The WKB Approximation: An application to the alpha decay

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1 uthor: Departament de Física Fonamental, Universitat de Barcelona dvisor: Joan Parellada bstract: In this paper, the WKB approximation is presented in full detail, including a derivation of the connection formulas particular case is studied which applies the WKB approximation to alpha particle decay in various isotopes of Uranium nuclei, and is then used to find the alpha energy values using data from numerical computations in published literature Finally a comparison between the WKB approximation to published studies is made, concluding a relationship between alpha decay energy and its parent nuclear radius may exist I INTODUCTION The Wentzel-Kramers-Brillouin WKB pproximation was first introduced in quantum mechanics in 96, although it had been developed earlier This approximation is important since at the beginning of the development of quantum mechanics, physicists around the world were attempting to solve the Schrödingers equation In 98 Gamow used the pproximation to theoretically describe alpha decay for the first time In this project an overview of the WKB approximation is presented, including a detailed example of how the WKB method can be used to solve the one-dimensional Scrödinger problem Problems in three-dimensions can often be reduced to one-dimensional D problems by separation of angular and radial variables if the geometry of the problem allows, further anding the usefulness of the WKB method lpha decay energy from various Uranium isotopes will be calculated using the WKB approximation and compared to values from literature II THE WKB PPOXIMTION We will consider the One-Dimensional Scrödinger equation, since the three-dimensional Schrödinger equation can often be reduced to the solution of a onedimensional equation by separating the angular and radial variables Thus, we have: m ψ x + V x ψ x = Eψ x We will consider a solution of the form is x ψ x = N By substituting equation into we find is x + S x = F x, Electronic address: pilarpujolclosa@gmailcom where F x = m E V x The WKB method consists of anding the function S as a power series of, S = S 0 + S + S + 4 Introducing equation 4 in and equating terms of equal powers of : S 0 x = F x is 0 x + S 0 x S x = 0 is x + S x + S 0 x S x = 0 5 which can be easily solved recursively one by one just by integration and substitution Thus: S 0 = ± x F x dx + C0 6 ewriting the second ression in 5 yields S x = is 0 x S 0 x = i ln S 0 x 7 The approximation essentially consists of taking only the first two terms of the ansion 4 and neglecting the others We will study to what extent this approximation is valid and consider the first neglected term: S x = is x S x S 0 x 8 One may notice that the approximation will provide us with a good solution when 4 is a rapidly convergent series ssuming that V x is slowly varying with x as well as S 0, S, it is reasonable to consider that S 0 when S S 0 With the observation that if S is small its second derivative will also be small Using the integral forms of S and S we have seen, we see that this condition translates into: F x F x 9 Or equivalently, V x E V x 0 m E V x

2 By introducing a local wavelength, it becomes: λ π λ F x 4π V x E V x Therefore, when the relative difference of the potential energy and the wavelength is small the pproximation provides us with good results This does not happen in what we will call the linear turning points LTP, where E V = 0 It is important to note that since the pproximation will only be acceptable when the potential is slowly and smoothly varying, as is the case when not near an LTP WKB Solution in the classically allowed regions We will study how the solution of the WKB will behave in the classically allowed regions, as limited by the returning points x a and x b, but we may be able to consider these points in the limit In these regions E > V, hence F x is positive If we define: κ a x F x = m E V x The ressions for S 0 x and S x will now be: S 0 x = ± x X κ a x dx + C S x = i ln κ a x + C, where X x a, x b nd the WKB solution will therefore be: ψ x = κa x i x X κ a x dx B + κa x i x X κ a x dx 4 Where and B are arbitrary constants that will be found when applying the boundary conditions lternatively, the solution may be written as: x ψ x = κa x sin B κ a x dx + C + κa x cos X x X κ a x dx + C 5 B WKB Solution for classically forbidden regions Now we will study how the solution will behave in the classically forbidden region, which will be again limited by the returning points In this region E < V, hence F x is negative nalogously we define κ f x F x= The ressions for S0 x and S x will be the same as, but replacing κ a κ f and constants The WKB solution will then become: ψ x = κf x x X κ f x dx B + κf x x X κ f x dx 6 gain, and B are arbitrary constants that will be found when applying the boundary conditions, and alternately written ψ x = κf x sin i x X κ f x dx + C + III B κf x cos i x X κ f x dx + C CONNECTION FOMULE 7 To this point we have studied how the solution of Schrödinger s equation may be written in classically allowed regions and in the classically forbidden regions However we know that the solution that the WKB approximation method provides it is not accurate near the LTPs, since λ becomes infinite at these points It is no trivial matter finding a general solution valid for every point in the space Furthermore, in order to find the arbitrary constants of the described ressions we will introduce now what it is commonly known as Connection Formulae that will enable us to enlarge the WKB solution along the linear turning points and construct an extended solution everywhere in the space There are several authors that approach this problem differently, yet nearly all of them consider the potential to have a linear and slowly varying behavior near the LTP Then, the Schrödingers equation can be solved exactly at this points by introducing a new solution involving the Bessel function of order ± This new solution will asymptotically approach the WKB solution found previously on both sides It is in this last step that the solutions are matched with arbitrary constants of the WKB approximation, generating a continuous solution across any LTP boundary Thus lets us consider that V x has a linear behaviour in the LTPs: V x V x r + V x r x x r = E + V x r x x r, 8 where x r is a Linear Turning Point LTP, and where we have used F x mv x r x x r The Schrödinger s equation around a LTP will then be: m ψ x + V x r x x r ψ x = 0 9 Doing the following change of variables will simplify drastically our equation and its solution s = x x r α / m α = V x d ds α mv x r s Our equation will now look like: { ψ s sψ s = 0 if V x r > 0 ψ s + sψ s = 0 if V x r < 0 ψ s = 0 0 One may notice that if ψ s is the solution for the first equation in the system, ψ s will then be the solution for the second one Undergraduate Thesis Barcelona, June 06

3 iry s Functions and the solution The general solution for the ψ s sψ s = 0 is ψ s = ai s + bbi s being i s and Bi s the iry s functions, which are particular solutions s s / s / i s = I / I / 9 s s / s / Bi s = I / + I /, where I ν x are the Bessel functions One may recall that when x, which happens when 0, the iry s functions have an assymptotic behavior and may written in terms of sinus and cosinus Therefore, the general solution of ψ s sψ s = 0 that we want to consider is a ψ s = sin /4 π s s/ + π 4 b + cos /4 π s s/ + π if s < 0 4 By substitution, we find ψ s sin /4 s ψ s s /4 e s/ + C s/ + B s e s/ /4 + B cos /4 s s/ + C 6 Finally by comparison with 4 it is obvious that the WKB solution for a linear potential matches the dominant assimptotic terms of the exact solution nd provided that ψ x and ψ x have to match the solution in each side of the LTP, we can therefore obtain the relationship between the unknown coefficients =, B = B and C = π 4 obtaining: ψ x = ψ x = κa x sin xr κ f x x x κ a x dx + π 4 κ f x dx xr + B κa x cos xr x κ a x dx + π 4 + B κf x x κ f x dx xr 7 s we have been mentioning all along this will be valid as long as the potential in the LTP is linear, thus if a, b x r narrow enough that satisfies V x V x r +V x r x x r and wide enough that the WKB solution is acceptable in the limiting points a and b ψ s = a πs/4 s/ + b πs /4 s/ if s > 0 B Solution at the right of a classically allowed regions When the LTP is at the right of a classically allowed region V x r > 0 and the exact solution of ψ s sψ s = 0 is 4 We have already seen how the WKB solution will be at both sides of the LTP, by using equations 5 and 6 and by changing the integration limits properly Furthermore, near the LTPs we have that κ a x = m m V x r x r x κ f x = V x r x x r 4 nd by using the change of variables described before in 0 we have κ a = α s / and κ f = α s / Therefore, xr x κ a x dx = s/ x x r κ f x dx = s/ 5 C Solution at the left of a classically allowed region When the LTP is at the left of a classically allowed region, V x r < 0 The exact solution satisfies ψ s + sψ s = 0 and can be written as ψ s = ai s + bbi s Likewise what we have previously done, a ψ s sin πs /4 s/ + π 4 a ψ s /4 π s + s/ nd the WKB solution being: ψ x = ψ x = b cos πs /4 s/ + π 4 + b s/ κf x xr x κ f x dx + B κf x xr x κ f x dx if s > 0 if s < 0 8 κa x sin x κ a x dx + C xr + B κa x cos x κ a x dx + C xr 9 gain by using the change described in 0 we find now that κ f = α s / and κ a = α s / By substitution in the WKB solution 9 we have that ψ s s /4 s/ + B s /4 s/ ψ s s /4 sin s/ + C + B s /4 cos s/ + C 0 Finally, we find the relationship between the constants comparing it with 8, =, B = B and C = π 4 We have described a complete method to find the solution of the Schrödingers equation everywhere in the space Undergraduate Thesis Barcelona, June 06

4 IV WKB PPLICTIONS ND THE LPH DECY The WKB approximation can be used to calculate the bound state energy levels of a one-dimensional potential well To study the penetration of a potential barrier, potential wells with several turning points We will study with deep detail the alpha decay, we will find the energy values of the alpha particle in different Uranium isotopes We will assume that the alpha particle is confined by a Coulomb barrier in a potential well of radius as in We will represent the potential of the strong nuclear binding as a simple potential well V 0, if r < V r = Ze, if r > 4πε 0 r For simplicity we assume we have s-waves and that we can write Ψ r = u r, where the wavefunction is: r sinkr, if r < e r u r = N κf x dx + Be r κf x dx, if r b κf r C i r κa b κ a x dx if r > b, where K = m E + V 0 In Fig a constant energy of 0 < E < V m ax will intersect the approximation potential twice, thus creating three regions Now we study the matching relations at r = b: So if we are in region II, r b we will use the formulas described for when we are at the left of a classically allowed region and s will be negative Notice that b κ f x dx + r b κ r x dx = r If we do the following changes of variables γ = b the WKB will then be κf r κ f x dx κ f x dx = Be γ B = e γ, 4 e b r κ f x dx + B e b r κ f x dx 5 nd with the change of variables defined in 0, and being the solution of u p s = ai s + bbi s, we have: u W KB s s /4 s/ a u p s /4 π s s/ + b + B s /4 s/ s/ 6 Thus a = πb and b = π Now on the other side of point b, in region III, thus s > 0 and using equation 8: u p s = ia + b e iπ/4 e i/s/ + ia + b e iπ/4 e i/s/ πs /4 u W KB = C i s/4 s/ 7 Thus we have that ia + b = 0 and b = C πe iπ/4, a = ic πe iπ/4 Therefore, = Ce iπ/4 B = ie iπ/4 eγ = ic e iπ/4 Be γ = Ce iπ/4 = ib e γ 8 FIG : Four energy potentials are plotted versus the Uranium nuclear radius: Coulombic, Wood-Saxon, total potential, and WKB approximation as in eq By continuity of the wavefunction and its derivative at r = : sink = + B cosk = κf κa K B By substitution in 8 we have K + tank = ie γ κ a tank = + B K 9 B κ a tank Kκa 40 The right-hand side of the last equation is extremely small Furthermore, the first term of the left-hand side Undergraduate Thesis 4 Barcelona, June 06

5 can be neglected since for high Coulomb barriers and lowlying states K κ a 7 The real part of the eigenvalue will almost be the same as in the case of an infinite square well: tank 0 K = nπ n N nd by the definition of K we find: nπ E α = m V 0 4 Notice that the constant well depth V 0 has an electrostatic component and a strong force component V ESULTS Formula 4 has been used to find the energy of the emitted alpha particles for different Uranium isotopes with even atomic number and orbital quantum number l = 0 The values have been compared with the ones obtained by numerical computation In Segiel, et al, the strong interaction is approximated by a Wood-Saxon potential where the nuclear radius is approximated by = / 5fm using 4 instead of the atomic number, the skin parameter is taken to be 08fm, the values of the Wood-Saxon potential have been numerically adjusted to obtain values of the ejected alpha particle consistent with erimental values For equation 4, we use the numerically-adjusted Wood-Saxon potentials in, which were obtained with a radial quantum number n = 0 With these values we find the first positive alpha particle energies at n = 0 as well Negative energy values are incompatible with the described system since we have only considered E > 0 For n < 0 the Wood-Saxon potential is weaker and smaller values must be used In Table I, the energy values of the alpha Isotope E α 4MeV E α MeV E αmev E αmev U U U U U U U U TBLE I: Energy of the alpha particle of different Uranium Isotopes and different radi particle are shown with different values for in calculating as -4, -, and, in Columns I, II, and III, respectively, and then compared with known values from in Column IV The erimental values in match the the values in because the Wood-Saxon potential was a free parameter that could be adjusted earranging 4, we find what the Wood-Saxon should be in our case, which is shown in Table II versus our adjusted parameters for Isotope V 0 4MeV V 0 MeV V 0MeV values MeV U U U U U U U U TBLE II: Wood-Saxon potential energy for different Uranium Isotopes and different radii VI CONCLUSIONS The WKB pproximation has been studied and applied to alpha decay since it has not been reviewed in any of the courses taken during the Physics Bachelor program lthough is a quite rough numerical calculation and WKB is just an approximation, all the results are in the same range and agree well with erimental results This analysis is limited by the initial Wood-Saxon values used from source dditionally, the effect on the nuclear radius should be studied to a more precise value to better understand the interaction between the four nucleons which become the alpha particle as it escapes and the parent nucleus cknowledgments I would like to ress my sincere gratitude to my advisor Prof Parellada, and also to Prof Salvat because without their guidance and assistance this project would have never been accomplished I am also grateful to my family and friends for their support not only during the development of this project but also during the years I have been studying Barry Holstein, Understanding alpha decay, mj Phys 64, PB Siegel, and D Duarte, potential model for alpha decay, m J Phys 78, IE Nuclear Data Services nd etrieved June 07, 06, from Undergraduate Thesis 5 Barcelona, June 06