is the atomic number and R0 is the average radius of nucleus given by

... (10)

RESULTS AND DISCUSSION

Reduced Transition Probabilities

The reduced transition probabilities are important for the structural information regarding the nucleus. The boson number is counted as the number of collective pairs of valence nucleons and it represents the pair of valence nucleons. We have calculated the boson number, transition level and downward electric quadrupole reduced transition probabilities B(E2)[arrow down] for the yrast state bands 8+[arrow right]6+, 6+[arrow right]4+, 4+[arrow right]2+ and 2+[arrow right]0+ of even-even 120-126Te isotopes (Table 1). A correlation exists between the nuclei that show identical spectra and their valence proton numbers (Np) and neutron numbers (Nn). The numbers of valance proton Np and neutron Nn have a total N = (Np + Nn)/2 = nx + nv bosons. To find the boson number, the 132Sn doubly-magic nucleus is taken as an inert core. The reduced transition probabilities of ... transitions of even-even 120,122,124,126Te isotopes are calculated using known B(E2)[arrow down] experimental data from the ... transition. The calculated results of B(E2)[arrow down] values are also compared with the previous experimental results [32-35].

Table 2 shows the calculation results of upward electric quadrupole reduced transition probabilities B(E2)[arrow up] of the ground state bands ... of even-even 120'26Te isotopes. The intrinsic quadrupole moments Qo and deformation parameter were simply calculated from B(E2) values using the restrictive assumption about the rigid shape of a nucleus and were compared to the available values [29]. The calculated results for B(E2)[arrow up], Q0 and values were found to be consistent with those given in the literature [31-35].

R4/2 Classifications

The dynamical symmetries U(5), SU(3) and O(6) are grouped into classes. Within each class the ratio of energy levels of the first 4+ and first 2+, i.e. ... or R4/2, can be used to classify the even-even nuclei [36-37]. The yield values of R4/2 is equal to 2.00, 2.5 and 3.33 for harmonic vibrator U(5), g-unstable O(6) and axially symmetric rotor SU(3) respectively. The variation of ... versus even neutron numbers of Te isotopes for the reported experimental values of U(5), O(6) and SU(3) limits [32-35] is shown in Figure 1. We identify U(5) symmetry in even 120-126Te isotopes because their R4/2 values are ~ 2.00.

Systematic Reduced Transition Probabilities B(E2)

Using equation (1), the effective charge (a2 ) of IBM-1 is determined by normalising the experimental data ... of each isotope. From the known experimental values of transition ..., we calculate the value of parameter a22 for each isotope and use this value to calculate transitions 4+[arrow right]2+, 6+[arrow right]4+ and 8+[arrow right]6+. Values of the fitted parameter ... with error indicate the square of effective boson charge and are presented in Table 1. Figure 2 shows theoretical and experimental values of B(E2) in W.u., plotted as a function of transition level. Calculated reduced transition probabilities using IBM-1 as a function of transition level of ground-state band are distinctly separated for different numbers of neutrons, and the B(E2) values increase in an approximately linear fashion with the increasing neutron number for any specific transition. The results of the present work are in good agreement within experimental errors [32-35].

According to the U(5) limit, B(E2; 41[arrow right] 21)/ B(E2; 21[arrow right] 01) = 2(N -1)/N < 2 [27], and these ratios are equal to 1.43±0.23 and 1.34±0.40 for 124Te and 126Te respectively. The 2(N-1)/N values are equal to 1.75, 1.71, 1.67 and 1.60 for 120Te, 122Te, 124Te and 126Te respectively. Therefore, the present calculations of U(5) limit are confirmed by B(E2) ratios as B(E2; 41[arrow right] 21)/ B(E2; 21[arrow right] 01) = 2(N-1)/N < 2 [27]. A good agreement between the calculated and experimental values indicates that Te isotopes obey U(5) limit.

Quadrupole Moments and Deformation Parameter (β)

The quadrupole moment (Q) is an important property for nuclei and is defined as the deviation from the spherical charge distribution inside the nucleus. The intrinsic quadrupole moments Q0 are calculated using Eq. (2) for even-even 120-126Te nuclei and are shown in Figure 3. They are simply calculated from B(E2) values using the restrictive assumption about the rigid shape of a nucleus and are compared with previous values [31] and found to be in good agreement.

The deformation parameters (β) of nuclei with proton Z=52 and even neutron N= 6 8-74 are obtained using Eq. (9) and presented in Table 2 and Figure 4. The calculated deformation parameters using the restrictive assumption about the rigid shape of a nucleus are compared with those given in the literature [31]. From Table 2 and Figure 4, it is noted that the deformation parameter increases with increasing neutron number and the present calculation results are consistent with those from the literature [31].

CONCLUSIONS

Downward electric quadrupole reduced transition probabilities B(E2)[arrow down] for the yrast state bands from 8+[arrow right]6+, 6+[arrow right]4+, 4+[arrow right]2+ and 2+[arrow right]0+ of even-even 120-126Te isotopes by IBM-1 have been reported. Using the restrictive assumption about the rigid shape of the nucleus, the calculated quadrupole moments and deformation parameters are found to be consistent with previous results.

The calculated upward electric quadrupole reduced transition probabilities B(E2)t from ... of even-even 120'26Te isotopes are in good agreement with the adopted values. The analytic IBM-i calculation of B(E2) values of even-even `20'26Te has been performed in U(5) character.

ACKNOWLEDGEMENTS

The authors thank King Abdulaziz University for facilities support.

References

REFERENCES

1. F. Iachello and A. Arima, "Boson symmetries in vibrational nuclei", Phys. Lett. B, 1974, 53, 309-312.

2. A. Arima and F. Iachello, "Collective nuclear states as representations of a SU(6) group", Phys. Rev. Lett., 1975, 35, 1069-1072.

3. R. Kumar, S. Sharma and J. B. Gupta, "Character of quasi-bands in 150Sm using IBM", Am. J. Phys., 2010, 3, 150-154.

4. Z. Z. Li, F. Y. Liu, L. Y. Zhu, Y. S. Li and L. Niu, "Positive parity states in even-even Te isotopes", Commun. Theor. Phys., 2000, 33, 405-410.

5. S. J. Robinson, W. D. Hamilton and D. M. Snelling, "Levels and transitions in `24Te following thermal-neutron capture by `23Te and the beta decay of oriented `24Sb ", I Phys. G. Nuci. Phys., 1983, 9, 961-984.

6. A. Küçükbursa, K. Manisa, M. Çiftçi, T. Babacan and Ý. Maraþ, "IBM-1 calculations on the even-even 122-128Te isotopes", Math. Comput. Appl., 2005, 10, 9-17.

7. D. G. Ghija, G. Cata-Danil, D. Bucurescu, I. Cata-Danil, M. Ivascu, C. Mihai, G. Suliman, L. Stroe, T. Sava and N. V. Zamfir, "124Te and the E(5) critical point symmetry", Int. J. Mod. Phys. E, 2008, 17, 1453-1466.

8. V. Lopac, "Semi-microscopic description of even tellurium isotopes", Nucl. Phys. A, 1970, 155, 513-525.

9. E. Degrieck and G. V. Berghe, "Structure and electromagnetic properties of the doubly even Te isotopes", Nucl. Phys. A, 1974, 231, 141-158.

10. A. Subber, W. D. Hamilton, P. Park and K. Kumar, "An application of the dynamic deformation model to the tellurium isotopes", J. Phys. G: Nucl. Phys., 1987, 13, 161-175.

11. M. Sambataro, "A study of Cd and Te isotopes in the interacting boson approximation", Nucl. Phys. A, 1982, 380, 365-382.

12. J. Rikovska, N. J. Stone and W. B. Walters, "Dynamical symmetries in even-even Te nuclides", Phys. Rev. C Nucl. Phys., 1987, 36, 2162-2165.

13. S. Inan, N. Türkan and I. Maras, "The investigation of 130,132Te by IBM-2'', Math. Comput. Appl., 2012, 17, 48-55,

14. I. M. Ahmed, "The U(5)-O(6) properties of 102Pd, 106 Cd, 124Te and 128Xe nuclei'', Int. J. Mod. Phys. E, 2013, 22, 1350074.

15. M. Délèze, S. Drissi, J. Kern, P. A. Tercier, J. P. Vorlet, J. Rikovska, T. Otsuka, S. Judge and A. Williams, "Systematic study of the mixed ground-state and 'intruder' bands in 110, 112, 114Cd", Nucl. Phys. A, 1993, 551, 269-294.

16. P. E. Garrett, K. L. Green and J. L. Wood, "Breakdown of vibrational motion in the isotopes 110-116Cd", Phys. Rev. C, 2008, 78, 044307.

17. P. Van Isacker, S. Pittel, A. Frank and P. D. Duval, "Microscopic study of configuration mixing in the interacting boson model", Nucl. Phys. A, 1986, 451, 202-218.

18. I. Inci, D. Bonatsos and I. Boztosun, "Electric quadrupole transitions of the Bohr Hamiltonian with the Morse potential", Phys. Rev. C, 2011, 84, 024309.

19. D. Bonatsos, P. Georgoudis, D. Lenis, N. Minkov and C. Quesne, "Bohr Hamiltonian with deformation-dependent mass term", Phys. Lett. B, 2010, 683, 264-271.

20. L. Próchniak, P. Quentin and M. Imadalou, "Collective properties of stable even-even Cd isotopes" Int. J. Mod. Phys. E, 2012, 21, 1250036.

21. I. M. Ahmed, H. Y. Abdullah, S. T. Ahmed, I. Hossain, M. K. Kasmin, M. A. Saeed and N. Ibrahim, "The evolution properties of even-even 100-110Pd nuclei", Int. J. Mod. Phys. E, 2012, 21, 1250101.

22. H. Y. Abdullah, I. Hossain, I. M. Ahmed, S. T. Ahmed, M. K. Kasmin, M. A. Saeed and N. Ibrahim, "Electromagnetic reduced transition properties of even-even 104-112Cd isotopes", Indian J. Phys., 2013, 87, 571-574.

23. I. Hossain, M. A. Saeed, N. N. A. M. B. Ghani, H. Sa'adeh, M. Hussein and H. Y. Abdullah, "Electromagnetic reduced transition properties of the ground state band of even-even 102-106Pd isotopes by means of interacting boson model-I", Indian J. Phys., 2014, 88, 5-9.

24. I. Hossain, H. Y. Abdullah, I. M. Ahmad and M. A. Saeed, "B(E2) value of even-even 108-112Pd isotopes by interacting boson model-1", Chinese Phys. C, 2014, 38, 024103.

25. I. Hossain, I. M. Ahmed, F. I. Sharrad, H. Y. Abdullah, A. D. Salman and N. Al-Dahan, "Yrast states and B(E2) values of even 100-102Ru isotopes using interacting boson model (IBM-1)", Chiang Mai J. Sci., 2015, 42, 996-1004.

26. K. H. Bhatt, C. W. Nestor Jr. and S. Raman, "Do nucleons in abnormal-parity states contribute to deformation?", Phys. Rev. C, 1992, 46, 164-170.

27. O. Scholten, F. Iachello and A. Arima, "Interacting boson model of collective nuclear states III. The transition from SU(5) to SU(3)", Ann. Physics, 1978, 115, 325-366.

28. M. A. El-Khosht, "Transition rates in level structure of doubly even ytterbium isotopes", IL Nuovo Cimento, 1993, 106, 875-883.

29. T. Venkova and W. Andrejtscheff, "Transition strengths B(E2) in the yrast bands of doubly even nuclei", Atom. Data Nucl. Data Tabl., 1981, 26, 93-136.

30. K. Schreckenbach, A. Mheemeed, G. Barreau, T. von Egidy, H. R. Faust, H. G. Börner, R. Brissot, M. L. Stelts, K. Heyde, P. van Isacker, M. Waroquier and G. Wenes, "The importance of intruder states in 114Cd", Phys. Lett. B, 1982, 110, 364-368.

31. S. Raman, C. W. Nestor Jr. and P. Tikkanen , "Transition probability from the ground to the first-excited 2+ state of even-even nuclides", Atom. Data Nucl. Data Tabl., 2001, 78, 1-128.

32. K. Kitao, Y. Tendow and A. Hashizume, "Nuclear data sheets for A = 120", Nucl. Data Sheets, 2002, 96, 241-390.

33. T. Tamura, "Nuclear data sheets for A = 122", Nucl. Data Sheets, 2007, 108, 455-632.

34. J. Katakura and Z. D. Wu, "Nuclear data sheets for A = 124", Nucl. Data Sheets, 2008, 109, 1655-1877.

35. J. Katakura and K. Kitao, "Nuclear data sheets for A = 126", Nucl. Data Sheets, 2002, 97, 765-926.

36. F. Iachello and A. Arima, "The Interacting Boson Model", Cambridge University Press, Cambridge, 1987, pp.52-54.

37. A. Shelley, I. Hossain, F. I. Sharrad, H. Y. Abdullah and M. A. Saeed, "Yrast states and electromagnetic reduced transition properties of Te by means of interacting boson model-1'', Probl. Atom. Sci. Technol., 2015, 97, 38-43.

© 2016 by Maejo University, San Sai, Chiang Mai, 50290 Thailand. Reproduction is permitted for noncommercial purposes.