J. Nig. Soc. Phys. Sci. 2 (2020) 250–256 Journal of the Nigerian Society of Physical Sciences Alpha decay half-lives of 171−189Hg isotopes using Modified Gamow-like model and temperature dependent proximity potential W. A. Yahya∗ Department of Physics and Materials Science, Kwara State University, Malete, Kwara State, Nigeria Abstract The alpha decay half-lives for 171−189Hg isotopes have been computed using the Gamow-like model (GLM), modified Gamow-like model (MGLM1), temperature-independent Coulomb and proximity potential model (CPPM), and temperature-dependent Coulomb and proximity potential model (CPPMT). New variable parameter sets were numerically calculated for the 171−189Hg using the modified Gamow-like model (termed MGLM2). The results of the computed standard deviation indicates that the modified Gamow-like model (MGLM2) and the temperature- dependent Coulomb and proximity potential model give the least deviation from available experimental values, and therefore suggests that the two models (MGLM2 and CPPMT) are the most suitable for the evaluation of α-decay half-lives for the Hg isotopes. DOI:10.46481/jnsps.2020.139 Keywords: Alpha decay, half-life, Gamow-like model, proximity potential, Radioactive decay Article History : Received: 26 September 2020 Received in revised form: 04 November 2020 Accepted for publication: 05 November 2020 Published: 15 November 2020 c©2020 Journal of the Nigerian Society of Physical Sciences. All rights reserved. Communicated by: B. J. Falaye 1. Introduction Alpha decay, first discovered in 1899 by Rutherford, is one of the crucial decay modes for heavy nuclei [1]. It is one of the important decay modes to describe nuclear structure [2] and it can be successfully described by quantum theory [3]. Alpha decay (α-decay) is a crucial decay mode that provides infor- mation about the nuclear structure and stability of heavy and superheavy nuclei [4]. It is important in the identification of new heavy and super heavy nuclei (SHN) [5], and in the study of nuclear structure and nuclear force [6]. Investigations of the α-decay half-lives have been carried out both theoretically and ∗Corresponding author tel. no: +2348036289110 Email address: wasiu.yahya@gmail.com (W. A. Yahya ) experimentally via various approaches. Some of the theoreti- cal models that have been employed to study the α-decay half- lives are the fission-like model [7], the generalised liquid drop model [8, 9, 10], the effective liquid drop model [11], the modi- fied generalized liquid drop model [6, 12, 13], the Coulomb and proximity potential model [14, 15, 16], the Gamow-like model [1, 4], and the preformed cluster model [17, 18]. Royer [19] developed an analytical formula to calculate the α-decay half-lives by applying a fitting procedure on a set of 373 nuclei. The universal decay law has also been developed by Qi et al. [20, 21]. They introduced the new universal decay law (UDL) to study α and cluster decay modes. The authors made use of the α-like R-matrix theory and the microscopic mecha- 250 W. A. Yahya / J. Nig. Soc. Phys. Sci. 2 (2020) 250–256 251 nism of the charged-particle emission. A generalization of the Viola-Seaborg formula was given by Ren et al. [22] for cluster radioactivity half-lives. Modified versions of the Ren formu- las, called new Ren A and new Ren B, have also been presented [23]. The new Ren A included nuclear isospin asymmetry while new Ren B included both nuclear isospin asymmetry and angu- lar momentum. Gharaei and Zanganeh [24] have studied the half-lives of cluster decay for some isotopes using temperature dependent proximity potential (with the prox. 2010 and prox. Zheng po- tentials). They found that the width and height of the Coulomb barrier in the temperature-dependent proximity potential are less than its temperature-independent version. Recently, the modi- fied Coulomb and proximity potential model was employed to study the α-decay of 186−218Po [25]. Zdeb et al. [1] proposed a phenomenological model which is based on the Gamow the- ory for the calculation of α-decay half-lives. In the Gamow-like model, the square well potential is chosen as the nuclear poten- tial, while the potential of a uniformly charged sphere is taken as the Coulomb potential. In Ref. [4], the authors presented the α decay half-lives of nuclei with Z > 51 (up to Z = 120) using a modified form of the Gamow-like model. Most of the isotopes of mercury (Hg) are radioactive with half-lives less than a day. Seven of the isotopes are stable. The Hg radioisotope with the longest half-life is 194Hg, with a half- life of 444 years. Some of the applications of Hg isotopes are: in medicine, in nuclear gyroscopes and magnetometres. The radioisotope 197Hg, for example, is useful for diagnostics of kidney and cerebral diseases [26]. The α-decay half-lives of some Astatine isotopes have been studied in Ref. [27] using a modified Coulomb and proximity potential model with one adjustable parameter. The α-decay half-lives of some mercury isotopes have also been studied in Ref. [28] using about 25 dif- ferent versions of the nuclear potential. In this study, the Gamow-like model, its modified version (the modified Gamow-like model), the temperature-independent Coulomb and proximity potential model (CPPM) and the temperature- dependent Coulomb and proximity potential model (CPPMT) have been employed to calculate the α-decay half-lives of 171−189Hg isotopes. The results are compared with the available experi- mental data. The article is organised as follows. The modi- fied Gamow-like model, the temperature-independent Coulomb and proximity potential model (CPPM) and the temperature- dependent Coulomb and proximity potential model (CPPMT) are described in Section 2 for the calculation of α-decay half- lives. The results are presented and discussed in Section 3 while the conclusion is given in Section 4. 2. Theory 2.1. Modified Gamow-like model In the modified Gamow-like model, the interaction potential between the alpha particle and daughter nucleus is given by [4]: V (r) = { −V0, 0 ≤ r ≤ R VH (r) + V`(r), r ≥ R (1) where V0 is the depth of the square well, the Hulthen type of screened electrostatic Coulomb potential VH = aZ1Z2e2 ear − 1 (2) the centrifugal potential V`(r) = ( ` + 12 )2 ~2 2µr2 (3) Z1 and Z2 are the proton numbers of the α particle and daughter nucleus, respectively, a is the screening parameter, and ` is the orbital angular momentum that the α particle takes away. The radius of the spherical square well is computed by adding the radii of both the α particle (A1) and the daughter nucleus (A2): R = r0 ( A 1 3 1 + A 1 3 2 ) (4) where r0 is a constant, an adjustable parameter. The α decay half-life is calculated using [1, 4]: T 1 2 = ln 2 λ 10h. (5) Here, h is the decay hindrance factor due to the effect of an odd-neutron and/or an odd-proton. The value of h is zero for even-even nuclei. In Ref. [4], the values of a, r0, and h were determined to be: a = 7.8 × 10−4, r0 = 1.14 fm, h = 0.3455. (6) For odd-odd nuclei, hnp = 2h. The decay constant λ is obtained via: λ = νP (7) where the penetration probability P is given by P = exp [ − 2 ~ ∫ b R √ 2µ (V (r) − Ek) dr ] . (8) Here, µ = mA1 A2/(A1 + A2) is the reduced mass of the daugh- ter nucleus and the α particle, m is the nucleon mass, Ek = Qα(A − 4)/A is the kinetic energy of the emitted α particle. The classical turning point b is obtained through the condition V (b) = Ek. In this model, the frequency of assault on the potential bar- rier is evaluated using ν = ( G + 32 ) ~ 1.2πµR20 , (9) 251 W. A. Yahya / J. Nig. Soc. Phys. Sci. 2 (2020) 250–256 252 where the parent nucleus radius R0 is obtained via R0 = 1.28A 1/3 − 0.76 + 0.8A−1/3. (10) The main quantum number G is obtained using G =  22 N > 126 20 82 < N ≤ 126 18 N ≤ 82 . (11) 2.2. The Coulomb and Proximity Potential Model (CPPM) The total interaction potential between the emitted α parti- cle and the daughter nucleus in the CPPM contains (for both the touching configuration and for separated fragments) the nu- clear, the Coulomb and the centrifugal terms [29]: VT (r) = Vprox(z) + VC (r) + ~`(` + 1) 2µr2 (12) where µ is the reduced mass of the interaction system and ` is the angular momentum. The Coulomb potential VC (r) is de- fined as: VC (r) = Z1Z2e 2  1 r for r ≥ RC 1 2RC [ 3 − ( r RC )2] for r ≤ RC . (13) Here, Z1 and Z2 are, respectively, the charge numbers of the daughter and emitted nuclei. The radial distance RC = 1.24 (R1 + R2). The term Vprox(z) represents the proximity potential and z = r − C1 − C2 is the distance between the near surfaces of the fragments, where r is the distance between the fragment cen- tres [16]. The presence of the proximity potential causes a re- duction in the height of the potential barrier. The proximity potential has been used in the preformed cluster model by Ma- lik et al. [30]. The proximity potential Vprox can be obtained by calculating the strength of the nuclear interactions between the daughter and emitted α particle: Vprox(z) = 4πbγR̄Φ ( z b ) MeV (14) where the term bγR̄ depends on the geometry and shape of the two nuclei, and the mean curvature radius R̄ is given as R̄ = C1C2 C1 + C2 . (15) The Süsmann central radii of fragments C1 and C2 are com- puted using Ci = Ri 1 − ( b Ri )2 + · · ·  (16) where the diffuseness of nuclear surface b ≈ 1 fm and Ri are given by Ri = 1.28A 1/3 i − 0.76 + 0.8A −1/3 i fm (i = 1, 2). (17) The universal function Φ ( � = zb ) is given in the form [24]: Φ(�) = { − 1 2 (� − 2.54) 2 − 0.0852 (� − 2.54)3 � ≤ 1.2511 −3.437 exp (−�/0.75) � ≥ 1.2511 (18) The nuclear surface energy coefficient γ is defined as γ = 1.460734 [ 1 − 4 ( N − Z N + Z )2] MeV/fm2 (19) where N and Z denote the neutron and proton numbers of the parent nucleus, respectively. The Prox. 2010 potential has been used in this work. According to the WKB approximation [24, 31, 32], the pen- etration probability P of the emitted α nucleus through the po- tential barrier is calculated using: P = exp [ − 2 ~ ∫ Rout Rin √ 2µ [V (r) − Q] dr ] (20) where the classical turning points Rin and Rout are determined from V (Rin) = V (Rout) = Q (21) and the reduced mass µ is calculated using µ = mA1 A2/A, where m is the nucleon mass, A1 and A2 denote the mass num- bers of the emitted and daughter nuclei, respectively, and A is the parent nucleus mass number. The α-decay half-life is then computed using T1/2 = ln 2 νP (22) where the assault frequency ν has been taken to be 1020 s−1. 2.2.1. Temperature-Dependent Proximity Potential The thermal effects are studied by using the temperature de- pendent forms of the parameters R, γ and b. They are given by [24]: Ri(T ) = Ri(T = 0) [ 1 + 0.0005T 2 ] fm (i = 1, 2) (23) γ(T ) = γ(T = 0) [ 1 − T − Tb Tb ]3/2 (24) b(T ) = b(T = 0) [ 1 + 0.009T 2 ] (25) where Tb is the temperature associated with near Coulomb bar- rier energies and b(T = 0) = 1. In this work, we have adopted an alternative form of the temperature dependent surface en- ergy coefficient in the form γ(T ) = γ(0) (1 − 0.07T )2 [33]. The temperature T (in MeV) can be obtained from E∗ = Ekin + Qin = 1 9 AT 2 − T (26) where E∗ is the excitation energy of the parent nucleus and A is its mass number, and Qin denotes the entrance channel Q-value of the system. The kinetic energy of the emitted α particle Ekin is obtained from Ekin = ( Ad/Ap ) Q. (27) 252 W. A. Yahya / J. Nig. Soc. Phys. Sci. 2 (2020) 250–256 253 3. Results and Discussions The α-decay half-lives of the Mercury isotopes (Z = 80) within the mass range 171 ≤ A ≤ 189 have been calculated us- ing the Gamow-like model (GLM), modified Gamow-like model (MGLM1) using the variable parameters of Ref. [4], the temperature- independent Coulomb and proximity potential model (CPPM), and the temperature-dependent Coulomb and proximity poten- tial model (CPPMT). The proximity 2010 potential has been employed in the CPPM and CPPMT calculations. We have also calculated the α-decay half-lives of the isotopes using the modi- fied Gamow-like model (MGLM2) with new parameter values. The new values for the parameters a, r0, and h in the modi- fied Gamow-like model (MGLM2) were obtained through least squares fitting procedure. The values of the three adjustable parameters obtained for the Hg isotopes are r0 = 1.2457fm, h = 0.1891, a = −2.159 × 10−3 (28) The database have been taken from the NUBASE2016 [34, 35, 36]. The reaction Qα value has been computed via [29]: Qα = ∆MP − ∆MD − ∆Mα + k ( ZεP − Z ε D ) (29) where ∆MP, ∆Mα, and ∆MD denote the mass excesses of the parent nucleus, the alpha particle, and the daughter nucleus, re- spectively. The term k ( ZεP − Z ε D ) denotes the screening effect of atomic electrons [37]; k = 8.7 eV , ε = 2.517 for Z ≥ 60, and k = 13.6 eV , ε = 2.408 for Z < 60 [38]. The angular momentum ` are obtained from the selection rule given by [23, 39, 40]: ` =  δ j for even δ j and πd = πp δ j + 1 for odd δ j and πd = πp δ j for odd δ j and πd , πp δ j + 1 for even δ j and πd , πp . (30) Here δ j = ∣∣∣ jp − jd∣∣∣, where jd , πd , jp, πp are the spin and parity values of the daughter and parent nuclei, respectively. The alpha-decay half-lives computed for the 19 Hg isotopes( 171−189Hg ) are shown in Table 1. The first four columns show, respectively, the mass number (A), the calculated Qα values, the calculated temperature and the experimental α-decay half-lives (Expt.) ( log [ T1/2(s) ]) . The last five columns of the Table show the computed α-decay half-lives using the GLM, MGLM1, MGLM2, CPPM, and CPPMT, respectively. All the models give reason- able values of the half-lives when compared to the available experimental results. The root mean square standard deviation σ has been evalu- ated using the formula: σ = √√√ 1 N N∑ i=1 [( log10 T Theory 1/2,i − log10 T Expt 1/2,i )2] (31) where T Theory1/2,i are the half-lives obtained using the five models and T Expt1/2,i are the experimental half-lives. The standard devi- ation have been calculated in order to compare the agreement between the experimental half-lives and theoretically calculated half-lives using the various models. The computed standard de- viations (σ) using the different models are shown in Table 2. The second column of the Table shows the calculated standard deviations for the Hg isotopes. From the Table, the MGLM2 has the least standard deviation with a value of 0.5013 followed by the GLM with a standard deviation value of 0.5885, less than the standard deviation of CPPM. The MGLM1, CPPM, and CPPMT have respective standard deviation values of 0.6130, 0.6866, and 0.5949. One observes that the CPPMT has a lower standard deviation value compared with the CPPM. This shows the importance of using temperature-dependent form of the prox- imity potential model. All the models give standard deviation values less than 0.7. However, among the five models, the GLM and MGLM2 seem to be the most suitable for the determination of the α-decay half-lives of the Hg isotopes. It should be noted these Hg isotopes were studied in Ref. [28] using about various proximity potential models. However, the authors considered experimental values of 171−177Hg only. This makes it difficult to compare the standard deviations. 90 95 100 105 110 -10 -5 0 5 10 15 Neutron Number (N) lo g [ T 1 /2 (s )] 171-189 Hg Expt GLM MGLM1 MGLM2 CPPM CPPMT Figure 1. Comparison of the calculated α-decay half-lives of Hg isotopes be- tween the various models and experiment. 90 95 100 105 110 4.5 5 5.5 6 6.5 7 7.5 8 Neutron Number (N) Q ( M e V ) 171-189 Hg Figure 2. Plot of the calculated Qα values against Neutron number (N) for the Hg isotopes. The calculated half-lives log [ T1/2(s) ] for the Hg isotopes using the five models have been plotted against the neutron 253 W. A. Yahya / J. Nig. Soc. Phys. Sci. 2 (2020) 250–256 254 Table 1. Calculated α-decay half-lives, log [ T1/2(s) ] , of Hg (Z = 80) using GLM, MGLM1, MGLM2, CPPM, CPPMT. log [ T1/2(s) ] A Qα(MeV ) T (MeV) Expt. GLM MGLM1 MGLM2 CPPM CPPMT 171 7.7011 0.9218 -4.1549 -3.4908 -3.3518 -3.7442 -3.9665 -3.7652 172 7.5571 0.9107 -3.6364 -3.3033 -3.5380 -3.7354 -3.8178 -3.6243 173 7.4111 0.8994 -3.0969 -2.6643 -2.5327 -2.8705 -3.1276 -2.9279 174 7.2661 0.8882 -2.6990 -2.4462 -2.6902 -2.8297 -2.9492 -2.7573 175 7.1051 0.8761 -1.9957 -1.7283 -1.8488 -2.1095 -2.4407 -2.2496 176 6.9301 0.8630 -1.6467 -1.3748 -1.6321 -1.6953 -1.8640 -1.6738 177 6.7686 0.8508 -0.8246 -0.6132 -0.5042 -0.6952 -1.0459 -0.8497 178 6.6105 0.8387 -0.5237 -0.2744 -0.5467 -0.5267 -0.7492 -0.5606 179 6.3935 0.8229 0.1461 0.7478 0.5927 0.5200 0.0658 0.2534 180 6.2916 0.8142 0.7321 0.9145 0.6241 0.7394 0.4550 0.6418 181 6.3175 0.8135 1.1249 1.0050 1.0923 1.0300 0.6013 0.7944 182 6.0288 0.7930 1.8947 1.9638 1.6562 1.8607 1.5193 1.7046 183 6.0717 0.7935 1.9049 1.9747 1.8009 1.8322 1.3184 1.5035 184 5.6951 0.7672 3.4442 3.4213 3.0873 3.4225 2.9941 3.1775 185 5.8061 0.7723 2.9129 3.1016 2.9081 3.0392 2.4621 2.6457 186 5.2376 0.7327 5.7140 5.6765 5.2962 5.8501 5.2691 5.4498 187 5.2628 0.7324 7.9777 5.7360 5.7394 6.1107 5.3940 5.5811 188 4.7405 0.6945 8.7218 8.5148 8.0682 8.9273 8.1268 8.3040 189 4.6699 0.6876 9.1818 9.1566 9.3338 10.0638 9.1191 9.3085 Table 2. Calculated root means square standard deviation (σ) using the different models. Model σ ( Hg ) GLM 0.5885 MGLM1 0.6130 MGLM2 0.5013 CPPM 0.6866 CPPMT 0.5949 90 95 100 105 110 0.65 0.7 0.75 0.8 0.85 0.9 0.95 Neutron Number (N) T ( M e V ) 171-189 Hg Figure 3. Plot of the calculated Temperature (T in MeV) against Neutron num- ber (N) for the Hg isotopes. 90 95 100 105 110 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 Neutron Number (N) ∆ T 1 /2 GLM MGLM1 MGLM2 CPPM CPPMT Figure 4. Plot of the ∆T against Neutron number (N) for the Hg isotopes using the different models. number in Figure 1. From the Figure, the half-lives can be seen to increase with increase in neutron number for the Hg isotopes. Aside N = 107 (corresponding to A = 187), the mod- els give very good descriptions of the half-lives. It should be noted that near double magicity of the parent nucleus 202Hg (Z = 80, N = 122) was suggested in Ref. [28]. Figure 2 shows plots of the Qα values against neutron number for the Hg isotopes while the calculated temperature of the Hg plotted against the neutron number in Figure 3. The temperature values decrease with increase in neutron number which is equivalent to increase in the half-lives. That is, the calculated temperature is inversely proportional to the α-decay half-lives. 254 W. A. Yahya / J. Nig. Soc. Phys. Sci. 2 (2020) 250–256 255 The difference between the theoretical and experimental α- decay half-lives have been computed using ∆T1/2 = log10 [ T theor1/2 /T expt 1/2 ] . (32) This factor ( ∆T1/2 ) has been plotted for all the models used in this work in Figure 4. It can be observed that most of the points are near zero and within ±0.5. The Figure shows that the MGLM2 model gives the lowest ∣∣∣∆T1/2∣∣∣ values while the CPPM model gives the highest ∣∣∣∆T1/2∣∣∣ values. This agrees with the results shown in Table 2. 4. Conclusion In this work, the α-decay half-lives of Hg isotopes in the mass range 171 ≤ A ≤ 189 have been studied using the Gamow- like model (GLM), modified Gamow-like model (MGLM1), the temperature-independent Coulomb and proximity potential model (CPPM), and the temperature-dependent Coulomb and proximity potential model (CPPMT). The Prox. 2010 proxim- ity potential has been employed for the CPPM and CPPMT calculations. For the set of isotopes, new parameter values were obtained through a least squares scheme using the modi- fied Gamow-like model (termed MGLM2). It is shown that the modified Gamow-like model (MGLM2) with new local param- eter values and the GLM are the most suitable methods (among the methods considered here) for calculating the α-decay half- lives for the Hg isotopes. In general, all the models give α- decay half-lives that are in good agreement with the experimen- tal data with the maximum standard deviation values less than 0.7. References [1] A. Zdeb, M. 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