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The estimation of phonon-induced contributions to the second-order spin-lattice coefficients for Gd3+ in crystals with fluorite structure
Physica B 190 ( 1993) 237-240 North+Holland SDI: 092 I -4526( 92)00453- 1
The estimation of phonon-induced order spin-lattice coefficients for structure*
contributions to the secondd3+ in crystals with
W. Pastusiak instit~rte
of Pkysics,
Technical
University
of Stczecin,
Poland
Received 26 July 1991 Final re+ed 2 November 1992
Phonon-induced contributions arising from the one-phonon second-order mechanism to the second-order coupling coefficients K:” (I’= E,, T,,) for Gd” in crystals with the fluorite structure arc estimated. It was found that the contributions resulting from the second-order mechanism do not exceed 10 ’ cm- ’ and are much smaller than the K:!’ coefticients dctermincd experimentally as well as the contributions resulting from the first-order mechanism.
Section 1 The application of a static external perturbation (hydrostatic or uniaxial stress, electric field etc.) to a crystal sample doped with magnetic ions Icads to changes in ligand positions with respect to equilibrium ones. At the same time the displacements of ligands are also caused by heat lattice vibrations. Estimating the amplitude of vibrations for an optic mode with a frequency of about lo’3 s-’ for CaF, we came to the conclusion, that the amplitude introduced by a static perturbation is at least two orders of magnitude smaller than the amplitude and wave!ength of dynamic vibrations. Thzse two perturbations. static and dynamic ones, result in a change of the energetical spec-
Cm-espondence
nical University
W. Pastusiak, Institute of Physics. Tcchof Szczecin. Aleja Piastciw 17, 711.710
to:
Szczecin, Poland. * This work was partially E”30. 0921 4SXi/93/$(16.00
supported
mdcr
@ 1993 - Elsevicr
project
Science
No. 1081
Publishers
trum of a dopdnt. This change, following Calve and Oseroff [l-3) can be described by a phenomenological Mamiltonian t’or the spin-lattice coupling:
ere
(2) and c $‘( T) are the Kth order spin-lattice coefficients at temperature T and G$.‘(RL) corresponds to the value of C!?for a ‘rigid lattice’. K-k;.’ is a parameter related to phonon-induced onntr;hmltinnc tt_r _rFji. (T) parameter. I$.: is k”“..~.,“U..V..V ._ the ____ {-!” combination of tensor a symmetrized linear UCh~StCP operators normalized according to et al. [4] and GY!.~ is a symmetrized component of the strain tensor. r is an irreducible representation of a cubic group and y numbers the basis functions of this r; presentation, p distinguishes repeating r representations.
B.V. Al\ rights racrvcd
W. Pastusiak / On phonon.induced contributions to spin-lattice coefficients
238
Section 2
In our previous two papers [5,6] we have presented a calculation method allowing the determination of the resulting phonon-induced contributions to the t?{K) coefficients. The idea of these calculations is based on the classification of the parameters describing dynamic interactions according to irreducible representations for normal (or symmetrized) vibrations of a lowsymmetry complex (a paramagnetic ion plus its ligands) and then on the decomposition of each of these parameters into the components corresponding to a coupling of an ion with ligand vibrations in the absence of static deformation, in the presence of a linear static deformation, quadratic static deformation etc. It seems that due to only slight static deformations one can, to a good approximation, use as a normal coordinates of a low-symmetry complex, symmetryadapted linear combinations of normal coordinates for a high-symmetry (undeformed) complex. The decomposition of parameters describing dynamic interaction allows the classification of particular terms in the Hamiitonian according to the powers of static deformations and the construction, in a harmonic approximation, of the first-order two-phonon mechanism and secondorder mechanisms which, like Hamiltonian (1) are linear in a static deformation [7]. In the first-order mechanism the interaction Hamiltonian is linear in static deformation and quadratic in dynamic deformation. In the considered second-order mechanism the Hamiitonian linear in the dynamic deformation and the Hamiltonian linear in both dynamic and static deformations are present. In the model in which the crystal field is m n d 1 1~,~ 1~ t~,rl ..... a t t t'~.. u
lal~ It~.ff
lI ~titttt ~ ,,~ t 1~.,%,
~V ,i ;UhI ~ -¢~l. . tl ;k ,l U-I, . ,l -; D
[Q1
[ ¢L)J ~
,I.k[ .l l.~ ,
.(:;.~. 111 31.-
mechanism leads to two-phonon proccsscs whereas the second-order mechanism leads to one-phonon processes (or two-phonon lie ones o r d e r
[91). In paper [7], based on crude estimations we suggested that the phonon-induced contributions to the second-order coefficients /(~!:~ is predominantly caused rathcr by the two-phonon
first-order mechanism and not by second-order mechanisms suggested by other authors (e.g. refs. I2,101). It is the aim of this letter to report the estimation of the contribution from the onephonon second-order mechanism to the secondorder coefficients K]? ) (F = Eg, T2g ) for Gd 3+ in crystals with the fluorite structure using the formalism presented previously [5].
Section 3
The temperature-dependent factor in relation (3.23) of ref. [5] is h
B~'a^(T) = ~ bn'aa.J 2Mtoj cth(htoj/2kT).
(3)
J
We would like to stress here that the 6'AA indices in the quantities B~.a~(T ) and b~.a^.j do not determine their transformation properties but they number them only (unfortunately, in paper [5] we omitted the index A in the quantity b,~'~A.;). Analyzing further the phonon part let us note two experimental facts: (i) the experimentally observed temperature dependence of the second-order spin-lattice coefficients may be described by relation (1); (ii) the spin-lattice coupling is very well described in the Einstein model with a frequency o) = 2 x 1013s -! [31. Correspondingly to the above statement and the argumentation given by Huang [11-13] this phonon part can be written as B,s.,~(T) = (h/6txw) cth(h~o/2kT) ,
(4)
independently of 6'AA indices. Inserting c q (4) into (3.23) of ref. [51 we can easily obtain the ana!vtical form for the secondorder cocffidcn~.s b~l?3 or K~t".~ (for the relations bztween the coefficients with and without bar see appendix A of ref. [5]). We can then note that t!~e obtaiped expression will be proportional to thc factor
ii,,6tzco ~ ,~
(5)
W. Pusrusiak
I On phonon-induced
which has the dimension of the square of amplitude. Because in the presented model we are interested in the displacements of ligands from their equilibrium positions we can treat p as a mass of a ligand.
Section 4
contributions
coefcicipmrs
23sr
The matrix elements required tions [S] have the form (‘&,,I
c b,(k’, k)W”“‘*““/%$) - k’.k
in the calcula-
.
(10)
Among these matrix elements only three will be essential: b,(O,
Let us now analyze the ‘electronic’ part in eq. (3.23) of ref. [S]. From the bj-symbol appearing in eq. (3.23) we conclude that
to spin-lattice
2)&f7
b2(09%-4~7,2
bZ(L
s2 I93-4
“PI
1 “W-d-..
1 ‘D)
1 “P)
,
,
(11)
l)~(“~7,zl”s)~(“~~l~P).
For the case of Gd”* the values of b,(O, 2) and For the f-electrons K,, K, = 2,4,6 and we should note that in the contributions to the K:!’ coefficients the products of the coefficients (dcfined in ref. [5] and calculated in the point charge model of the crystal field) can participate:
b,( 1, 1) [ 141 with the values of R + +, R + _ and R _ _ for the optimized Hartrec-Fock-Slater
orbitals [ 151 are equal to b,(O, 2) = -2.240
6,(1,1)
also with I(, and/or K, # 2. However, most importantly the contribution should arise from the second-order coefficients (K, = K, =2) because their values are the highest with respect to the ratio r,,lR’ (r. is the electronic radius and R’ is the metal-ligand distance). Therefore in our calculation we include only these products in expression (7) for which K, = K, = 2. The ground state I%&,,) for an ion with the 4f’ electronic configuration will be next approximated by three LS states contributing dominantly to the formation of 1*Y7,?):
Among the excited states we shall include only the “PJ multiplet (with J =3/2, 5/2 and 7/2) which is lowest and we shall approximate it using the same LS states as for the ground state:
= -1.958x
x
lOWI7cm’ ,
(12)
lo+cmZ.
The needed values of the cncrgy differ-e WI:, f, for the “9, muftrptete and &j’:,lI “i) and &P, 17) coefficients arc listed in ref. [ 141.
Thus, using relation (3.23) of ref. [S] appropriate numerical data we can obtain following vab &es for the second-order K’,‘,’ K 7’ coefficients per unit square dynamical placement:
and the and dis-
K(.?/$ r.!&
(13)
= 4 .32 cm-’ A” (Z'lR'")
,
ectriccharge of 4 ation amplitude ~CI ion at optic frequency 2 X 10” s10 -’ A2 and the metal-hgand distance> fluorides are not less then 2.3 A. from relation (13) and (13) tions of the second-order met
240
W. Pastusiak / On phonon-induced contributions to spin-lattice coefficients
second-order coefficients within the approximations carried out here do not exceed 10 -4 cm -I whereas the values determined experimentally are about 10 -2 cm -~ [3]. The contributions from the first-order mechanism are equal to
K~2~/u2= _35.54 cm_ 1 ~3 Ktr2' / u 2g
(Z2]R,5)
2= +35.54cm-"/~3 (Z2/R,5)
'
"
(15)
(16)
and, taking into account their dependence on the metal-ligand distance, they are more than about two orders in magnitude greater than those resulting from the second-order mechanism and have the correct signs.
References [1] R. Calvo, R.A. lsaacson and Z. Sroubck, Phys. Rev. 177 (1969) 484. [2] S.B. Oseroff and R. Calvo, Phys. Rev. B 5 (1972) 2474. [3] S.B. Oseroff, Phys. Lett. 41 A (1972) 387. [4] H.A. Buckmaster, R. Chatterjee and Y.H. Shing, Phys. Stat. Sol. A 13 (1972) 9. [5] W. Pastusiak, Physica A 154 (1989) 307. [6] W. Pastusiak, Acta Phys. Pol. A 73 (1988). [7] W. Pastusiak, Phys. Star. Sol. B 126 (1984) K77. [8] J.H. Van Vleck, J. Chem. Phys. 7 (1939) 72. [9] R. Orbach and H.J. Stapleton, in: Electron Paramagnetic Resonance, ed. S. Geschwind (Plenum Press, New York, 1972). [I0] J.M. Baker, J. Phys. C 12 (1979) 4039. [11] Chao-Yuan Huang, Phys. Rev. 154 (1967) 215. [12] Chao-Yuan Huang, Phys. Rev. 158 (1967) 280. [13] Chao-Yuan Huang, Phys. Rev. 159 (1967) 683. [14] B.G. Wybourne, Phys. Rev. 148 (1966) 317. [15] A. Rosen and J.T. Waber, Int. J. Quantum Chem. 8 (1974) 127.
The estimation of phonon-induced order spin-lattice coefficients for structure*
contributions to the secondd3+ in crystals with
W. Pastusiak instit~rte
of Pkysics,
Technical
University
of Stczecin,
Poland
Received 26 July 1991 Final re+ed 2 November 1992
Phonon-induced contributions arising from the one-phonon second-order mechanism to the second-order coupling coefficients K:” (I’= E,, T,,) for Gd” in crystals with the fluorite structure arc estimated. It was found that the contributions resulting from the second-order mechanism do not exceed 10 ’ cm- ’ and are much smaller than the K:!’ coefticients dctermincd experimentally as well as the contributions resulting from the first-order mechanism.
Section 1 The application of a static external perturbation (hydrostatic or uniaxial stress, electric field etc.) to a crystal sample doped with magnetic ions Icads to changes in ligand positions with respect to equilibrium ones. At the same time the displacements of ligands are also caused by heat lattice vibrations. Estimating the amplitude of vibrations for an optic mode with a frequency of about lo’3 s-’ for CaF, we came to the conclusion, that the amplitude introduced by a static perturbation is at least two orders of magnitude smaller than the amplitude and wave!ength of dynamic vibrations. Thzse two perturbations. static and dynamic ones, result in a change of the energetical spec-
Cm-espondence
nical University
W. Pastusiak, Institute of Physics. Tcchof Szczecin. Aleja Piastciw 17, 711.710
to:
Szczecin, Poland. * This work was partially E”30. 0921 4SXi/93/$(16.00
supported
mdcr
@ 1993 - Elsevicr
project
Science
No. 1081
Publishers
trum of a dopdnt. This change, following Calve and Oseroff [l-3) can be described by a phenomenological Mamiltonian t’or the spin-lattice coupling:
ere
(2) and c $‘( T) are the Kth order spin-lattice coefficients at temperature T and G$.‘(RL) corresponds to the value of C!?for a ‘rigid lattice’. K-k;.’ is a parameter related to phonon-induced onntr;hmltinnc tt_r _rFji. (T) parameter. I$.: is k”“..~.,“U..V..V ._ the ____ {-!” combination of tensor a symmetrized linear UCh~StCP operators normalized according to et al. [4] and GY!.~ is a symmetrized component of the strain tensor. r is an irreducible representation of a cubic group and y numbers the basis functions of this r; presentation, p distinguishes repeating r representations.
B.V. Al\ rights racrvcd
W. Pastusiak / On phonon.induced contributions to spin-lattice coefficients
238
Section 2
In our previous two papers [5,6] we have presented a calculation method allowing the determination of the resulting phonon-induced contributions to the t?{K) coefficients. The idea of these calculations is based on the classification of the parameters describing dynamic interactions according to irreducible representations for normal (or symmetrized) vibrations of a lowsymmetry complex (a paramagnetic ion plus its ligands) and then on the decomposition of each of these parameters into the components corresponding to a coupling of an ion with ligand vibrations in the absence of static deformation, in the presence of a linear static deformation, quadratic static deformation etc. It seems that due to only slight static deformations one can, to a good approximation, use as a normal coordinates of a low-symmetry complex, symmetryadapted linear combinations of normal coordinates for a high-symmetry (undeformed) complex. The decomposition of parameters describing dynamic interaction allows the classification of particular terms in the Hamiitonian according to the powers of static deformations and the construction, in a harmonic approximation, of the first-order two-phonon mechanism and secondorder mechanisms which, like Hamiltonian (1) are linear in a static deformation [7]. In the first-order mechanism the interaction Hamiltonian is linear in static deformation and quadratic in dynamic deformation. In the considered second-order mechanism the Hamiitonian linear in the dynamic deformation and the Hamiltonian linear in both dynamic and static deformations are present. In the model in which the crystal field is m n d 1 1~,~ 1~ t~,rl ..... a t t t'~.. u
lal~ It~.ff
lI ~titttt ~ ,,~ t 1~.,%,
~V ,i ;UhI ~ -¢~l. . tl ;k ,l U-I, . ,l -; D
[Q1
[ ¢L)J ~
,I.k[ .l l.~ ,
.(:;.~. 111 31.-
mechanism leads to two-phonon proccsscs whereas the second-order mechanism leads to one-phonon processes (or two-phonon lie ones o r d e r
[91). In paper [7], based on crude estimations we suggested that the phonon-induced contributions to the second-order coefficients /(~!:~ is predominantly caused rathcr by the two-phonon
first-order mechanism and not by second-order mechanisms suggested by other authors (e.g. refs. I2,101). It is the aim of this letter to report the estimation of the contribution from the onephonon second-order mechanism to the secondorder coefficients K]? ) (F = Eg, T2g ) for Gd 3+ in crystals with the fluorite structure using the formalism presented previously [5].
Section 3
The temperature-dependent factor in relation (3.23) of ref. [5] is h
B~'a^(T) = ~ bn'aa.J 2Mtoj cth(htoj/2kT).
(3)
J
We would like to stress here that the 6'AA indices in the quantities B~.a~(T ) and b~.a^.j do not determine their transformation properties but they number them only (unfortunately, in paper [5] we omitted the index A in the quantity b,~'~A.;). Analyzing further the phonon part let us note two experimental facts: (i) the experimentally observed temperature dependence of the second-order spin-lattice coefficients may be described by relation (1); (ii) the spin-lattice coupling is very well described in the Einstein model with a frequency o) = 2 x 1013s -! [31. Correspondingly to the above statement and the argumentation given by Huang [11-13] this phonon part can be written as B,s.,~(T) = (h/6txw) cth(h~o/2kT) ,
(4)
independently of 6'AA indices. Inserting c q (4) into (3.23) of ref. [51 we can easily obtain the ana!vtical form for the secondorder cocffidcn~.s b~l?3 or K~t".~ (for the relations bztween the coefficients with and without bar see appendix A of ref. [5]). We can then note that t!~e obtaiped expression will be proportional to thc factor
ii,,6tzco ~ ,~
(5)
W. Pusrusiak
I On phonon-induced
which has the dimension of the square of amplitude. Because in the presented model we are interested in the displacements of ligands from their equilibrium positions we can treat p as a mass of a ligand.
Section 4
contributions
coefcicipmrs
23sr
The matrix elements required tions [S] have the form (‘&,,I
c b,(k’, k)W”“‘*““/%$) - k’.k
in the calcula-
.
(10)
Among these matrix elements only three will be essential: b,(O,
Let us now analyze the ‘electronic’ part in eq. (3.23) of ref. [S]. From the bj-symbol appearing in eq. (3.23) we conclude that
to spin-lattice
2)&f7
b2(09%-4~7,2
bZ(L
s2 I93-4
“PI
1 “W-d-..
1 ‘D)
1 “P)
,
,
(11)
l)~(“~7,zl”s)~(“~~l~P).
For the case of Gd”* the values of b,(O, 2) and For the f-electrons K,, K, = 2,4,6 and we should note that in the contributions to the K:!’ coefficients the products of the coefficients (dcfined in ref. [5] and calculated in the point charge model of the crystal field) can participate:
b,( 1, 1) [ 141 with the values of R + +, R + _ and R _ _ for the optimized Hartrec-Fock-Slater
orbitals [ 151 are equal to b,(O, 2) = -2.240
6,(1,1)
also with I(, and/or K, # 2. However, most importantly the contribution should arise from the second-order coefficients (K, = K, =2) because their values are the highest with respect to the ratio r,,lR’ (r. is the electronic radius and R’ is the metal-ligand distance). Therefore in our calculation we include only these products in expression (7) for which K, = K, = 2. The ground state I%&,,) for an ion with the 4f’ electronic configuration will be next approximated by three LS states contributing dominantly to the formation of 1*Y7,?):
Among the excited states we shall include only the “PJ multiplet (with J =3/2, 5/2 and 7/2) which is lowest and we shall approximate it using the same LS states as for the ground state:
= -1.958x
x
lOWI7cm’ ,
(12)
lo+cmZ.
The needed values of the cncrgy differ-e WI:, f, for the “9, muftrptete and &j’:,lI “i) and &P, 17) coefficients arc listed in ref. [ 141.
Thus, using relation (3.23) of ref. [S] appropriate numerical data we can obtain following vab &es for the second-order K’,‘,’ K 7’ coefficients per unit square dynamical placement:
and the and dis-
K(.?/$ r.!&
(13)
= 4 .32 cm-’ A” (Z'lR'")
,
ectriccharge of 4 ation amplitude ~CI ion at optic frequency 2 X 10” s10 -’ A2 and the metal-hgand distance> fluorides are not less then 2.3 A. from relation (13) and (13) tions of the second-order met
240
W. Pastusiak / On phonon-induced contributions to spin-lattice coefficients
second-order coefficients within the approximations carried out here do not exceed 10 -4 cm -I whereas the values determined experimentally are about 10 -2 cm -~ [3]. The contributions from the first-order mechanism are equal to
K~2~/u2= _35.54 cm_ 1 ~3 Ktr2' / u 2g
(Z2]R,5)
2= +35.54cm-"/~3 (Z2/R,5)
'
"
(15)
(16)
and, taking into account their dependence on the metal-ligand distance, they are more than about two orders in magnitude greater than those resulting from the second-order mechanism and have the correct signs.
References [1] R. Calvo, R.A. lsaacson and Z. Sroubck, Phys. Rev. 177 (1969) 484. [2] S.B. Oseroff and R. Calvo, Phys. Rev. B 5 (1972) 2474. [3] S.B. Oseroff, Phys. Lett. 41 A (1972) 387. [4] H.A. Buckmaster, R. Chatterjee and Y.H. Shing, Phys. Stat. Sol. A 13 (1972) 9. [5] W. Pastusiak, Physica A 154 (1989) 307. [6] W. Pastusiak, Acta Phys. Pol. A 73 (1988). [7] W. Pastusiak, Phys. Star. Sol. B 126 (1984) K77. [8] J.H. Van Vleck, J. Chem. Phys. 7 (1939) 72. [9] R. Orbach and H.J. Stapleton, in: Electron Paramagnetic Resonance, ed. S. Geschwind (Plenum Press, New York, 1972). [I0] J.M. Baker, J. Phys. C 12 (1979) 4039. [11] Chao-Yuan Huang, Phys. Rev. 154 (1967) 215. [12] Chao-Yuan Huang, Phys. Rev. 158 (1967) 280. [13] Chao-Yuan Huang, Phys. Rev. 159 (1967) 683. [14] B.G. Wybourne, Phys. Rev. 148 (1966) 317. [15] A. Rosen and J.T. Waber, Int. J. Quantum Chem. 8 (1974) 127.