Eur. Phys. J. A (2016) 52: 323 THE EUROPEAN DOI 10.1140/epja/i2016-16323-7 PHYSICAL JOURNAL A Regular Article – Theoretical Physics Proton emission half-lives within a Gamow-like model A. Zdeb1,a,M.Warda1, C.M. Petrache2, and K. Pomorski1 1 Katedra Fizyki Teoretycznej, Uniwersytet Marii Curie-Sklodowskiej, ul. Radziszewskiego 10, 20-031 Lublin, Poland 2 Centre de Sciences Nucl´eaires et Sciences de la Mati`ere,Universit´e Paris-Sud and CNRS/IN2P3, Universit´e Paris-Saclay 91405 Orsay, France Received: 14 July 2016 / Revised: 7 September 2016 Published online: 26 October 2016 c The Author(s) 2016. This article is published with open access at Springerlink.com Communicated by F. Nunes Abstract. Proton emission is described using a model which has previously given good results in the description of α and cluster radioactivity. The simple phenomenological formalism, based on the Gamow theory for alpha decay, is now extended by including the centrifugal term. The model contains only one parameter: the effective nuclear radius constant. Its value was once found for alpha and cluster emitters. A good agreement with the experimental half-lives for proton radioactivity is achieved without any additional fitting procedures to the data for proton emission. 1 Introduction plicity. It is a quasi-classical method based on the WKB approximation where the penetrability of the barrier is The decay of 53Co from its isomeric state was the first ex- studied [18]. perimental observation of proton radioactivity [1, 2]. The Recently, a simplified version of the Gamow theory proton emission from the ground state of mother nuclei was successfully applied to determine half-lives for alpha with masses in the range 110 ≤ A ≤ 150 was measured and cluster emission in ref. [19]. This method has much in the early 1980s [3]. This decay mode is typical for the deeper physical basements and conserve the simplicity of odd-Z emitters beyond the proton drip line. The associ- the Viola-Seaborg approach. Now, we would like to show ated half-lives range from 10−6 s to a few seconds. The the model potentialities to describe the various decays investigation of this process provides information on the modes even though the inclusion of the omitted micro- structure of very neutron-deficient nuclei which are far scopic effects can give more precise values. Thus we ap- from the β-stability region (shell structure, coupling be- plied the same model to describe proton emission from tween bound and unbound nuclear states). It also allows spherical nuclei by keeping a fixed value of a single ad- one to obtain spectroscopic properties of the nucleus due justable parameter. The main aim of this paper is to show to the unpaired proton. that using a very simple, uniform formalism, one can eval- Up to now several attempts have been made to de- uate half-lives for the three different decay modes: alpha, scribe this phenomenon theoretically, which have been dis- cluster and proton radioactivity. The obtained agreement cussed in review papers [4–6]. The distorted wave Born ap- with experimental data clearly shows that this model de- proximation (DWBA) [7] and the two-potential approach scribes properly the physics of various decays (although (TPA) [8] represent rather sophisticated methods which omitted microscopic effects can give more precise values). allow the decay width to be found by analyzing the cor- The model can also be used for rapid evaluations of the or- relations between initial and final state wave functions in der of magnitude of the lifetimes for different decay modes the interaction potential. The correlations between proton using only the Q-values and angular momentum l of the emission half-lives and the deformation of the mother nu- emitting state. It is a useful tool to globally guide a sur- cleus were widely discussed in refs. [9–11]. However, these vey of the known experimental masses and conceive future sophisticated methods often include phenomenological in- measurements. put, fitted parameters [4] or rough approximations [9,10]. Another phenomenological attempt with the Viola- Seaborg–type formula for alpha decay and cluster radioac- 2Model tivity [12,13] was used to describe proton emission [14–17]. This is one of the most popular formalisms due to its sim- Analogous to alpha and cluster radioactivity, proton emis- sion might be understood in terms of the quantum- a e-mail: [email protected] mechanical tunneling of the emitted particle through Page 2 of 6 Eur. Phys. J. A (2016) 52: 323 The proton emission half-life formula takes the follow- V(r) V C ing simple form: Vl p ln 2 T1/2 = , (6) VC +Vl νP where ν is the frequency of assaults of the proton against the potential energy barrier. In a first approximation one Ep may assume that it is given by the harmonic oscillator frequency present in the Nilsson potential [20] Rin Rout r 41 hν =¯hω MeV. (7) A1/3 -V0 The only parameter of the model is the effective nu- Fig. 1. Schematic plot of the potential energy as a function clear radius constant r0. It is closely related to the liquid of the distance between the nucleus and emitted proton. The drop model parameter [21, 22], but it cannot be used to dotted lines correspond to the Coulomb (VC ) and centrifugal determine the size of a nucleus. It accumulates all the (Vl)terms. secondary effects not included in the simplified picture (e.g.: surface diffuseness, shape deformation, nuclear in- a one-dimensional barrier. The penetration probability teraction, preformation factor in case of alpha and cluster calculated using the WKB approximation is given by emission). 2 Rout P =exp − 2μ(V (r) − Ep)dr , (1) ¯h Rin 3Results − where μ =(mpmA−1)/(mp + mA−1) 938.3(A The comparison between the logarithms of experimen- 1)/A MeV/c2 is the reduced mass of the emitted proton tal and calculated half-lives obtained using the presented from the mother nucleus of mass number A and Ep is the model for proton emission is shown in fig. 2 and table 1. In kinetic energy of the proton, extracted from the experi- this work we considered 27 proton emitters (both ground mental Q-value. Rin is the classical inner turning point, and excited state emitters) from 113Cs to 177Tl, for which equal to the radius of the spherical square well in which all necessary input data are available. We have excluded the proton is trapped before emission, that is from the analysis some isotopes used in other elabora- π 1/3 1/3 tions for which T1/2, Q-value, branching ratio or J are Rin = r0 A1 + A2 , (2) available but measured without reasonable accuracy. The best accuracy in proton emission half-lives data reproduc- where A =1andA = A − 1 for proton emission. R 1 2 out tion is obtaineed for nuclear radius constant r0 =1.20 fm is the exit point from the potential barrier which is deter- (see fig. 3). This value is very close to the one adjusted mined by the condition V (R )=E , taking quite huge out p to alpha and cluster radioactivity (r0 =1.21 fm [19]). We values from 60 fm up to 110 fm. found that the difference between the root-mean-square In the description of proton emission the centrifugal deviation for r0 =1.20 and 1.21 fm is negligible. Thus we barrier plays a more important role than in alpha decay, mostly because the proton carries a non-vanishing angular momentum. The impact of the centrifugal term on the th exp potential barrier cannot be neglected as it reduces the 2 tunneling probability and increases the half-life. Thus the 1 proton-nucleus potential should consist of a Coulomb VC 0 and a centrifugal V part l -1 2 / s ) ZpZde -2 VC (r)= , (3) p 1/2 r -3 h¯2l(l+1) ( T Vl(r)= 2μr2 , (4) 10 -4 log -5 where l is the angular momentum carried by the valence -6 proton which should be taken from experiment. The po- tential energy V (r) shown in fig. 1 is given by I Eu Cs Tm Tm Tm Ta Ta Ta Re Re Ir Ir Au Au Tl Ho* Tm* Tm* Lu* Lu* Ta* Re* Ir* Au* Au* Tl* 109 131 113 145 146 147 155 156 157 160 161 166 167 170 171 177 141 146 147 150 151 156 161 166 170 171 177 −V0 0 ≤ r ≤ Rin, V (r)= (5) VC (r)+Vl(r) r>Rin. Fig. 2. The comparison between the logarithms of proton emission half-lives calculated using formula (6) with r0 = Note that close to Rout, Vl is negligible in comparison with 1.21 fm and the experimental data [23]. The r.m.s. of theo- VC . retical data is 0.46. Eur. Phys. J. A (2016) 52: 323 Page 3 of 6 Table 1. The comparison between the logarithms of the cal- 0.6 culated (r0 =1.21 fm) and the experimental half-lives for pro- best fit value best fit value ton emission. The Q-value and the experimental half-lives are 0.55 cal for ptoton emission for and cluster taken from ref. [23]. The log10(T1/2/s) values given in brackets emission [19] refer to the results obtained after the replacement of lexp by 0.5 the value found in our analysis. The emission from the excited state is marked by a star. 0.45 “ exp ” “ cal ” T T r.m.s. deviation Nucleus lQ[keV] log 1/2 log 1/2 10 s 10 s 0.4 109I 0 [23] 819.5 (1.9) −3.99 −5.27 (−4.25) 113 − − 0.35 Cs 2 [23] 973.5 (2.6) 4.78 5.67 1.12 1.16 1.20 1.24 1.28 131 Eu 2 [23] 947.0 (5.0) −1.70 −2.70 (−1.70) r0 [fm] 145 − − Tm 5 [23] 1736.0 (7.0) 5.50 5.55 Fig.