TThe Coulomb Phase Shift Revisited
J. C. A. Barata, ∗ L. F. Canto, † and M. S. Hussein ‡ Instituto de F´ısica, Universidade de S˜ao Paulo,C.P. 66318, 05314-970 S˜ao Paulo, SP, Brazil Instituto de F´ısica, Universidade Federal do Rio de Janeiro,C.P. 68528, 21941-972 Rio de Janeiro, RJ, BrazilandCentro Brasileiro de Pesquisas F´ısicas (CBPF),Rua Xavier Sigaud, 150, 22290-180 Rio de Janeiro, RJ, Brazil
We investigate the Coulomb phase shift, and derive and analyze new and moreprecise analytical formulae. We consider next to leading order terms to the Stirlingapproximation, and show that they are important at small values of the angularmomentum l and other regimes. We employ the uniform approximation. The use ofour expressions in low energy scattering of charged particles is discussed and somecomparisons are made with other approximation methods. PACS numbers: 25.60.Pj, 25.60.GcKeywords: Coulomb scattering, phase shifts, semiclassical approximation
I. INTRODUCTION
The customary procedure to deal with charged particle scattering is to partial wave theamplitude and identify the Coulomb phase shift from which scattering information can beobtained, by adding to the Coulomb amplitude, the contribution from whatever other short-range potential. In many applications, the asymptotic form of large angular momentumis employed for the Coulomb phase shift. In this paper we revisit the derivation of thisasymptotic and derive next to leading order correction to the usual WKB from. ∗ Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected] a r X i v : . [ m a t h - ph ] M a y First, we recall the form of the Coulomb phase shift [1] e iσ l = Γ(1 + l + iη )Γ(1 + l − iη ) , (1)where Γ is Euler’s gamma function, l , a non-negative integer, is the angular momentum andthe real parameter η is the so-called Sommerfeld parameter, which is inversely proportionalto the square root of the scattering energy.In this paper we will present simple formal proofs of some asymptotic approximations forthe gamma function, like Stirling series and Gudermann series, and we will discuss severalmethods for approximately computing the phase shift, Eq. (1), from these asymptoticapproximations in various regimes. We will compare these results with the correspondingones obtained from other methods for computing phase shifts, like the WKB and the eikonalapproximations. We will also present proofs of some exact relations for the phase shifts σ l which are more or less known in the literature. For instance, using Eq. (1) we will easilyshow that σ l = σ + l (cid:88) m =1 tan − (cid:16) ηm (cid:17) , (2)and that σ = − γη − ∞ (cid:88) m =0 (cid:20) tan − (cid:18) ηm + 1 (cid:19) − ηm + 1 (cid:21) , (3)valid for all η ∈ R , from which we obtain for | η | < σ = − γη − ∞ (cid:88) k =1 ( − k ζ (2 k + 1)2 k + 1 η k +1 , (4)where ζ is Riemann’s zeta function and γ is Euler’s constant.Using our asymptotic expansions we show that σ (1)0 = 12 tan − ( η ) + η (cid:16) ln (cid:16)(cid:112) η (cid:17) − (cid:17) − η
12 (1 + η ) (5)is an excellent approximation for the exact expression (3). We also show numerically thatthis approximation is very good even for small values of η . More generally, we show that σ (1) l = (cid:18) l + 12 (cid:19) tan − (cid:18) ηl + 1 (cid:19) + η (cid:18) ln (cid:16)(cid:112) ( l + 1) + η (cid:17) − (cid:19) − η (cid:0) ( l + 1) + η (cid:1) (6)approximates (2) very well for (cid:112) ( l + 1) + η “large”. II. ASYMPTOTIC PROPERTIES OF THE GAMMA FUNCTION
We start with the well-known integral representation for Euler’s gamma function, validfor Re( z ) > − z ) = (cid:90) ∞ dt t z e − t . (7)For (1) we will take z = l + iη . Writing t z = e z ln t , we can use the saddle point method toevaluate the integral Γ(1 + z ) = (cid:90) ∞ dt e − t + z ln( t ) . (8)Expanding the exponent of the integrand around its extremum, at t = z and keeping termsup to second order, − t + z ln( t ) (cid:39) − z + z ln( z ) − z ( t − z ) , (9)Eq. (8) becomes Γ(1 + z ) (cid:39) e − z z z (cid:90) ∞ dt e − ( t − z ) / z . (10)For Re( z ) = l (cid:29) z ) = η >
0, the Gaussian integral can be easily evaluated as (cid:90) ∞ dt e − ( t − z ) / z = (cid:90) ∞−∞ dt e − ( t − z ) / z = √ πz (11)and using this result in the previous equation, we obtainΓ(1 + z ) (cid:39) √ π e − z z z +1 / . (12)This is the well-known Stirling’s approximation for the gamma function, whose validitycan be established for the whole complex plane, except the non-positive real axis, i.e., in C − = C \ ( −∞ , | z | . See, e.g., Ref. [2].On the real line, Stirling’s approximation (12) has a long history having been first pre-sented by de Moivre around 1730, who found an approximation for n !, valid for “large” n ,in the form n ! (cid:39) K n n + e − n , for some unspecified constant K . In the same year, Stirlingproved that K = √ π using Wallis product formula for π . This approximation becameknown as Stirling’s approximation for n ! and quite soon generalizations for Euler’s gammafunction on the positive real line became available. Stirling’s approximation is very usefulin Statistics and Probability Theory because if offers a very good approximation for “large”factorials. For small values of n , however, correcting factors are in order. Stirling foundcorrections for the approximation that bears his name for n ! in terms of an asymptotic(but not convergent!) series in 1 /n that became known as Stirling’s series. Further anal-ysis of those corrections for Euler’s gamma function Γ( z ), valid on the complex half-planeRe( z ) >
0, have been performed by Binet in 1839 (Ref. [12]). Another important contri-bution was made by Gudermann in 1845 (Ref. [13]), who found another expression for thecorrecting factors in terms of a convergent expansion of another kind. The most importantcontribution to the study of corrections to Stirling’s approximation on the complex planewas the work of Stieltjes, dated of 1889 (Ref. [14]), who generalized Stirling’s approximationand Gudermann’s corrections to the whole complex plane, excluding the negative real axis(i.e., to C − ). For a more detailed account of the developments on the complex plane, see[2] and [11]. For some recent contributions to the correcting factors to factorials, see [15]and [16]. This last reference contains a list of historical results on corrections on Stirling’sapproximation.Let us briefly describe the ideas behind Gudermann’s corrections and present Stirling’sseries. Since Euler’s gamma function satisfies Γ( z + 1) = z Γ( z ), we get from (12) theapproximation Γ( z ) (cid:39) √ π e − z z z − / . (13)However, (13) contrasts with the expression obtained from (12) itself by replacing z by z − z ) (cid:39) √ π e − ( z − ( z − z − / . (14)Since both expressions (13) and (14) are only valid for | z | very large, there is no practicaldifference between them. Nevertheless, one can better deal with this situation by seeking anexact representation for Γ in the whole region C − (and not only for “large” | z | ) in the formΓ( z ) = √ π e − z z z − / e µ ( z ) , (15)and fixing the correction factor e µ ( z ) by imposing the relation Γ( z + 1) = z Γ( z ). A simplecomputation reveals that this condition implies that µ has to satisfy the functional equation µ ( z ) − µ ( z + 1) = (cid:18) z + 12 (cid:19) ln (cid:18) z (cid:19) − . (16)Moreover, the validity of (13) for “large” | z | leads to the condition lim | z |→∞ µ ( z ) = 0. Thisallows to a solution for (16). Indeed, it follows immediately from (16) that for any positiveinteger n one has µ ( z ) − µ ( z + n ) = n − (cid:88) m =0 (cid:20)(cid:18) z + m + 12 (cid:19) ln (cid:18) z + m (cid:19) − (cid:21) . (17)Hence, the condition lim | z |→∞ µ ( z ) = 0 implies, in particular, that lim n →∞ µ ( z + n ) = 0 andwe get µ ( z ) = ∞ (cid:88) m =0 (cid:20)(cid:18) z + m + 12 (cid:19) ln (cid:18) z + m (cid:19) − (cid:21) . (18)This series is known as Gudermann series. According to [2], it was first obtained by thatauthor in 1845 [13] for real and positive z and the generalization for z ∈ C − was obtainedby Stieltjes in 1889 [14].The series (18) converges for all z ∈ C − and µ can be bounded by | µ ( z ) | ≤
112 1cos ( ϕ/
2) 1 | z | , (19)with z = | z | e iϕ , z ∈ C − (see [2] or [11]). On the real line, very precise upper and lowerapproximants of the Γ-function can be obtained from (15) and (18), valid also for smallvalues of the argument (see [15] and [16]).Notice that, by (19), for | z | (cid:39)
1, the contribution of the factor e µ ( z ) to (15) is lower thanci. 9% for ϕ (cid:39) ϕ (cid:39) π/
2. Hence, for the sake of precision, it isrelevant in that range to consider corrections to Stirling’s approximation (12).With (15), and since Γ( z + 1) = z Γ( z ), we can also writeΓ( z + 1) = √ π e − z z z +1 / e µ ( z ) . (20)Hence, e µ ( z ) acts as a correcting factor for both (12) and (13). Beyond the Gudermann series(18), the function µ can be represented in many other forms. One of the most useful of themis the so-called Stirling series: µ ( z ) = ∞ (cid:88) n =1 B n (2 n − n z n − , (21)where B k is the k -th Bernoulli number. One has to say that the series on the r.h.s. of (21)is an asymptotic series in C − , but is not convergent! See again [2]. Therefore, it is not to beseen as a Laurent expansion for µ around z = 0. In fact, z = 0 is not an isolated singularityof µ , but a branch point, as we see from (18). Since (21) is an asymptotic series we can getgood approximations for µ in C − by truncating it at some finite value of the index n andtaking 1 / | z | small enough. The first terms of (21) are µ ( z ) = 112 1 z − z + 11260 1 z + · · · . (22)For the correcting factor e µ ( z ) , this gives e µ ( z ) = 1 + 112 1 z + 1288 1 z − z + · · · . (23)Therefore, we may writeΓ( z ) = √ π e − z z z − / (cid:20) z + 1288 1 z − z + · · · (cid:21) , (24a)Γ( z + 1) = √ π e − z z z +1 / (cid:20) z + 1288 1 z − z + · · · (cid:21) . (24b)These approximations are also known as Stirling’s series for the gamma function. Theyare asymptotic (but not convergent!) expansions for Γ, valid for z ∈ C − and 1 / | z | “small”.Although (24) are asymptotic but not convergent approximations for Γ, they can in somepractical situations be more useful than the representation (15) or (20) with the convergentGudermann expansion (18). III. STIRLING’S SERIES. A FORMAL DERIVATION
The Stirling’s series (24) leads to more accurate estimates of the phase-shifts (1) thanStirling’s approximation (12), even when the value of l is not “large”. There are many proofsof Eqs. (21) or (24) in the real or in the complex domain (see, e.g., Ref. [2, 3]), but they areall rather involved. We will present now a simple derivation of Eq. (24b) (see also [4]).Consider the curve C in the complex t -plane parametrized by s ∈ ( −∞ , ∞ ) and satisfyingthe parabolic functional mapping [4, 6, 7], t ( s ) − z ln (cid:0) t ( s ) (cid:1) − A ( z ) = s , with A ( z ) = z − z ln z and with t (0) = z . Defining ω ( t ) = t − z ln( t ) , we can write ω (cid:0) t ( s ) (cid:1) = A ( z ) + s . One can choose t ( s ) satisfying t ( s ) → t ( s ) (cid:39) e − s z for s → −∞ and t ( s ) (cid:39) s + i Im( z ) ln (cid:16) s (cid:17) for s → + ∞ . By a continuous deformation of the t -integration curve in (7)from the positive real axis to C and by carefully extending the integration to infinity, onecan write, by Cauchy’s theorem,Γ(1 + z ) = (cid:90) C dt t z e − t = e − z + z ln z (cid:90) ∞−∞ e − s / dt ( s ) ds ds. The derivative dt/ds can be written as a power series in s using the mapping equation, dtds = ∞ (cid:88) k =0 a k s k . (25)It is a simple matter to evaluate the coefficients a k , by repeated differentiation of the mappingequation. We calculate below the first two terms, a , and a (higher order terms can bederived similarly). Calling ω (cid:0) t ( s ) (cid:1) ≡ v ( s ), we write dv ( s ) ds = (cid:16) − zt (cid:17) dtds = s,d v ( s ) ds = (cid:16) − zt (cid:17) d tds + zt (cid:18) dtds (cid:19) = 1 ,d v ( s ) ds = (cid:16) − zt (cid:17) d tds + 2 zt d tds dtds − zt (cid:18) dtds (cid:19) = 0 ,d v ( s ) ds = (cid:16) − zt (cid:17) d tds + 4 zt d tds dtds + 3 zt (cid:18) d tds (cid:19) − zt d tds (cid:18) dtds (cid:19) + 6 zt (cid:18) dtds (cid:19) = 0 . We now evaluate these equations at the extremum point, defined by the condition (cid:20) dω ( t ) dt (cid:21) t = 0 , (26)which is t = z and s = 0. The above four equations lead to the results (cid:20) dtds (cid:21) t = t = √ z, (cid:20) d tds (cid:21) t = t = 16 √ z and (cid:20) d ds (cid:21) t = t = 136 z / . (27)Thus the coefficients a k above are, a = (cid:20) dtds (cid:21) t = t = √ z, (28) a = 12! (cid:20) d tds (cid:21) t = t = 112 √ z , (29) a = 14! (cid:20) d tds (cid:21) t = t = 14! 136 z / . (30)Using the Gaussian integral formula, (cid:90) ∞−∞ ds e − s / s n ds = √ π ( n − , (31)for n even, the full integral in the Γ function can be written down in the form of the Stirlingseries above, namely, Γ( z + 1) = √ π e − z z z ∞ (cid:88) k =0 (2 k − a k . (32)Thus, evaluating the coefficients a k and inserting above, we getΓ( z + 1) = √ π e − z z z +1 / (cid:20) z + 1288 z + · · · (cid:21) . (33)The use of the quadratic mapping analysis of integrals and the generation of appropriateasymptotic series in general was demonstrated in, e.g., [7] for the case of the Gamow integralemployed in nuclear astrophysics. This integral has the general form, I ( a ) = (cid:90) ∞ dxS ( x ) exp (cid:16) − x − ax / (cid:17) , (34)where the function S ( x ) is usually a slowly varying function of x and can be written as asum S ( x ) = (cid:80) ∞ k =0 S k x k , which would then result in a series representation of the Gamowintegral similar to what was done for the Γ function above.We now use the above results to evaluate the Coulomb phase-shifts (1). IV. AN EXACT EXPRESSION FOR THE PHASE SHIFT AND FIRSTAPPROXIMATIONS
From (1) and (15), and using the identityln (cid:18) x + iyx − iy (cid:19) = 2 i tan − (cid:16) yx (cid:17) , (35)valid for x, y ∈ R with x >
0, we get after some elementary computations the followingexact expression for the phase shift: σ l = σ (0) l + M l, η , (36)where we define σ (0) l ≡ (cid:18) l + 12 (cid:19) tan − (cid:18) ηl + 1 (cid:19) + η (cid:16) ln (cid:16)(cid:112) ( l + 1) + η (cid:17) − (cid:17) , (37)and M l, η ≡ i (cid:0) µ (1 + l + iη ) − µ (1 + l − iη ) (cid:1) . Below, we will use the series expansion (18) tofind closed expressions for M l, η , for σ l and, in particular, for σ . Before we proceed let usmake some comments about some useful approximation we can obtain from (36)–(37).Using (19) with | z | = (cid:112) ( l + 1) + η and ϕ = tan − (cid:0) ηl +1 (cid:1) , one finds the bound | M l, η | ≤ (cid:16) l + 1 + (cid:112) ( l + 1) + η (cid:17) . Hence, for l (cid:29) | η | (cid:29) M l, η to (36) can be neglected and we canrestrict as a first approximation to σ (0) l .For l = 0, for instance, one gets from (37) the approximation σ (cid:39) σ (0)0 = 12 tan − ( η ) + η (cid:16) ln (cid:16)(cid:112) η (cid:17) − (cid:17) (38)with an error bounded by (cid:16) (cid:112) η (cid:17) − . For | η | (cid:29)
1, Eq. (38) gives the approximation σ (cid:39) π η (cid:0) ln( η ) − (cid:1) . (39)For | η | l +1 (cid:28) l (cid:29) σ l (cid:39) η ln( l + 1) . (40) V. CLOSED EXPRESSION FOR THE PHASE SHIFT
Now we will try to find closed expressions for σ l and σ . According to (1), using the factthat Γ( z + 1) = z Γ( z ) for all z ∈ C − , we have e iσ l = Γ(1 + l + iη )Γ(1 + l − iη ) = ( l + iη ) · · · (1 + iη )( l − iη ) · · · (1 − iη ) Γ(1 + iη )Γ(1 − iη ) = ( l + iη ) · · · (1 + iη )( l − iη ) · · · (1 − iη ) e iσ . Therefore σ l = σ + i (cid:80) lm =1 ln (cid:16) m + iηm − iη (cid:17) and using (35), one has σ l = σ + l (cid:88) m =1 tan − (cid:16) ηm (cid:17) . (41)Eq. (41) is a remarkable expression, since it shows that σ l differs from σ by a finite sum.Let us now find a more explicit expression for σ . According to (36)–(37), σ = 12 tan − ( η ) + η (cid:16) ln (cid:16)(cid:112) η (cid:17) − (cid:17) + M , η , (42)Now we analyze M , η ≡ i (cid:0) µ (1 + iη ) − µ (1 − iη ) (cid:1) more closely. According to (18) (withthe change of summation variable m → m − M , η = 12 i ∞ (cid:88) m =1 (cid:34) (cid:18) m + iη + 12 (cid:19) ln (cid:18) m + iη (cid:19) − (cid:18) m − iη + 12 (cid:19) ln (cid:18) m − iη (cid:19) (cid:35) . (43)0After simple rearrangements and using we can write (43) as M , η = 12 i ∞ (cid:88) m =1 (cid:34)(cid:18) m + 12 (cid:19) (cid:18) ln (cid:18) m + 1 + iηm + 1 − iη (cid:19) − ln (cid:18) m + iηm − iη (cid:19)(cid:19) + iη (cid:32) ln (cid:0) ( m + 1) + η (cid:1) − ln (cid:0) m + η (cid:1) (cid:33)(cid:35) . (44)Using (35) and defining A m ≡ i ln (cid:18) m + iηm − iη (cid:19) = tan − (cid:16) ηm (cid:17) and B m ≡ η (cid:0) m + η (cid:1) we can write (44) as M , η = lim N →∞ M N , η , where M N , η = N (cid:88) m =1 (cid:20)(cid:18) m + 12 (cid:19) ( A m +1 − A m ) + B m +1 − B m (cid:21) . (45)Now, N (cid:88) m =1 ( B m +1 − B m ) = B N +1 − B (46)and N (cid:88) m =1 (cid:18) m + 12 (cid:19) ( A m +1 − A m ) = N (cid:88) m =1 (cid:20)(cid:18) ( m + 1) + 12 (cid:19) A m +1 − (cid:18) m + 12 (cid:19) A m (cid:21) − N (cid:88) m =1 A m +1 = (cid:18) N + 1 + 12 (cid:19) A N +1 − (cid:18) (cid:19) A − N (cid:88) m =1 A m +1 . (47)Hence, collecting the results, we have M , η = lim N →∞ (cid:34)(cid:18) N + 32 (cid:19) tan − (cid:18) ηN + 1 (cid:19) + η (cid:16) ( N + 1) + η (cid:17) − N (cid:88) m =1 tan − (cid:18) ηm + 1 (cid:19)(cid:35) −
32 tan − ( η ) − η (cid:16) η (cid:17) = η −
32 tan − ( η ) − η (cid:16) η (cid:17) + lim N →∞ (cid:34) η ln( N + 1) − N (cid:88) m =1 tan − (cid:18) ηm + 1 (cid:19)(cid:35) . (48)Now, adding and subtracting η (cid:80) Nm =0 1 m +1 to the terms in brackets, whose limit is beingtaken in (48), we get η ln( N +1) − N (cid:88) m =1 tan − (cid:18) ηm + 1 (cid:19) = η (cid:34) ln( N + 1) − N (cid:88) m =0 m + 1 (cid:35) − N (cid:88) m =0 (cid:20) tan − (cid:18) ηm + 1 (cid:19) − ηm + 1 (cid:21) +tan − ( η ) . With this, (48) becomes M , η = η (1 − γ ) −
12 tan − ( η ) − η (cid:16) η (cid:17) − ∞ (cid:88) m =0 (cid:20) tan − (cid:18) ηm + 1 (cid:19) − ηm + 1 (cid:21) , (49)1 ησ π / FIG. 1: The graph of σ /π according to the exact formula (50) for 0 ≤ η ≤
4. Notice that σ vanishes at η (cid:39) . where γ ≡ lim N →∞ (cid:104)(cid:80) Nm =0 1 m +1 − ln( N + 1) (cid:105) is Euler’s constant γ (cid:39) . . . . .Inserting (49) into (42) we finally get, after some trivial cancelations, σ = − γη − ∞ (cid:88) m =0 (cid:20) tan − (cid:18) ηm + 1 (cid:19) − ηm + 1 (cid:21) , (50)By recalling the Taylor expansion of tan − about 0,tan − ( x ) = ∞ (cid:88) k =0 ( − k k + 1 x k +1 , (51)we can write (50) for | η | < σ = − γη − ∞ (cid:88) m =0 ∞ (cid:88) k =1 ( − k k + 1 (cid:18) ηm + 1 (cid:19) k +1 = − γη − ∞ (cid:88) k =1 ( − k ζ (2 k + 1)2 k + 1 η k +1 , where ζ is Riemann’s zeta function.In Figure 1 we plot σ for 0 ≤ η ≤
4. As we discuss below, for larger values σ followsvery closely the behavior dictated by the asymptotic approximations (38) or (39). In Figure1 we see that σ vanishes at η = 0 and η (cid:39) . η is coincidentallyquite close to the value of √
2, which was obtained in [17] in connection with the scatteringof identical charged particles (Fermions or Bosons). The Mott cross section, at this criticalvalue of η = √
2, was predicted to be isotropic over a broad range of angles around 90 o .2 VI. THE PHASE SHIFT AND STIRLING’S SERIES
The above asymptotic formulae (37), (38), (39) and (40) will be discussed further belowwithin the low-energy, large- η , large- l case of the WKB approximation, and the high-energy,small- η , large- l case of the eikonal approximation. In this section we use Stirling series forthe gamma function (24) to further improve those approximations.A first order correction to Eq. (37) can be easily evaluated using Stirling’s series (22)–(24).For this purpose, we rewrite Eq. (1) in the form e iσ l = e iσ (0) l F ( z ) , (52)with z = 1 + l + iη and σ (0) l given in (37), where F ( z ) = e µ ( z ) − µ ( z ∗ ) = exp (cid:18) (cid:18) z − z ∗ (cid:19) − (cid:18) z ) − z ∗ ) (cid:19) + · · · (cid:19) . (53)where the ∗ symbol refers to complex conjugation. The first order approximation for F ( z )is F (1) ( z ) ≡ exp (cid:18) (cid:18) z − z ∗ (cid:19)(cid:19) =: e i ∆ σ (0) l . (54)Evaluating the above expression, we find∆ σ (0) l = − η (cid:0) ( l + 1) + η (cid:1) . (55)Therefore, the first order approximation to the Coulomb phase-shifts is σ (1) l = σ (0) l + ∆ σ (0) l = (cid:18) l + 12 (cid:19) tan − (cid:18) ηl + 1 (cid:19) + η (cid:18) ln (cid:16)(cid:112) ( l + 1) + η (cid:17) − (cid:19) − η (cid:0) ( l + 1) + η (cid:1) . (56)Figure 2 shows Coulomb phase-shifts versus l .From (56) we get, in particular, the first order correction to (38) due to Stirling’s series: σ (1)0 = σ (0)0 + ∆ σ (0)0 = 12 tan − ( η ) + η (cid:18) ln (cid:16)(cid:112) η (cid:17) − (cid:19) − η
12 (1 + η ) . (57)It is interesting to compare the approximation (57) to our exact expression for σ givenin (3). Figure 3 we plot the relative error σ (1)0 − σ σ for values of the Sommerfeld parameterbetween 0 and 5. It shows that σ (1)0 is an excellent approximation for σ , with relative errorsbelow 1%, even for “small” values of η , except, perhaps, near η (cid:39) .
81, where σ vanishes.3 FIG. 2: Coulomb phase-shifts as functions of l . Results normalized with respect to π are given fortwo values of the Sommerfeld parameter. η FIG. 3: The relative error σ (1)0 − σ σ for 0 ≤ η ≤
5. The sharp peak around η (cid:39) .
81 is due the thevanishing of σ at that point. VII. DISCUSSION OF THE RESULTS FOR STIRLING’S SERIES
The lowest orders approximations (in 1 /z ) of the previous section are supposed to workfor | z | (cid:29)
1, which means l (cid:29) η (cid:29)
1. They are, in fact, very accurate, even whenthese conditions are not well satisfied. A first illustration of this fact is presented in Figure4, where we compare the exact phase-shifts (solid line) with the lowest order approximation σ (0) l (open circles) for a small η , as functions of l . They are very close. The only exception4 l σ l ( d e g . ) exact phase-shifts large l approximation η = 0.1 FIG. 4: The approximate Coulomb phase-shifts σ (0) l and σ (1) l for η = 0 .
1, as functions of l . Fordetails see the text. is the case of l = 0, where they are quite different.For larger values of η , the agreement is much better. This can be seen in Table I, wherewe compare the two approximations of the previous section with the exact Coulomb phase-shifts. Note that the σ (1) l is very accurate even for l = 0 and η = 0 . σ (0) l =0 and σ (1) l =0 . The comparison indicates that the usuallarge- η approximation, σ (0) l =0 , is very poor. The situation is completely different with theimproved approximation, σ (1) l =0 , which includes the first order correction of Stirling’s series.In this case, one gets accurate results for any value of the Sommerfeld parameter, η . VIII. COMPARISON WITH OTHER APPROXIMATIONS
We turn now to well known approximations used to calculate the phase shifts, motivatedby the physical conditions. The first such approximation is the WKB one invoked to considerscattering under semi-classical conditions of short local wave lengths. In this approximation,5 η l σ (0) l /π σ (1) l /π σ exact l /π l , for two values of the Sommerfeld parameter. one finds for the phase shift the following expression [1], σ l = lim r →∞ (cid:34)(cid:90) rr l dr (cid:48) k l ( r (cid:48) ) − (cid:90) rr (0) l dr (cid:48) k (0) l ( r (cid:48) ) (cid:35) , (58)where k l ( r ) and k (0) l ( r ) are the local wave numbers in the presence and absence of thepotential, respectively, (cid:126) k l ( r ) = p l ( r ) = (cid:115) µ (cid:18) E − V ( r ) − (cid:126) ( l + 1 / µr (cid:19) , (59) (cid:126) k (0) l ( r ) = p (0) l ( r ) = (cid:115) µ (cid:18) E − (cid:126) ( l + 1 / µr (cid:19) , (60)where µ , here, is the reduced mass. The radii, r l and r (0) l , are the classical turning pointsdefined by p l ( r l ) = 0, and p (0) l ( r (0) l ) = 0, respectively. The phase shift, now a function ofthe energy (or the asymptotic wave number k ), and the semi-classical angular momentum, λ = l +1 /
2, can be evaluated once the potential is given. In the case of point-charge Coulombscattering, the result of such a calculation results in the following expression for the WKBphase shift function δ ( λ, k ) = σ ( λ, k ), σ ( λ, k ) = 12 η ln (cid:2) η + λ (cid:3) + λ sin − (cid:34) η (cid:112) η + λ (cid:35) − η. (61)6 η -0.1-0.0500.05 σ / π σ l = σ l = σ l =0exact FIG. 5: Coulomb s-wave phase-shifts as a function of the Sommerfeld parameter. Exact values arecompared with the approximations discussed in the text. where the Sommerfeld parameter η is related to the asymptotic wave number, k , by, η = ka ,where a is half the distance of closest approach for head-on, zero impact parameter, collision.The above expression should be compared to our Eq. (37). Clearly the major differenceresides in the introduction of the semi-classical angular momentum variable λ in the formerequation Eq. (61). This change is in fact necessary when the WKB form of the wave functionis invoked [8], to guarantee the presence of a classical turning point for s-waves (now theangular momentum is not zero but rather (cid:126) / r = z + b , with b being the impact parameter, kb = λ ,we find for the eikonal Coulomb phase shift, σ ( b, k ), the following, σ ( b, k ) = − η (cid:90) ∞−∞ dz √ b + z . (62)The integral in the above equation diverges at both extremes. Ref. [9] introduced ascreening function that renders the integral finite. The screening function considered is7 F ( r ) = Θ( a − r ), where the screening length a is of very large, and taken to be a (cid:29) b . Then, σ ( b, k ) = − η (cid:18) a + 2 √ a − b a − √ a − b (cid:19) , (63)which, with a (cid:29) b , reduces to σ ( b, k ) = η ln (cid:18) b a (cid:19) = η ln λ − η ln(2 ka ) . (64)The above form is similar to Eq. (40), except for the change l → λ and the screening term,which depends only on energy and not on λ . Glauber [9] considered also and exponentialscreening function of the form F ( r ) = exp − r/a , and found the following limiting expressionfor the Coulomb phase, σ ( b, k ) = η (cid:20) ln b a − γ (cid:21) , (65)where γ is Euler’s constant, γ = 0 . . . . . Again, the main feature of Eq. (40) ofa logarithmic dependence on b , and thus λ is maintained. A Gaussian shaped screeningfunction F ( r ) = exp [ − r /a ] was considered by [10], and the result found for the phaseshift is, σ ( b, k ) = η (cid:20) ln b a − γ (cid:21) . (66)Once again the main feature of Eq. (40) is maintained, namely, the logarithmic dependenceon b. Clearly the discussion above demonstrates that the best route to follow to obtainwell behaved and defined asymptotic forms for the Coulomb phase shift is to rely on theexact expression, Eq. (1), and use the uniform approximation method of Stirling’s series toobtain the correct asymptotic form of the Γ function. IX. QUANTUM AND CLASSICAL COULOMB DEFLECTION FUNCTIONS
An important theoretical entity which enters in any semi-classical treatment of scatteringis the deflection function. In the continuous λ = l + 1 / σ ( λ, k ) with respect to λ [1]. Thus,Θ( λ ) = 2 dσ ( λ, k ) dλ . (67)8Using the WKB expression, Eq. (61), we obtain the well known classical Rutherford deflec-tion function, Θ( λ ) = 2 tan − ηλ . (68)The above should be compared to the exact ”quantum” deflection function obtained from σ l of Eq. (2), which can be re-written as a recursion formula, viz, σ l = σ l − + tan − ηl . (69)Clearly one can define the derivative as merely the difference, σ l − σ l − , and thus the quantumdeflection function, Θ q ( l ) is, Θ q ( l ) = 2[ σ l − σ l − ] = 2 tan − ηl . (70)The quantum and classical deflection functions agree if one makes the change l → λ , asexpected. X. CONCLUSIONS
In this paper we have revisited the literature on the point Coulomb phase shift, andsharpened the applicability and range of validity of several of the approximations usuallyemployed to calculate it . In particular, the phase shift at zero angular momentum, σ ( η ), as a function of the Sommerfeld parameter, η , is calculated exactly and a simpleanalytic expression for it valid for small and large η is found. This is useful whenevaluating the phase shift as a function of angular momentum, l , using the exact formula, σ l ( η ) = σ ( η ) + (cid:80) lm =0 tan − ( η/m ), of Eq. (2). Acknowledgments
This work was supported in part by the CNPq, FAPESP and the MCT-National Instituteof Quantum Information. [1] See, e.g., D. M. Brink,
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