A program to compute the angular coefficients of the relativistic one-electron hyperfine structure parameters

Computer Phy~s Communtcat,ons Computer Physics Commumcation.s 9(I (1995) 381 387

FI.%.'V1ER

A program to compute the angular coefficients of the relativistic one-electron hyperfine structure parameters Sophie Kr6ger a, Martin Kr6ger b • Op¢uches lnsutut, Technischt Umtaenfflat Berhrg llardenbergstragOe36, D-10623 Berfin, Germany b Inantut fur 7"heoretuche Phy.~ik, Techniache Unwena~t Berlin, ltardenber~rmlJ¢ 36, D-IO6Zi Berlin, (;ermany

Recctved 23 F'cbniary 1995

Abstrllt't The C--'hfs program calculates the angular coefficients a t,t, and ~ t ,t, of the relativistic one-electron hyperfmc structurc parameters for the case of pure Sl.-coupling or, alternatively, the case of intermediate coupling. The calculation is based on SL-basis states and, optionally, mixing coefficients of the electronic wave functions of atoms. The dcrivation of the necessary matrix elements is based on the effective tensor operator formalism; explicit expressions are cited. Chfs is the first program which is able to computc the angular cocfficicnts for electronic configurations with up to four electron shells. ~

'

Atomic structure; Ilypcrfmc structure; Tensor algebra; A factor; B factor; Complcx atoms

PROGRAM SUMMARY "Iitle o/program: Chh

Cata/ot¢~ number: ADBV Program obcamab/e from CPC Program Library, Queen's University of BeLfast, N Ireland (see applicanon form in this i.s6ue)

CPC Program Library sublmTgram, used: Catalogue number TueI Refs m CPC ACQB P SIIFJ.I. CF'P 1 (1969) 15 ACRN A NEW I) SHF.I.I. CFP 6 (1973) 88 ACRY F SHEI.I. CF'P 8 (197,1) 246

Key words' atormc structurc, hyperfinc structure, ten.sot algcbra, A factor. B factor

l.*censm& /xotxston.s: nonc

Computers: 44~6PC, SGI, DF.C; In.naUatums: F77L-3 compiler (on PC) Opemnng systems under wh~ch the pcogram has been rested: DOS, UNIX /~grantmkt#~/anguag¢ used. USANSI Fortran 77 Memo~ reqtamd to er.ecut.e wuh ryp~a/data. 1 Mw No. o/ btu m a ~m~" 32 No. of lutes m dismbuled program, including te.rt data. etc.. 4508

Nan,~reof 9ky~cal 9rob~m The atomic hypcr6nc structu:c splmmg is characterized by thc hypcffmc intcraction constants A and B. Within thc cffccln, c tensor operator formalism [1] thcsc h y l x r f n e mtcrac'~on constants can bc cxprc~scd as a linear combinaUon of prc~luc~ between thc rclath,,Lstic onc-clc~ron hypc~nc structun: parameters al_l*, and b~l~, and corrcapondmg angular coefficients arm*' and ~f~,, re.specuvcly [2] The problem is to calculate thc angular cocf~cicnts for the ca.se of pure SL,-coupling and, alternatively, the case of intermediate cou.

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382

S K.tog~. M. ~

/ Computer ~ s

piing for elcctronic configuration with up to four open electron shells MeO,oe of,o~,o,, The program Ch/s calculates thc angular coefficients a**tt' and /3~;i' directty by evaluating explicit expressmns for the matrix elements, which have been dem,ed by the method of Racah algebra from effect/re Hamiltonums of hyperfine structure. The program Chfs allows two types of couplings to computc the mamx elements: ['/'t. ~'2)~12. (~l'3. ~N'~,]9') as derived in [3] and [(~',, ~2N',2. ~'319~,3L 9',N'), see ref. [4]. In the case of intermediate coupling the angular coeffi. cients are obtained by sammmg up a set of matrix elements, with fine stng'turc mixing coefficients as weighting factors. SesmOaons on O~ c

~

of O~ ~ , n

A configuration may have up to mammal four open electron (sub)shells. Because of the limited availability of fractional

Commumcanons 90 (1995) 381 387

parentage routines [5 71, the program C'hh ~ restricted to handle s-, p-, d-, and f-shcUs and at moat two panicles m higher sheUs. Typical ~nnmg

tune: s o m e

minutes on 486PC

Rtpf't~r e ~ - ~

[I] P.GH. Sandars and J. Beck, Ptoc. R. Soc. London, Set. A 289 (1%5) 97. [2] WJ. (~ilds, Case Stud. At. Phys. 3 (1973) 215. [3] H.-D. Kronfeldt, G. lrdemz S. Kr6ger and J.-F. Wyatt, Phys. Rev A 4gl (1993) 4500. [4] O. Klemz, S. Kr6ger and H.-D Kronfeldt, m prcparation. [5] DC.S. All~n. Comp. Phys. Comm. ! (1%9) 15. [6] A.T. Chivcrs, Comp. Phys. Comm. 6 (1973) 88 [7] D C S . Allison and ./E. McNulty. Comp. Phys. Comm.

$ (1974) 246.

Long W R I ' I T ~ U P

1. I n t r o d u c t i o n In early years hyper~ne structure has been studied with the emphasis on obtaining information on the nuclear structure [1]. In the last twenty years the electronic aspect has b e c o m e the main concern in optical hyperfinc investigations. Traditionally the hyperfine structure is analyzed by a fit o f the one-electron hyperfine structure parameters hyperfinc constants A and B using the effective tensor operator formalism [2]. T h e knowledge of these one-electron hyperfine structure parameters offers the possibility to find gcneral trends and systematics in the variation of electronic hyperfine quantities in different configurations within one element or over a series o f elements by progressing through an atomic shell. Systematic comparisons of experimental and theoretical hyper~ne structure data for g r o u n d configurations of m a n y atoms have been presented by IAndgren and Ros~n [3] and for several types of configurations o f 4d- and 5d-shell atoms by Biittgenhach [4] and of the lanthanides by Pfeufer [5]. All these investigations arc restricted to low-lying configurations. Just recently one starts to analyse also higher, m o r e complex configurations, see for example [7,8]. For some atoms and configurations experimental data in form of hypcrfine constants A and B is available. But in most o f cases the composition of the wave functions are unknown and it is difficult to calculate the angular part of one-electron hyperfine structure parameters. T h e present p r o g r a m has been written in o r d e r to c o m p u t e the angular m o m e n t u m part. T h e r e exist some unpublished p r o g r a m s for this purpose. However, not one o f those deals with configurations up to four o p e n electron shells. T h e p r o g r a m Chfs has already been used in refs. [7-.9] for the analysis o f the hyperfine structure of the lanthanides G d I and Er I.

2. Theory T h e atomic hyperfine structure leads to a splitting of fine structure energy levels, which in most cases is described to a sufficient degree o f accuracy by the magnetic dipole hyperfine structure constant A and

S. tOf,gt.r. M. grC,a~ / Computtr Phra~ C

"

90 (1995) 38t 3a7

383

the electric quadrupole hyperfine structure constant B. Within the effective tensor operator formalism [2] it is possible to decompose both hyperfine constants into relativistic one-electron hyperfine structure parameters ak,t k', b~l k, using non-relativistic SL-states. The effective tensor operators are single-electron operators. The llamiltonian for the magnetic dipole interaction is written, as in [6], Hh"(M1)

°1 i i = ~N t[ a .,t,

l~atn2,t,(sCa))(, 1) + a,,t,s 1o , ] "1,

(])

where the sum on i extends over all electrons of the atom, i and s are the electronic angular momentum and spin operators for a single electron, I is the nuclear angular momentum operator and (set2)) tl~ is the rank-one irreducible tensor operator formed by coupling the spin s with the second-rank spherical operator C c2) according to the standard definition of the tensor product. The first term of the operator H m (M 1) represents the field caused by orbital motions of the electrons. "I'he second term describes the dipole field due to the spin motions of the electron. The last term is of pure relativistic origin and represents the contact interaction between the nucleus and the electron spin. The Hamiltonian for the electric quadrupole interaction reads 2 r,2) v" N H~.(E2) = r;' O~" "-'

02 b.,,,

21,(1, + 1)(21,

(-~,--1)~,--~3)

U(02~2 .

V

:

~",',"J"' , , o , a ~.

b~],,U,11~

,

(2)

where the subscript n refers to the nucleus, i to the ith electron, !, is the electronic angular momentum, Q the electric quadrupole moment of the nucleus and U(k'k') * are double unit tensor operator with the rank k,, k t and k of the operator in spin, orbital, and combined spaces, respectively [6]. Of these U(k'k') k operators, U (°2~2 has the tensor properties of the operator E C~2) in the non-relativistic quadrupole hamiltonian, and the other two terms are entirely relativistic in origin. By means of the effective tensor operator formalism the hyperfine constants A and B are expressed as a linear combination of products between one-electron hyperfine parameters a~} k, and b~t k' and of corresponding angular coefficients a~t*' and /3.*tk', respectively: A =

~

10 10

12-12

Ol 01

(3)

,~13j.13

a11~.11

(4)

l.., o t , t t a n t + c t n t u n t + a t n t a . t , nl

B = T__.Hnl '~°2~'°2 ~1

+ bPnl t~nl + t"nl ~'nl ,

nl

where the sum has to be taken over all open electron sheiks n/. The one-electron hyperfme structure parameters can be interpreted as free parameters, which are assumed to be constant for all the levels of a configuration. In general, they are determined by fitting the experimental hyperfine constants A and B. The method of extrapolation leads to predictions of the quantitative hyperfine structure of all leveks of a configuration, based on the determination of a small number of hyperfme levels of the same configuration. It is also useful to compare the resulting parameter values with those of other elements in order to establish dependencies of the parameters on the nuclear charge numbers and to extrapolate parameter values to elements and configurations, which have not yet measured. If an atomic state ~ is not purely SL-couplod, it can be represented as a linear combination of a set of SL-basis states @i:

~'= F_.m,¢',. i

(5)

384

S. Kr6g~, M Kriigrt / Computer Physws Commuair~ttons 90 (1995) 381 387

where the m, denote the (real valued) mixing coefficients, which may be obtained, e.g., from calculations of the fine structure. Using Fxl. (5), matrix elements of an arbitrary operator H arc calculated by ( q ' l H I q ' ) = ~m,ms<~,lal~s).

(6)

The matrix elements ( ~ , i H I ~ s) have been evaluated with H = Hh,(M1) and H = Hhf,(E2) for configurations with up to four open electron shells for two relevant types of coupling, . . h'~ and I({[(nlI1)N:oISlLI,(n212)N:a2,f~2L2]S12L12,

(r13/3)

N~

. N, a3~3-/.3,ISI3LI3, (tl4/4) ot,t.'41.4)S-/.]),

(8) where the (4),1 and I~s) are at the .same configuration. IStplicit expressions are given in the appendices of Ref. [7] and [9], respectively. Moreover, the tlamilton operators Hhf,(M1) and Hhf,(E2) can produce matrix elements between different configurations. In general, they are not taken into consideration to the hyperfine structure, because their contribution to the hypcrfine constants A and B is small and the number of parameters for the fit increases considerably. The program Chfs does not calculate the matrix clement.s between different configurations. 3. Program organization The Chfs program calculates matrix elements (~illlhr,(M1)l~ J) and (~,lHbu(/'.'2)l~ s) according to the explicit expressions above cited for all given basis states (~,1 and I ~s), at a time for a specific value of total electronic angular momentum quantum number J and a single configuration. For pure SI.-coupling only the diagonal elements (i = j ) arc calculated. In the case of intermediate coupling all elements are calculated, then multiplied with the corresponding mixing coefficients m, and m s and finally summed up in order to determine (q' HI'/'). For each '/' the matrix elements are named after the *, with the greatest value for m, in the composition. While reading the input data, for shells with electrons having an angular momentum quantum number 1 < I ~ 3 and a number of electrons in this shell N > 2 the CPC Program Library subroutines for coefficients of fractional parentage [10-12] are called. These routines have been slightly modified to be compatible with the modern Fortran language. The program Chfs is restricted by the available fractional parentage routines to p-, d-, and/-sub-shells and at most two particles in higher shells. 4. Global variables and constants

Global variables and constants are listed in Tables 2 and I, respectively. 5. Input data

,All input data is given by an ASCII file, whose name is interrogated by a prompt. This file must contain the following declarations given in the following order: • •

title (up to 80 characters) total electronic angular momentum quantum number J multiplied by 2 (integer)

S. I 0 ~ .

M K n ~ r / Computer Piomcs Commumcanona 90 (199.S) 381 387

383

Table 1 Declaration of constants. "I'hc parameters nmob and mnoq can be changed m the source code Name

Mcaning

mnos mn~ mnoq mnop

rna~ max. rmm max.

• •

•

•

number number number number

of electron shells of basis .states o f s t a t ~ in qucsttons of parent states

l.Am~tat~on

Actual value

4 no no 210

4 100 20 210

number of open electron shclls nos for cach electron shell i 1..nos one linc with: principal and azimuthal quantum number and number of electrons in this shell n,, I,, N, in the format (I2, AI, I3). The I, must be given a.s small lettcrs (s, p, d,..) only if number of clectron shells (nos) is greatcr than 2: coupling type icplt(1 or 2), where '1' corresponds to {[(nlll)~":atStLt, (n212)N2a2S2L2]S12Lt2, (n;l;)'~',a3S3l.~}SI3) or i([(nxlt)~':alS1L. (n21z)~'2a2SzL2]St2 l-x2, [(n~l.~)~"'a~S31.~, (nJ4)~"a4s41.4]S~41.~}S13), respectively; '2' corresponds to t{(ntl 1)N:atS 1l.t,[(nzl 2)'vL (n313)'~'3ct~S3L 3]a2S 2 L 2 ]$12 LI2}SLJ ) or [({[(n ~1t)'~'~a t S~ L t, (n2l 2)'s''a 2S 2 Lz ]St2 l.t2, (n ~l~)'V'cr~S~I.~}S t ~L t.~, (n~l~)N'a~S~ I.~)SLJ ), respectively number of basis states nob

Tablc 2 Global variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Name

Type

nos nob noq tc/~t l nq I

int. int. mt. int. mt. mt. int.

nmoa mnos

rift

lilt.

~trlO$

sg

real int. rot. real rot. real mt. mr. rot. int. int. rcal rcal rcal real

mn~s, rmtob

/g

iv ~coupl/2 I.coupl/2 end I end n p s p I p iv p p max

aOl, alO. a12 b02, bl l, hi3

Dtrnen~on

mnos, mnob mno& mnob mnob, mno~ mnoa, mnob mnos, mnob mm~, mnob mnos. msu~ nmos, mnob, mnop maos, nmob, mnop nmos, mnob, mnop mnos. m s t o b . . m o p mnob, m n o q ma~os, m.noq ~s, mnoq

• of shell a n d actual basis state. u of shcll, actual basis state and actual parent state.

' of shell, actual basis statc and actual s ~ t c in question.

Meaning number of clcctron shells (! 4) number of basis states ( <; nmob) n. of states m questions ( ~ mnoq, <; nob) type of coupling (1,2) 2 x total angular momentum q.n.J principal quantum number n angular momentum q.n I(0 ~ 1 ~ ( n . 1)) number of electrons N(nn .<;(41 ¢ 2)) spin quantum number" angular momentum q.n. t scmority q.n. of core state' resulting spin q n. of thc fir,st/second coupling" resulting angular momentum q.n. of the t'Lt~t/second coupling' total spin q.n." total angular momentum q.n." number of parent states" spin q.n. of parent slate' angular momentum q n. of parent statcb semonty q n of parcnt state b coef. of fract, parcntagc b rniling coefficicnts 01 O(~l),

10

O~l

at02 QII

12¢

• ¢~fa t

i~c

S. g.nTg~, M. gr6g~ / compm~ e~acs Commamcazmets9O H995) 38t. 387

386

•

for each basis states one line with the following data ('SL-blocks') depending on the n u m b e r of shells nos and the type o f coupling icplt:

nos

ieplt

1 2 3 4

1 2 1

4

2

2S I + 1 L l [ m d l] 2 S l + 1 L l [ i n d l ] 2 S 2 + 1L2[md212S + 1L 2 S 1 + 1 L l [ i n d l ] 2 S 2 + 1L2[/nd212S12 + 11.12 2S 3 + 1 L 3 [ m d 3 ] 2 S + 1L 2S 1 + 1 L l [ / n d 2 ] 2 S 2 + 1L2[ind2]2S 3 + 1L3[md3]2Sz3 + 1L232S + 1L 2 S l + 1 L l [ i n d l ] 2 S + I L 2 [ / n d 2] 2S12 + 1Lt2 2S3 ~- 1L3[ind3]2S 4 + 1L4[/nd 4] 2S34+ 1 L 3 4 2 S * - I L S l + l l . l [ i n d ~ ] 2 S 2 + 1L2[/nd 2] 2Sa2 + 1Ll2 2S~ + 11.~[ind312S~3 + 1L13 2 S 4 + 1L4[ind4]2S + I L

w h c r c 2S, + 1L,[ind,] describes the statc of thc ith clectron shcll, 2S,j ~ 1 L,j thc intcrmcdiatc statc, and 2S + 1L the final statc. 2S, + 1 is an intcgcr, L, a capital lcttcr; rod, an additional n u m b e r to distinguish tcrms with samc values o f 2s, x L. For these n u m b e r s the conventions as tabled in Rcf. [13] must be used; if in Rcf. [13] a state !;'~' 2s.. XL, has no additional number, rod, = 1 has to be set. If IV, = 1 thcn 2S, + 1 /.,[mar,] must not be given. A SL-block has no spacings, between thc blocks one blank is required. The n u m b e r of S L - b l o c k s depends on nos and the N,'s. It ranges from 0 - for a single electron (nos = 1, N~ = 1) nob blanc lines havc to be includcd - to 7 - for 4 shclls with cach of them 2 or m o r c electrons (nos = 4, N, > 2). All the nob lines are o f same structure. • • •

mixing (yes) or no mixing (no, is equivalent to E N D O F I N P U T ) n u m b e r of states in question noq(_< nob) mixing coefficients: for each basis state one line (same succession as above) with the mixing coefficients m , ( i = 1..noq) for cach state in question. Hence, a noq x nob matrix is requested.

"Each input block ' . ' must bc p r c c e d c d by a linc which may contain comments.

References [I] H. Kopfermann, Nuclear M o m c n m (Academic, New York, 1958).

{2] P.GH. Sandars and J. Beck. Proc. R. Soc. London, Scr A 289 (1965) 97. [3] I. IJndgren and A Rein, Case Stud. At. Phys. 4 (1974) 93. [4] S. B6ttgenbach, Hypcrfmc structure m 4<1-and 5d-shell atoms (Spnnger, Berlin, 1982). [5] V. Pfcukr, 7~ Phys. D 4 (1987) 351. [6] W.J. Child& C.as¢ Stud. At. Phys. 3 (1973) 215. [7] H.-D Ka'onfcldt, G. Klemz, S. Ka'6gcr, and J. -F. Wyatt, Phys. Rev. A 48 (1993) 4500. [8] H - D Kronfeldt. D. Ashkcnasi. S Kr6ger, and J.-F Wyatt. Phys Yg"r.48 (1993) 688. [9] G. Klcraz, S. Kr6ger and li.-D Kronf©ldt. in preparation. [10] D.CS Alhson. C.omput. Phys. Commun. 1 (1969) 15 [11] A.T. Chivcrs. Comput. Phys. Commun. 6 (1973) 88. [12] D.CS. Allison and J. F. Mc Nulty, Comput. Phys. Commun 8 (197,1) 246. [13] C W Nielson and G. F. Koster. Spectroscopic Coefficients for the P% d" and f" configurations

S. Kr~e~. M. Kr6ger / Comp~r ! ~

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