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Effective operators of the spin correlated crystalline electric field
Volume 76A, number 2
PHYSICS LETTERS
17 March 1980
EFFECTIVE OPERATORS OF THE SPIN CORRELATED CRYSTALLINE ELECTRIC FIELD J.M.DIXON Physics Department, The University of Warwick, Coventry CV4 7AL, England
and R. CHATTERJEE Physics Department, The University of Calgary, Calgary, Alberta T2NJN4, Canada
Received 14 January 1980
The effective operators of the spin correlated crystal field have been formed by coupling the exchange with the crystal field. The exchange field is a scalar in spin space so the effective orrators will still have the same rank as that of the crystal field but will give non-zero matrix elements between the states ~ of the half filled shells, contrary to the zero value obtained for a pure crystal field tensor.
From the table of the “Spectroscopic coefficients for the ~ d7? and f’~configurations” by Nielson and Koster [1], it is well known that for the half-filled shell ions the reduced matrix elements of the crystal field tensor operator Ck (k = 2,4, 6) between the 2ShiL~states vanish. Newman [2] proposed an exchange-coupled crystal field operator which is called the spin correlated crystal field VsCCf. Andriessen et al. [3] calculated this “Newman effect” using relativistic wave functions, using a diagrammatic techniqueand applied it in quadrupole analysis. In this paper we will generate the effective operators of Vsccf elaborating the schematic approach of Judd [4]. The crystal field tensor B~(C~)~ is replaced by Vsccf=B~flk~
(1)
where each tensor component (C~)~ of the single electron tensor Ck, of the crystal field, is mixed with the exchange interaction term S• %. The B~’sare the parameters of the crystal field and the Ilk’s are constants. We will apply the Racah formalism to express eq. (1) in the form of an effective operator. Using the formulae (32.5) and (32.2) from Yutsis et al. [5], eq. (1) is written as VsccfB~\/~TIlk[c/~X
ES? X S,1]0]~.
(2)
From Yutsis et al., formula (34.6), this may be rewritten as
Vsccf =
~
(_1)kB~Ilk(2 2~ 1)1/2 [p(lk)k’ X
SjJ~,
(3)
where w~’’[S1XC/’J’~’
and
k’=k+l,k,k—l.
(4)
The matrix elements of Vsccf, from the Wigner—Eckart theorem, are given by
147
Volume 76A, number 2
I =
(~M
PHYSICS LETTERS
17 March 1980
2S’41(L’)ii~’)
(5)
~)
~(_l)k’B~Ilk (~k~ ~)“2
The reduced matrix elements can be written, using Yutsis et al., eq. (33.3) as
R
=
<~ 2S+lL~Il[Wf~”
X
s7] k~ 1~P 2s’+l(L’)
>
=
an”
2S’~(L~’)j.,IiS~lI a’ 2S’+l (L’)~~) (_l)~~~’~( X
+2k 1)h/2t,
The reduced matrix element of sJ is diagonal so that a” R=(_l)~J’+k[S’(S’+ l)(25’+ l)(2k+
l)]1/2
=
a’, J”
=
~
i,).
J’, S” = S’ and L”
(6) =
L’ [6]. Thus
k) JI
The 6/symbol in eq. (7) may be evaluated using Rotenberg’s eq. (2.23) [7] and the properties of the 6/symbol. We are specifically interested in those cases where k is even and 11’) VsCCf is hermitian. the double tensors tok’be k with k even so For we drop the term when hermitian + k(7) + k’ must and this cannot occur for W~ = k. From 1eqs. and (5)be weeven therefore find I=
(_i)J~
(J
~::~) B~Ilk[S’(S’
+
1)(25’ +
1)]
1/2(i)k+2(J+J)
~ [(J+J’+k+2)(J’_J+k+1)(J_J’+k+1)(J+J’_k)1<
[
2J’(2J’+1)(2J’+2)(2k+l)(k+1)
a
25~1LIlw(1k)k+1
a
12S’+lLI )
( )~ (8)
1(~~~’ + k+ 1)(J’-~J+k)(J—J’+ k)(J+J’ 2J’(2J’+l)(2J’+2)k(2k+1)
—
k+ 1)11/2< 2S+lL iiw~1~1ii ‘2S’+1(L1 a a )~
The reduced matrix elements of the double tensors W (lk)k ~ can be easily evaluated from the table of Chatterjee etal. [8]. We are currently trying to use the above analysis to calculate zero field splittings for 4f7 and 3d5 using a specific physical mechanism to obtain 71krn References [1] C.W. Nielson and G.F. Koster, Spectroscopic coefficients for the p’1, d~?and f’~configurations (MIT Press, Cambridge, MA, 1963). [2] D.J. Newman, Chem. Phys. Lett. 6 (1970) 288. [3] J. Andriessen, 1). Van Ormondt, SN. Ray and T.P. Das, J. Phys. Bli (1977) 2601. [4] B.R. Judd, Phys. Rev. Lett. 39 (1977) 242. [5]A.P. Yutsis, I.B. Levinson and V.V. Vanagas, Mathematical apparatus of angular momentum, TransL A. Sen and R.N. Sen (NSF, Washington, DC, 1962) by the Israel program for Scientific Translation, Jerusalem. [6] B.G. Wybourne, Spectroscopic properties ofrare earths (Interscience, 1965) p. 20, eq. (2.34). [7] M. Rotenberg, R. Bivins, N. Metropolis and J.K. Wooten Jr., The 3/ and 6/ symbols (MIT, 1959). [8] R. Chatterjee, M.R. Smith and H.A. Buckmaster, Can. J. Phys. 54 (1976) 1228.
148
PHYSICS LETTERS
17 March 1980
EFFECTIVE OPERATORS OF THE SPIN CORRELATED CRYSTALLINE ELECTRIC FIELD J.M.DIXON Physics Department, The University of Warwick, Coventry CV4 7AL, England
and R. CHATTERJEE Physics Department, The University of Calgary, Calgary, Alberta T2NJN4, Canada
Received 14 January 1980
The effective operators of the spin correlated crystal field have been formed by coupling the exchange with the crystal field. The exchange field is a scalar in spin space so the effective orrators will still have the same rank as that of the crystal field but will give non-zero matrix elements between the states ~ of the half filled shells, contrary to the zero value obtained for a pure crystal field tensor.
From the table of the “Spectroscopic coefficients for the ~ d7? and f’~configurations” by Nielson and Koster [1], it is well known that for the half-filled shell ions the reduced matrix elements of the crystal field tensor operator Ck (k = 2,4, 6) between the 2ShiL~states vanish. Newman [2] proposed an exchange-coupled crystal field operator which is called the spin correlated crystal field VsCCf. Andriessen et al. [3] calculated this “Newman effect” using relativistic wave functions, using a diagrammatic techniqueand applied it in quadrupole analysis. In this paper we will generate the effective operators of Vsccf elaborating the schematic approach of Judd [4]. The crystal field tensor B~(C~)~ is replaced by Vsccf=B~flk~
(1)
where each tensor component (C~)~ of the single electron tensor Ck, of the crystal field, is mixed with the exchange interaction term S• %. The B~’sare the parameters of the crystal field and the Ilk’s are constants. We will apply the Racah formalism to express eq. (1) in the form of an effective operator. Using the formulae (32.5) and (32.2) from Yutsis et al. [5], eq. (1) is written as VsccfB~\/~TIlk[c/~X
ES? X S,1]0]~.
(2)
From Yutsis et al., formula (34.6), this may be rewritten as
Vsccf =
~
(_1)kB~Ilk(2 2~ 1)1/2 [p(lk)k’ X
SjJ~,
(3)
where w~’’[S1XC/’J’~’
and
k’=k+l,k,k—l.
(4)
The matrix elements of Vsccf, from the Wigner—Eckart theorem, are given by
147
Volume 76A, number 2
I =
(~M
PHYSICS LETTERS
17 March 1980
2S’41(L’)ii~’)
(5)
~)
~(_l)k’B~Ilk (~k~ ~)“2
The reduced matrix elements can be written, using Yutsis et al., eq. (33.3) as
R
=
<~ 2S+lL~Il[Wf~”
X
s7] k~ 1~P 2s’+l(L’)
>
=
an”
2S’~(L~’)j.,IiS~lI a’ 2S’+l (L’)~~) (_l)~~~’~( X
+2k 1)h/2t,
The reduced matrix element of sJ is diagonal so that a” R=(_l)~J’+k[S’(S’+ l)(25’+ l)(2k+
l)]1/2
=
a’, J”
=
~
i,).
J’, S” = S’ and L”
(6) =
L’ [6]. Thus
k) JI
The 6/symbol in eq. (7) may be evaluated using Rotenberg’s eq. (2.23) [7] and the properties of the 6/symbol. We are specifically interested in those cases where k is even and 11’) VsCCf is hermitian. the double tensors tok’be k with k even so For we drop the term when hermitian + k(7) + k’ must and this cannot occur for W~ = k. From 1eqs. and (5)be weeven therefore find I=
(_i)J~
(J
~::~) B~Ilk[S’(S’
+
1)(25’ +
1)]
1/2(i)k+2(J+J)
~ [(J+J’+k+2)(J’_J+k+1)(J_J’+k+1)(J+J’_k)1<
[
2J’(2J’+1)(2J’+2)(2k+l)(k+1)
a
25~1LIlw(1k)k+1
a
12S’+lLI )
( )~ (8)
1(~~~’ + k+ 1)(J’-~J+k)(J—J’+ k)(J+J’ 2J’(2J’+l)(2J’+2)k(2k+1)
—
k+ 1)11/2< 2S+lL iiw~1~1ii ‘2S’+1(L1 a a )~
The reduced matrix elements of the double tensors W (lk)k ~ can be easily evaluated from the table of Chatterjee etal. [8]. We are currently trying to use the above analysis to calculate zero field splittings for 4f7 and 3d5 using a specific physical mechanism to obtain 71krn References [1] C.W. Nielson and G.F. Koster, Spectroscopic coefficients for the p’1, d~?and f’~configurations (MIT Press, Cambridge, MA, 1963). [2] D.J. Newman, Chem. Phys. Lett. 6 (1970) 288. [3] J. Andriessen, 1). Van Ormondt, SN. Ray and T.P. Das, J. Phys. Bli (1977) 2601. [4] B.R. Judd, Phys. Rev. Lett. 39 (1977) 242. [5]A.P. Yutsis, I.B. Levinson and V.V. Vanagas, Mathematical apparatus of angular momentum, TransL A. Sen and R.N. Sen (NSF, Washington, DC, 1962) by the Israel program for Scientific Translation, Jerusalem. [6] B.G. Wybourne, Spectroscopic properties ofrare earths (Interscience, 1965) p. 20, eq. (2.34). [7] M. Rotenberg, R. Bivins, N. Metropolis and J.K. Wooten Jr., The 3/ and 6/ symbols (MIT, 1959). [8] R. Chatterjee, M.R. Smith and H.A. Buckmaster, Can. J. Phys. 54 (1976) 1228.
148