intensities of crystal spectra of rareearth ions.pdf

7/27/2019 Intensities of Crystal Spectra of RareEarth Ions.pdf

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Intensities of Crystal Spectra of RareEarth IonsG. S. OfeltCitation: J. Chem. Phys. 37, 511 (1962); doi: 10.1063/1.1701366View online: http://dx.doi.org/10.1063/1.1701366View Table of Contents: http://jcp.aip.org/resource/1/JCPSA6/v37/i3Published by theAIP Publishing LLC.Additional information on J. Chem. Phys.Journal Homepage: http://jcp.aip.org/Journal Information: http://jcp.aip.org/about/about_the_journalTop downloads: http://jcp.aip.org/features/most_downloadedInformation for Authors: http://jcp.aip.org/authors

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T H E J O U R N A L OF C H E M I C A L P H Y S I C S V O L U M E 3 7 , N U M B E R 3 A U G U ST I , 1962

Intensities of Crystal Spectra of Rare-Earth Ions*G. s. OFELT

The Johns Hopkins University, Baltimore, Maryland(Received February 26, 1962)

Magnetic and electric dipole transitions between levels of the 4jx configuration pe rturbed by a staticc ~ ~ s t a ~ l i n e field are ~ r e a t e d . The expression obtained for the pure-electronic electric-dipole transition probability mvolves matnx elements of an even-order unit tensor between the two 4jx states involved in the transition. The contributions to th e transition probability from interactions, via the crystalline field, with thenlj,94P-1, 4j


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512 G. S. OFELTRestricting to interactions within the 4/" configuration, the energy Ea of the ath level, is defined by

where the eigenvector I f>a) may be expressed as alinear combination of the unperturbed states corresponding to an arbitrary coupling scheme. We shallconsider all calculations in the SL coupling scheme sothat the transformation can be expressed as

where the summation extends over the set of quantumnumbers aSLJ J. and the a(a; aSLJ J. ) are real.Following Condon and Shortley5 we define the linestrength between two levels a and b asS(a, b) =}: I (a IV 1,8) 1, (5)all

where V may be either the electric (P), or magnetic(M) dipole-moment operator, and a and f3 are thecomponents of states a and b, respectively. The magnetic dipole-moment and electric-dipole-moment operators are given by

andM= -e(2w)-1}:(L i +2 S i )i

P=-e}:ri,i

(6)

respectively, where e is the electronic charge of theelectron and }J. is the mass of the electron.We adopt the irreducible tensorial description ofRacah,6 and use the phase convention of Fano andRacah.78 The tensorial set of components of a vectorV is given by

In this notation, suppressing the constants -e/2}J.cand -e here and throughout the remainder of thecalculations, Eqs. (6) become

MIl] = }:Lill]+2SP],i

PIl]=i"rC 11 ].L..J 't, " ,i (6')where CP] is a tensorial operator normalized like theLegendre polynomials7 and the subscript i indicates

6 E. U. Condon and G. H. Shortley, The Theory of AtomicSpectra (Cambridge University Press, London, 1935).6 G. Racah, Phys. Rev. 63, 367 (1943); 76, 1352 (1949).7 U. Fano and G. Racah, Irreducible Tensorial Sets (AcademicPress Inc., New York, 1959).8 The phase of the V symbol has, however, been converted tobe compatible with the 3-j symbol [M . Rotenberg, R. Bivins,N. Metropolis, J. K. Wooten, The 3-j and 6-j Symbols (Tech

nology Press, Cambridge, Massachusetts, 1959) J.

that it is a one-particle operator. The line strengthbecomesS(a,f3)=(1 (a l V1[!] 1(3)12+ I (a l V_1111 1(3)IZ)

+(1 (a I Volll I 3) 12), (7)where VII] may be either M[I] or P[!]. Since the radiation from ions within the crystal is polarized we separatethe line strengths into two terms as indicated by theparentheses in Eq. (7), denoting the first term by S"and the second by S", corresponding to u polarized and7r polarized for electric-dipole transitions, respectively(for magnetic-dipole transitions the u and 7r areinterchanged) ,

(8)Thus, calculations will be made for the pth componentof the dipole operator, the results being substitutedinto Eq. (8) depending upon which type of transitionis to be considered. The transition probabilities arerelated to the line strengths through the Einsteincoefficients A and B,5spontaneous emission

absorption

where /I is the frequency in cm -I between the two levelsconsidered, and the radiation integrated over a sphereso that any angular effects may be neglected.

III. MAGNETIC-DIPOLE TRANSITIONSThe theory for magnetic-dipole transitions is well

known,4.59 and therefore only the application to thecrystal field problem need be given. As the magneticdipole operator is of even parity, to a first approximation we restrict interactions to the states of the 4fxconfiguration, the I f>a) of Eq. (4). The pth componentof the line strength for a transition between the levelsa and b is given bySp(a, b) = I }:a(a; aSLJ J.)a(b; a'S'L' J' J.')

(jxaSLJJ. IMpll] Ifxa'S'L'J'J.')IZ, (10)where the summation extends over the sets of quantumnumbers aSLJ Jz and a'S'L' J' J / .

In our notation the matrix elements of Eq. (10) are(jxaSLJJ.I Mp[l] Ifxa'S'L' J' J /)

=o(a, a')o(L, L')o(S, S')i( l ) J - J . ~ ,9 L. J. F. Broer, C. J. Gorter, and J. Hoogschagen, Physica 11,231 (1945).



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CRYSTAL SPECTRA OF RARE - EARTH IONS 513whereml:=g(SLI)[J(J+l)J!( I 1 I) for I '=I ,

-I. p Iz'ml:= ( [ (S+L+1)LPJ[P- (L - S)2J)t

. ( I 1 ( J -,1) for I'= I -1, (11)-I. p I .and g(SLI) is the Lande g factorg(SLI)

= 1+[I(I+1) -L(L+1) +S(S+1) J/2I(I+1).IV. ELECTRIC-DIPOLE TRANSITIONS

The matrix elements of the electric-dipole operatorbetween two states a and {3 can be nonzero only if aand {3 do not have the same parity. As the free-ion statesof the fX configurations have the same parity, states ofopposite parity have to be mixed into the wave functions associated with at least one of the two levelsconsidered. This may be done by vibrational interaction or by the odd order terms of the static-potentialexpansion (see Sec. II). The former has been the usualapproach for the transition elements lO and also hasbeen applied to the 4p configuration311 for the vibrationelectronic transitions, bu t in an individual interactionprocedure. We shall restrict our considerat"on to thepure electronic transitions, disregarding any group oftransitions that have been experimentally determinedto be the electronic-vibrational type. Thus, the oddparity part of the crystalline potential shall be considered as the only mechanism for mixing the parity ofthe states.The excited, opposite-parity configurations havequalitatively the same features as to electrostatic,spin-orbit, and crystal-field splitting as the 4fx configurations, although the magnitude of these interactions are by no means similar. We shall consider thestates of these configurations under the transformationto the SL-coupling scheme, such that the eigenvectorsmay be expanded analogous to Eq. (4). Thus, the

state I 3) of an excited, opposite-parity configurationat an energy E{3 is given byI 3) = La({3; a1SlL1I1I,J Ia1SlL1Id'I) ' (12)where the summation extends over the set of quantumnumbers a1SlL1I 1'1"

We now wish to consider the transition between twolevels of the 4fx configuration of energy Ea and Ebwith eigenvectors I pa) and I pb) expanded as in Eq.(4). With first-order perturbation theory the mixedparity eigenvector IXa) corresponding to the energyEa is given byIXa)= I pa)+ 2 )Ea-E,8)-I({31 Voddk I pa) I 3), (13)

,8with a similar expression for the level b.The matrix element between the two mixed-paritystates 1 Xa) and 1 Xb) of the dipole-moment operatorP[I] is

(Xa I p[l] 1Xb)= L(Eb-E,8)-I(rpa 1P[l] 1 3),8X (31 Vodd k 1 pb)+ L (Ea-E,8)-l

,8X ({31 p[l] 1 pb)({31 Vodd k 1 pa)+, (14)

where the two terms containing matrix elements of p[l]between states of the same parity have been omittedas they are identically zero. The remainder of thecalculation is done in the notation of Sec. II and usingthe following notation[ IJ= (21+1). (15)

A straightforward application of the methods ofRacah6 is made using the states I 3) of the form4f",-lnl(SILII1) 12 and the potential (Vkq) of the form

Vkq= (i ) kAkqCP]+ i)kBkqC_q[k] (q2::0) ,where the constants Akq and Bkq may be complex,subject to the restriction that

Bkq= (-1)qA kq*in order that Vkq is hermitian.For clarity, the calculations will be carried outusing Bkq with substitution of ( -1 ) qAkq* at the finish.


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514 G. S. OFELTwhere 1/; represents the set of quantum numbers aSLand the quantum numbers of the parents of the pstates are indicated by 1/;2. The summation index Arepresents the sets of quantum numbers asLJ J z,a'S'L' J' J/, SlLlJ lJ z1 , and 1/;2, The radial wavefunction for the electron in the nl state is r-1R(nl).The one-particle tensorial operator is CP) and itsreduced matrix element between the states I and f is

(J II C;['l Ill) ( 1) ,,'" "7 ([ I ] ) 'G : ~ )The approximation is now made that

(Ea-E{3)=(Eb-E{3)=t1Enl (17)for all (3 states of the 4p- 1nl configuration that are

connected in Eq. (16). This is not to say that theentire width of the 4fx-1nl configuration is small compared to the distance from the average of the energiesof the two states I f>a) and I f>b) of the 4fx configurationbut rather only the connected states of the 4fx-1nlconfiguration are considered degenerate. I f intermediatecoupling in this configuration is large, the entire configuration has to be considered degenerate. Similarly,if the intermediate coupling in the 4fx configuration isnot too large, and we restrict to 1 f states of the highermultiplets, the number of connected 4fx-1nl states isreduced. Once this (t1E) approximation is made, thecoefficients A ({3; 1/;1) of Eq. (16) may be set equal tounity and the sum over (3 omitted.Summing over Ll in Eq. (16), we obtain a morecompact expression


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516 G. S. OFELTI:lEnd 9, I:lEnd, and I:lEnu above the two 4fx states considered. In this approximation l/(I:lEnl) forms adecreasing sequence with increase of n. Using thefollowing notation

Rk(nl) ==J (4f)R(nl)rkdrJ (4f)R(nl)rdr, (22)and writing explicitly only the 1and n dependent partof Eq. (20'), summing over the principal quantumnumber we have


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CRYSTAL SPECTRA OF RARE-EARTH IONS 517elements7 when summed over l giving


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CRYSTAL SPECTRA OF RARE-EARTH IONS 519With the aid of relation (2.19) of reference 8, noticingthat (_1)2J1= (-1)", we obtain

I 3 1 1W= L:CY] 3 Lz :') ( - I ) " ' + J ' " " / ~ [ ' J Y Z ,y

k Lwhere

Y= (J,J/

and

Z ~ L : [ J ' f 1Jl J' kApplying relation (3.22)expressed as r '= (_1)2r 1 r

(I-4)

J 1 J 1J' S L'k L Y

of reference 8 Z may be

:)1: y :). (I-5)JThen performing the summation over y in Eq. (I-4)we findW= (_1)x+J'-J/-L-L'-IL( 1)-r[r]Y

, (I-6)

where this 12-j symbol corresponds to the "twisted"Mobius strip.14,1.I f the 12-j symbol of Eq. (I-6) is expressed in termsof 6-j symbols, with the introduction of a summationindex t, the r dependent terms may be collectedl' J t)L[rJYr k 1 r

and the summation over r may be performed with theaid of relation (2.19) of reference 8. The resultingexpression obtained for W is

(J, J t)1-17 = ( _ 1 ) , * 8 - J e q+L2 L[tJ

t J. ' -J z (q+p)

-C : - ( : + J I ~ , ~ j l : :' ;J1: : ;f (I-7)

Entering Eq. (I-7) into Eq. (1-2) we obtain therequired expression for M by comparison with Eq.: - ~ ( - l ) ' - " - ~ ' L : [ t J l k 1 t)

t 3 3 I

( J' J t )(k Jz' - J . (q+p) q (JxaSLJ II U[ t ) II fXa' SL' J'). (1-8)

APPENDIX IIContributions to the electric-dipole transition probability from interactions with states of the nd94fx+lconfiguration may be considered. We write the reducedmatrix element of C[k) as

M = (dlO (1 S) , fx (azSzL2); SLJ II C[k) II d9(2D) ,f"'+l(az' S lLz') ; SlLdl). (II-1)

Expanding these states with the aid of coefficients offractional parentage following Racah,6 M may beexpressed asM = (lO(x+ 1i.L (dlOeS) {I d9(2D (f, f X (1/;2); 1/;2' I jx+I1/;2')

83L3 2Dd) IS, 1/;2; SLJ II C[kJ II (2Df)1/;a, I/;z; SlLdl. 3) L3, Lz I2, (3L2) Lz') (Ll).H) S3, s21 !, (!S2) S2') (81). (II-2)

Expressing the recoupling coefficients in terms of 6-jsymbols, reducing the matrix element in Eq. (II-2) tothe single-particle matrix element, and relating thecoefficient of fractional parentage to the standard formwe obtainM=o(S, SI)O(I/;, 1/;2) (x+l)t(f"'1/; i}f"'+11/;2')

. ( -1 ) J+L+Ll+L 2'+t( [J] [J J[LJ[L1]) ( [ S2'J[Lz'] ) t' ([SJ[LJ)-!lJ J 1 kIlL Ll k)(dIICPJIIf),

Ll L S 2 3 L z' (II-3)where the summation over S3 and La has been performed and algebraic expressions have been used for the6-j symbols that contain a zero argument.The coefficient of fractional parentage in Eq. (II-3)

may be inverted,6 which gives the resulting expressionforMM=o(S, SI)O(I/;, 1/;2) (14-x)t(P3-"1/;2' 1}j14-xl/;)

(_1)J-S+Ll([J][J1J[LJ[LlJ)!. /J J 1 kjlL Ll k jed II cpJ Ilf). (II-4)Ll L S 2 3 Ll



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520 G. S. 0 ( E L TIn the electric-dipole transition calculation matrixelements enter as the product

(11-5)Therefore, these matrix elements, expressed in the formof equation (11-4), enter into the calculation to formreduced matrix elements of U[ t ] between the almostclosed shellp4-x states. With the aid of the relationship

T H E J O U R N A L OF C H E M I C A L P H Y S I C S

for reduced matrix elements of U[ t ] (t even)( f x ~ II U[ t ] II F ~ ' ) = - ( f 1 4 - x ~ II U[ t ] II f I 4 - x ~ ' ) ,

and inspecting the phase of Eq. (11-4) according tothe product of expression (11-5), we find that theangular part from the nd94fx+l interaction is of thesame form and phase as that from the 4F-1ndinteraction.

V O L U M E 3 7 , N U M B E R 3 A U G U S T I , 1962

Sequential Substitution Reactions on B10H 10-2 and B12H 12-2ROALD HOFFMANN AND WILLIAM N. LIPSCOMB

Department of Chemistry, Harvard University, Cambridge 38, Massachusetts(Received April 4, 1962)

Predictions are made of the order of sequential electrophilic substitution on the BlOH lO-2 and B12H12-2polyhedral ions. Simple LCAO-MO calculations are used and resonance and inductive effects are treatedseparately.

I N comparison with boron hydrides and their ionsthe closed polyhedral Bl oHlO-2 of D4d symmetryI2and B12H12-2 of icosahedral symmetry3 are so extraordinarily stable that they can be regarded as pseudoaromatic in character. They have also been reported4to form a very interesting series of substitution derivatives, as yet incompletely identified, of the typeBlOH lO_ nX n-2 where O::;n::;lO, and X=F, Cl, Br, or I.This indication that a large number of sequentiallysubstituted derivatives can be prepared has led us tomake predictions of reactivities of the B10H lO-2 andB12H12-2 ions in an effort to include resonance andinductive effects in unsubstituted and substitutedpolyhedral ions as formulated by an extension of ourLCAO-MO descriptions of B10H lO-2 and B12H12-2. Weare encouraged by the initial prediction, verified byexperiment2.4 that apparent electrophilic attack by D+appears to produce preferential exchange in BlOH10-2at the two apices, which are more negatively chargedthan the eight other B atoms.Formulated in terms of a specific example, the question investigated here is the following: Suppose thatattack by a halogen is electrophilic and that successive

1 W. N. Lipscomb, M. F. Hawthorne, and A. R. Pitochelli,J. Am. Chern. Soc. 81, 5833 (1959).2A. Kaczmarczyk, R. Dobrott, and W. N. Lipscomb, Proc.Natl. Acad. Sci. 48, 729 (1962).3 J. Wunderlich and W. N. Lipscomb, J. Am. Chern. Soc. 82,4427 (1960).4 W. H. Knoth, H. C. Miller, D. C. England, G. W. Parshall,J. c. Sauer, and E. L. Muetterties, J. Am. Chern. Soc. 84, 1056(1962) .6 R. Hoffmann and W. N. Lipscomb, J. Chern. PhY5. (to bepublished) .

TABLE I. Atom-atom polarizabilities in 10-3 units ofelectronic charge/eV.

-5 2 0 0o -5 2 11o 11 -5 211 0 011 0 0o 11 11o 11 1111 0 011 0 11o 11 011 0 11o 11 0

11 11o 0o 0-5 2 1111 -52o 0o 1111 0o 011 11o 1111 0

o1111oo-5 2o1111oo11

o1111o11o-5 2oo1111o

11 11o . 0o 1111 0o 011 11o 0-5 2 1111 -5 2o 0o 1111 0

o 11 011 0 11o 11 011 0 1111 11 0o 0 1111 11 0o 0 11o 11 0-5 2 0 11o -5 2 011 0 -5 2-2B 1 0 ~ 0 surface

-179 2424 -17935 435 435 435 44 354 354 354 35

-87 0o -8 721 121 121 121 11 211 211 211 21

40 40 40 40444 4-121 17 3 1717 -121 17 33 17 -121 1717 3 17 -12120 20 0 020 0 0 20o 0 2020o 20 20 0

-2BlOHI0 in

4402020oo_12117317

44020oo2017-121173

440oo2020317-12117

440o2020o17317-12112 12 12 12 0 0 0 0o 0 0 0 12 12 12 12-8 0 15 -1 15 19 19 0 015 -80 15 -1 19 0 0 19-1 15 -80 15 0 0 19 1915 -1 15 -8 0 0 19 19 019 19 0 0 -8 0 15 -1 1519 0 0 19 15 -80 15 -1o 0 19 19 -1 15 -80 15o 19 19 0 15 -1 15 -80

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