Generalized Fermi-Dirac integrals — FD, FDG, FDH

Computer Physics Communications 39 (1986) 181—185 North-Holland, Amsterdam

GENERALIZED FERMI-DIRAC INTEGRALS

181

FD, FDG, FDH

-

L.W. FULLERTON IMSL, Houston, TX 77086, USA

and G.A. RINKER Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA Received 19 August 1985

PROGRAM SUMMARY Title of subroutines: FD, FDG, FDH

Keywords: plasma physics, Fermi statistics, transport coefficients

Catalogue number: AADU Program obtainable from: CPC Program Library, Queen’s University of Belfast, N. Ireland (see application form in this issue) Computers: Cray-i, X-MP; Installation: Los Alamos National Laboratory

Nature of physical problem Generalized Fermi—Dirac integrals are evaluated. These integrals arise in applications of statistical mechanics to Fermi systems. Method of solution Chebyschev polynomial representations are provided.

Programming language used: Fortran 77 High speed storage required: 3084 words No. of bits in a word: 64 Peripheralrequired: output device for error messages No. of lines in combined program and test deck: 2119

Restriction The argument a can be any positive or negative real number. The indices ~s and v are restricted to certain half-integral values, as discussed in the text. Typical running times Average computation time is approximately 20 s s per evaluation on a Cray-i.

OO1O-4655/86/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

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LONG WRITE-UP In the quantum-mechanical treatment of electrical and thermal transport in plasmas, one encounters integrals with the Fermi statistical weight [exp( t a) + 1] 1 The normal Fermi—Dirac integral is defined by —

F(a)=J

dt

______

o 1 + et~ This integral appears in many applications. Accurate and efficient rational approximations have been obtained by Cody and Thacher [1] for ~ = 1/2, 1/2 and 3/2. Implementations of series expansions have been published by Bañuelos et al. [2]. Two related integrals appear in the plasma transport theory of Lampe [3], which is exploited in other work [4,5]. The logarithmic Fermi—Dirac integral is defined by —

Generalized Fermi — Dirac integrals

the package. They should be reset for the local environment before use. Most systems provide more sophisticated error handling than does XERR, which can be replaced. Use of the routines The routines are defined by FD(v, a) = FDG(p, a)

=

FDH( i’,

a) = H~(a).

All arguments are real. Legal values of a are cc
—

FDH: tPln(1+ea_l) Gv(a)=f dt o 1+e’’~ The double Fermi—Dirac integral is defined by H~~(a) =,~

dt~+

el_a ~

1

+

es_~

The present subroutines evaluate these functions for selected values of the indices v and ~i. For zero indices, the functions are elementary and are evaluated by FD, FDG and FDH in terms of the function ALNREL, which is accurate essentially to full machine precision. For nonzero indices, these integrals were evaluated by numerical quadrature or asymptotic expansion and fitted piecewise with Chebyschev polynomials. The subroutines FD, FDG and FDH evaluate these polynomials. All fits have maxium relative error <10_b. Several required subroutines are included. These are ALNREL, a natural logarithm function which provides good relative accuracy for arguments near 1: ALNREL(x) = ln(1 + x); CSEVL and INITS, used to evaluate the Chebyschev polynomial; R1MACH, which sets the machine constants; and XERR, which handles error returns. The constants provided in R1MACH are set for Cray machines, representing the only machine-dependent code in

p. 0.0 1.0 2.0 3.0

v 0.0,0.5,1.5,2.5 —0.5,0.5,1.5 —0.5,0.5 —0.5

Error returns are specified by the variables

S~

ds

~t,

G~(a),

IXER, JXER and KXER, set in internal data statements in each routine. IXER controls large negative a underfiows, JXER controls large positive a overflows and KXER controls illegal indices. As provided, IXER = 1 causes once-only warning messages to be printed, and JXER = KXER = 2 are fatal. A driver is provided which will reproduce the test run output if the routines are successfully implemented. The first, second and third output pages represent results from FD, FDG and FDH, respectively, for listed values of the arguments. —

References [1]

W.J.

Cody and H.C. Thacher, Jr., Math. Comput. 21 (1967) 30. [2] A. Bañuelos, R.A. Depine and R.C. Mancini, Comput. Phys. Commun. 21 (1981) 315. [3] M. Lampe, Phys. Rev. 170 (1968) 306, 174 (1968) 276. [4] W.B. Hubbard and M. Lampe, Astrophys. J. Suppl. Ser. 18 (1969) 297. [5] GA. Rinker, Phys. Rev. B31 (1985) 4220.

LW. Fullerton, GA. Rinker

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Generalized Fermi— Dirac integrals

TEST RUN OUTPUT FD XNU XNU XNU XNU XNU XNU XNU XNIJ XNU XNU XNU XNU XNU XNLJ XNU XNU XNU XNU XNU XNU XNU XNU XNU XNU XNLJ XNLJ XNU XNU XNU XNU XNLI XNU XNU XNU XNU XNU XNU XNU XNU XNU XNU XNU XNU XNU XNU XNU XNU XNU XNU XNU XNU XNU XNU XNU XNU XNU XNU XNU XNU XNU

ALPHA ALPHA ALPHA ALPHA ALPHA ALPHA ALPHA ALPHA ALPHA ALPHA ALPHA ALPHA ALPHA ALPHA ALPHA ALPHA ALPHA ALPHA ALPHA ALPHA ALPHA ALPHA ALPHA ALPHA ALPHA ALPHA ALPHA ALPHA ALPHA ALPHA ALPHA ALPHA ALPHA ALPHA ALPHA ALPHA ALPHA ALPHA ALPHA ALPHA ALPHA ALPHA ALPHA ALPHA ALPHA ALPHA ALPHA ALPHA ALPHA ALPHA ALPHA ALPHA ALPHA ALPHA ALPHA ALPHA ALPHA ALPHA ALPHA ALPHA

ED ED ED ED ED ED ED FD ED ED ED ED ED ED ED ED ED FD ED ED ED ED ED ED ED ED ED ED ED ED ED ED ED ED FD ED ED ED ED ED ED ED FD FD FD ED ED ED ED FD FD FD ED ED ED ED FD ED ED ED

= = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =

-0.5 -0.5 -0.5 -0.5 -0.5 -0.5 0.0 0.0 0.0 0.0 0.0 0.0 0.5 0.5 0.5 0.5 0.5 0.5 1.0 1.0 1.0 1.0 1.0 1.0 1.5 1.5 1.5 1.5 1.5 1.5 2.0 2.0 2.0 2.0 2.0 2.0 2.5 2.5 2.5 2.5 2.5 2.5 3.0 3.0 3.0 3.0 3.0 3.0 3.5 3.5 3.5 3.5 3.5 3.5 4.0 4.0 4.0 4.0 4.0 4.0

-0.50000000000000E+03 -0.30000000000000E+03 -O.10000000000000E+03 0.10000000000000E+03 0.30000000000000E+03 0.50000000000000E+03 -0.50000000000000E+03 -0.30000000000000E+03 -0.I0000000000000E+03 0.I0000000000000E+03 0.30000000000000E+03 0.50000000000000E+03 -0.S0000000000000E+03 -0.30000000000000E+Q3 -0.I0000000000000E+O3 0.I0000000000000E+03 0.30000000000000E+03 0.50000000000000E+03 -0.50000000000000E+03 -0.30000000000000E+03 -0.I0000000000000E+03 0.I0000000000000E+03 0.30000000000000E#03 0.50000000000000E+03 -0.50000000000000E+03 -0.30000000000000E+03 -0.10000000000000E+03 0.10000000000000E+03 0.30000000000000E+03 0.50000000000000E+03 -0.50000000000000E+03 -0.30000000000000E+03 -0.10000000000000E+03 0.I0000000000000E+03 0.30000000000000E+03 0.50000000000000E+03 -0.50000000000000E+03 -0.30000000000000E+03 -0.I0000000000000E+03 0.I0000000000000E+03 0.30000000000000E+03 0.50000000000000E+03 -0.50000000000000E+03 -0.30000000000000E+03 -0.I0000000000000E+03 0.I0000000000000E+03 0.30000000000000E+03 0.50000000000000E+03 -0.50000000000000E+03 -0.30000000000000E+03 -0.10000000000000E+03 0.10000000000000E+03 0.30000000000000E+03 0.S0000000000000E+03 -0.S0000000000000E+03 -0.30000000000000E+03 -0.10000000000000E+03 0.I0000000000000E+03 0.30000000000000E+03 0.50000000000000E+03

0.12627982887284-216 0.91249473087853-130 0.65936629888812E-43 0.19999177177235E+02 0.34640857860135E+02 0.44721285985046E+02 0.71245764067413-217 0.51482002224121-130 0.37200759760209E-43 0.I0000000000000E+03 0.30000000000000E+03 0.50000000000000E+03 0.63139914436169-217 0.45624736543746-130 0.32968314944276E-43 0.66674892047782E+03 0.34641491008360E+04 0.74535967074708E+04 0.71245764059861-217 0.51482002218663-130 0.37200759756265E-43 0.50016449339552E+04 0.45001644930909E+05 0.12500164492132E+06 0.94709871653790-217 0.68437104815285-130 0.49452472416172E-43 0.40024673301505E+05 0.62358102722621E+06 0.22361231501689E+07 0.14249152813359-216 0.10296400444735-129 0.74401519519774E-43 0.33366232013306E+06 0.90009869606358E+07 0.41668311602493E+08 0.23677467915193-216 0.17109276205082-129 0.12363118104954E-42 0.28612555474329E+07 0.13363671653276E+09 0.79864168370125E+09 0.427474584A0346-216 0.30889201334399-129 0.22320455856072E-42 0.25049359388038E+08 0.20254441433507E+10 0.15626233710133E+11 0.82871137703640-216 0.59882466718125-129 0.43270913367583E-42 0.22279819771718E+09 0.31185889659944E+11 0.31059718153885E+12 0.17098983376169-215 0.12355680533781-128 0.89281823424445E-42 0.20065842818497E+10 0.48617766655842E+12 0.62508224903645E+13

183

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L. W. Fullerton, GA. Rinker

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Generalized Fermi— Dirac integrals

FDG XNU XNU XNU XNU XNU XNU XNU XNU XNU XNU XNU XNU XNU XNU XNU XNU XNU XNU XNU XNU XNU XNU XNU XNU XNU XNLJ XNU XNU XNU XNU XNU XNU XNU XNU XNU XNU XNU XNU XNU XNU XNU XNU XNU XNU XNU XNU XNU XNU XNU XNU XNU XNU XNU XNU XNU XNU XNU XNU XNU XNU XNU XNU XNU

ALPHA ALPHA ALPHA ALPHA ALPHA ALPHA ALPHA ALPHA ALPHA ALPHA ALPHA ALPHA ALPHA ALPHA ALPHA ALPHA ALPHA ALPHA ALPHA ALPHA ALPHA ALPHA ALPHA ALPHA ALPHA ALPHA ALPHA ALPHA ALPHA ALPHA ALPHA ALPHA ALPHA ALPHA ALPHA ALPHA ALPHA ALPHA ALPHA ALPHA ALPHA ALPHA ALPHA ALPHA ALPHA ALPHA ALPHA ALPHA ALPHA ALPHA ALPHA ALPHA ALPHA ALPHA ALPHA ALPHA ALPHA ALPHA ALPHA ALPHA ALPHA ALPHA ALPHA

EDG EDG EDG EDG EDG FDG EDG FDG EDG FDG EDG EDG EDG EDG FDG EDG FOG EDG EDG FDG EDG EDG EDG EDG FDG FOG FDG EDG EDG FDG FDG FOG FDG EDG EDG EDG EDG EDG FDG FDG EDG EDG EDG FDG FDG FDG FDG FDG EDG EDG EDG EDG FOG EDG FOG EDG FDG FDG EDG FDG FDG EDG FOG

= = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =

0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 2.5 2.5 2.5 2.5 2.5 2.5 2.5 2.5 2.5 2.5 2.5 2.5 2.5 2.5 2.5 2.5 2.5 2.5 2.5 2.5 2.5

-0.S0000000000000E+03 -O.45000000000000E+03 -0.40000000000000E+03 -0.35000000000000E+03 -0.30000000000000E+03 -0.25000000000000E+03 -0.20000000000000E+03 -0.15000000000000E+03 -0.10000000000000E+03 -0.50000000000000E+02 0.00000000000000E+00 0.50000000000000E+02 0.I0000000000000E+03 0.15000000000000E+03 0.20000000000000E+03 0.25000000000000E+03 0.30000000000000E+03 0.35000000000000E+03 0.40000000000000E+03 0.45000000000000E+03 0.50000000000000E+03 -0.50000000000000E+03 -0.45000000000000E÷03 -0.40000000000000E+03 -0.35000000000000E+03 -0.30000000000000E+03 -0.25000000000000E+03 -0.20000000000000E+03 -0.15000000000000E+03 -0.I0000000000000E+03 -0.50000000000000E+02 0.00000000000000E+00 0.50000000000000E+02 0.I0000000000000E+03 0.15000000000000E+03 0.20000000000000E+03 0.25000000000000E+03 0.30000000000000E+03 0.35000000000000E+03 0.40000000000000E+03 0.45000000000000E+03 0.50000000000000E+03 -0.50000000000000E+03 -0.45000000000000E+03 -0.40000000000000E+03 -0.35000000000000E+03 -0.30000000000000E+03 -0.25000000000000E+03 -0.20000000000000E+03 -0.15000000000000E+03 -0. 10000000000000E+03 -0.50000000000000E+02 0.00000000000000E+00 0.50000000000000E+02 0.10000000000000E+03 0.15000000000000E+03 0.20000000000000E+03 0.25000000000000E+03 0.30000000000000E+03 0.35000000000000E+03 0.40000000000000E+03 0.45000000000000E+03 0.50000000000000E+03

0.15904427612621-434 0.42752964496261-391 0.11492497672586-347 0.30893179995959-304 0.83044486712351-261 0.22323330826479-217 0.60007728257091-174 0.16130780296933-130 0.43361427027056E-87 0.11656059528504E-43 0.17011027222851E+00 0.47141316136710E+04 0.26666727255397E+05 0.73484741613018E+05 0.15084948931342E+06 0.26352317648138E+06 0.4156~222863049E+06 0.6111~740542623E+06 0.85333336354071E+06 0.11455130140524E+07 0.14907120121274E+07 0.11928320707359-434 0.32064723366533-391 0.86193732529170-348 0.23169884992878-304 0.62283365023264-261 0.16742498116902-217 0.45005796184870-174 0.12098085220563-130 0.32521070264549E-87 0.87420446448340E-44 0.15374373755973E+00 0.10102779997334E+06 0.11428750307165E+07 0.47240379002518E+07 0.12929977970975E+08 0.28234650387169E+08 0.53446170354322E+08 0.91670639631499E+08 0.14628575027204E+09 0.22092039966769E+09 0.31943832274272E+09 0.14910400886902-434 0.40080904215432-391 0.10774216568099-347 0.28962356246347-304 0.77854206293193-261 0.20928122649922-217 0.56257245241285-174 0.15122606528446-130 0.40651337838055E-87 O.10927555808023E-43 0.21888376138496E+00 0.28069923494294E+07 0.63494998156420E+08 0.39367342900178E+09 0.14366697988211E+10 0.39214870482003E+10 0.89077053815917E+10 0.17824859643043E+11 0.32507960499295E+11 0.55230119073527E+11 0.88732889962145E+11

L. W. Fullerton, GA. Rinker / Generalized Fermi — Dirac integrals

FDH XNU XMU XNLJ XMU XNLI XMU XNU XMU XNU XMU XNU XMU XNU XMU XNU XMLJ XNU XMU XNU XMU XNU XMU XNU XMU XNU XMU XNU XMU XNU XMU XNU XMU XNU XMU XNU XMU XNU XMU XNU XMU XNU XMU X~’JU XML) XNU XMU XNLI XML) XNU XMU XNU XMU XNU XML) XNU XMU XNLJ XMU XNU XMU XNU XML) XML) XML) XNU XMU XML) XML) XNU XMU XNU XML) XNL) XML) XNU XML) XNL) XML) XNU XMU XML) XML) XML) XMU XNU XML) XML) XMU XML) XMU XNU XML) XML) XML) XNU XML) XNU XML) XNL) XML) XNU XMU XML) XMU XML) XML) XNU XML) XML) XML) XNU XML) XML) XML) XNU XML) XNU XMU XML) XML) XNL) XMU XNL) XMU XNU XML) XNU XML) XNU XML) XML) XML)

ALPHA ALPHA ALPHA ALPHA ALPHA ALPHA ALPHA ALPHA ALPHA ALPHA ALPHA ALPHA ALPHA ALPHA ALPHA ALPHA ALPHA ALPHA ALPHA ALPHA ALPHA ALPHA ALPHA ALPHA ALPHA ALPHA ALPHA ALPHA ALPHA ALPHA ALPHA ALPHA ALPHA ALPHA ALPHA ALPHA ALPHA ALPHA ALPHA ALPHA ALPHA ALPHA ALPHA ALPHA ALPHA ALPHA ALPHA ALPHA ALPHA ALPHA ALPHA ALPHA ALPHA ALPHA ALPHA ALPHA ALPHA ALPHA ALPHA ALPHA ALPHA ALPHA ALPHA ALPHA ALPHA ALPHA

FDG FOG FOG FOG FOG FOG FOG FOG FOG FOG FDG FOG FDG FOG FDG FOG FOG FOG FOG FOG FDG FOG FOG FOG FOG FDG FOG FOG FOG FOG FOG FOG FOG FOG FOG FOG FOG FOG FOG FOG FOG FOG FOG FOG FOG FOG FOG FOG FOG FDG FOG FOG FOG FOG FOG FDG FOG FOG FOG FOG FOG FOG FOG FOG FDG FDG

= = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =

-0.5 -0.5 -0.5 -0.5 -0.5 -0.5 -0.5 -0.5 -0.5 -0.5 -0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 -0.5 -0.5 -0.5 -0.5 -0.5 -0.5 -0.5 -0.5 -0.5 -0.5 -0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 -0.5 -0.5 -0.5 -0.5 -0.5 -0.5 -0.5 -0.5 -0.5 -0.5 -0.5

1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0 3.0 3.0 3.0 3.0 3.0 3.0 3.0 3.0 3.0 3.0 3.0

-0.50000000000000E+03 -0.40000000000000E+03 -0.30000000000000E+03 -0.20000000000000E+03 -0.I0000000000000E+03 0.00000000000000E+00 0.I0000000000000E+03 0.20000000000000E+03 0.30000000000000E+03 0.40000000000000E+03 0.50000000000000E+03 -0.50000000000000E+03 -0.40000000000000E+03 -0.30000000000000E+03 -0.20000000000000E+03 -0.10000000000000E+03 0.00000000000000E+00 0.10000000000000E+03 0.20000000000000E+03 0.30000000000000E+03 0.40000000000000E+03 0.50000000000000E+03 -0.50000000000000E+03 -0.40000000000000E+03 -0.30000000000000E+03 -0.20000000000000E+03 -0.10000000000000E+03 0.00000000000000E+00 0.I0000000000000E+03 0.20000000000000E+03 0.30000000000000E+03 0.40000000000000E+03 0.50000000000000E+03 -0.50000000000000E+03 -0.40000000000000E+03 -0.30000000000000E+03 -0.20000000000000E+03 -0.10000000000000E+03 0.00000000000000E+00 0.I0000000000000E+03 0.20000000000000E+03 0.30000000000000E+03 0.40000000000000E+03 0.50000000000000E+03 -0.50000000000000E+03 -0.40000000000000E+03 -0.30000000000000E+03 -0.20000000000000E+03 -0.10000000000000E+03 0.00000000000000E+00 0.10000000000000E+03 0.20000000000000E+03 0.30000000000000E+03 0.40000000000000E+03 0.50000000000000E+03 -0.50000000000000E+03 -0.40000000000000E+03 -0.30000000000000E+03 -0.20000000000000E+03 -0.10000000000000E+03 0.00000000000000E+00 0.10000000000000E+03 0.20000000000000E+03 0.30000000000000E+03 0.40000000000000E+03 0.500000000000008+03

0.79522138083090-434 0.57462488377369-347 0.41522243366610-260 0.30003864136085-173 0.21680713518976E-86 0.767715414O9641E~00 0.80032716227970E+05 0.45259473745431E+06 0.12471334590865E+07 0.25600657069322E+07 0.44722094379382E+07 0.27832748323737-434 0.20111870928217-347 0.14532785175523-260 0.10501352445613-173 0.75882497301846E-87 0.340889608~0830E+00 0.19058526330919E+07 0.21553014248900E+08 0.89082586559716E+08 0.24381828487085E+09 0.53240938483110E+09 0.26838721600606-434 0.19393589825601-347 0.14013757134959-260 0.10126304145010-173 0.73172408119903E-87 0.38311560057664E+00 0.88955293298859E+08 0.20116998702415E+10 0.12471794631592E+11 0.45513221463438E+11 0.12422968370746E+12 0.17097259687947-433 0.12354435001194-346 0.89272823238645-260 0.64508307892896-173 0.46613534066025E-86 0.18162026039970E+01 0.57208354974802E+07 0.64668330721052E+08 0.26726483331906E+09 0.73148114922252E+09 0.15972649089874E+10 0.70575897548863-434 0.50997958435001-347 0.36850990987929-260 0.26628429420821-173 0.19241633248124E-86 0.94111420873146E+00 0.14836572573158E+09 0.33534455921465E+10 0.20788019352454E+11 0.75858855935393E+11 0.20705557185969E+12 0.52782819153122-433 0.38140726660819-346 0.27560389034833-259 0.19915064820504-172 0.14390573598349E-85 0.59149187902167E+01 0.44542750224221E+09 0.10062201399898E+11 0.623691929292478+11 0.22758710524485E+12 0.62118512018059E+12

185