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A Potential Model for Alpha Decay D A potential model for alpha decay D. Duarte and P. B. Siegel California State Polytechnic University Pomona, Pomona, California 91768 ͑Received 28 February 2010; accepted 3 May 2010͒ We calculate the half-life and energy of alpha decay using a simple potential model consisting of the sum of the electrostatic Coulomb potential plus a Woods–Saxon form to represent the alpha-nucleus strong interaction. The calculation extends the standard treatment of alpha decay and gives students experience in fitting a theoretical model to experimental data. © 2010 American Association of Physics Teachers. ͓DOI: 10.1119/1.3432752͔ I. INTRODUCTION potential energy between a point particle of charge 2e and a uniformly charged sphere of radius w with total charge of Alpha decay is discussed in all undergraduate modern ͑Z−2͒e, physics and introductory quantum mechanics courses, usu- ally as the first example of tunneling in quantum systems. It 3w2 − r2 has the historical significance of being one of the first tests of 2͑Z −2͒e2 ͑r Յ w͒ 1 2w3 the nonrelativistic Schrödinger equation. In this article, we V = ͑1͒ present an extension of the textbook treatment of alpha decay em Ά 2͑Z −2͒e2 · ͑r Ͼ w͒, to the modeling of experimental data: The calculation of the r half-life ␶ and the energy of the emitted alpha particle E␣ using an ␣-nucleus potential. where w is the effective radius and Z is the atomic number of The use of computers has made it possible for undergradu- the nucleus. For the strong interaction, we use the Woods– ate students to obtain numerical solutions to complicated 5 Saxon form, which is a common choice for the nucleon- mathematical problems and, in particular, to perform model- nucleus potential, ing of real data. By tackling a research-type problem, stu- dents experience the difficulty of reconciling theory with experiment.2 V0 V ͑r͒ = ͑ ͒/ , ͑2͒ The quantitative analysis of alpha decay in undergraduate strong e r−w a +1 texts consists of treating the problem as barrier penetration in one dimension.3 It is shown that the decay constant where a is the “surface thickness.” For the uranium isotopes, −1/2 ␭=ln2/␶ varies approximately as e−E␣ . Some textbooks we fix the parameter a to be 0.8 fm. With a fixed, the poten- −1/2 tial V͑r͒ has two free parameters, V and w, which can be include a graph of log ␭ versus E␣ for a wide range of 0 3 isotopes that undergo alpha decay. The linear plot supports adjusted so that the calculated values of E␣ and ␶ agree with the barrier penetration model and the alpha-nucleus potential experiment. used in the calculation. In more advanced courses, a similar There are two important radii which satisfy V͑r͒=E␣. 4 approach is done using the WKB approximation. These are the radius r1 where the “tunnel” begins and the ͑ ͒ ͑ ͒ In this article, we discuss a numerical calculation of ␶ and radius r2 where the tunnel ends: V r1 =V r2 =E␣ with Ͻ E␣ for the even isotopes of uranium using a potential model r1 r2. for the interaction. The parameters of the potential are varied to fit the experimental data. To do this calculation, we will need to consider some details of alpha decay that are not part of the standard textbook treatment, such as the value of the III. SOLVING THE SCHRÖDINGER EQUATION radial quantum number of the decaying state, reasonable val- ues for the size and strength of the potential, and the prob- To solve for the energies of the bound alpha particle states, ͑ ͒ ability that neutrons and protons cluster to form an alpha we use the Schrödinger equation with the potential V r .We particle in the decaying state. We start by describing the po- will also show that the energy of the quasibound decaying tential model and the methods used in calculating E␣. Then state of the alpha particle can be determined using the same the more difficult calculation of the half-life is presented. method. We consider alpha decay transitions where the initial This article is structured so as to provide a systematic analy- and final nuclear spins are zero, Ji =Jf =0, and the decay sis that can done by undergraduate physics majors. takes place in the ᐉ=0 channel. With the usual substitution R͑r͒=u͑r͒/r for the radial coordinate, the Schrödinger equa- tion becomes II. POTENTIAL MODEL ប2 d2u͑r͒ − + V͑r͒u͑r͒ = Eu͑r͒͑3͒ For spin zero nuclei, the interaction between an alpha par- 2m dr2 ticle and the nucleus can be modeled by a spherically sym- ͑ ͒ ᐉ ͑ ͒ ͑ ͒ ͑ ͒ metric potential, V r =Vem+Vstrong, where Vem is the electro- for =0. Because R 0 is finite, u r=0 =0. To solve Eq. 3 static contribution and Vstrong represents the strong numerically, we employ a simple finite difference approxi- interaction. The electrostatic contribution is taken to be the mation for the second derivative,6 949 Am. J. Phys. 78 ͑9͒, September 2010 http://aapt.org/ajp © 2010 American Association of Physics Teachers 949 This article is copyrighted as indicated in the article. Reuse of AAPT content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 74.217.196.212 On: Sun, 29 Jun 2014 19:28:50 Table I. Experimental data for the even uranium isotopes. The width ⌫ Table II. The first quasibound alpha particle energy E␣ in terms of the radial ͑ ͒ ␶͑ ͒ ⌫ ͑ ͒ប/␶ ᐉ MeV is related to the half-life s as =ln 2 . quantum number n for orbital angular momentum =0. V0 is the potential strength for the Woods–Saxon potential. A value of w=8 fm was used. ␶ ⌫ Isotope E␣ exp exp V0 E␣ 238 17 −39 U 4.27 1.41ϫ10 3.23ϫ10 n ͑MeV͒ ͑MeV͒ U236 4.57 7.39ϫ1014 6.21ϫ10−37 U234 4.86 7.73ϫ1012 5.94ϫ10−35 2 Ϫ45.2 3.7 U232 5.41 2.17ϫ109 2.16ϫ10−31 3 Ϫ49.2 3.9 U230 5.99 1.80ϫ106 2.55ϫ10−28 4 Ϫ53.4 5.1 U228 6.80 3.28ϫ104 1.40ϫ10−26 5 Ϫ59.2 6.2 U226 7.72 3.5ϫ10−1 1.3ϫ10−21 6 Ϫ66.3 7.1 U224 8.62 9ϫ10−4 5ϫ10−19 7 Ϫ74.9 7.9 U222 9.50 1ϫ10−6 5ϫ10−16 8 Ϫ84.7 8.6 9 Ϫ95.8 9.2 10 Ϫ108.2 9.8 2m ͓ ͔ ͓ ͔ ͓ ͔ ⌬2 ͑ ͓ ͔ ͒ ͓ ͔ ͑ ͒ u i +1 =2u i − u i −1 + 2 V i − E u i , 4 ប phase shift through resonance. However, at the resonance ͑ ͒/ ͑ ͒ 7 where r͓i͔, u͓i͔, and V͓i͔ are arrays for r, u͑r͒, and V͑r͒ with energy E␣, the ratio u r2 u r1 is a minimum, with the −20 i an integer: r͓i͔=i⌬, u͓i͔=u͑r͓i͔͒, and V͓i͔=V͑r͓i͔͒. The value of this ratio being as small as 10 . Thus, we can determine the resonance energy E␣ by using the boundary step size ⌬ is taken to be 0.01 fm. To solve Eq. ͑4͒ for a condition that u͑r ͒ is a minimum. In practice, we do not particular energy, we set u͓0͔=0 and u͓1͔=1. We can calcu- 2 need to iterate out as far as r . We can iterate Eq. ͑4͒ out to late u͓i͔ for arbitrary i by iteration because u͓i+1͔ is ex- 2 a value of r inside the tunnel between r and r , which is pressed in terms of u͓i͔ and u͓i−1͔ in Eq. ͑4͒. 0 1 2 appropriate for the desired accuracy of E␣. For the uranium The energy of a bound state alpha particle E␣ Ͻ0 can be 6 series, a value of r0 of 20 fm gives an accuracy of E␣ of calculated using the shooting method. For E␣ Ͻ0, −17 10 MeV. This shooting method of calculating E␣ for the ͑ →ϱ͒→ u r 0. We start with an energy E0 slightly greater quasibound states of the Schrödinger equation gives a value ͑ ͒ ӷ than V0 and iterate Eq. 4 out to a radius r0, where r0 r1. that is more accurate than the measured value. ͑ ͒ϵ The value of u r0 uE is saved. Then the energy is incre- ⌬ ͑ ͒ mented by E and Eq. 4 is iterated again out to r0. The new ͑ ͒ϵ value of u r0 uE+⌬E is compared to uE. If the product IV. RADIAL QUANTUM NUMBER AND ENERGY Ͼ uEuE+⌬E 0, the process is repeated with the same value of CALCULATION ⌬ Ͻ E.IfuEuE+⌬E 0, we know that E␣ is between E and ⌬ ͑ ͒ E+ E because u r0 has changed sign. The process is con- An important consideration for the alpha decay calculation ͑ ͒ tinued with a value of ⌬E equal to ⌬E=−⌬E/2. As this is the approximate strength of V0 in Eq. 2 , which deter- energy stepping process is repeated, the energy converges to mines the radial quantum number n of the quasibound ᐉ=0 ͉ ͑ ͉͒ the value of E for which u r0 has a minimum. state. As the magnitude of V0 is increased, more bound states We can use this same method to find energies E␣ Ͼ0 for can be fitted into the well. The energy E␣ of the decaying quasibound states. For a given energy EϾ0, the function state is greater than zero, and we assume that the decaying u͑r→ϱ͒ equals the Coulomb wave function for ᐉ=0 shifted state is the first state with energy greater than zero.
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