Proton radioactivity half-lives with nuclear asymmetry factor

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A. Soylu, F. Koyuncu, G. Gangopadhyay, V. Dehghani and S. A. Alavi. Proton radioactivity half-lives with nuclear asymmetry factor[J]. Chinese Physics C.
A. Soylu, F. Koyuncu, G. Gangopadhyay, V. Dehghani and S. A. Alavi. Proton radioactivity half-lives with nuclear asymmetry factor[J]. Chinese Physics C. shu
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Proton radioactivity half-lives with nuclear asymmetry factor

  • 1. Department of Physics, Nigde Ömer Halisdemir University, Nigde 51240, Turkey
  • 2. Program of Opticianry, Gölhisar Vocational School of Health Services, Burdur Mehmet Akif Ersoy University, 15400, Gölhisar, Burdur, Turkey
  • 3. Department of Physics, University of Calcutta, Kolkata 700009, India
  • 4. Department of Physics, University of Sistan and Baluchestan, Zahedan, Iran

Abstract: The dependence of the proton decay half-lives on nuclear asymmetry parameter has been investigated using WKB method and two types of empirical formulas. Using single-folding formalism with asymmetry dependent nuclear radius and surface diffuseness of nuclear matter, the nuclear potential and consequently the half-life are functions of the asymmetry factor. Despite small values of asymmetry in neutron deficient proton emitters, noticeable changes in half-lives are observed. Addition of an asymmetry parameter term to the two forms of empirical formulas leads to a reduction in the rms error for ground state and isomeric transitions. Noticeable reduction of about 43% is obtained for isomeric transition in the second form of empirical formula. Considering ground state transitions in two categories, odd-even and odd-odd emitters, and adopting deformation and asymmetry dependent empirical formulas, the rms decreases remarkably. The lowest values of rms errors, viz. 0.1492, 0.2312, and 0.1999, are obtained for the mentioned empirical formulas of ground state transitions of odd-even and odd-odd emitters and all isomeric transitions, respectively.

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    I.   INTRODUCTION
    • Proton decay is one of the interesting and relatively recent topics of nuclear decay. Finding proton decay half-lives may be of great importance for knowing decaying feature of nuclei [1], determining the nuclear deformation, studying the behavior of single-particle resonances close to the drip line [2], constraining the density slope of the symmetry energy [3], and termination of the astrophysical rp-process [4]. Very informative introduction to the history of proton decay, experimental techniques and methods, and theories of proton emission have been provided in a number of review papers [5-9].

      Nuclear deformation and asymmetry parameter are two important properties in the study nuclear decay. In different theoretical models for calculation of proton decay half-lives the nuclear deformation has been included directly through explicit dependence of the Coulomb and nuclear potentials [10-13]. Moreover, the nuclear asymmetry parameter has been included in nuclear potential and spectroscopic factor [14-17]. The proton decay empirical formula is basically a function of the Q-value of proton decay and atomic number of daughter nuclei. The formula may comprise the nuclear properties of parent or daughter nuclei, such as orbital angular momentum and deformation parameters, through microscopic basements or introduction of new modification terms. The dependence of the empirical half-life on these additional terms may be justified based on microscopic or macroscopic theoretical calculations. The first empirical formula in this regard introduced as an implicit function of deformation and momenta [1]. An empirical formula as a function of deformation has been given in Ref. [18]. The effect of deformation and Q-value on empirical formula have been investigated in our previous work [19]. This motivated us to assess the effect of deformation and asymmetry parameter, separately as well as together, on two forms of empirical formulas. Using single-folding potential [20, 21] and adopting asymmetry parameter dependent densities, the influence of asymmetry parameter on calculated half-lives can be evaluated.

      In Sec. II, the theory related to our calculations are given in two parts, theoretical calculations and empirical formulas. Results of our calculation are displayed in Sec. III. Finally, the conclusion and summary are presented in Sec. IV.

    II.   THEORY
    • In order to show the dependence of the theoretical half-life on asymmetry parameter, this section has been divided into two main parts: theoretical half-life and empirical half-life.

    • A.   Theoretical half-life

    • The effective interaction potential between proton and daughter nuclei $ V_{\rm eff} $(r), includes the attractive nuclear potential $ V_N(r) $, the repulsive Coulomb potential $ V_{\rm C}(r) $, and repulsive centrifugal term $ V_l(r) $,

      $ V_{\rm eff}(r) = V_N(r)+V_{\rm C}(r)+V_l(r), $

      (1)

      where r is relative distance between the proton and the daughter nucleus and l is the angular momentum carried by the proton.

      The nuclear potential can be calculated through single folding model

      $ V_{N}(r) = \int {\rm d}\vec{r}_1 \rho(\vec{r}_1) v(s) , $

      (2)

      where $ s = |\vec{r}_1-\vec{r}| $ is the relative distance between interacting nucleon pair. The effective nucleon-nucleon potential Yukawa (M3Y)-Reid-type interaction with zero-range exchange contribution is given by [22]

      $ v(s) = 7999 \frac{{\rm e}^{-4s}}{4s}-2134 \frac{{\rm e}^{-2.5s}}{2.5s}-276\delta(s). $

      (3)

      Here, $ \rho(\vec{r}_1) $ is the matter density distributions of daughter nucleus, which is given by two-parameter Fermi distribution

      $ \rho(r) = \frac{\rho_0}{1+\exp\left[\dfrac{r-R}{a}\right]}. $

      (4)

      The adopted parameters for radii and surface diffuseness parameters are $ R = 1.07 A^{1/3} ({\rm fm}) $ and $ a = 0.54\; ({\rm fm}) $ [23]. Using the following asymmetry parameter dependent formulas for proton and neutron radii $ R_p $, $ R_n $, and corresponding surface diffuseness $ a_p $, $ a_n $ [24],

      $ \begin{aligned}[b] &R_p = 1.249 A^{1/3}-0.5401-0.9582\; I, \\ &R_n = 1.2131 A^{1/3}-0.4415+0.8931 \; I,\\ &a_p = 0.4899-0.1236 \;I,\\ & a_n = 0.4686+0.0741 \; I, \end{aligned} $

      (5)

      where $I = \dfrac{N-Z}{A}$ is asymmetry parameter of daughter nuclei, two separate asymmetry dependent proton and neutron matter density distributions $ \rho_p $, $ \rho_n $ are obtained and, consequently, the nuclear potential is obtained as

      $ V_N(r) = V^n_N(r)+V^p_N(r) = \int {\rm d}\vec{r}_1 \rho_n(\vec{r}_1) v(s)+\int {\rm d}\vec{r}_1 \rho_p(\vec{r}_1) v(s), $

      (6)

      where $ V^n_N $ and $ V^p_N $ are the contributions of neutron and proton to the total nuclear potential. As for the different proton and neutron density distributions and the introduction of the asymmetry term in the folding potential, there have already been quite a few studies in the alpha or heavier cluster decay cases [25, 26].

      The Coulomb potential $ V_{\rm C}(r) $ is determined through single-folding potential Eq. (2) by replacing matter distribution with charge distribution and M3Y nucleon-nucleon potential with repulsive proton-proton potential $v(s) = \dfrac{e^2}{4\pi\epsilon_0}\dfrac{1}{s}$.

      The proton decay half-life can be calculated through WKB approximation. The two necessary conditions of WKB for centrifugal term and nuclear potential, respectively, are the Langer modification and Bohr$ - $Sommerfeld quantization condition [27]. So, the effective potential with these modifications is written as

      $ V_{\rm eff}(r) = \eta V_N(r)+ V_{\rm C}(r)+\dfrac{\hbar^2\left(l+\dfrac{1}{2}\right)^2}{2\mu r^2} ,$

      where $ \eta $ is quantization factor.

      In Ref. [28] the asymmetry dependence of nuclear potential has been investigated by adding Lane potential [29] to the nuclear potential $ V_N(r)+V_N^{\rm Lane}(r) $. The Lane nuclear potential approximately can be written as $V_N^{\rm Lane}(r) = -I_d \dfrac{V_N(r)}{2}$ [30]. Therefore, the total nuclear potential is obtained as $\left(1- \dfrac{I_d}{2}\right)V_N(r)$ and asymmetry is appeared in the coefficient of nuclear potential. Hence by considering quantization condition the asymmetry dependence is eliminated.

      The proton decay half-life is calculated as

      $ T_{1/2} = \frac{\ln 2}{\nu P S_p}, $

      (7)

      where $ \nu $, P, $ S_p $ are the assault frequency, tunneling probability, and spectroscopic factor, respectively. Using WKB approximation, the assault frequency and tunneling probability are given as

      $ \begin{aligned}[b] &\nu = \frac{\hbar}{2\mu}\left[\int^{r_2}_{r_1}\dfrac{{\rm d}r}{\sqrt{\dfrac{2\mu}{\hbar^2}|Q-V_{\rm eff}(r)}|} \right]^{-1}, \\ &P = \left[1+\exp\left(\frac{2\sqrt{2\mu}}{\hbar}\int^{r_3}_{r_2}{\rm d}r \sqrt{V_{\rm eff}(r)-Q}\right)\right]^{-1} , \end{aligned} $

      (8)

      where Q, is the Q-value of proton decay, $ \mu $ is the reduced mass, and $ r_i $ refers to the turning points.

      Spectroscopic factor is a crucial quantity for theoretical calculations and different microscopic methods have been used and adopted for its calculations. For example, Lalazissis et al. [31] have used the Relativistic Hartree-Bogoliubov approach to calculate the spectroscopic factor for proton decay. In Ref. [14] the spectroscopic factor of 27 proton emitters have been calculated through relativistic mean field approach (RMF) theory combined with the BCS method with the force NL3. We have followed a similar approach. As pointed out in Ref. [31], the spectroscopic factor for proton decay from a particular level in the parent nucleus may be estimated by the non-occupancy of the level in the daughter nucleus.

    • B.   Empirical half-life

    • The empirical formulas, in spite of their common structure being based on the explicit dependence of the atomic number and inverse square-root Q-value, can be modified by adding new terms as functions of the basic structure of the emitter or daughter nuclei. Further, the formulas may be an explicit or implicit functions of angular momentum l. The reduced half-life $ T_{1/2}^{\rm red} $, as an implicit function of l, $ f(l) $, is given as [18],

      $ \log_{10} T_{1/2}^{\rm red} = a\mu^{1/2}Z_d Q^{-1/2}+b\mu^{1/2}Z_d+c+d|\beta_2^3|, $

      (9)

      where $T_{1/2}^{\rm red}\equiv T_{1/2}\,{\rm e}^{-f(l)}$.

      A recently introduced formula, with explicit and strict dependence on angular momentum, is written as [32]

      $ \log_{10} T_{1/2} = [al+b]Z_d^{0.8} Q^{-1/2}+[cl+d]. $

      (10)

      In order to include the effect of quadruple deformation and asymmetry parameter in the two main formulas Eqs. (9) and (10), we propose to include new terms as function of $ \beta_2^4 $ and I, of daughter nuclei, in the following form

      $ \log_{10} T_{1/2}^{\rm red} = a\mu^{1/2}Z_d Q^{-1/2}+b\mu^{1/2}Z_d+cI+d\beta_2^4+e, $

      (11)

      and

      $ \log_{10} T_{1/2} = [al+b]\,Z_d^{0.8} Q^{-1/2}+[cl+d]+eI+f\beta_2^4. $

      (12)

      In Ref. [18] the third power of $ \beta_2 $ has been employed. In our previous work [19] the power was considered as an adjustable parameter and for similar formula, but with different terms, approximately the fourth power was obtained for all transitions. Here, because of the difference of the adopted formulas and adding new asymmetry dependent term, we also checked its value as an adjustable parameter and found best fitting for values around four. Therefore, in order to reduce the number of the adjustable parameters we fixed the fourth power of $ \beta_2 $ in both new empirical formulas.

      The root-mean-square $ {\rm (rms)} $ error of the decimal logarithm of the proton decay half-life is evaluated as

      $ \sigma = \sqrt{\frac{1}{n}\sum\limits_{i = 1}^{n}[\log_{10} T_{1/2}^{\rm exp.}(i)-\log_{10} T_{1/2}^{\rm cal.}(i)]^2}. $

      (13)
    III.   RESULTS AND DISCUSSION
    • The theory and empirical formulas introduced in the previous section are applied to evaluate the effect of asymmetry parameter on calculations. The adopted experimental half-lives $ \log_{10} T_{1/2}^{\rm exp} $ and quadruple deformation parameters have been taken from [33, 34] and [35], respectively. There are in total, 44 proton emitting nuclei as shown, in Table 3 of Ref. [19]. In Ref. [24] the asymmetry dependent nuclear matter densities have been determined for spherical nuclei. So, in first part of present work the single-folded nuclear potential has been calculated with these spherical nuclear matter densities.

      The asymmetry parameters of parent ($ I_p $) and daughter nuclei ($ I_d $) have been plotted in Fig. 1 (a) and (b) for ground state and isomeric transitions, respectively. Despite neutron deficiency of proton emitters, an increasing trend is observed in figures and heavier emitters have larger asymmetry number. We find very small differences between $ I_p $ and $ I_d $.

      Figure 1.  (color online) Asymmetry parameter of parent and daughter nuclei for (a) g.s (b) isomeric transitions. The numbers correspond to the nuclei in Table 1.

      Using asymmetry parameter dependent densities in Eq. (4), nuclear potential and evidently decay half-life are the functions of this factor. Fig. 2 (a) and (b) show the deviation between calculated half-lives with and without inclusion of asymmetry factor for ground state and isomeric transitions. This comparison is independent of the value of spectroscopic factor $ S_p $. As can be seen, although the asymmetry parameter is very small, it causes noticeable changes in calculated half-lives. For ground state and isomeric transitions the absolute value of variations of the decimal logarithm of half-lives lie between 0.1 to 0.7 and 0.05 to 0.25, respectively.

      Figure 2.  (color online) Difference between calculated half-lives with and without nuclear asymmetry parameter.

      As already mentioned, spectroscopic factors for proton decay can be estimated by the non-occupation probability of the relevant level in the daughter nucleus. These values have been calculated in the RMF+BCS formalism. The computer code DIZ [36], extended for odd particle numbers in Ref. [37] has been employed. We have employed two standard parameters NL1 [38] and NL3 [39] for RMF calculation. A constant pairing gap approach has been assumed with the values of the pairing gap being estimated from the experimental odd-even mass difference, if available, or its corresponding estimate from semi-microscope binding energy values. Obtained values for spectroscopic factors are presented in Table 1. Except in some cases for isomeric transition, NL1 and NL3 give similar results. Many of the daughter nuclei are found to be nearly spherical at low energy, these have been indicated in the table.

      No. Parent $ S_p({\rm NL1}) $ $ S_p({\rm NL3}) $ No. Parent $ S_p({\rm NL1}) $ $ S_p({\rm NL3}) $
      1 $ ^{109} $I 0.838 0.789 1 $^{141{\rm m}}$Ho 0.505 0.658
      2 $ ^{112} $Cs 0.584 0.753 2 $^{146{\rm m}}$Tm 0.923 0.905
      3 $ ^{113} $Cs 0.571 0.741 3 $^{147{\rm m}}$Tm 0.331 0.508
      4 $ ^{117} $La 0.743 0.744 4 $^{150{\rm m}}$Lu 0.702 0.817
      5 $ ^{121} $Pr 0.911 0.907 5 $^{151{\rm m}}$Lu 0.622 0.737
      6 $ ^{130} $Eu 0.697 0.852 6 $^{156{\rm m}}$Ta* 0.496 0.401
      7 $ ^{131} $Eu 0.678 0.844 7 $^{159{\rm m}}$Re* 0.387 0.303
      8 $ ^{135} $Tb 0.868 0.829 8 $^{161{\rm m}}$Re* 0.376 0.272
      9 $ ^{140} $Ho 0.745 0.645 9 $^{165{\rm m}}$Ir* 0.245 0.163
      10 $ ^{141} $Ho 0.747 0.650 10 $^{166{\rm m}}$Ir 0.771 0.620
      11 $ ^{144} $Tm 0.558 0.418 11 $^{167{\rm m}}$Ir 0.789 0.641
      12 $ ^{145} $Tm 0.918 0.898 12 $^{170{\rm m}}$Au* 0.137 0.086
      13 $ ^{146} $Tm 0.923 0.905 13 $^{171{\rm m}}$Au* 0.43 0.087
      14 $ ^{147} $Tm 0.685 0.596 14 $^{177{\rm m}}$Tl* 0.056 0.083
      15 $ ^{150} $Lu 0.753 0.671 15 $^{185{\rm m}}$Bi* 0.986 0.985
      16 $ ^{151} $Lu 0.737 0.648
      17 $ ^{155} $Ta* 0.509 0.402
      18 $ ^{156} $Ta* 0.459 0.651
      19 $ ^{157} $Ta* 0.681 0.812
      20 $ ^{159} $Re* 0.678 0.730
      21 $ ^{160} $Re* 0.402 0.549
      22 $ ^{161} $Re* 0.609 0.768
      23 $ ^{164} $Ir* 0.339 0.475
      24 $ ^{166} $Ir 0.576 0.713
      25 $ ^{167} $Ir 0.745 0.845
      26 $ ^{170} $Au* 0.339 0.339
      27 $ ^{171} $Au* 0.531 0.660
      28 $ ^{176} $Tl* 0.296 0.408
      29 $ ^{177} $Tl* 0.291 0.414

      Table 1.  Spectroscopic factor of proton emitters. Daughters of the nuclei marked with an '$ * $' are nearly spherical at low excitation energy.

      Using assault frequency and tunneling probability as functions of asymmetry parameter and calculated values of spectroscopic factor, the proton decay half-lives are obtained. The difference between calculated half-lives and experimental values for ground state and isomeric transitions can be observed in Fig. 3(a) and (b), respectively. Obtained half-lives are in good agreement with experiment.

      Figure 3.  (color online) Difference between calculated half-lives and experimental data for (a) ground state and (b) isomeric transitions. M3Ynp (NL1) and M3Ynp (NL3) stand for calculated half-lives with asymmetry parameter and spectroscopic factor with RMF+BCS model.

      As second part of this paper the adjustable parameters of empirical formulas and corresponding rms error have been given in Tables 2 and 3. Table 2 presents the fitted parameters for the empirical formula for $ \log_{10} T_{1/2}^{\rm red} $, as given in Eq. (11). Four categories have been considered for evaluation of the nuclear deformation and asymmetry parameter. Categories (I) to (IV), correspond to the empirical formula $ \log_{10} T_{1/2}^{\rm red} $ without the deformation and asymmetry terms, with only the deformation term, with only the asymmetry term, and with both the deformation and asymmetry terms, respectively. For each category, we have presented the fitted parameters and corresponding rms values for the ground states, the isomeric states, and all transitions. Furthermore, for ground state and all transitions, the corresponding data for odd-even and odd-odd proton emitters have been given in the second vertical block of the table, separately. First category, as the simplest form of $ \log_{10} T_{1/2}^{\rm red} $ with minimum number of adjustable parameters, gives reasonable rms error values. For this case the rms error increases slightly for odd-even nuclei while noticeable decrease, about 0.1, is obtained for odd-odd nuclei. In comparison with category I, lower values of rms errors are obtained for ground state, isomeric, and all transitions in category II. In this category the rms errors decrease for both odd-even and odd-odd divisions. The formula of category III gives lower values of rms error in comparison with category I. These results show that including the deformation or asymmetry terms can decrease the rms errors for different forms of transitions, odd-even and odd-odd ground state, isomeric, or all transitions. By considering both correction terms completed form of $ \log_{10} T_{1/2}^{\rm red} $ with lowest values of rms error are obtained in category IV. The calculated rms errors are lower than 0.3 for ground state (odd-even and odd-odd) and isomeric transitions.

      Transition Parameters $ \sigma $ Parameters $\sigma^{{\rm o-e/o-o}}$ Category
      Ground (0.432, −2.138, −13.701) 0.4578 (0.430, −2.371, −11.644) 0.4678
      (0.439, −1.668, −18.073) 0.3507
      Isomeric (0.384, −0.664, −23.242) 0.3505 (I)
      (0.409, −1.955, −13.698) 0.4997
      All (0.413, −1.833, −14.985) 0.4822 (0.422, −1.485, −18.578) 0.3943
      Ground (0.434, −1.791,89.210, −16.903) 0.3593 (0.428, −1.910, 116.136, −15.609) 0.2669
      (0.444, −1.462, 52.092, −20.184) 0.3152
      Isomeric (0.377, −0.702, 77.508, −22.606) 0.2999 (II)
      (0.407, −1.5091, 115.878, −16.877) 0.3330
      All (0.414, −1.509, 94.610, −18.013) 0.3899 (0.427, −1.251, 57.189, −20.987) 0.3667
      Ground (0.464, −0.819, −42.578, −24.286) 0.4176 (0.481, −0.210, −70.55, −28.747) 0.3458
      (0.498,1.132, −80.948, −40.760) 0.2944
      Isomeric (0.345, −2.464, 39.942, −8.140) 0.2870 (III)
      (0.405, −2.197, 6.836, −11.832) 0.4982
      All (0.409, −2.031, 5.643, −13.447) 0.4812 (0.484, 1.535, −85.279, −42.966) 0.3306
      Ground (0.433, −1.816, 0.881,90.028, −16.714) 0.3593 (0.425, −1.987, 2.939, 119.431, −15.009) 0.2668
      (0.489, 0.714, −66.927, 17.012, −37.520) 0.2919
      Isomeric (0.353, −2.001,29.3195, 31.020, −11.902) 0.2828 (IV)
      (0.399, −2.044, 13.012, 118.035, −13.383) 0.3249
      All (0.407, −1.944, 12.666, 97.60, −14.658) 0.3838 (0.479, 1.325, −77.883, 12.827, −41.385) 0.3295

      Table 2.  Parameters for the empirical formula for $ \log_{10} T_{1/2}^{\rm red} $ in Eq. (11), for ground state (29 Nuclei), isomeric (15 Nuclei), and all transitions. The first to fourth sets of data, categories I to IV, belong to the without deformation and asymmetry, just with deformation, just with asymmetry, and with deformation and asymmetry, respectively.

      Transition Parameters $ \sigma $ Parameters $\sigma^{\rm o-e/o-o}$ Category
      Ground (0.032, 0.825, −0.151, −26.545) 0.3505 (0.047, 0.819, −0.540, −26.219) 0.3091
      (0.022, 0.852, 0.202, −27.686) 0.2564
      Isomeric (0.120, 0.337, −2.404, −14.061) 0.4220 (I)
      (0.032, 0.840, −0.144, −26.903) 0.4183
      All (0.023, 0.845, 0.109, −27.176) 0.3976 (0.002, 0.935, 0.761, −30.084) 0.2549
      Ground (0.014, 0.908, 0.374, −29.085, 65.348) 0.2680 (0.024, 0.896, 0.091, −28.630, 75.488) 0.1555
      (0.006, 0.937, 0.663, −30.197, 35.209) 0.2321
      Isomeric (0.142, 0.211, −2.782, −11.848, 215.944) 0.2783 (II)
      (0.008, 0.935, 0.545, −29.881, 88.050) 0.3142
      All (0.005, 0.933, 0.641, −29.88, 73.00) 0.3286 (-0.010, 1.002, 1.107, −32.069, 37.513) 0.2317
      Ground (0.011, 0.936, 0.446, −29.073, −10.909) 0.3161 (0.020, 0.949, −29.116, −14.129) 0.2386
      (0.014, 0.892, 0.415, −28.659, −2.604) 0.254
      Isomeric (0.235, −0.221, −5.431, −1.379, 28.833) 0.2414 (III)
      (0.026, 0.869, 0.020, −27.423, −4.809) 0.4091
      All (0.019, 0.867, 0.221, −27.596, −3.029) 0.3945 (-0.004, 0.967, 0.927, −30.859, −2.448) 0.2530
      Ground (0.011, 0.924, 0.456, −29.407, −2.271, 60.099) 0.2667 (0.019, 0.923, 0.222, −29.139, −3.944, 66.096) 0.1492
      (0.009, 0.919, 0.564, −29.790, 1.780, 38.753) 0.2312
      Isomeric (0.213, −0.123, −4.736, −3.954, 20.271, 116.391) 0.1990 (IV)
      (0.009, 0.933, 0.534, −29.856, 0.620, 89.305) 0.3140
      All (0.008, 0.916, 0.559, −29.578, 37.856, 81.051) 0.3215 (-0.005, 0.974, 0.964, −31.484, 3.142, 45.284) 0.2293

      Table 3.  Same as Table 2, but for $ \log_{10} T_{1/2} $, Eq. (12).

      The adjustable parameters of the empirical formula $ \log_{10} T $ in Eq. (12), and corresponding rms error have been presented in Table 3. The categories and divisions in this table are the same as in Table 2. Similar behaviors are observed in this table, that is inclusion of deformation and asymmetry terms reduce the error, and combining both terms could decrease the rms error remarkably.

      Moreover, comparison between two empirical formulas shows that, except for the isomeric transition formula in first category, using Eq. (12) for $ \log_{10} T $, we obtain smaller rms error values for all formulas of categories and divisions. In most of cases this reduction about 0.1.

      In order to evaluate the contribution of nuclear deformation and asymmetry parameter terms in the empirical formula for $ \log_{10} T $, the difference between experimental half-lives and corresponding values calculated by means of the parameters of categories I, II, and III, given in Table 3, have been plotted in Fig. 4 (a)-(c). Figures (a), (b), and (c) show the deviations for odd-even and odd-odd ground state transitions and isomeric transitions, respectively. As can be seen in most of cases the difference decreases by addition the deformation term or asymmetry term. The role of the asymmetry term in reduction of the rms of isomeric transitions is apparent. Similar results are obtained for $ \log_{10} T_{1/2}^{\rm red} $, which have not been presented here.

      Figure 4.  (color online) Difference between calculated and experimental half-lives with parameters of Table 3. The numbers in figures (a) and (b) correspond to the odd-even and odd-odd nuclei in Table 1.

      Figure 5 (a)-(c) shows the difference between experimental data and calculated half-lives of ground state odd-even, odd-odd, and isomeric transitions with lowest rms, category IV, for both the empirical formulas. These figures clearly display that inclusion of both deformation and asymmetry terms effectively reduces the deviation between empirical results and experiment as well as indicates a crucial role of the deformation and asymmetry terms. Very good agreement between theory and experiment is observed, especially for the second form of empirical formula $ \log T $, which reveals the reliability of these modified formulas for prediction of the proton decay half-lives.

      Figure 5.  (color online) Difference between calculated and experimental half-lives for the formula with the lowest rms error, category IV.

    IV.   SUMMARY AND CONCLUSION
    • In this paper the effect of the nuclear asymmetry parameter on calculated half lives and empirical formulas of 44 proton emitters have been investigated. The WKB method and single-folding nuclear potential with inclusion of the Bohr-Sommerfeld quantization condition have been employed for calculation of the tunneling probability and assault frequency. Spectroscopic factors have been calculated in the RMF formalism using two standard parameter sets, NL1 and NL3. Adding Lane nuclear potential to the effective potential gives the total nuclear potential as an explicit function of the asymmetry parameter. Imposing quantization condition causes this dependence to disappear. Therefore, by inclusion of the asymmetry parameter dependent radius and surface diffuseness in the nuclear density expression, the normalized nuclear potential through quantization condition, tunneling probability, assault frequency, and decay half-life are obtained as functions of the asymmetry parameter. Because of the neutron deficient nature of the proton emitters, the asymmetry parameter is a relatively small quantity. However, since it directly affects the surface diffuseness, the remarkable changes of calculated half-lives, as obtained here, can be anticipated.

      The remarkable contribution of the asymmetry parameter in theoretical calculations justifies the explicit presence of a correction term as a function of asymmetry parameter in empirical formulas. The dependence of two forms of empirical formulas on the asymmetry term have been investigated in different categories of transitions and specialized divisions of odd-even and odd-odd ground state and all transitions. The inclusion of this modification could reduce the rms errors in all cases and for isomeric and odd-even and odd-odd ground state transitions the reduction is very apparent, especially for the second type of empirical formula in which the rms error reaches to about 0.2 (corresponding to a 43% relative reduction). The data has been fitted for quadruple deformation term separately and, as expected, very good results have been obtained. Therefore, the presence of both deformation and asymmetry terms in empirical formulas have been evaluated and the lowest rms error values have been achieved for both formula. The rms error has been came down to less than 0.2 for the isomeric and ground state odd-even transitions and better agreement between theory and experiment is observed for second type of formula with explicit dependence on angular momentum, deformation, and asymmetry parameter.

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