Numerical calculation of Coulomb wave functions for repulsive Coulomb fields

[

6.D

[

NuclearPhysics 43 (1963) 31--44; (~) Nor/h-HollaTcd Publishing Co., AmsIerdam Not to be reproduced by photoprint or microfilm without written permission from the p u n i s h e r

NUMERICAL CALCULATION OF C O U L O M B WAVE FUNCTIONS FOR REPULSIVE COULOMB FIELDS H. F. LUTZ and M. D. KARVELIS Lawrence Radiation Laboratory, University of California, Livermore, California Received 29 October 1962 Abstraet: A description of the numerical calculation of Coulomb functions used in scattering and reaction calculations is given. A table of Coulomb functions for selected values of Coulomb parameter r/ over the range of 1 to 25 is also presented. 1. Introduction In scattering and reaction calculations one often makes use of the fact that asymptotically the radial wave function is given by

~ ~ F~+CdG,+iF~],

(1)

where Ct = 1 [ee~Ot2/

I]

(2)

and 6l is the phase shift, induced by the nuclear interaction, relative to the Coulomb phase. In practice one determines the C~ by equating the logarithmic derivatives of the numerical solution of the radial Schr6dinger equation and the asymptotic expression. The Coulomb functions Ft and Gt are for a particular value of 1, the two real, linearly independent solutions of the differential equation {d2 2 r /+ d [ 1p 2 p

l(l+l)']ltFl(tl'P] 1 = O. p2 ] / [Gt(r/, p)

(3)

The function F t is regular at the origin f,(tl, 0) = 0.

(4)

"Ihe function G~ is irregular at the origin. Asymptotically we have V,(~, p) -~ sin 0,,

(5)

p-*oO

Gt(~l, P) --* cos 0,, p--~ oo

31

(6)

32

r I . F. L U T Z

AND

M. D.

KARVELIS

where 0 t = p-~/In 2p-

(½ln)+at(rl),

(7)

al(r/) = arg F(irl + l + 1) = arg F(i~l + 1) + ~ arctg

.

(8)

k=l

Obviously, accurate computation of the Coulomb functions is essential in scattering and reaction calculations. The purpose of this report is to describe the method used in such calculations at this laboratory. The method is a slight variation of that used by Tubis J) to generate his table of I = 0 Coulomb functions. Our code calculates the Coloumb functions, their derivatives and a~(~) for l = 0, 1, 2 . . . . lma~. In sect. 2 are given the equations used in the calculations. Sect. 3 consists of tables of Coulomb functions and their derivatives along with the phases. It is hoped that these will be of use to people who wish to check the accuracy of their Coulomb function calculations. 2. Description of Computation

The body of literature concerning Coulomb wave functions is now quite extensive. It is well summarized in refs. 2-4). The equations we present now arejustthose used in the computations. Figs. 1 and 2 give the schemes used for the calculations. The idea is that, except for F o and F~ with p < 1 when we can use the power series solution, we solve eq. (3) by the Taylor series method s) obtaining accurate starting values by various power series, asymptotic expansions and integral representations. For Fo and F~ with p < 1 we use B0 = B1

B~+, -

1

(. + 1)(. + 2)

=

1,

(9)

r/p,

(10)

{2rlpB,-p2B,_l},

Co = ] / 2m/ re 2~- 1 ' B

=

n = 1, 2 . . . .

(11) 02) (13)

~Bk, k=0

S = ~ kBk.

(14)

k=l

Actually these series are terminated according to

B---L--~ '

rrB---~-~

< 10_s

(15)

COULOMBWAVEFUNCTIONCALCULATIONS

33

Fo = pCoB,

(16)

F'o = C o ( B + S).

(17)

T o obtain F o and Fg in the range 1 < p < 2r/ or p > 1 but r / < 4 we employ the Taylor series solution o f (3) starting at p = 1 where Fo(r/, 1) and Fg(r/, 1) are given by the above power series.

If pSI,use power series ~ Ifor~starting p<2-r] alp=
m/nt If ~_>4,useTaylor

series development starting

at p =2 r/

Fo or F I o p=l Value atp=l obtained power series

by

p:2"q

Value at p:2"qoLqa;ned byasymptotic expansion

Fig. 1. Scheme used for calculating 1 = 0 regular Coulomb function and its derivative.

I f "~< 2 , use Taylor series development starting at p = 4

~.

If "q>Z,qseTaylor

series development

Go or

G~ p=4

p=27/

Value a t p = 4 obtained integral representation

by

by 71<4and by asymptotic

Value atp=2"q obtained inlegral representation if expansion if

"q>4

Fig. 2. Scheme used for calculating l = 0 irregular Coulomb function and its derivative. The same formulae (and coding) apply to both the F o and Go Taylor series development

Go = Fo(~, p), or 60(7, p),

al = ApF'o(~, p),

or

ApG'o(~, p),

(18) (19) (20)

34

n,

F. L U T Z

AND

M. D .

KARVELIS

pn(n + 1)a.+ 1 + Ap(n 2 - n)a, + (p - 2r/)(ap)2a._a + (Ap)3 a. -2 = O,

n = 2,3,4,...,

Fo(q, p + ap)

oo

or

= ~ a., Go(g, p + Ap) .=o

F'o(q, p + Ap)

or

1

(22)

o~

= -- ~1n6..

G(., p + Ap)

(21)

(23)

ap

The Taylor series are terminated by means of the same criteria used for the power scries~ If/5 is the starting point and Pmax the desired point, we determine the interval Ap and the number of times the above expansion must be applied by the following method: Is Ipm=x- l < 0.29. If the answer is NO, divide by 5 and test the result. If N is the number of times we apply this test, then (24)

Ap - Pmax-P

5(n- 1) ,

where number of times = 5(u- 1).

(25)

In order to obtain the starting points for F o and F~ when q >= 4 and p > 2q we use the following asymptotic expansions along the transition line Fo(q ' p=2q)=0.7063326373g~ {1_0.04959570165 q~

0.00888888889 q2

0.002455199181 _ 0.0009108958061 _ 0.0002534684115 + ...} ~ ~4 ~

(26) '

+ 0.0003174603174 F~(q, p = 2q) = 0.4086957323r/-~ '11+ ~0.,728~ -0.172826039 ~2

+

0.003581214850 0.0003117824680 0.0009073966427 t rtU ~ + q4 + r/~l" + . . . .j

(27)

We now turn our attention to the calculation of the l = 0, irregular Coulomb function Go and its derivative. We use the Taylor series development as given in eqs. (18) through (25) and therefore need only point out how the starting points are obtained. If r/ =>_4 we use the asymptotic expansions along the transition line

35

COULOMB WAVE FUNCTION CALCULATIONS

_{ • 0.04959570165_0,00888888889 Go(r/, P = 2t/) = 1.223404016r/° 1+ t/~ t/2 0.002455199181 +

t/~

0.0009108958061 --

t/4

0.0002534684115 +

G~(t/, p = 2t/) = -0.7078817734t/ -1~ l 1

t/~

~/

(28)

+ "" "~ '

0.172826039 + 0.0003174603174 r/§ t/2

0.003581214850t/~ + 0.0003117824680t/4 -- 0.0009073966427t/1~ + . . . .i

(29)

There are two situations in which we employ the integral representations described below. First, if t/ < 2, we use them to calculate Go(t/, 4), G~(t/, 4). Secondly, if2 < t/ < 4, the starting points Go(r/, p = 2t/) and G~(t/, p = 2t/) are calculated by means of the integral representations. These are

fo exp (-p~, +2t/arctg ~)d~- fo° 1 - t g h 2 4) sin (p tgh 4-2t/~)d~, (30) - fo ~ exp ( - p ~ + 2 t / a r c t g ~)d~- footgh ~ ( 1 - t g h 24) cos (0 tgh ~-2t/~)d4.

go = g; =

(31) The cutoffs for the integrals in (30) and (31) are determined by the following criteria: p~eutoff--2t/= 22 for integrals with exponential terms, 4cutoff = 12, for integrals with trigonometric terms. The domain between 4 = 0 and ~ = ( c u t o f f is divided into 5 intervals and the 16point Gaussian quadrature formula 6) is applied to each. The function Go and its derivative G~ are related to these integrals by

Go = AoPgo, =

(32)

Ao[' apl go+

,

(33)

where Ao = 1/vl-e-2~. 2rot/

(34)

At this point we have the l = 0 values for F,, F;, G, and Gj. T h e l ~ 0 values are obtained by means of recurrence relations 7)

Gl = (l +r/2)-~ {(l +t/) Go-G'o} .

(35)

36

H. 1a. LUTZ AND M. D. KARVELIS

G',=(l+t/2)~Go- (lp+rl) G1.

(36)

For l > 2,

l[(l+l)Z+t/2]*=Gt+l=

(2/+1) It/+ 1(I+1) 1 Gl-(l+l)[lZ+t/2]~G,_t, p _1

(37)

G~ = ~- (/2+t/2)~G~_ 1 -

(38)

+t/ Gz •

If we are interested in F~ out to some maximum l value, say l 1, we start with the greater of l = P + 10 or l = p + 10 and arbitrarily set Ft = 1,

Fl+l = 0.

The ff~ are then generated down to 1 = 0 by Ft_l=

~

(/2-t-t/2)-~t {(21+1)It/+

These are related to the desired

-l(l+l)q - p _a ffl_l[(l_l_l)2+1?2]~ff,+t}

"

(39)

Ft by F, = 1_F,,

(40)

C(

where = (F ° G1 _ F 1 Go)( 1 +q2)}.

(41)

The derivatives F~' obey the equation ,

1

When all the functions are generated, we test to see how closely the Wronshian relation F; G , - F~ G~ = 1 (43) is obeyed. The phases err(t/) are computed by means of the two following equations: tro(t/) = - r/+ (½t/) In (r/2 + 16) + ~ arctg (¼t/)- [arctg t/+ arctg (½t/)+ arctg (½t/)] q II+A 12(t/2+16)

/'/2-48 1 r/4--160r/2+l_280 -] (r/2+16) 2 +Tws(16+t/2) 4 j ,

cr,+l(q) = a,(q) + arctg (/+-~) .

(44)

(45)

COULOMB WAVE FUNCTION CALCULATIONS

37

3. Tables

The following tables give the Coulomb functions, their derivatives and phases for ~/ = 1, 2, 3, 4, 5, 10, 25 at p = 1, 5, 10. The parameter lmax is 10. The numbers are printed out in a floating point format:

O.XXXXXX E Y Y , where YY is the power of ten that multiplies O.XXXXX.,Y. C o u l o m b functions Eta = 1.0 F 0.227526E--00 0.751782E--01 0.147859E--01 0.208650E--02 0.229829E--03 0.207618E--04 0.158945E--05 0.105570E--06 0.619156E--08 0.325082E--09 0.154489E--10

FP 0.348734E--00 0.171414E--00 0.470870E--01 0.863072E--02 0.117394E--02 0.126420E--03 0.112466E--04 0.851515E--06 0.560849E--07 0.326781E--08 0.170669E--09

R h o m a x = 1.0 G 0.204310E 0.378287E 0A26414E 0.659452E 0.472226E 0.430922E 0.478400E 0.626003E 0.943597E 0.161018E 0.306853E

01 01 02 02 03 04 05 06 07 09 l0

L m a x = 10 GP --0.126360E --0.467637E --0.273742E --0.206492E --0.193898E --0.219264E --0.290645E --0.442313E --0.760364E --0.145767E --0.308303E

01 01 02 03 04 05 06 07 08 10 11

SIG --0.301640E--00 0.483758E--00 0.947405E 00 0.126916E 01 0.151413E 01 0.171153E 01 0.187668E 01 0.201858E 01 0.214293E 01 0.225359E 01 0.235326E 01

00 00 00 00 00 01 01 02 02 03 03

SIG --0.301640E--00 0.483758E--00 0.947405E 00 0.126916E 01 0.151413E 01 0.171153E 01 0.187668E 01 0.201858E 01 0.214293E 01 0.225359E 01 0.235326E 01

GP --0.433259E--00 0.189706E--00 0.825556E 00 0.820211E 00 0.210494E--00 --0.452671E--00 --0.761766E 00 --0.722864E 00 --0.581425E 00 --0.596134E 00 --0.100582E 01

SIG --0.301640E--00 0.483758E--00 0.947405E 00 0.126916E 01 0.151413E 01 0.171153E 01 0.187668E 01 0.201858E 01 0.214293E 01 0.225359E 01 0.235326E 01

C o u l o m b functions ]Eta = 1.0 F 0.684937E 00 0.109288E 01 0.118637E 01 0.904223E 00 0.526623E 00 0.247940E--00 0.979764E--01 0.333656E--01 0.998521E--02 0.266565E--02 0.642377E--03

FP --0.723642E 00 --0.342810E--00 0.154145E--00 0.406603E--00 0.379097E--00 0.239524E--00 0.117459E--00 0.474928E--01 0.164008E--01 0.495229E--02 0.132996E--02

R h o m a x = 5.0 G --0.898414E 00 --0.401136E--00 0.382961E--00 0.109154E 01 0.170862E 01 0.266645E 01 0.503203E 01 0.118176E 02 0.332762E 02 0.108419E 03 0.399408E 03

L m a x = 10 GP --0.510805E --0.789186E --0.793149E --0.615090E --0.668919E --0.145729E --0.417388E --0.131498E --0.454918E --0.173720E --0.729796E

C o u l o m b functions Eta = 1.0 F 0.477560E--00 --0.223323E--00 --0.963616E 00 --0.982018E 00 --0.237216E--00 0.678562E 00 0.122030E 01 0.126336E 01 0.993021E 00 0.645434E 00 0.361647E--00

FP 0.841142E 00 0.921027E 00 0.424848E-- 00 --0.393795E--00 --0.858051E 00 --0.716907E 00 --0.247646E-- 00 0.167860E--00 0.354649E--00 0.346527E--00 0.250841E--00

R h o m a x = 10.0 G 0.942873E 00 0.103974E 01 0.481304E--00 -- 0.494008E-- 00 --0.110724E 01 --0.966422E 00 --0.284331E--00 0.516868E 00 0.119169E 01 0.177543E 01 0.253646E 01

L m a x = 10

FI. F. LUTZ AND M. D. KARVELIS

38

C o u l o m b functions Eta = 2.0 F 0.288981E--01 0.113531E--01 0.247506E--02 0.370022E--03 0.422121E--04 0.390203E--05 0.303569E--06 0.204021E--07 0.120737E--08 0.638408E--10 0.305121E--11

FP

R h o m a x = 1.0 G

0.613082E--01 0.305590E--0| 0.863051E--02 0.161790E--02 0.223743E--03 0.243929E--04 0.219050E--05 0.167073E--06 0.110692E--07 0.648065E--09 0.339828E--10

Lmax = 10 GP

0.980033E 0.193257E 0.664964E 0.346115E 0.245671E 0.222226E 0.244855E 0.318386E 0.477396E 0.811050E 0.153985E

01 02 02 03 04 05 06 07 08 09 11

SIG

--0.138126E --0.360629E --0.172158E --0.118917E --0.106682E --0.117356E --0.152732E --0.229420E --0.390570E --0.743078E --0.156238E

02 02 03 04 05 06 07 08 09 10 12

0.129646E--00 0.123680E 01 0.202219E 01 0.261020E 01 0.307384E 01 0.345435E 01 0.377610E 01 0.405440E 01 0.429938E 01 0.451805E 01 0.471544E 01

00 00 00 00 01 01 02 02 03 03 04

0.129646E--00 0.123680E 01 0.202219E 01 0.261020E 0! 0.307384E 01 0.345435E 01 0.377610E 01 0.405440E 01 0.429938E 01 0.45180~E 01 0.471544E 01

0.129646E--00 0.123680E 01 0.202219E 01 0.261020E 01 0.307384E 01 0.345435E 01 0.377610E 01 0.405440E 01 0.429938E 01 0.451805E 01 0.471544E 01

C o u l o m b functions Eta = 2.0 F 0.114334E 01 0.993506E 00 0.721278E 00 0.431316E--00 0.214821E--00 0.907954E--01 0.331700E--01 0.106430E--01 0.303994E--02 0.781667E--03 0.182653E--03

FP

R h o m a x = 5.0 G

0.293796E--00 0.370866E--00 0.395241E--00 0.320534E--00 0.202959E--00 0.104255E--00 0.448461E--01 0.165561E--01 0.534671E--02 0.153340E--02 0.395309E--03

L m a x = 10 GP

0.794445E 0.108148E 0.149688E 0.204869E 0.309409E 0.562984E 0.125053E 0.330003E 0.100578E 0.346982E 0.133555E

00 01 01 01 01 01 02 02 03 03 04

SIG

--0.670489E --0.602828E --0.566180E --0.795991E --0.173180E --0.454934E --0.132404E --0.426234E --0.152054E --0.598643E --0.258437E

C o u l o m b functions Eta=

2.0

Rhomax=

10.0

Lmax=

10

F

FP

G

GP

-0.106161E 01 -0.865739E 00 -0.341607E--00 0.402871E--00 0.103985E 01 0.131965E 01 0.123272E 01 0.937122E 00 0.606981E 00 0.344139E--00 0.173936E--00

--0.293531E--00 --0.555775E 00 --0.814411E 00 --0.800003E 00 --0.485440E--00 --0.677316E--01 0.240491E--00 0.358316E--00 0.328633E--00 0.235587E--00 0.142231E--00

--0.399307E--00 --0.746983E 00 --0.111148E 01 --0.112488E 01 --0.683295E 00 0.257419E--01 0.742937E 00 0.134479E 01 0.190626E 01 0.268906E 01 0.420848E 01

0.831559E 00 0.675598E 00 0.277509E--00 --0.248446E--00 --0.642690E 00 --0.759099E 00 --0.666274E 00 --0.552907E 00 --0.615404E 00 --0.106496E 01 --0.230787E 01

SIG

COULOMB WAVE FUNCTION CALCULATIONS

39

C o u l o m b functions Eta = 3.0 F 0.287511E--02 0.128912E--02 0.319784E--03 0.526025E--04 0.642324E--05 0.623719E--06 0.503372E--07 0.347960E--08 0.210531E--09 0.113324E--10 0.549608E--12

FP

R h o m a x = 1.0

Lmax = 10

G

0.742389E--02 0.393543E--02 0.t20475E--02 0.241833E--03 0.352428E--04 0.399790E--05 0.370148E--06 0.289168E--07 0.195301E--08 0.116150E--09 0.617043E--11

GP

0.765515E 0.144312E 0.466093E 0.228787E 0.154878E 0.135002E 0.144492E 0.183596E 0.270202E 0.452069E 0.847401E

02 03 03 04 05 06 07 08 09 10 1l

SIG

--0.150148E --0.335170E --0.137116E --0.849232E --0.707070E --0.737952E --0.924103E --0.134814E --0.224233E --0.419082E --0,868104E

03 03 04 04 05 06 07 09 10 11 12

0.105335E 0.230240E 0,328519E 0,407059E 0,471409E 0,525451E 0,571816E 0,612305E 0,648182E 0.680357E 0.709503E

00 00 01 01 01 02 02 03 03 04 05

0.105335E 0.230240E 0.328519E 0.407059E 0.471409E 0.525451E 0.571816E 0.612305E 0.648182E 0.680357E 0.709503E

0.105335E 0.230240E 0.328519E 0.407059E 0.471409E 0.525451E 0.571816E 0.612305E 0.648182E 0.680357E 0.709503E

01 01 01 01 01 01 01 01 01 01 01

C o u l o m b functions Eta = 3.0 F 0.488823E--00 0,395790E--00 0.262224E--00 0.144437E--00 0.673085E--0l 0.269901E--01 0.945643E--02 0.293381E--02 0.815245E--03 0.204898E--03 0.469694E--04

FP

R h o m a x = 5.0 G

0~312638E--00 0.279268E--00 0.215294E--00 0.139742E--00 0.762181E--0l 0o353105E--01 0.140999E--01 0.492360E--02 0o152321E--02 0.422228E--03 0.105890E--03

Lmax = 10 GP

0.207880E 0.233008E 0.294526E 0.431887E 0.755136E 0.157678E 0.386636E 0.109193E 0.349493E 0.125152E 0.496154E

01 01 01 01 01 02 02 03 03 04 04

SIG

--0.716179E --0.882493E --0.139538E --0.274496E --0.630603E --0.164221E --0.480993E --0.157603E --0.573631E --0.230151E --0.101049E

01 01 01 01 01 01 01 01 01 01 01

C o u l o m b functions Eta= F 0.660104E 00 0.869033E 00 0.115596E 01 0.134435E 01 0,132711E 01 0.111951E 01 0.818626E 00 0.526415E 00 0.301494E--00 0.155476E--00 0.728622E--01

3.0

Rhomax=

10.0

Lmax=

10

FP

G

GP

--0.701802E 00 --0.606570E 00 --0.398452E--00 --0.112884E--00 0,154261E--00 0.316204E--00 0.351156E--00 0.296542E--00 0.207956E--00 0.126049E--00 0.676010E--01

--0.106014E 01 --0.916629E 00 --0.580970E 00 --0.639466E--01 0.531958E 00 0.109488E 01 0.159957E 01 0.215139E 01 0.301085E 01 0.469926E 01 0.836286E 01

--0.387802E--00 --0.510911E 00 --0.664827E 00 --0.738486E 00 --0.691685E 00 --0.584004E 00 --0.535412E 00 --0.687715E 00 --0.124007E 01 --0.262204E 01 --0.596554E 01

SIG 01 01 01 0l 01 0l 01 01 01 01 01

H. F. LUTZ AND M. D. KARVELIS

40

C o u l o m b functions Eta = 4.0 F 0.252241E--03 0.124151E--03 0.345733E--04 0.630037E--05 0.835264E--06 0.865097E--07 0.734513E--08 0.528684E--09 0.330513E--10 0.182752E--11 0.906394E--13

FP 0.749334E--03 0.419261E--03 0.139317E--03 0.303206E--04 0.473374E--05 0.567903E--06 0.550042E--07 0.445686E--08 0.310151E--09 0.189086E--10 0.102565E--11

R h o m a x = 1.0

Lmax = 10

G 0.740124E 0.132576E 0.397137E 0.180781E 0.114629E 0.946749E 0.969839E 0.118905E 0.169923E 0.277423E 0.509428E

GP 03 04 04 05 06 06 07 09 10 11 12

--0.176564E --0.357720E --0.129210E --0.717197E --0.547580E --0.534435E --0.635181E --0.889109E --0.143105E --0.260151E --0.526817E

SIG 04 04 05 05 06 07 08 09 11 12 13

0.230970E 0.363552E 0.474266E 0.566996E 0.645536E 0.713010E 0.771810E 0.823725E 0.870089E 0.911912E 0.949963E

01 01 01 02 02 02 03 03 04 04 05

0.230970E 0.363552E 0.474266E 0.566996E 0.645536E 0.713010E 0.771810E 0.823725E 0.870089E 0.911912E 0.949963E

00 00 00 00 00 00 00 01 01 01 02

0.230970E 0.363552E 0.474266E 0.566996E 0.645536E 0.713010E 0.771810E 0.823725E 0.870089E 0.911912E 0.949963E

01 01 01 01 01 01 01 01 0t 01 01

C o u l o m b functions Eta = 4.0 F 0.132272E--00 0.106029E--00 0.690499E--01 0.372765E--01 0.170289E--01 0.670533E--02 0.231195E--02 0.707348E--03 0.194201E--03 0.483019E--04 0.109722E--04

FP 0.118389E--00 0.100047E--00 0.713689E--01 0.430152E--01 0.220650E--01 0.973800E--02 0.374316E--02 0.126831E--02 0.383017E--03 0.104107E--03 0.256895E--04

R h o m a x = 5.0 G 0.501443E 0.585295E 0.802947E 0.130241E 0.249190E 0.556677E 0.143282E 0.419383E 0.137962E 0.504862E 0.203692E

L m a x = 10 GP

01 01 01 02 02 02 03 03 04 04 05

--0.307173E --0.390737E --0.618314E --0.117974E --0.264353E --0.682901E --0.200555E --0.661758E --0.242832E --0.982162E --0.434486E

SIG 01 01 01 01 01 01 01 01 01 01 01

C o u l o m b functions Eta = 4.0 F 0.139921E 01 0.137350E 01 0.128963E 01 0.112340E 01 0.889737E 00 0.635282E 00 0.408652E--00 0,237623E--00 0.125547E--00 0,606150E--01 0,268941E--01

FP 0.739288E--01 0.137755E--00 0.234052E--00 0.314486E--00 0.343100E--00 0.313552E--00 0.245889E--00 0.168544E--00 0.102459E--00 0.558951E--01 0.276326E--01

R h o m a x = 10.0 G 0.356646E--00 0.523407E 00 0.817049E 00 0.117699E 01 0.156161E 01 0.199265E 01 0.259152E 01 0.363158E 01 0.568278E 01 0.100223E 02 0.197644E 02

L m a x = 10 GP --0.695815E --0.675481E --0.627134E --0.560666E --0.521740E --0.590605E --0.887733E --0.163250E --0.332740E --0.725566E --0.168758E

SIG 01 01 01 01 01 01 01 01 01 01 01

COULOMB WAVE FUNCTION CALCULATIONS

41

C o u l o m b functions Eta = 5.0 F 0.204130E--04 0.107599E--04 0.330102E--05 0.662425E--06 0.955940E--07 0.106314E--07 0.957402E--09 0.723476E--10 0.470964E--11 0.269402E--12 0.137510E--13

FP

R h o m a x = 1.0

L m a x = 10

G

0.676154E--04 0.395272E--04 0.141172E--04 0.332471E--05 0.558529E--06 0.714018E--07 0.729676E--08 0.618444E--09 0.446951E--10 0.281335E--11 0.156815E--12

GP

0.808522E 0.138687E 0.387711E 0.I63640E 0.966657E 0.750448E 0.729353E 0.855451E 0.117769E 0.186284E 0.332955E

04 05 05 06 06 07 08 09 11 12 13

--0.222053E --0.419853E --0.137127E --0.688296E --0.481300E --0.436598E --0.488623E --0.650956E --0.100567E --0.176657E --0.347520E

SIG 05 05 06 06 07 08 09 10 12 13 14

0.381590E 0.518930E 0.637959E 0.740997E 0.830602E 0.909142E 0.978616E 0.104064E 0.109650E 0.114721E 0.I19358E

02 02 02 02 03 03 03 04 05 05 06

0.381590E 0.518930E 0.637959E 0.740997E 0.830602E 0.909142E 0.978616E 0.104064E 0.109650E 0.114721E 0.119358E

00 00 00 00 00 01 01 01 01 02 02

0.381590E 0.518930E 0.637959E 0.740997E 0.830602E 0.909142E 0.978616E 0.104064E 0.109650E 0.114721E 0.119358E

01 01 01 01 01 01 01 02 02 02 02

C o u l o m b functions Eta = 5.0 F 0.276730E--01 0.222679E--01 0.145803E--01 0.790966E--02 0.362592E--02 0.143073E--02 0.493803E--03 0.151120E--03 0.414805E--04 0.103117E--04 0.234077E--05

FP

R h o m a x = 5.0

Lmax = 10

G

0.303600E--01 0.253123E--01 0.176751E--01 0.104105E--01 0.522849E--02 0.226637E--02 0.858329E--03 0.287326E--03 0.859135E--04 0.231622E--04 0.567692E--05

GP

0.181928E 0.217253E 0.309034E 0.520518E 0.102840E 0.235593E 0.618256E 0.183710E 0.611569E 0.225942E 0.918679E

02 02 02 02 03 03 03 04 04 05 05

--0.161756E --0.202065E --0.311227E --0.579185E --0.127499E --0.325749E --0.950446E --0.312437E --0.114410E --0.462259E --0.204409E

SIG 01 01 01 01 01 01 01 02 02 02 02

C o u l o m b functions Eta= F 0.917945E 00 0.853237E 00 0.733285E 00 0.578595E 00 0.416880E--00 0.273629~--00 0.163698E--00 0.894814E--01 0.448545E--01 0.207045E--01 0.883841E--02

5.0

FP 0.331032E--00 0.329112E--00 0.317541E--00 0.287347E--00 0.238351E--00 0.t79114E--00 0.121551E~00 0.746165E--01 0.416031E--01 0.211751E--01 0.989074E--02

Rhomax=

10.0

G 0.160852E 0.170871E 0.190376E 0.220379E 0.266752E 0.344707E 0.486973E 0.763287E 0.133029E 0.256293E 0.541403E

Lmax = 10 GP

01 01 01 01 01 01 01 01 02 02 02

--0.509273E --0.512525E --0.539321E --0.633861E --0.873621E --0.139817E --0.249287E --0.481063E --0.995566E --0.220868E --0.525561E

SIG 01 01 01 01 01 01 01 02 02 02 02

H. F. LUTZ AND M. D. KARVELIS

42

C o u l o m b functions Eta = 10.0 F 0.369658E--10 0.233736E--10 0.965255E--11 0.273501E--11 0.559671E--12 0,868710E--13 0.105912E--13 0.105263E--14 0.875123E--16 0.622244E--17 0.385365E--18

FP 0.171725E--09 0.114392E--09 0.516147E--10 0.162701E--10 0.372509E--11 0.644582E--12 0.874335E--13 0.959676E--14 0.875545E--15 0.679018E--16 0.456083E--17

R h o m a x = 1.0 G 0.308814E 0.464436E 0.103085E 0.326900E 0.142483E 0.820860E 0.603024E 0.547443E 0.598011E 0.768676E 0.114114E

Lmax = 10 GP

10 10 11 11 12 12 13 14 15 16 18

--0.127057E --0.200525E --0.484776E --0.171162E --0.838119E --0.542742E --0.446364E --0.450900E --0.544397E --0.768277E --0.124439E

SIG 11 11 11 12 12 13 14 15 16 17 19

0.138029E 0.152740E 0.166474E 0.179268E 0.t91171E 0.202242E 0.212546E 0.222147E 0.231107E 0.239487E 0.247341E

02 02 02 02 02 02 02 02 02 02 02

06 06 06 06 07 07 07 08 08 09 10

0.138029E 02 0.152740E 02 0.166474E02 0.179268E 02 0.191171E 02 0.202242E 02 0.212546E 02 0.222147E 02 0.231107E 02 0.239487E 02 0.247341E 02

03 03 03 03 03 04 04 04 05 05 05

0.138029E 0.152740E 0.166474E 0.179268E 0.191171E 0.202242E 0.212546R 0.222147E 0.231107E 0.239487E 0.247341E

C o u l o m b functions Eta = 10.0 F 0.172074E--05 0.143828E--05 0.100868E--05 0.597824E--06 0.302297E--06 0.131831E--06 0.501385E--07 0.168122E--07 0.502129E--08 0.134837E--08 0.328291E--09

FP 0.309760E--05 0.262278E--05 0.188697E--05 0.115886E--05 0.612109E--06 0.280464E--06 0.112503E--06 0.398767E--07 0.126019E--07 0.358082E--08 0.922004E--09

R h o m a x = 5.0 G 0.167631E 0.197932E 0.275166E 0.447962E 0.847755E 0.184909E 0.460594E 0.129820E 0.410346E 0.144245E 0.559538E

Lmax = 10 GP

06 06 06 06 06 07 07 08 08 09 09

--0.279360E --0.334240E --0.476633E --0.804377E --0.159142E --0.365162E --0.960971E --0.286887E --0.961673E --0.358572E --0.147462E

SIG

C o u l o m b functions Eta = 10.0 F 0.162627E--02 0.146468E--02 0.118948E--02 0.873136E--03 0.581223E--03 0.352231E--03 0.195156E--03 0.992909E--04 0.465911E--04 0.202487E--04 0.818344E--05

FP 0.170605E--02 0.155055E--02 0.128316E--02 0.967104E--03 0.665444E--03 0.419076E--03 0.242261E--03 0.128964E--03 0.634312E--04 0.289242E--04 0.122691E--04

R h o m a x = 10.0 G 0.307861E 0.338444E 0.408749E 0.541618E 0.785893E 0.124570E 0.215086E 0.403279E 0.818391E 0.179150E 0.421614E

L m a x = 10 GP

03 03 03 03 03 04 04 04 04 05 05

--0.291917E --0.324312E --0.399764E --0.545390E --0.820739E --0.135694E --0.245409E --0.483342E --0.103214E --0.237953E --0.598872E

SIG 02 02 02 02 02 02 02 02 02 02 02

COULOMB WAVE FUNCTION CALCULATIONS

43

C o u l o m b functions Eta= F 0.188944E--28 0.141402E--28 0.798223E--29 0.344211E--29 0.115277E--29 0.305308E--30 0.651194E--31 0.113821E--31 0.165690E--32 0.203849E--33 0.212969E--34

25.0

FP 0.137350E--27 0.105092E--27 0.615752E--28 0.279852E--28 0.997094E--29 0.282492E--29 0.646191E--30 0.121189E--30 0.189128E--31 0.249077E--32 0.282668E--33

Rhomax=

1.0

G 0.377443E 0.494167E 0.842920E 0.185612E 0.521155E 0.183929E 0.803617E 0.428101E 0.274000E 0.207801E 0.114114E

Lmax=

10

GP 28 28 28 29 29 30 30 31 32 33 18

--0.255055E --0.340473E --0.602551E --0.139613E --0.416701E --0.157355E --0.738197E --0.422759E --0.290779E --0.236654E 0.559521E

SIG 29 29 29 30 30 31 31 32 33 34 33

0.562540E 0.577848E 0.592757E 0.607271E 0.621393E 0.635127E 0.648479E 0.661457E 0.674068E 0.686320E 0.698223E

21 21 22 22 22 22 22 23 23 24 24

0.562540E 0.577848E 0.592757E 0.607271E 0.621393E 0.635127E 0.648479E 0.661457E 0.674068E 0.686320E 0.698223E

16 16 16 16 16 17 17 17 17 18 18

0.562540E 0.577848E 0.592757E 0.607271E 0.621393E 0.635127E 0.648479E 0.661457E 0.674068E 0.686320E 0.698223E

02 02 02 02 02 02 02 02 02 02 02

C o u l o m b functions Eta = 25.0 F 0.702710E--21 0.62t926E--21 0.487503E--21 0.338958E--21 0.209495E--21 0.115410E--21 0.568517E--22 0.251317E--22 0.100079E--22 0.360448E--23 0.117896E--23

FP 0.214799E--20 0.190926E--20 0.151012E--20 0.106365E--20 0.668491E--21 0.375758E--21 0.189426E--21 0.859104E--22 0.351724E--22 0.130457E--22 0.440005E--23

R h o m a x = 5.0 G 0.237083E 0.266700E 0.337278E 0.478906E 0.762098E 0.135598E 0.269008E 0.593186E 0.144893E 0.390638E 0.115813E

L m a x = 10 GP

21 2l 21 21 21 22 22 22 23 23 24

--0.698338E --0.789023E --0.100649E --0.144741E --0.234156E --0.424990E --0.862644E --0.195129E --0.489990E --0.136049E --0.415975E

SIG 02 02 02 02 02 02 02 02 02 02 02

C o u l o m b functions Eta = 25.0 F 0.154513E--15 0.142455E--15 0.121137E--15 0.950619E--16 0.689074E--16 0.461916E--16 0.286753E--16 0.165120E--16 0.883508E--17 0.440121E--17 0.204531E--17

FP 0.313909E--15 0.290279E--15 0.247946E--15 0.196012E--15 0.143459E--15 0.973068E--16 0.612449E--16 0.358207E--16 0.194998E--16 0.989708E--17 0.469204E--17

R h o m a x = 10.0 G 0.161777E 0.175026E 0.204812E 0.259083E 0.353998E 0.521908E 0.829224E 0.141778E 0.260440E 0.513119E 0.108229E

Lmax = 10 GP

16 16 16 16 16 16 16 17 17 17 18

--0.318553E --0.345500E --0.406298E --0.517733E --0.714228E --0.106545E --0.171626E --0.298052E --0.557039E --0.111824E --0.240640E

SIG 02 02 02 02 02 02 02 02 02 02 02

44

I t . F. L U T Z

AND

M. D .

KARVELIS

The authors gratefully acknowledge the work of Mrs. K. Oliver who participated in the initial programming of this calculation. This work was performed under the auspices of the U.S. Atomic Energy Commission. References 1) A. Tubis, Los Alamos Scientific Laboratory, AEC LA 2150 (1958) 2) M. H. Hull Jr. and G. Breit, in Handbuch der Physik, ed. by S. Fliigge (Springer-Verlag, Berlin, 1959) Vol XLI/1, pp. 408-465 3) C. E. Froberg, Revs. Mod. Phys. 27 (1955) 399 4) M. Abramowitz, Table of Coulomb wave functions (U.S. National Bureau of Standard, Washington, D. C., 1952). Applied Mathematics Series No. 17, Vol. I 5) M. Abramowitz and P. Rabinowitz, Phys. Rev. 96 (1954) 77 6) A. N. Lowan, N. Davids and A. Levenson, Amer. Math. Soc. Bull. 48 (1942) 739 7) T. Stegun and M. Abramowitz, Phys. Rev. 98 (1955) 1851