Calculation of g-tensor of rare-earth ions with account of isotropic reduction of orbital momentum Текст научной статьи по специальности «Физика»

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aänetic Resonance in Solids

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Short cite this: Magn. Reson. Solids 19, 17208 (2017)

Calculation of ^-tensor of rare-earth ions with account of isotropic reduction of orbital momentum

A.S. Dudalov, O.V. Solovyev*

Kazan Federal University, Kremlevskaya 18, Kazan 420008, Russia

*E-mail: oleg.solovyev@mail.ru

(Received December 5, 2017 ; revised December 21, 2017; accepted December 24, 2017; published December 28, 2017)

g-tensor components for a Kramers doublet of an impurity ion doped into dielectric crystal are expressed through coefficients of expansion of the doublet wavefunctions in the basis of full momentum of the impurity ion taking into account isotropic reduction of orbital momentum in Zeeman energy due to covalence. Mixing of terms and multiplets of the impurity ion is considered rigorously in the expansion of doublet wavefunctions. The obtained expressions can be useful in calculations and analysis of g-factors dependence on the impurity ion Hamiltonian parameters. The derived formulas are applied to calculation of g-factors of the ground Kramers doublet of Ce3+ ion doped in LiYF4 crystal. It is shown that considering reduction of orbital momentum of the 4f electron of the Ce3+ ion in Zeeman energy can significantly improve agreement with experimentally measured g-factors for this compound, available in literature.

PACS: 75.10.Dg, 76.30.-v

Keywords: g-factors, orbital momentum reduction, LiYF4:Ce3+ 1. Introduction

This research was stimulated by theoretical investigation of 4f crystal field parameters and g-factors of the ground Kramers doublet of Ce3+ ion doped in LiYF4 crystal which we conducted earlier [1]. Impurity Ce3+ ions substitute for Y3+ ions in LiYF4 crystal in sites with S4 point symmetry, in the nearest surrounding of the Y-site there are eight fluorine ions which form two deformed tetrahedrons. Safe values of crystal field parameters for the Ce3+ ion in LiYF4 crystal are absent in literature due to difficulties in measuring of 4f crystal field energies for this compound. However, g-factors for the ground level of the Ce3+ ion in LiYF4 crystal are known in literature [2]. In [1] we considered the Zeeman energy

Hze =Mb (2S + L)H, (1)

where H is magnetic field, /uB is the Bohr magneton, S and L are dimensionless spin and orbital angular momentum of the Ce3+ ion 4f electron. We found that no reasonable set of 4f crystal field parameters can satisfactorily fit the g-factors values of the Ce3+ ion ground Kramers doublet measured in [2], even if we rigorously consider mixing of the ^5/2 and 2F//2 Ce3+ ion multiplets by crystal field. We suggested in [1] that it is necessary to consider reduction of the orbital momentum of the 4f electron in the Zeeman energy due to covalency effects, to achieve better agreement with experiment. Let us note, as an argument in favor of considering covalency effects, that the effective ionic radius of the Ce3+ ion is the largest for trivalent lanthanide ions, it amounts 1.143 A and is bigger than the ionic radius of the Y3+ ion (1.019 A) [3].

In the present study we consider the following rather general case: an impurity ion with an odd number of electrons in lattice crystal field, which can be of an arbitrary point symmetry, but energy level under consideration is a Kramers doublet. We aim to consider rigorously mixing of impurity ion electronic states by crystal field, therefore we consider expansion of the Kramers doublet wavefunctions , \ y2) in the basis of full momentum taking into account all terms and multiplets of the impurity ion electronic configuration:

\k) = I aJjML\ySLJM), (2)

ySLJM

k = e\k = I (-1)J+M afLM\ySLJM), (3)

ySLJM

where d is the time reversal operator; quantum number y allows to differ terms with equal spin and orbital momentum quantum numbers S and L; J and M are quantum numbers of the full momentum of the ion with the usual meaning; complex coefficients aM satisfy the normalization condition. In many papers only mixing of electronic states within the same multiplet is considered; for example, as for LiYF4:Ce3+ crystal, only mixing of states within the ground 2F5/2 multiplet was considered in [4,5].

We consider the Zeeman energy

H Ze =^mH, (4)

m = gS S + k L, (5)

where k is isotropic reduction factor for the orbital momentum, gS is taken as 2.0023 in calculations. Approximation of isotropic reduction factor was introduced in [6] for Tm2+ ion doped in CaF2 crystal. A more rigorous analysis of covalent binding by molecular orbital method leads to different reduction factors for orbital momentum matrix elements on wavefunctions, transforming according to different irreducible representations of the impurity ion point symmetry. For example, as for Tm2+ ion in CaF2 crystal (cubic point symmetry), two reduction factors have been introduced and calculated in [7,8]. In the case of strong covalent binding (nd transition metal ions) reduction factors can sufficiently differ from 1 and from each other for different matrix elements of orbital momentum. However, for the 4f electrons of impurity rare-earth ions covalent binding is small, with reduction factors exceeding 0.95, therefore, we can neglect difference between their values and consider isotropic reduction of orbital momentum (5) as an approximation in estimation of its matrix elements.

In the basis of the Kramers doublet states (2), (3) the Zeeman energy can be represented by the effective spin Hamiltonian

HZf =^b I HagaSf (6)

a,(

where a and (3 denote Cartesian axes; Seff is the effective spin operator with S = \J2 ; ga/} are

components of the g-tensor, which are determined by matrix elements of Cartesian components of the m operator (5) on the Kramers doublet states as follows:

g ax = 2Re( k\mM), gay =-2Im (k1 \mM), g az = 2( k | ma |k ) . (7)

As follows from the above formulas, the set of coefficients ayM determines Kramers doublet wavefunctions (2), (3) and, consequently, values of g-tensor components (7). We find it useful to express g-tensor components explicitly through the arM coefficients both for calculative reasons and as a tool for analysis of g-tensor dependence on the impurity ion Hamiltonian parameters.

Thus, the purpose of the present study is to establish formulas that would express g-tensor components for a Kramers doublet with account of orbital momentum reduction through the aM coefficients; as an example of application the derived formulas are used in calculation of g-factors of the ground Kramers doublet of the Ce3+ ion in LiYF4 crystal.

The paper is organized as follows. In section 2 we derive formulas for gap expressed through the ayM coefficients that determine the wavefunctions (2), (3) of the Kramers doublet under

consideration. In section 3 a particular case of tetragonal local symmetry for the impurity ion is considered. In section 4 a particular case of electronic configuration nll of the impurity ion is considered. Finally, in section 5 we calculate g-factors for a specific system - the Ce3+ ion in LiYF4 crystal - and compare results with the known experimental values [2].

2. Formulas for g-tensor components expressed through ayM coefficients

For the sake of generality let us consider matrix elements of Cartesian components of the m operator (5) between the state \ yx) (2) and an arbitrary state )) defined by the following expansion with a set

of coefficients byM :

|)=X bM\rSLJM). (8)

ySLJM

We find it convenient to carry out such transformations that allow to calculate the least possible number of different matrix elements in the right part of the following equation (note that states of the same term ySL appear in matrix elements according to selection rules):

(H ma\)= X aMM'b]Su\ySLJM\ma\ySLJM'). (9)

ySLJMJ' M'

For example, it is sufficient to consider only the case J > J' in (9), as simultaneous substitutions J — J', M — -M' give the term under the sign of summation in (9)

(-1)J'J+M+MJmJm (ySLM\ma\ySLJM'). (10)

To obtain (10) one should use the equation

( ySLJM | m\SLJ 'M') = (-l)J+J'-M -M'+1 (ySLJ', -M '| m\SLJ, -M), (11)

which is actually valid for any time-odd Hermitian operator. Utilizing (10) we obtain for the z component of operator m

(12)

Y k ))_ I [aÏÏ'bÏÏirSLJM^ ySLJM) +

ySLJM

+ (<% + aJJ-\- m Jm )(ySLJM\mz\ ySL, J -1, M> ]. Property (11) allows to transform the following sum for the x and y components of operator m I CjJMm1 ( ySLJM | mx, y | ySLJ M ') = I (CJ^ {ySLJM\mXy\ySLJ ',M-1) +

{J,J '}eD, M ,M ' {J, J '}eD,M

{ , } , , { , } , (13)

+ (-1)JjJ' CJ-M,1-M {ySLIM\mJySLJ',M -1)'),

where CrJJ^MM: are some numerical coefficients, quantum numbers J, J' take values from some fixed set D. Let us apply (13) to calculation of matrix elements ( y1 | mx y )) : for J = J ' in (9) we apply (13) with

CySL = a ySL*bySL • (14)

JJMM' "JM UJM'1

for J > J ' in (9) we apply (13) with

CySL _ aySL*bySL +(-1)J'-J+M+M' aySL % ySL (15)

JJ'MM' _ UJM UJ'M' 1J UJ M ' UJ,-M> V1J/

according to (10). Thus we obtain, omitting quantum numbers ySL in the matrix elements for brevity:

Calculation of g-tensor of rare-earth ions with account of isotropic reduction of orbital momentum

k ky )) = I [ayL J-1(JM|mjJ,M -1) + ay% J-M (JM\mjJ, M -1)'

ySLJM

(ar(M-1 " JU-M jm ))k y\j -1, M -1) +

■ ((,M-i*bJSM - JMJ1-M ))JMI mx,„ J -1,M -1)' '

Note that formulas (12) and (16) are valid for Cartesian components of any time-odd Hermitian vector operator with matrix elements diagonal in quantum numbers ySL . For time-even Hermitian vector operator analogous formulas can be derived, but terms, which include coefficients with indices -M or 1 - M (or both), will appear with the opposite signs.

Calculating matrix elements diagonal in J in (12) and (16) we can substitute ma by equivalent operator

gL (SLJ ) J a , (17)

where gL (SLJ) is the Lande g-factor modified with account of orbital momentum reduction:

v 1, v , vS (S +1)-L(L +1)

« L (SLJ ) =^ (S + k )+(SS - k ) ( 2J)J +((L ) -

(18)

Calculating matrix elements in (12), (16) between multiplets with different J, we can substitute ma by equivalent operator (gS - k)Sa. On the basis of the standard formula of algebra of irreducible tensor operators [9] (with Wigner notation for 3j- and 6j-symbols)

L - M f J 1 J -

(ySLJM I S{11 ySL, J -1,M ') = (-1)2

-Mq M

j J 1 J -1 |SL S

^(2J +1)(2J -1) •

(19)

■<S|| S(1) ||S>

we can write

yySLJM \S21 ySL, J -1, M) = 2 A (SLJ )j2 - M2, (ySLJM IS+ I ySL, J -1, M -1) = -11 (SLJ )(J + M - 1)(J + M ),

1(SLJ ) =

(L + S + J + 1))S - L + J))L - S + J))L + S - J +1)

J2 (2 J +1)2 J -1)

(20) (21) (22)

Thus we can write explicit expressions for matrix elements of ma operators with the use of coefficients gL (SLJ) (18) and à(SLJ) (22):

m. )) = I [ gL (SLJ)Ma%J + ^ ¿(SLJ)JW J*b?XM + JM )], (23)

ySLJM

(k |mx W)

i my W)

= X Ï2 gL (SLJ ) J (J +1)-M (M -1) (a(L-1 ± JLM JM )-1 - k )•

ySLJM L2 4 (24)

■Ä(SLJ )(J+M - 1)(J+M ) (aZ*bTXM_x - a^M *b]% ± a^b^ + afLM )].

Here and further in this section upper signs, if any, are taken for x-axis, lower signs - for j-axis.

Now let us consider matrix elements of ma operators on the Kramers doublet states (2), (3). Expressions for diagonal matrix elements can be obtained from (23), (24) by simply putting bJ) = aJ):

Wlmz|w) = £ k{SLJ)M\ajMf + ( -k)A(SLJ))J2 -M2 Re(a^J))],

ySLJM

(25)

(w \mx |w)

(k \my \Wi)

= Z

ySLJM

gL (SLJ) J (J +1)-M (M -1) Re (aJfaJ,)

Im

(26)

1(gs - k)A(SLJ))(J+M - 1)(J+M) R (aJML*aJ-^,M-i - a^

Note, that (w2|ma | w2)= -Wi|ma |w,) for the time-odd operator. Expressions for the non-diagonal matrix elements on the states |wi) (2) and | w2) (3) of the Kramers doublet can be obtained from (23), (24) by putting by) = (-1)J+M afL_M':

(Wi\mz IW2) = Z

ySLJ

2 Z(-1)J+Mg L (SLJ )May))LayISL

-Z(-1)J+M (gS -k)^(SLJ)JMa!MLay/X_m ],

(27)

(k K IW2)*

-i (W1 |my |W2>*

SLJ

(-1)J gL (SLJ )^(-1)2,/ J (J +1) +1 / 4 ( + aJ%22) + +2 Z (-1)MVJ(J +1)-M(M-1)(a(, + a^J-1 )) + (28)

m>1/2 7

+ Z(-1)J+M (gS - kMSLJ)(J+M - 1)(J + M) ± ay*^,))

M

Of course, (w2|m|wj = (W1 |m|W2) . Let us also note for completeness that matrix elements for my in (24), (26), (28) can be obtained directly from matrix elements of the mx operator by rotating corresponding wavefunctions by the angle of -n/2 around z-axis; in the basis | ySLJM) this is equivalent to transformation of the wavefunctions expansion coefficients aJ) ^ ayJM exp (iM n/2), byy)L ^ byML exp (iM n/ 2).

Calculated matrix elements of the ma operators (25)-(28) on the Kramers doublet states (2), (3) allow to find components gap of the g-tensor according to (7). Rotating Cartesian axes and making

unitary transformation in the two-dimensional space of the Kramers doublet states we can always bring the g-tensor to diagonal form (we refer to diagonal components as g-factors). In the following section we show that for a particular case of tetragonal symmetry of the impurity ion Hamiltonian.

3. Formulas for g-factors in the case of tetragonal crystal field

The derived formulas (25)-(28) for g-tensor components may be greatly simplified in the case of tetragonal symmetry of the lattice crystal field. If z-axis is chosen as the tetragonal symmetry axis, and if expansions (2), (3) of the Kramers doublet wavefunctions k), |w2) are obtained by diagonalization of Hamiltonian, which includes spin-orbit and Coulomb interactions and interaction

M

with crystal field, in the basis of full momentum \ySLJM}, then coefficients ajM in (2), (3) can be nonzero only for M values that differ by each other by an amount of 4p, here and further p is integer. Therefore, as seen from (26), (27), k | mz 11//2) = 0, k | mx y | k) = 0 and, consequently, according to (7):

gxz gyz gzx gzy .

(29)

Note, that (29) will be valid for z-axis being the symmetry axis of the second order already.

Moreover, by appropriate numbering of k), k) states we can always obtain nonzero coefficients ayM in (2) only for M = 1/2 + 2p0 + 4p , where p0 equals either 0 or 1 (note that expansion (3) of k) will contain terms only with M = -1/2 + 2p0 +4p in this case). Therefore, in (28) the second terms in all three parentheses will vanish, leading to equality k \mx |k2) = ik \my k). The latter means that by appropriate choice of complex phase of the

wavefunction k) the g-tensor will be directly brought to diagonal form with components gxx = gyy = g±, gzz = g||, where g-factors derived from (25), (28) equal

ySLJp

-.ySL

AJ ,1/2+ 2 p0 + 4 p

g|| = 2 I g l (SL/)(1/2 + 2 p0 + 4p)\a

-(gs -k)4(SLJ)\J2-(1/2 + 2po + 4p)2 Re(aJSL:

,1/2+ 2p0 + 4p aj-1,1/2+2p0 + 4p

ySL

(30)

I

ySLJ

ySL 2 4J ,1/2

gL (SLJ(J +1) +1/4 aJ

+2 I p(J +1) +1/4 -(2p0 + 4p)2

p>-p0/2

ySL aySL

4J ,1/2+2 p0 +4 puJ ,1/2-2 p0 -4 p

■I(gs -k)(SLJ))(J + 2p0 + 4p)2 -1/4 a',,2. , 4pa

ySL aySL

J,1/2+ 2p0 + 4p J-1,1/2-2pa -4p

(31)

4. g-tensor components in the case of impurity ion electronic configuration nl1

In the case of electronic configuration nl1 of the impurity ion there is only one term L = l, S = 1/2

with two multiplets J = l ± 2. Wavefunctions |J,M) of the multiplets components can be expressed

through the one-electron wavefunctions \ml,ст), where ml and a denote magnetic and spin quantum

numbers of the nl electron (below a values are denoted simply as '+' and '-'), with the use of tabulated Clebsch-Gordan coefficients:

l ± 2,M)=1 ^2,l a ml

2, l, l ± 2, M^j\m, a =

Vl ± M +1/2

2l +1

M -1, +\±-JwM^~2

2 2l +1

(32)

M+2,- v

Coefficients gL (SLJ) (18) and A (SLJ) (22) for configuration nl1 take the form (here and further in this section uninformative indices SL are omitted)

l+11=gkl2ki, gL ( l -1V gS +2k (l+1), 41+11=Oiizkl

2 ) 2l +1 LI 2 1 2l +1 I 2 ) 2l +1

(33)

p

Let us consider the example of an nf electron (l = 3). Let us provide only formulas for the matrix elements |mz(25) and \mx(28) (in the case of tetragonal symmetry these matrix elements directly define the g-factors as has been shown in the previous section):

(k K ki) = _ X

_ M

(gs + 6k )M

-(gs - 8k)M

(gS - k))49 - 4M2 Re

(34)

A"

a7 a5

_,M -,M

V 2 2 7

(kii m ik)*=-i4

2 _ 2 ai 1 -a7_ _1 V 2'2 2' 2 7

M >1/2

4 (gs +6k)

EHf"2 (gs + 6k)63 - 4M (M -1)

V

+ (gs - 8k)35 - 4M(M -1) X(-1)M-2 (gs -k))4(3 + M)2 -1

a ^

3 (gs -8k)

2 2 aH -ai-1

V 2'2 2' 2 7

f

a _ a_ a_ a_

_'M _'1-M _'-M _'M-1 2 2 7

V 2 2 (

a 5 a5 - a5 a5

5'M 5'1-M 5'-M 5'M-1

V 2 2 2

/

A"

(35)

a_ a5 + a _ a5

_'M 5'1-M _'-M 5'M-1

V 2 2 2 2 7

5. g-factors of the ground Kramers doublet of the Ce3+ ion in LiYF4 crystal

Spectrum of the Ce3+ ion 4f1 configuration consists of _ Kramers doublets - ^5/2 (ground) and 2Fm multiplets splitted by S4 symmetry crystal field induced by LiYF4 lattice. As has already been mentioned in introduction, g-factors for the ground level of the Ce3+ ion in LiYF4 crystal have been found in [2] from EPR measurements:

gexp = 2.737' gf = 1.4_5.

(36)

We consider an effective Hamiltonian H of the impurity Ce3+ ion consisting of spin-orbit interaction Hamiltonian

HSO =#SL, (37)

where ^ is a 4f electron spin-orbit coupling constant, and Hamiltonian of interaction with the S4 symmetry crystal field (in crystallographic axes)

Hcf = B2C02) + B4C04) + B4C44) + B-44C-4) + Bo6C06) + bC + B^C-?,

(38)

where Bp are crystal field parameters, satisfying the equation Bp =(-l) Bpk; C(kp) are components

of one-electron spherical tensor operators C(p).

Looking ahead, we may state that it follows from calculations that the largest contribution to

3+ 5 5

wavefunctions of the ground 4f Kramers doublet of the Ce ion in LiYF4 comes from the —, ±—

states (see also expansion (32)). Let us also note that the ground doublet corresponds to irreducible representations T5, T6 of the double 54 point group in Bethe notation. According to section 3 we

choose the wavefunction (2) to contain the state (this corresponds to p0 = l in section 3;

M

Calculation of g-tensor of rare-earth ions with account of isotropic reduction of orbital momentum wavefunction k) (3) then contains the

5 5 '

—, - — ) state), which is mixed by crystal field interaction

HCF (38) only with the states

7,—) and

2 2

7 3^

-, - — ). Thus, it is sufficient to numerically

5 - 3 2, 2/

diagonalize a matrix 4 x 4 of the Ce3+ ion Hamiltonian H = HSO + HCF with the following matrix elements, which we give here for completeness:

— —

2,2

H

— —

2,2

= -24— B 21

J_ 21

B0

— -3 ?-2

H

--) = -24+—B0 -1 B04, 2 2 3— 0 7 0

7 —

2,2

H

2,2=4 _L B0 - 77 B4 + -2—L B6

2 2 2 21 0 77 0 42L 0

7 - 3 ? - 2

H

2, -2=-4+7 B02 - 3. b4 - .n „¡.

2 2 2 7 0 77 0 143 0

— —

2,2

H

— - 3 l, - 2

V14 21

B44

— — ï,ï

H

7 — ï,ï

46 „2 10V6 „4

-B02--21 0 231

B04

— — ï,î

H

7 - ^ ï - 2

473— .

231

B

B 3

143 ^ \2, 2

H

7 —

2,ï

WÎT

231

B4

-B0,

42L 0

1^/10— 42L

B6

— - 3 2 - 2

H

7 - 3 2 - 2

VÎÔ „2 8^10 . ^yjîô

3—

B -

BU +—-—B06, /—,— 231 0 143 0 \2 2

H

7 - 3 2 - 2

V2T0

77

I2 B46. (3L)

Numerically diagonalizing the matrix with the elements (39) we obtain the wavefunction k) (2) of the ground Kramers doublet of the Ce3+ ion in LiYF4; bringing k) to the form, suitable for comparison with [1], we may write

I \ 1 W / = — 1 ' N

— — ï,ï

a

— - 3 2 - 2

P

7 —

2,2

7

7 - 3V ^

l, - 2

4

N= ,1+a

P+172.

(40)

Utilizing the formulas (34), (35) for wavefunction (40) we obtain the following expressions for g-factors of the ground Kramers doublet of the Ce3+ ion in LiYF4 with account of orbital momentum reduction:

g|l = 7N2((gS +6k)(5(( -3|y|2) + (-gS +8k)(5 -3\af ) + 4(gs -k)((6Re( + Vi0Re(ay'), (41)

7 N2

(gS + 6k )2j3fî7 + (gS - 8k )4—a + (gS - k )(430ap + 42^

(42)

If we put gS = 2, k = 1, then (41), (42) transform to the formulas that we obtained earlier in [1] but

with the opposite signs of P and 7 coefficients, since

-2,M ) states have been taken in [1] with the

opposite signs as compared to (32); this also results in the opposite signs of matrix elements cross — 7

between J = 2 and J = 2 states in (3L) as compared to [1]. If we additionally put P = 7 = 0, then we

obtain the case, investigated earlier in [4,—], when mixing of the 2Fy2 and 2F7/2 multiplets by crystal field is not considered.

We use the following crystal field parameters that were exploited in [10] in modeling of the interconfigurational 4f-—d absorption spectrum of the Ce3+ ion in LiYF4 (in cm-1):

B02 = 360, B04 = -1400,

B44 =-1240 + i •7—1, B06 = -67.2, B46 =-10L— + i -4—8.

(43)

No direct fitting of g-factors or 4f crystal field energies has been fulfilled in [10], but signs and orders of magnitude of parameters (43) are in agreement with parameters obtained in literature for trivalent rare earth ions doped in LiYF4 crystal from fitting experimental data (see, for example, [11]). We also use value £ = 625 cm-1 from [10] for the spin-orbit coupling constant, this value also being in agreement with the common in literature (for example, 615 cm-1 in [11] and 628 cm-1 in [12]). Diagonalization of matrix (39) with the Ce3+ ion Hamiltonian parameters given above leads to the following g-factors of the ground Kramers doublet of the Ce3+ ion, calculated without taking orbital momentum reduction into account [1] (i.e. for k = 1 in (41), (42)):

g„= 2.845, g±= 1.551, (44)

with a root-mean-square deviation from experimental values (36) equal to 0.132. Calculations show [1] that large B+4 values ( B+4|« 2000 cm-1) could, in principle, solve the problem of this

discrepancy with experiment, but such large values are at least one and a half times greater than expected, as follows from trends of parameters commonly used in literature for trivalent rare earth ions doped in double fluoride crystals [11, 13], and do not seem reasonable.

In order to improve agreement with experiment we consider reduction of the orbital momentum of the 4f electron of the Ce3+ ion. For k = 0.97 the obtained formulas (41), (42) lead to the following g-factors of the ground Kramers doublet of the Ce3+ ion in LiYF4:

gll = 2.734, g±= 1.487, (45)

with a root-mean-square deviation from experimental values (36) equal to 0.013, that is 10 times less than without taking orbital momentum reduction into account (44).

Thus, we have shown that considering reduction of the orbital momentum of the Ce3+ ion 4f electron in the Zeeman energy could improve agreement with g-factors obtained from experiment. However, the result (45) should not be considered as final yet. First of all, one must keep in mind that a more thorough determination of crystal field parameters in LiYF4:Ce3+ remains a problem. Correct 4f energy levels for this compound should be provided. Recently infrared absorption spectrum of the LiYF4:Ce3+ crystal corresponding to 2F5/2 ^ 2F7/2 intra-4f transitions have been measured at 10 K [14], providing energies of the excited 4f (2F7/2) levels. Another attempt [15] was made to determine several of the Ce3+ excited 4f crystal field levels from analysis of low-temperature 5d-4f luminescence spectrum in LiYF4:Ce3+ crystal: microscopic modeling of vibrational structure of the luminescence spectrum based on the exchange charge model [16] allowed to distinguish electron-vibrational peaks observed in the spectrum from zero-phonon lines corresponding to transitions from the lowest 5d state to different 4f levels of the Ce3+ ion. The total of five known for the present moment excited 4f energy levels need to be reproduced by modeling of crystal field in LiYF4:Ce3+, consistent with modeling of g-factors.

6. Conclusions

Analytical expressions have been derived for g-tensor components of a Kramers doublet of an impurity ion doped into dielectric crystal with account of orbital momentum reduction due to covalence: g-tensor components are expressed explicitly through coefficients of expansion of the doublet wavefunctions in the basis of full momentum of the impurity ion. The obtained expressions can be useful in calculations and analysis of g-tensor dependence on the impurity ion Hamiltonian parameters. Additionally, a particular case of tetragonal symmetry of crystal field and a case of the electronic configuration nl1 of the impurity ion have been considered. For completeness, expressions are provided for matrix elements of the impurity ion magnetic moment on arbitrary wavefunctions defined by expansions in the basis of full momentum with account of orbital momentum reduction.

The derived formulas have been used in calculation of g-factors of the ground Kramers doublet of the Ce3+ ion doped in LiYF4 crystal. It was shown that taking into account reduction of the orbital

momentum of the Ce3+ ion 4f electron in the Zeeman energy can significantly improve agreement with g-factors measured experimentally. Further progress can be achieved by going beyond the approximation of isotropic reduction factor, estimation of reduction of orbital momentum matrix elements by molecular orbital method, and, on the other hand, by consistent modeling of 4f crystal field energies. Safe values of 4f crystal field parameters for the Ce3+ ion in LiYF4 crystal are absent in literature, though certain progress have been recently achieved in determining crystal field energies from measurements for this compound. A thorough investigation of possible 4f crystal field parameters in LiYF4:Ce3+ in the context of both g-factor values of the ground Kramers doublet and 4f energy levels, established in literature for the present moment, is planned to be conducted in future.

Acknowledgements

This work was performed according to support of the RFBR Grant 17-02-00403. References

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