Formulas for Angular-Momentum Barrier Factors Version II

Transcription

1 BNL PREPRINT BNL-QGS brfactor1.tex Formuas for Anguar-Momentum Barrier Factors Version II S. U. Chung Physics Department, Brookhaven Nationa Laboratory, Upton, NY March 19, 2015 abstract A detaied description is given of the formuas for anguar-momentum barrier factors and their properties. Expicit formuas for the barrier factors are given for up to 9. under contract number DE-AC02-98CH10886 with the U.S. Department of Energy

2 1 Introduction In this note, we have worked out a simpe method of evauating the Batt-Weisskopf centrifuga-barrier factors for arbitrary. We start with the formuas given by von Hippe and Quigg[1]. The functions of interest for our purposes are the spherica Hanke functions of the first kind xh (1) (x) = i exp [i(x π 2 ) ] k=0 ( 1) k (+k)! k!( k)! (2ix) k (1) and the recursion reation ( 2+1 x ) h (1) (x) = h (1) 1 (x)+h(1) +1 (x) (2) We show ater in this note that the desired barrier factors are, for rea x 0, 1 f (x) = xh (1) (x) with x = p/p R (3) The expression for barrier factors comes with a parameter p R, which is traditionay taken to be equa to 1 fermi or p R = GeV/c. With the use of the formua, we give expicit formuas for the barrier factors for up to 9. A Fortran program has been written, which can be used to cacuate f (x) for arbitrary integer vaues of and for any vaues of x. A coection of figures for f (x) for various vaues of and x is given at the end of this note. In the Appendix, we retrace much of the formuas regarding the properties of the Hanke functions as given by von Hippe and Quigg[1]. 2 Anguar-Momentum Barrier Factors For our purposes, it is convenient to introduce another function g (x) via xh (1) (x) = ( ) ie ix g (x) (4) x 1

3 where g (x) = a k ( ix) k, k=0 a k = (+k)! 2 k k!( k)!, a 0 = 1 (5) The functions g (x) are poynomias in ( ix) of order with integer coefficients. The barrier factors (3) can be rewritten f (x) = x g (x) 1 (6) Thus we find, again with x = p/p R, f (x) = x x 0 a and f (x) = 1 (7) x We see that the functions f (x) indeed possess the desired properties of being the barrier factors. Note that the functions f (x) have the right threshod behavior of x, and, in addition, they have the desired property of being damped to a constant (=1) for arge x. After a, the barrier factors p have been introduced to overcome the singuarities at p = 0, associated with the wave functions as described by the spherica harmonics Ym(Ω). So the ampitude shoud not bow up as p when p. The coefficients a k are integers, and they are tabuated beow for 9: {k} {00} {10} {11} {20} {21} {22} {30} {31} {32} {33} a k {k} {40} {41} {42} {43} {44} {50} {51} {52} {53} {54} {55} a k {k} {60} {61} {62} {63} {64} {65} {66} a k {k} {70} {71} {72} {73} {74} {75} {76} {77} a k {k} {80} {81} {82} {83} {84} {85} {86} {87} {88} a k

4 {k} {90} {91} {92} {93} {94} {95} {96} {97} {98} {99} a k The functions g (x) for 9 are g 0 (x) = 1 g 1 (x) = ix+1 g 2 (x) = x 2 3ix+3 g 3 (x) = ix 3 6x 2 15ix+15 (8) g 4 (x) = x 4 +10ix 3 45x 2 105ix+105 g 5 (x) = ix 5 +15x ix 3 420x 2 945ix+945 g 6 (x) = x 6 21ix x ix x ix and g 7 (x) = ix 7 28x 6 378ix x ix x ix g 8 (x) = x 8 +36ix 7 630x ix x ix x ix (9) g 9 (x) = ix 9 +45x ix x ix x ix x ix The recursion reation for the functions g (x) is, from (2) and (4), g +1 (x) = (2+1)g (x) x 2 g 1 (x) (10) It can be shown that the poynomias in (8) and (9) satisfy the recursion reation. With the hep of (8) and (9), we can write down the expicit form of f (x) f (x) = x { [ b R (x) ] 2 +x 2 [ b I (x) ] 2 } 1/2 with x = p/p R (11) The functions b R (x) and bi (x) are tabuated beow for 9: 3

5 b R (x) bi (x) x (2x 2 5) x x 4 45x (2x 2 21) 5 15(x 4 28x 2 +63) x 4 105x x 6 210x x (x 4 60x ) 7 7(4x 6 450x x ) x 6 378x x x 8 630x x 4 9(4x 6 770x x ) x (x 8 308x x 4 x 8 990x x x ) x Both b R (x) and bi (x) are poynomias in x2. For = even, b R (x) is a poynomia of order (/2) in x 2, and b I (x) is a poynomia of order (/2 1) in x2. For = odd, both b R (x) and b I (x) are poynomias of order ( 1)/2 in x2. For any, the factor given in (11), i.e. [ ] 2 ] b R (x) +x 2[ 2 b I (x) is a poynomia of order in x 2. This is to be expected, since f (x) goes to 1 as x. It is instructive to study the behavior of the functions f (x) under two specia conditions of x: (a) x = and (b) x = 1. The behavior of f (x) under these conditions is iustrated in Figs. 1, 2, 3 and 4. The function f (x) is shown in Fig. 1 for 30 and again in x= Fig. 2 for up to 100 on og scae; it is seen that the function for 1 decreases sowy as a function of, approaching f (x) 0 as. a The behavior of the function is x= further iustrated in a tabuar form beow: f (x) x= a The imiting vaue of f (x) as is assumed to be equa to 0. x= 4

6 The decrease in f (x) as is extremey sow. An approximate formua is, since x= f (x) = 1/ 2 for x = = 1, (n) (n) 2 (12) f (x) x= This formua is correct to within 2% for Consider next the behavior of f (x), if we set x = 2. It can be shown that f (x) x=2 So f (x) approaches rapidy to a constant : x=2 [ ] f (x) exp( ) x=2 (13) accurate to within 1% for 1. What may be worth noting here is that f (x) is a sowy decreasing function of x, if we set x = and et, but that it rapidy becomes a constant, if we set x = 2. The behavior of f (x) with x = 1.0 is shown as a function of in Fig. 3 (inear scae for the y axis) for 8 and again in Fig. 4 (og scae for the y axis) for 30. It is seen that f (1.0) drops off rapidy as increases, even though the traditiona barrier factor has been fixed to 1, i.e. x = 1. For exampe, f (1.0) for = 5. It is instructive to examine this phenomenon further by setting x = 0.50: it can be shown in this case that f (0.50) for = 5. But the conventiona barrier factor x aone amounts to , so we see that the factor g (0.50) 1 contributes an additiona damping factor of [see (6)]. An approximate formua is f (x) 1 exp[ ( 1)( /)] (14) x=1 2 accurate to within 5% for We gain further insight from a study of the coefficient a by using an approximate formua for the factorias, i.e. the Stiring s formua n! ( n n 2πn e) 5

7 1.0 x= f (x) Figure 1: The behavior of f (x) as a function of = {0,30}. x= 1.0 x= f (x) Figure 2: The behavior of f (x) as a function of = {1,100}. x= 6

8 1.0 f (1.0) Figure 3: The behavior of f (1.0) as a function of = {0,8}. n { f (1.0) } Figure 4: The behavior of n{f (1.0)} as a function of = {0,30}. 7

9 We see that a 2 ( ) 2 for 1 (15) e We give a comparison of the correct and approximate vaues of a in a tabuar form beow correct a k approx. a k correct a k approx. a k The approximation is exceent for a 1. We see that, from (7) and (15), f (x) 1 ( e ) x = 1 ( e ) ( x ) x for 1 (16) Itisworthnotingthatthefunctionsf (x)forsmavauesofxare highy suppressed because of the factor (x/) ; this is especiay true for sma x and for arge. The suppression x for x < 1 comes from the famiiar barrier factor of anguar momentum. The additiona suppression 1/ is ess famiiar; it is rooted in the Hanke functions empoyed here for the barrier factors. We have aready commented on this effect previousy, in the context of Figs 3 and 4. The approximate formua for a shown in (15) can aso be used to assess the onset of the pateau as x. It is cear that the required condition is ( ) 2 x a x x (c) = (2) 1/(2) e or x x (c) = ( ) 2 e dropping the factor (2) 1/(2) as it approaches 1 with increasing (the factor is aready ess than 1.20 for = 2). One may note that the critica point of x, i.e. x (c), increases ineary with. A tabe for x (c) is given beow x (c) (17)

10 Negecting the factor (2/e) in (17), we see that the the critica point is given simpy by x (c) =. This mnemonic is particuary usefu for arge vaues of. For some appications, it is convenient to define the Batt-Weisskopf centrifuga-barrier factors with a different normaization { } F (p) = h(1) (1) xh (1) (x) = h (1) 1/2 (1) 2, x = p/p xh (1) R (18) (x) 2 Note that F (p R ) = 1 by definition. We see here that the normaization has been chosen such that the fitted parameter for an wave refects its actua size at p = p R. We find, for 4, where z = x 2 = (p/p R ) 2. F 0 (p) = 1 2z F 1 (p) = z +1 13z F 2 (p) = 2 (z 3) 2 +9z 277z F 3 (p) = 3 z(z 15) 2 +9(2z 5) z F 4 (p) = 4 (z 2 45z +105) 2 +25z(2z 21) 2 (19) 3 Concusions We have worked out a convenient genera formua for cacuating the barrier factors. We have aso cacuated expicit forms for up to 9. It shoud be cear, from inspection of the expicit formuas, that it woud be impractica to work them out for arger vaues of. It is be better to compute the barrier factors for any with a computer program. For the purpose, we use the formuas (5) and (6). An integer-based Fortran program has been written to cacuate a k. It is aso quite easy to write a Fortran program which incorporates the recursion formua (10), using compex variabes. With this program, the expicit formua (11) has been independenty checked for a 9. We use the recursion reation (10) to 9

11 cacuate g (x) s for any integer vaue of > 10. If x is arge, e.g. x > 100, then the program can run into numbers that are out bounds, especiay if is aso arge. In order to circumvent this probem, we have introduced into the program an additiona branch with different formuas, which is invoked if x > x max = 100. The formuas are obtained by setting g (x) = g (x)/( ix), g (x) = a k ( ix) k, k=0 with the recursion reations g +1 (x) = i ( 2+1 x and the barrier factor takes on the form f (x) = a k = (+k)! 2 k k!( k)!, a 0 = 1 (20) ) g (x)+g 1 (x) (21) g (x) 1 One concusion which can be drawn in this regard woud be that the spherica Hanke functions as given in [1] are we suited for arge x but not for x near zero; the formuation as given in Section 2 has been motivated by the need to dea with the case when x approaches zero. The Fortran program, designed to hande a of vaues x and any, is avaiabe upon request. We summarize here the properties of barrier factors f (x) for a wide range of x and : f 0 (x) = 1, f 1 (x) = x x2 +1, f 2(x) = (22) x 2 (23) (x2 3) 2 +9x2 and f (x) 1 ( e ) ( x ) for 1 x 2 2 f (x) 1 exp[ ( 1)( /)] for 1 30 x=1 2 1 f (x) for x= (n) (n) 2 [ ] f (x) exp( ) for 1 x=2 f (x) 1 x (24) We remark that f (x) behaves ike a δ-functionin xas. Inthis regard, it is interesting to note that, if we set x = 2, f (x) ranges from 944 at = 1 to at =

12 and remain at as, i.e. it is neary constant for any 10. For a visua iustration of f (x) for a wide range of x and, the reader may consut a coection of the figures appended at the end of this note. Appendix The spherica Besse and Hanke functions satisfy the foowing differentia equation[2] [ d 2 dρ + 2 ] d (+1) +1 B 2 ρdρ ρ 2 (ρ) = 0 (A1) where B (ρ) can be any one of the spherica Besse functions of the first and the second kind, j (ρ) and n (ρ), and the spherica Hanke functions of the first and the second kind, h (1) (ρ) and h (2) (ρ), respectivey. They are defined by π j (ρ) = 2ρ J +1/2(ρ) π n (ρ) = 2ρ N +1/2(ρ) h (1,2) (ρ) = j (ρ)±in (ρ) (A2) Here J ν (ρ) and N ν (ρ) are the Besse functions of order ν of the first and the second kind, respectivey. Now we set B (ρ) = h (1) (ρ) = U (ρ)/ρ and convert the differentia equation (A1) to that invoving U (ρ) [ ] d 2 (+1) +1 U dρ2 ρ 2 (ρ) = 0, U (ρ) = ρh (1) (ρ) (A3) Withρ = pr, wesee that (A3) istheradia Schrödinger equation withthe potentia V(r) = 0 and U (ρ) is the outgoing-wave soution[3] such that [ ] U (ρ) 2 2 ( ) 2 a ρ 0 ρ ρ 0 e ρ 2 ρ 0 U (ρ) 2 [ ] 2exp 2( 1)( /) ρ=1 U (ρ) 2 = 1 ρ for 1 30 (A4) 11

13 See (7) and (14), and aso (A12) and (A16) beow. For a two-partice system with r denoting the partice separation, p becomes the breakup momentum in its center-of-mass system. Foowing von Hippe and Quigg[1], we define the semi-cassica impact parameter b = (+1)/p and write the differentia equation (A3) again 1 p 2 d 2 dr 2 U (pr) = [( ) 2 ] b 1 U (pr) = 0 r (A5) (A6) von Hippe and Quigg[1] identify U (pr) as the wave function which shoud hod outside an interaction radius R, i.e. r > R. As such the wave function is not dependent on the strong-interaction dynamics which must be taken into account for the region of r inside R. They define the transmission coefficient T (R/b) as a ratio of the probabiity density at r = to that at r = R, viz. so that U (pr) T (R/b) = U (pr) 2 r 2 r=r T 1 (R/b) = U (pr) 2 = (pr) 2 h (1) (pr) 2, R b = pr (+1) (A7) (A8) From(A4), we find that thedenomenator for T (R/b) above, forp = 1/R, is anexponentiay increasing function of ; this comes evidenty from the centrifuga barrier term (+1)/ρ 2 in theradiaschrödinger equation(a3). Wedefine thebarrierfactorf (x) tobethesquare-root of the transmission coefficient [ ] 1/2 1 f (x) = T (R/b) =, x = pr = p/p x h (1) R (A9) (x) The variabe x is simpy the ρ at a fixed interaction radius R (= 1fm) and thus it becomes a variabe in the breakup momenum p for the two-body system. We now expore the properties of the barrier factors for a wide range of x and. For x, the imiting form of the spherica Besse functions[2] are j (x) x (2 +1)!! (2 1)!! and n (x) x (A10)

14 where the doube factoria is defined as!! = ( 2)( 4) (A11) So for x, we can negect j (x) over n (x) and obtain And the barrier factor takes on the form xh (1) (x) i (2 1)!! x (A12) f (x) x (2 1)!! for x Comparing this with (7), we see that the doube factoria above is given by a = (2)! 2! (A13) = (2 1)(2 3)(2 5) = (2 1)!! ( ) (A14) 2 2 e where the atter approximation comes from the use of the Stiring s formua as given in (15). For vaues of x arge compared to, i.e. x, the spherica Besse functions[2] have the asymptotic behavior xj (x) +sin (x π ) 2 and xn (x) cos (x π ) 2 (A15) so that the spherica Hanke functions in turn have the imiting behavior xh (1,2) (x) ( i) exp [±i(x π ] 2 ) (A16) Thus the functions h (1) (x) and h (2) (x) behave asymptoticay as outgoing and incoming waves, respectivey. We concude that f (x) 1 for x (A17) which has aready been given in (7). Acknowedgment The author acknowedges many hepfu comments by R. Hackenburg. He is indebted to R. Longacre for his deep interest on this subject; we have had many enightening conversations on a wide range of issues reated to this topic. 13

15 References [1] F. von Hippe and C. Quigg, Phys. Rev. 5, 624 (1972). [2] J. D. Jackson, Cassica Eectrodynamics (Third Edition), John Wiey & Sons, Inc. (1999); Section 9.6. [3] A. Messiah, Quantum Mechanics (Voume I), John Wiey & Sons, Inc. (1961); Chapter IX, Section 7 and Appendix B, Section 6. 14

16 1.0 f (x) for = x 1.0 f (x) for = x 15

17 1.0 f (x) for = x 1.0 f (x) for = x 16

18 1.0 f (x) for = x 1.0 f (x) for = x 17

19 1.0 f (x) for = x 1.0 f (x) for = x 18

20 1.0 f (x) for = x 1.0 f (x) for = x 19