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Crystal potential model for the description of crystalline electric field effects in rare earth metals and intermetallics
Journal of Magnetism and Magnetic North-Holland, Amsterdam
Materials
61 (1986) 337-346
337
CRYSTAL POTENTIAL MODEL FOR THE DESCRIPTION OF CRYSTALLINE FIELD EFFECTS IN RARE EARTH METALS AND INTERMETALLICS V.G. ORLOV
+
The Niels Bohr Institute, Received
ELECTRIC
20 January
University
of Copenhagen, DK-2100 Copenhagen 0, Denmark
1986; in revised form 18 April 1986
A crystal potential model is suggested. It allows us to interpret crystalline electric field effects in optical, EPR, NMR, NGR and neutron spectroscopy measurements in rare earth metals and intermetallics. The crystal potential character and space distribution are discussed. The model is used for the theoretical interpretation of the effects of the crystalline field in the compound PrAl s
1. Introduction In most cases the presently existing crystalline electric field (CEF) models, i.e. the model of effective point charges (MEPC) [l] and the covalent model based on the method of molecular orbitals (see, for example, ref. [2]), do not allow us to determine unequivocally quantitative characteristics specifying the physical nature of the CEF effects in metals. Moreover, the use of the MEPC results in unphysical values of ion valences in metals (see ref. [3]). In the CEF theoretical description the most severe difficulties arise due to the presence of conduction electrons in metals. The problem of taking account of the conduction electrons’ contribution to the CEF has become aggravated in connection with the detection of large deviations of experimentally obtained coefficients of the CEF from those calculated in the MEPC for heavy rare earth metals (REM) and dilute solutions of REM in d-metals. Several equivalent methods for taking into account the conduction electrons’ contribution to the metal CEF have been suggested. Dixon [4] proposed to take account of valence and inner 5d- and 4f-electrons’ mixing in the OPW method. + Permanent address: I.V. Kurchatov Energy, Moscow 123182, USSR.
Institute
of Atomic
Eagles [5] suggested that the energy shift of the 4f-electrons and the sign variation of the CEF coefficients of the 4th order can be accounted for by the Coulomb and exchange interaction of the 4f- and Sd-electrons contributing to the metal CEF. Detailed atomic calculations [6,7] made it passible to determine the perturbation of outer (5% 5p) and inner shells of a RE-ion induced by the external CEF. The perturbed charge distribution of the ion shells creates within the ion an additional electrostatic field which “shields” the external CEF. As a result, so-called “shielding factors” a, (n = 2,4,6) appear in the CEF Hamiltonian. Despite the validity of the results of these studies, they may be used only qualitatively for the interpretation of experimental data on the CEF effects in metals, since calculations were made for a “free” RE-ion arranged in an external field with the CEF symmetry of a metal. In ref. [8] an attempt was made to take into account the shielding effect of conduction electrons on the CEF in RE metal materials. Using the APW method Schmitt [9] calculated the contribution of the conduction electrons to the CEF for a number of cubic RE-intermetallics, which turned out to be comparable with the ionic contribution. Adam et al. [lO,ll] noted that the difference
0304-8853/86/$03.50 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
338
V.G. Orlov / Crystal potential model for the description of CEF effects
between the form of the CEF potential and that of the Coulomb interaction can noticeably change the values of the crystalline field coefficients. In the present paper a model of the crystal potential (CPM) is suggested. It assumes that the RE-ion ground-state splitting is induced by an effective crystal potential (CP) V(r). The parameters of the CP V(r) can be either calculated from first principles or reconstructed by comparing experimental and theoretical data. V(r) has the crystal symmetry. As a result, the CEF Hamiltonian in the CPM yields the correct systematics of the split ground state levels of the RE-ion. A particular arrangement of levels is determined by the Hamiltonian coefficients which are expressed as derivatives of V(r). Thus, it becomes possible to investigate special features of the CP by using spectroscopic measurements. It will be shown below that the CP V(r) oscillates and rapidly decreases with distance. The latter property implies that the CEF effects are highly local. In the present paper the application of the model is illustrated by the interpretation of the crystalline field effects in the compound PrAl,.
the test charge, the potential y can be expanded in a Taylor series in the small parameter r/R,, i.e. the ratio of the f-orbit radius r = (x, y, z) to the distance to the ith ion Ri = (R,,, R,,, R,,): V(‘)=C~(IRi-r+C~(R;)
+c
i
av av -x+-y+,Iz
i ax
ay
ay
Hhex -eV(r)= CPM --
A,,(3z2
Let us formulate the basic aspects of the CPM and indicate its special characteristics as compared with the MEPC. Let the electrostatic CP V(r), with the symmetry of the crystal, act on a trial charge at a distance r from the origin. We assume that the CP V(r) is distributed nonuniformly over the crystal, i.e. it has sharp maxima at ionic sites. Thus, V(r) may be expressed as the sum: (1)
where y is a potential containing the contribution of both ions and conduction electrons (the character and space distribution of the y potential are discussed in the next section). For convenience, we shall treat the potential v as spherically symmetric with respect to Ri the coordinate of the i th ion. If an inner f-electron of the RE-ion is taken as
(2)
+ A,,r*z*
- r’) + A,,z* + A4,z4 + A,,x6
+A6*y6 + A63~6 + A,,x4y2
2. Crystal potential model
+*-.. i
With the aim of employing the CPM for the interpretation of the CEF effects in hexagonal pure REM and intermetallics of the RM, and RM, types (where R is an atom of a REM and M is a nontransition or transition metal) the further description of the CPM will be made for the case of hexagonal crystal symmetry. It is easy to generalize the model for a case of cubic or any other type of symmetry. For the case of hexagonal symmetry, the crystalline field Hamiltonian in the CPM frame has the following form:
+A,,r4
V(r>=CK(lR,-rl), i
I
i
+A66(~2
+ y2)*z2
+ A,,(x*
+ A6,x2y4 + y2)z4. (3)
The 6th order expansion is performed in the Hamiltonian (3) taking account of the fact that matrix elements of HcpM will be taken over the wave functions of f-electrons with an orbital momentum I = 3 [12]. The coefficients A,, are presented in the appendix (A.l)-(A.7). One can easily show with the aid of the formulae (A.6) and (A.7) that, due to the hexagonal symmetry of the Hamiltonian (3) there exist two relations between the coefficients A,, (WI = 1, 2, 4, 5): &i - A,, + i+ ( A64
-
A,,
A,, + A,, - +( A,, + A,,)
> =
0,
= 0.
(4)
(5)
In order to calculate matrix elements of H&& with the aid of the Stevens equivalent-operator method [13,14] it is necessary to rearrange the Hamiltonian (3) so as to obtain tensor operators
V.G. Orlov / Crystal potentialmodel for the description of CEF effects
from the respective powers of the electron coordinates *. Taking into account (4) and (5) it is possible to write the H/?&r as follows:
an be written as follows: ~20=~&42~+4~22~~2 + ,(
+A,(3524
+Al,,(x6
- 315z4r2 + 105z2r4 - 5r6) -
15x4y2 + 15x2y4 -y”),
cnn)(rn),
(7)
where an are the so-called “shielding factors” of the f-electrons by those of all the other ion shells [6,7], while (r”) stands for the average values of the nth power of the f-electron radius [15-171. The formula (7) should be treated only as an estimate due to the uncertainty in the extent to which it is per-n&sable to use the results of atomic calculations, obtained for “free” ions, for ions in a crystal. Thus, the final form of the CEF Hamiltonian for hexagonal symmetry in the CPM is as follows: H:FM
= j2,,02,
+ k:,O,,
+ g6,,06,
+ i6,,06,,
g40
=
&
(6)
where 2, and i,, (n = 2,4, 6), a,, are presented in the appendix (A.8)-(A.12). The first term i0 (A.8) in (6) is a scalar with respect to the electron coordinates. It will be omitted in the following analysis because it can only shift each crystal field level in an equivalent manner. It is easy to show that in the case of the C_oulomb potential v(r) = Z, e/ 1R i - r 1 the term A a in (6) should go to zero and the coefficients Al,,, (n = 2, 4, 6) and a66 go over into the corresponding coefficients A,, and A,, of the MEPC [l]. To obtain the final form of the Hamiltonian in the CPM, let us introduce three coefficients s, (n = 2, 4, 6) by which the Hamiltonian terms will be multiplied. One can make an approximate estimate of the values of s, employing the formula s, = (I-
A42
+
QA,,)S,
++1[A67+5A63-
- 30r2z2 + 3r4)
+&,(231z6
(8)
where O,, (n = 2, 4, 6) and q66 stand for Steven equivalent operators [l] and B,, (n = 2, 4, 6), j6,,
* The author is deeply grateful to Dr. J. Jensen for valuable suggestions on this point.
339
4(A61+A62)1s6}~
{ &443s4
+
+9(A6,
‘60
=
yJ&
k
[ i5A63
-
c9)
7A66
-
4A67
(10)
+A62)1S6},
[ A63
+
A66
-A,,
-
+(A61
+A6211
s6, (11)
‘66
=
YJi(A6,
-
(12)
A62)S6.
The values of the numerical coefficients (Ye, fiJ and yJ are calculated for the ground state of all the three-fold positively-charged RE ions and presented in refs. [1,18]. s, (n = 2, 4, 6) are three constants which can be approximately estimated employing the formula (7) and expressions for A nm are presented in the appendix (A.l)-(A.7).
3. Discussion of the nature and shape of the crystal potential If a test charge Zre ( Zr, is the test charge value, e is the electron charge) is placed at a point r in a metal then, according to the metal pseudopotential model [19,20], in the linear shielding approximation the electrostatic potential acting on the test charge from the metal ions and electrons can be written as follows:
1
V&(
x
J0
O” u,,(q) ((4)~dqNClq)q
Xsin(q]R;-r])dq
2+n*lR,--rl
1,
(14)
where II,(q) is a dipole [21], r(q) = 1 + (1 G( q))V( q)IIo( q) is a static dielectric permeability, V(q) = 4Te*/q*. For G(q) one can use, for
V.G. Orlov / Crystal potential model
340
for the description
example, the Vashista-Singwi (VS) approximation [22] or that of Geldart-Taylor (GT) [23]. U,: (q) is a Fourier component of the metal ion pseudopotential, for example, of the Animalu-Heine type [24]: 4az.e2 uz(q)
=
-
A
(I+ q2
-
-&
~1 cos(qr,)
05)
[ sin(qr,)
1
exp(
-5q4),
expression for the interion potential [19]: Yns( IR,-R,l)=
vii(
IR,-Rjl).
06)
But if we take an inner electron, for example, the RE-ion f-electron, as a test charge, Zre = -e, U,,(q) = 4e2/q2 and V,,(r) (13) and I&,( I R, r I) (14) will represent, respectively, the CP V(r) and the potential y( I R, -r I) from (1) in the linear shielding approximation: VLS(
where Z, is the metal ion valence, r0 and u are parameters of the pseudopotential, { = 0.03/16 K& KrO is the Fermi momentum of the metal at zero temperature and pressure. Up’ is a Fourier component of the potential energy of a test-charge interaction with the conduction electrons of a metal. If we taken one of the metal ions as a test charge, Zre = Z#e, Up” (q) = Uz (q) and I& ( 1Rj - R, I) (14) will reduce to the well-known
of CEF effects
=
I Ri-rl) zie
2
lR,-rl
- TIRi-rl
O”I&&) /0 P(4)
X&T(q) sin(qIR,-rlhh.
(17)
Note that formula (17) takes into account not only the shielding effect of the conduction electrons but also their direct contribution to the CEF of a metal, since the second term in the right-hand part of (17) takes account of the test charge interaction
V
v‘Oli
20
15 0.10
t
V 10
T
V
0.05 2
3
4
5
6
r--A
0
Fig. 1: Interion and crystal potentials for Al. Distances to the nearest ions in the fee structure comparison, the dependence of the Coulomb and crystal potentials on distance is given in the inset.
are denoted
by arrows.
For
KG. Orlov / Crystalpotentialmodelfor the descriptionof CEF effects
with metal ions via the conduction electrons. Unfortunately there do not exist reliable pseudopotentials for the RE-ions at the present time. Therefore, we will perform a qualitative study of the I& behaviour (17) using Al as an example, the pseudopotential parameters of which are wellknown (for example, for c(q) in the GT approximation [23] ‘a = 2.464 au, u = - 1.216 [25]). Fig. 1 shows the coordinate dependence of the inter-ion potential I$ (16) and the potential V& (17) for ions of Al 3+. It follows from fig. 1 that the behaviour of the potentials I$ and I& is very similar, i.e. they decrease sharply with distance, performing simultaneously oscillations which are due to the Friedel singularity [26] of the dielectric function c(q). From this there follow two significant conclusions: 1) The major contribution to the CP V(r) should be yielding by just a few coordination spheres of ions, since at larger distances the potential I& becomes small in its value and smooth in its shape. This results in a strong decrease with distance of derivatives of y( 1Ri - r I) through which the coefficients A,,,, of the CEF Hamiltonian (8) in the CPM are expressed. 2) The arrangement of ions in the crystal structure correlates with the position of minima of both V, and I$,. Consequently, the CP V(r) is structure-sensitive. Thus, studies of the CEF effects employing various spectroscopic methods may serve as source of information on the structure of matter. These measurements might be of particular importance in cases when conventional structural measurements are insufficient (for example, for amorphous materials). As has been shown by the analysis of the data on neutron-spectroscopic measurements in intermetallics of the RM,-type structure 1271 (the results for the RM,-type structure are presented in the next section), the qualitative conclusions made on the short range and oscillating character of the potential Vi( 1R, - r I) hold true for the CP in intermetallics based on REM. The difference in the shape of potentials y( IRj - r I) (for example, the position and depth of the first minimum) for nontransition, transition and RE-ions is determined, apparently by their valence, the value of
341
the ion radius (for RE-ions, for example, it is approximately twice as large as that of A13+) and the resonance character of pseudopotentials of transition and RE metal ions. It should be emphasized that the value and sign of the CEF parameters are determining by the shape of the potential vi ( IRi - r I) and the distance to the ion whose inner electrons are test charges reacting upon the CEF. Thus, one may assume that the generally-accepted subdivision of crystalline fields (see, for example, ref. [28]) into weak, as compared with the spin-orbit interaction, (for rare earths, actinides and their compounds), intermediate (for 3d-elements and their compounds and strong (for 4d- and Sd-elements and their compounds) is determined not only by the shielding of inner 4f- and Sf-shells of RE and actinide ions, but also by the fact that, due to the large ion radius, the RE and actinide ions in pure substances and compounds are separated from the neighbouring atoms to considerably greater distances, as compared with those to the nearest neighbours in pure 3d, 4d and 5d substances and compounds. Thus, 4f- and Sf-electrons are in regions of considerably weakened CP due to great distance to the neighbouring ions, in the vicinity of which the CP has sharp maxima. Note that the sign and magnitude of magnetocrystalline anisotropy in pure REM and their compounds, such as RM, and R2M1,, are, apparently determined to a great extent by the shape of the potential y( IR, - r I). In the work dealing with the magnetocrystalline anisotropy in compounds of SmCo, and Sm,Co,, (see, for example, refs. [29,30]) the authors employing the MEPC had to make a rather artificial assumption that the magnitude and sign of the CEF acting on the RE-ions were determined not by the nearest ions of Co (whose contribution was assumed to be zero) but only by he neighbouring RE-ions. This assumption stems from the fact that in the MEPC the shape of the potential y ( I Ri - r I) is assumed to be purely Coulombic, i.e. the potential v ( 1Rj - r I) in the MEPC decreases with distance very slowly and monotonically (as compared with 5 ( I Ri - r I) in fig. 1). The CPM allows us to avoid such artificial assumptions.
342
V. G. Orlov / Crystal potentialmodel for the descriptionof CEF effects
Interpretation of neutron spectroscopy measurements on PrAl, in the framework of the CPM
4.
The intermetallic compound PrAl,, with the hexagonal structure P4/mmc, is convenient for the investigation of CEF effects because the exchange interaction between the RE-ions is diminished due to the large interionic distances [27]. Owing to the fact that for the PrAl, compound the pseudopotentials of the A13+ and Pr3+ ions are unknown we shall make an attempt to reconstruct the CP space distribution in the PrAl, sample. Let us introduce a model potential which reproduces the main features of the vLs potential in fig. 1: 1) the law Z,e/R at R + 0; 2) the exponential decrease with the distance; 3) oscillations which correlate with the arrangement of the ions in the crystal:
08) where R = 1Ri - r 1, Z, is the valence of the i th ion. The pre-exponential factor in (18) provides the description of two oscillations within the range of which four nearest shells of ions in PrAl, are placed. This factor determines the magnitude of the &:,, coefficient of the CEF Hamiltonian (8), and the overall splitting of the CEF levels is controlled mainly by the screening parameter. Though the form of y (18) is quite general, because it allows the imitation of some of the nonlinear screening effects, still we cannot regard it as final due to the possible ambiguity caused by the large number of adjustable parameters. The parameters of K have been determined from the comparison of measured and calculated doubledifferential neutron cross-sections for PrAl,. An additional convenience of the RM,-type compound consists in the fact that all the crystallographical sites for the ions of the same type are equivalent. So all the parameters (Z,, p, q, s, R,, R’, R”) for the ions of the same type are identical and the space distribution of the crystal potential of the PrAl, compound is described by two sets of parameters. The coefficients A,, of the CEF
Hamiltonian (3) (see appendix) are expressed as derivatives of the potentials Vi. The matrix elements of the operators O,, for the wavefunctions of the Pr3+ ion (J = 4) split ground multiplet can easily by found using well known quantum-mechanical formulae [18,28]. The following values were taken for (r”) of the Pr3+ ion: (r2) = 1.086, (r4) = 2.822, (r6) = 15.726 au 1151. The best agreement between experimental and calculated results was achieved with the “shielding factors” a2 = 0.57, a4 = 0.84 and a6 = -0.64 (compare with the calculated values for the “free” Pr3+ ion [7]: a2 = 0.67, a4 = 0.03, a6 = -0.04). The mentioned possible ambiguity of the form of the potentials y (18) might cause certain variations of the values of the u,, factors. Still the considerable deviations of the a4 and a6 values from those for the “free” ion were necessary for any reasonable changes of the form of the V, potentials. The magnitude of the coefficients of the H&h (8) was the following: &o/aj = - 127 K, fi4,,/& = 17 K, fi6,,/vJ = 23.5 K and fi66/fi60 = 5. In fig. 2 the experimental [27] and the best theoretical double differential neutron cross sections for PrAl, are shown. The calculated cross sections correspond to the values of the parameters of the potentials V, from table 1. These potentials are shown in fig. 3 for comparison. The calculated level splittings and the wave functions for Pr3+ are shown in fig. 4a, b, together with the level scheme found in [27]. The theoretical cross sections utilized the results on the double scattering of cold neutrons on PrAl, samples [31]. The calculated cross sections are in satisfactory agreement with the experimental data except the range of energies near 11 meV where the calculated intensity is too low in comparison with the experimental one. It is interesting to note that the CPM revealed the high sensitivity of the CEF effects to the crystal structure parameters. The calculated level schemes in figs. 4a, b and the corresponding cross sections in fig. 2 for the temperatures 90 and 295 K were obtained with the assumption of the temperature independence of the potential parameters v but with account of the temperature expansion of the lattice parameters for PrAl, [27] (at T = 90
V. G. Orlov / Crystal potential model for the description of CEF effects
10 &
343
12
(meV)-
Fig. 2. Double differential neutron cross sections for PrAl 3: experimental data, 0, T = 90 K, 0, T= 295 K, the solid curves show the best theoretical results.
v 0.5
0.4
vi A13+ /
0.3
0.2
0.1
0
I
4s
I
A
6
5
r-0.1
-0.2 Fig. 3. Potentials V, (18) with parameters from the table 1. Distances from the Pr3+ Ion to the nearest ions in PrAI, are denoted by arrows.
V. G. Orlov / Crystal potential model for the description of CEF effects
344 Table 1 Parameters for PrAl,
of the potentials
Ion
V; (18) for the A13+ and Pr 3+ ions which were used for calculating
neutral
cross sections
Parameters
-7 Al Pr
+3 +3
P
s
4
1.2 2.6
2.3 6.9
1.91 2.07
K a=6.481 A, c=4.591 A and at T=295 K a = 6.511 A and c = 4.606 A). It follows from fig. 4 that the temperature expansion of the PrAl, lattice should decrease the energies of transitions between the CEF levels. This fact agrees with the temperature variations of the experimental cross sections in fig. 2 but needs more detailed experimental investigation. The most notable difference between the level schemes corresponding to the CPM (figs. 4a,b) and that found in ref. [27] (fig. 4c) consists in the magnitude of the IX-I, transition energy. The level scheme 4c was obtained [27] with arbitrary variations of all four parameters A,, of the Hamiltonian Hi&. It should be noted that the distance between the I, and I, levels could not be made in the CPM as large as in fig. 4c, even with considerable variations of the parameters of the potentials y. The form of the potential V, in fig. 3 corresponding to the Al 3+ ions is similar to that of the potential vLs for A13+ in fig. 1. The potential V, in fig. 3 for the Pr 3+ ions seems reasonable. It
meV
15.6
l-51 =
;;
the differential
=
10.9
0.2781’4>+0.961
10.8
0.70713>+0.7071-3>
l-4
_
7.5
0.70713>-0.7071-3>
r6
-
4.6
0
rl _
I T2>
r5,
=
14.8
r52
e
1o.5 10.3
l-3
R’
R”
(A)
(A)
1.733 1.661
1.437 2.570
2.581 3.918
reveals a deeper minimum and its repulsive part is placed at larger distances, in comparison with the v for A13+ ions. Similar tendencies were observed in the cases of the light nontransition metals and of the heavy d-metals [32].
Acknowledgements The author is deeply grateful to Dr. J. Jensen for valuable suggestions on the form of the crystalline electric field Hamiltonian and for discussion of the present work. The author thanks Professor A.R. Mackintosh and Drs. P.A. Alekseev and I.P. Sadikov for stimulating and fruitful discussions. The author acknowledges the hospitality of the NBI/NORDITA during his stay in Copenhagen.
Appendix We present expressions for the coefficients
r5,
=
15.6
0.813
l-3
-
14.3
0.70713>+0.7071-3>
1?4>+0.5821~2>
r52
_
9.3
0.582I-f4>-0.8131~2>
r4
_
7.2
r&
-
6.4
0.70713>-0.7071-3>
1+1>
r6
=
4.4
I?,
=
4.5
1+1>
lo>
r, -
0
r, _
0
IO>
a Fig. 4. The CEF level-splitting
b scheme for the ground
A,,
meV
meV
0.961/-+4>+0.2781+2>
R, 6)
C
state of Pr 3+ in PrAl,:
a) CPM, T = 90 K, b) CPM, T = 295 K,
C)
EPCM [27].
V. G. Orlov / Crystal potential model for the description of CEF effects
of the Hamiltonian A,, = ; c
(3):
(l/RJK(‘)
+(1/Rf)[3R;(2R;x+Rf,) + (RfJR?)
- lOR:( Rf,+6RfxRfy)
iI
+105 R;xR;y]yc4)
I [ x (
pq2) - 1 p) I Ri '
A,, = -eC [(l/Ri)v(‘)+
345
)
+(3/R;)[R;(R;-6Rfy)
+Y(~‘],
-R;Rf,(l2Rf-90R;,) + R;x(15R; - 140R;Y)] K(3)
A,, = -~c[(R~~/R4)~("+6(R~~/Ri) I
- (3/Rt0)[3R;(Rf-5R$)
x(R;-R;~)~~"
130RjR:,(R;-7R:,)
+(3/RT)(R;-6R;Rfx+5R:',) x (yc2)- (l/Ri) A,,=
vu’)]
,
+35 R;x(R'-9R;y)] (A-3)
-~cI(R~~/R4)(3R~~-R:,) x [ rc4; - (61~~)K(3) + (15/R;) x (y(2) - (l/Ri)
KC”)]
+( 3,'Rj)(R;/Rf,)
x[y'"- (3/Ri)(~(2)-(1/R,,~(1')]}, (A.4)
X[vlc4)-
x(vc2'- (l/Ri)y('))}.
(A-7)
In the formulae (A.l)-(A.7) cm) stand for the mth derivatives (m = 1, 2, . . . ,6) of the potential of the n th (n = 1, 2, . . .) ion coordination shell. One can get the coefficients A,, and A,, by replacing x by y or z in (A.6). The coefficients A 659 A,, and A,, can be written in an analogous way to (A.7). For the compound PrAl, the contributions of only four nearest shells of ions were taken into account. In writing down the Hamiltonian (3) account has been taken of the following properties of ion arrangement in the hexagonal structure:
(~/R~)K(~)
+( 15,‘R;)(~C2)-
(l,'Ri)~(l))], (A.51
~~~= - $C[(RP,/R~)J+~)+ 15(R;,/R;) x(
R;-
R;&(')+ 15(3R$4)-
10RfRfx
i
i
+7R;x)(R;x/R;)~(4)+(15/R~) x(R~-~~R~R~~+~~R;R;~-~~R~&(~~ -(45,'Rf')(R;-15R?R? +35R:R"IX -21Rfx)(vc2)-(l,R,&))], A,,=
-
gc{(
Rfx~fy/R:)p
The coefficients i. and A,,,,, of the Hamiltonian (6) can be expressed through the combinations of A nm (A.l)-(A.7) and powers of the f-electron radius r:
64.6)
A,=+A,, r2+(A41 -&A4,)r4 +f(A42+$A43)r2
+(R;x;R~)[R~(R;x+6R~y) -15RfxRf,]1/@)
+ & [15A,, + 6A,, + 8A,, +=(A,, +A,&?
G4.8)
V.G. Orlou / Crystal potential model for the description of CEF effects
346
i*,=A,,
+~A,,+&4,,+$4,&*
+~[A,,+~A,,-~(A,,+A,,)]~~, &,,
= &A,,
+ &
[ 15A,,
(A.9) -
+9(A,,
7A,,
-
4A,,
+A&*>
(A.10)
K60=~[A63+A66-A6,-~(A61+A62)1r
(A.ll) Al,, = t(&
-452).
(A.12)
References [l] M.T. Hutchings, Solid State Phys. 16 (1964) 227. [2] M. Faucher and D. Garcia, J. Less-Common Metals 93 (1983) 31. eds., [31 J.E. Crow, R.P. Guertin and T.W. Mikhalishin, Crystalline Electric Field and Structural Effects in f-Electron Systems (Plenum Press, New York, London, 1980). I41 J.M. Dixon, Solid State Commun. 12 (1973) 789. [51 D.M. Eagles, Z. Phys. B21 (1975) 171. Fl R.M. Stemheimer, M. Blume and R.F. Peierls, Phys. Rev. 173 (1968) 376. I71 P. Erdos and J.H. Kang, Phys. Rev. B6 (1972) 3393. 181 J.C. Duthie and V. Heine, J. Phys. F 9 (1979) 1349. [91 D. Schmitt, J. Phys. F 9 (1979) 1759. WI S.Adam, Gh. Adam and A. Corciovei, Phys. Stat. Sol. (b) 105 (1981) 85. IllI S. Adam, Gh. Adam and A. Corciovei, Phys. Stat. Sol. (b) 114 (1982) 85. WI V. Heine, Group Theory in Quantum Mechanics (Pergamon Press, New York, 1960). 1131 K.W.H. Stevens, Proc. Phys. Sot. A65 (1952) 209.
[14] B. Bleaney and K.W.H. Stevens, Rep. Prog. Phys. 16 (1953) 108. P51 A.J. Freeman and R.E. Watson, Phys. Rev. 127 (1962) 2058. [161 A.J. Freeman and J.P. Desclaux, J. Magn. Magn. Mat. 12 (1979) 11. 1171 W.B. Lewis, in: Proc. 16th Congress Ampere, ed. I. Ursu (Bucharest, 1971) p. 717. VI S.A. Altshuler and B.M. Kozyrev, Electron Paramagnetic Resonance of Compounds of Intermediate Group Elements (Nauka, Moscow, 1972). in the Theory of Metals v91 W.A. Harrison, Pseudopotentials (Benjamin, New York, Amsterdam, 1966). WI E.G. Brovman and Yu.M. Kagan, Usp. Fiz. Nauk 112 (1974) 369. VI D. Pines and P. Nozieres, The Theory of Quantum Liquids (Benjamin, New York, Amsterdam, 1966). WI P. Vashishta and K.A. Singwi, Phys. Rev. B6 (1972) 4883. and R. Taylor, Can. J. Phys. 48 (1970) v31 D.J.W. Geldart 167. ~241 A.O.E. Animalu and V. Heine, Phyl. Mag. 12 (1965) 1249. V.G. Vaks and A.B. Trefilov, J. Phys. F 1251 A.M. Bratkovsky, 13 (1983) 2517. WI J. Friedel, Phyl. Mag. 43 (1952) 153. 1271 P.A. Alekseev, I.P. Sadikov, Yu.L. Shitikov, I.A. Markova, O.D. Chistyakov, E.M. Savitsky and J.K. Kjems, Phys. Stat. Sol. (b) 114 (1982) 161. ResoWI A. Abragam and B. Bleaney, Electron Paramagnetic nance of Transition Ions (Clarendon Press, Oxford, 1970). and V.U.S. Rao, J. Solid State Chem. 6 v91 J.E. Greedan (1973) 387. 1301 S.C. Sankar, V.U.S. Rao, E. Segal, W.E. Wallace, W.G.D. Frederick and H.J. Carrett, Phys. Rev. Bll (1975) 435. (311 P.A. Alekseev and I.P. Sadikov, Preprint IAE-3117 (1979). and R. Taylor, eds., Handbook of lnter1321 A.M. Stoneham atomic Potentials 2. Metals (Theoretical Physics Division AERE Harwell, Oxfordshire, 1981).
Materials
61 (1986) 337-346
337
CRYSTAL POTENTIAL MODEL FOR THE DESCRIPTION OF CRYSTALLINE FIELD EFFECTS IN RARE EARTH METALS AND INTERMETALLICS V.G. ORLOV
+
The Niels Bohr Institute, Received
ELECTRIC
20 January
University
of Copenhagen, DK-2100 Copenhagen 0, Denmark
1986; in revised form 18 April 1986
A crystal potential model is suggested. It allows us to interpret crystalline electric field effects in optical, EPR, NMR, NGR and neutron spectroscopy measurements in rare earth metals and intermetallics. The crystal potential character and space distribution are discussed. The model is used for the theoretical interpretation of the effects of the crystalline field in the compound PrAl s
1. Introduction In most cases the presently existing crystalline electric field (CEF) models, i.e. the model of effective point charges (MEPC) [l] and the covalent model based on the method of molecular orbitals (see, for example, ref. [2]), do not allow us to determine unequivocally quantitative characteristics specifying the physical nature of the CEF effects in metals. Moreover, the use of the MEPC results in unphysical values of ion valences in metals (see ref. [3]). In the CEF theoretical description the most severe difficulties arise due to the presence of conduction electrons in metals. The problem of taking account of the conduction electrons’ contribution to the CEF has become aggravated in connection with the detection of large deviations of experimentally obtained coefficients of the CEF from those calculated in the MEPC for heavy rare earth metals (REM) and dilute solutions of REM in d-metals. Several equivalent methods for taking into account the conduction electrons’ contribution to the metal CEF have been suggested. Dixon [4] proposed to take account of valence and inner 5d- and 4f-electrons’ mixing in the OPW method. + Permanent address: I.V. Kurchatov Energy, Moscow 123182, USSR.
Institute
of Atomic
Eagles [5] suggested that the energy shift of the 4f-electrons and the sign variation of the CEF coefficients of the 4th order can be accounted for by the Coulomb and exchange interaction of the 4f- and Sd-electrons contributing to the metal CEF. Detailed atomic calculations [6,7] made it passible to determine the perturbation of outer (5% 5p) and inner shells of a RE-ion induced by the external CEF. The perturbed charge distribution of the ion shells creates within the ion an additional electrostatic field which “shields” the external CEF. As a result, so-called “shielding factors” a, (n = 2,4,6) appear in the CEF Hamiltonian. Despite the validity of the results of these studies, they may be used only qualitatively for the interpretation of experimental data on the CEF effects in metals, since calculations were made for a “free” RE-ion arranged in an external field with the CEF symmetry of a metal. In ref. [8] an attempt was made to take into account the shielding effect of conduction electrons on the CEF in RE metal materials. Using the APW method Schmitt [9] calculated the contribution of the conduction electrons to the CEF for a number of cubic RE-intermetallics, which turned out to be comparable with the ionic contribution. Adam et al. [lO,ll] noted that the difference
0304-8853/86/$03.50 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
338
V.G. Orlov / Crystal potential model for the description of CEF effects
between the form of the CEF potential and that of the Coulomb interaction can noticeably change the values of the crystalline field coefficients. In the present paper a model of the crystal potential (CPM) is suggested. It assumes that the RE-ion ground-state splitting is induced by an effective crystal potential (CP) V(r). The parameters of the CP V(r) can be either calculated from first principles or reconstructed by comparing experimental and theoretical data. V(r) has the crystal symmetry. As a result, the CEF Hamiltonian in the CPM yields the correct systematics of the split ground state levels of the RE-ion. A particular arrangement of levels is determined by the Hamiltonian coefficients which are expressed as derivatives of V(r). Thus, it becomes possible to investigate special features of the CP by using spectroscopic measurements. It will be shown below that the CP V(r) oscillates and rapidly decreases with distance. The latter property implies that the CEF effects are highly local. In the present paper the application of the model is illustrated by the interpretation of the crystalline field effects in the compound PrAl,.
the test charge, the potential y can be expanded in a Taylor series in the small parameter r/R,, i.e. the ratio of the f-orbit radius r = (x, y, z) to the distance to the ith ion Ri = (R,,, R,,, R,,): V(‘)=C~(IRi-r+C~(R;)
+c
i
av av -x+-y+,Iz
i ax
ay
ay
Hhex -eV(r)= CPM --
A,,(3z2
Let us formulate the basic aspects of the CPM and indicate its special characteristics as compared with the MEPC. Let the electrostatic CP V(r), with the symmetry of the crystal, act on a trial charge at a distance r from the origin. We assume that the CP V(r) is distributed nonuniformly over the crystal, i.e. it has sharp maxima at ionic sites. Thus, V(r) may be expressed as the sum: (1)
where y is a potential containing the contribution of both ions and conduction electrons (the character and space distribution of the y potential are discussed in the next section). For convenience, we shall treat the potential v as spherically symmetric with respect to Ri the coordinate of the i th ion. If an inner f-electron of the RE-ion is taken as
(2)
+ A,,r*z*
- r’) + A,,z* + A4,z4 + A,,x6
+A6*y6 + A63~6 + A,,x4y2
2. Crystal potential model
+*-.. i
With the aim of employing the CPM for the interpretation of the CEF effects in hexagonal pure REM and intermetallics of the RM, and RM, types (where R is an atom of a REM and M is a nontransition or transition metal) the further description of the CPM will be made for the case of hexagonal crystal symmetry. It is easy to generalize the model for a case of cubic or any other type of symmetry. For the case of hexagonal symmetry, the crystalline field Hamiltonian in the CPM frame has the following form:
+A,,r4
V(r>=CK(lR,-rl), i
I
i
+A66(~2
+ y2)*z2
+ A,,(x*
+ A6,x2y4 + y2)z4. (3)
The 6th order expansion is performed in the Hamiltonian (3) taking account of the fact that matrix elements of HcpM will be taken over the wave functions of f-electrons with an orbital momentum I = 3 [12]. The coefficients A,, are presented in the appendix (A.l)-(A.7). One can easily show with the aid of the formulae (A.6) and (A.7) that, due to the hexagonal symmetry of the Hamiltonian (3) there exist two relations between the coefficients A,, (WI = 1, 2, 4, 5): &i - A,, + i+ ( A64
-
A,,
A,, + A,, - +( A,, + A,,)
> =
0,
= 0.
(4)
(5)
In order to calculate matrix elements of H&& with the aid of the Stevens equivalent-operator method [13,14] it is necessary to rearrange the Hamiltonian (3) so as to obtain tensor operators
V.G. Orlov / Crystal potentialmodel for the description of CEF effects
from the respective powers of the electron coordinates *. Taking into account (4) and (5) it is possible to write the H/?&r as follows:
an be written as follows: ~20=~&42~+4~22~~2 + ,(
+A,(3524
+Al,,(x6
- 315z4r2 + 105z2r4 - 5r6) -
15x4y2 + 15x2y4 -y”),
cnn)(rn),
(7)
where an are the so-called “shielding factors” of the f-electrons by those of all the other ion shells [6,7], while (r”) stands for the average values of the nth power of the f-electron radius [15-171. The formula (7) should be treated only as an estimate due to the uncertainty in the extent to which it is per-n&sable to use the results of atomic calculations, obtained for “free” ions, for ions in a crystal. Thus, the final form of the CEF Hamiltonian for hexagonal symmetry in the CPM is as follows: H:FM
= j2,,02,
+ k:,O,,
+ g6,,06,
+ i6,,06,,
g40
=
&
(6)
where 2, and i,, (n = 2,4, 6), a,, are presented in the appendix (A.8)-(A.12). The first term i0 (A.8) in (6) is a scalar with respect to the electron coordinates. It will be omitted in the following analysis because it can only shift each crystal field level in an equivalent manner. It is easy to show that in the case of the C_oulomb potential v(r) = Z, e/ 1R i - r 1 the term A a in (6) should go to zero and the coefficients Al,,, (n = 2, 4, 6) and a66 go over into the corresponding coefficients A,, and A,, of the MEPC [l]. To obtain the final form of the Hamiltonian in the CPM, let us introduce three coefficients s, (n = 2, 4, 6) by which the Hamiltonian terms will be multiplied. One can make an approximate estimate of the values of s, employing the formula s, = (I-
A42
+
QA,,)S,
++1[A67+5A63-
- 30r2z2 + 3r4)
+&,(231z6
(8)
where O,, (n = 2, 4, 6) and q66 stand for Steven equivalent operators [l] and B,, (n = 2, 4, 6), j6,,
* The author is deeply grateful to Dr. J. Jensen for valuable suggestions on this point.
339
4(A61+A62)1s6}~
{ &443s4
+
+9(A6,
‘60
=
yJ&
k
[ i5A63
-
c9)
7A66
-
4A67
(10)
+A62)1S6},
[ A63
+
A66
-A,,
-
+(A61
+A6211
s6, (11)
‘66
=
YJi(A6,
-
(12)
A62)S6.
The values of the numerical coefficients (Ye, fiJ and yJ are calculated for the ground state of all the three-fold positively-charged RE ions and presented in refs. [1,18]. s, (n = 2, 4, 6) are three constants which can be approximately estimated employing the formula (7) and expressions for A nm are presented in the appendix (A.l)-(A.7).
3. Discussion of the nature and shape of the crystal potential If a test charge Zre ( Zr, is the test charge value, e is the electron charge) is placed at a point r in a metal then, according to the metal pseudopotential model [19,20], in the linear shielding approximation the electrostatic potential acting on the test charge from the metal ions and electrons can be written as follows:
1
V&(
x
J0
O” u,,(q) ((4)~dqNClq)q
Xsin(q]R;-r])dq
2+n*lR,--rl
1,
(14)
where II,(q) is a dipole [21], r(q) = 1 + (1 G( q))V( q)IIo( q) is a static dielectric permeability, V(q) = 4Te*/q*. For G(q) one can use, for
V.G. Orlov / Crystal potential model
340
for the description
example, the Vashista-Singwi (VS) approximation [22] or that of Geldart-Taylor (GT) [23]. U,: (q) is a Fourier component of the metal ion pseudopotential, for example, of the Animalu-Heine type [24]: 4az.e2 uz(q)
=
-
A
(I+ q2
-
-&
~1 cos(qr,)
05)
[ sin(qr,)
1
exp(
-5q4),
expression for the interion potential [19]: Yns( IR,-R,l)=
vii(
IR,-Rjl).
06)
But if we take an inner electron, for example, the RE-ion f-electron, as a test charge, Zre = -e, U,,(q) = 4e2/q2 and V,,(r) (13) and I&,( I R, r I) (14) will represent, respectively, the CP V(r) and the potential y( I R, -r I) from (1) in the linear shielding approximation: VLS(
where Z, is the metal ion valence, r0 and u are parameters of the pseudopotential, { = 0.03/16 K& KrO is the Fermi momentum of the metal at zero temperature and pressure. Up’ is a Fourier component of the potential energy of a test-charge interaction with the conduction electrons of a metal. If we taken one of the metal ions as a test charge, Zre = Z#e, Up” (q) = Uz (q) and I& ( 1Rj - R, I) (14) will reduce to the well-known
of CEF effects
=
I Ri-rl) zie
2
lR,-rl
- TIRi-rl
O”I&&) /0 P(4)
X&T(q) sin(qIR,-rlhh.
(17)
Note that formula (17) takes into account not only the shielding effect of the conduction electrons but also their direct contribution to the CEF of a metal, since the second term in the right-hand part of (17) takes account of the test charge interaction
V
v‘Oli
20
15 0.10
t
V 10
T
V
0.05 2
3
4
5
6
r--A
0
Fig. 1: Interion and crystal potentials for Al. Distances to the nearest ions in the fee structure comparison, the dependence of the Coulomb and crystal potentials on distance is given in the inset.
are denoted
by arrows.
For
KG. Orlov / Crystalpotentialmodelfor the descriptionof CEF effects
with metal ions via the conduction electrons. Unfortunately there do not exist reliable pseudopotentials for the RE-ions at the present time. Therefore, we will perform a qualitative study of the I& behaviour (17) using Al as an example, the pseudopotential parameters of which are wellknown (for example, for c(q) in the GT approximation [23] ‘a = 2.464 au, u = - 1.216 [25]). Fig. 1 shows the coordinate dependence of the inter-ion potential I$ (16) and the potential V& (17) for ions of Al 3+. It follows from fig. 1 that the behaviour of the potentials I$ and I& is very similar, i.e. they decrease sharply with distance, performing simultaneously oscillations which are due to the Friedel singularity [26] of the dielectric function c(q). From this there follow two significant conclusions: 1) The major contribution to the CP V(r) should be yielding by just a few coordination spheres of ions, since at larger distances the potential I& becomes small in its value and smooth in its shape. This results in a strong decrease with distance of derivatives of y( 1Ri - r I) through which the coefficients A,,,, of the CEF Hamiltonian (8) in the CPM are expressed. 2) The arrangement of ions in the crystal structure correlates with the position of minima of both V, and I$,. Consequently, the CP V(r) is structure-sensitive. Thus, studies of the CEF effects employing various spectroscopic methods may serve as source of information on the structure of matter. These measurements might be of particular importance in cases when conventional structural measurements are insufficient (for example, for amorphous materials). As has been shown by the analysis of the data on neutron-spectroscopic measurements in intermetallics of the RM,-type structure 1271 (the results for the RM,-type structure are presented in the next section), the qualitative conclusions made on the short range and oscillating character of the potential Vi( 1R, - r I) hold true for the CP in intermetallics based on REM. The difference in the shape of potentials y( IRj - r I) (for example, the position and depth of the first minimum) for nontransition, transition and RE-ions is determined, apparently by their valence, the value of
341
the ion radius (for RE-ions, for example, it is approximately twice as large as that of A13+) and the resonance character of pseudopotentials of transition and RE metal ions. It should be emphasized that the value and sign of the CEF parameters are determining by the shape of the potential vi ( IRi - r I) and the distance to the ion whose inner electrons are test charges reacting upon the CEF. Thus, one may assume that the generally-accepted subdivision of crystalline fields (see, for example, ref. [28]) into weak, as compared with the spin-orbit interaction, (for rare earths, actinides and their compounds), intermediate (for 3d-elements and their compounds and strong (for 4d- and Sd-elements and their compounds) is determined not only by the shielding of inner 4f- and Sf-shells of RE and actinide ions, but also by the fact that, due to the large ion radius, the RE and actinide ions in pure substances and compounds are separated from the neighbouring atoms to considerably greater distances, as compared with those to the nearest neighbours in pure 3d, 4d and 5d substances and compounds. Thus, 4f- and Sf-electrons are in regions of considerably weakened CP due to great distance to the neighbouring ions, in the vicinity of which the CP has sharp maxima. Note that the sign and magnitude of magnetocrystalline anisotropy in pure REM and their compounds, such as RM, and R2M1,, are, apparently determined to a great extent by the shape of the potential y( IR, - r I). In the work dealing with the magnetocrystalline anisotropy in compounds of SmCo, and Sm,Co,, (see, for example, refs. [29,30]) the authors employing the MEPC had to make a rather artificial assumption that the magnitude and sign of the CEF acting on the RE-ions were determined not by the nearest ions of Co (whose contribution was assumed to be zero) but only by he neighbouring RE-ions. This assumption stems from the fact that in the MEPC the shape of the potential y ( I Ri - r I) is assumed to be purely Coulombic, i.e. the potential v ( 1Rj - r I) in the MEPC decreases with distance very slowly and monotonically (as compared with 5 ( I Ri - r I) in fig. 1). The CPM allows us to avoid such artificial assumptions.
342
V. G. Orlov / Crystal potentialmodel for the descriptionof CEF effects
Interpretation of neutron spectroscopy measurements on PrAl, in the framework of the CPM
4.
The intermetallic compound PrAl,, with the hexagonal structure P4/mmc, is convenient for the investigation of CEF effects because the exchange interaction between the RE-ions is diminished due to the large interionic distances [27]. Owing to the fact that for the PrAl, compound the pseudopotentials of the A13+ and Pr3+ ions are unknown we shall make an attempt to reconstruct the CP space distribution in the PrAl, sample. Let us introduce a model potential which reproduces the main features of the vLs potential in fig. 1: 1) the law Z,e/R at R + 0; 2) the exponential decrease with the distance; 3) oscillations which correlate with the arrangement of the ions in the crystal:
08) where R = 1Ri - r 1, Z, is the valence of the i th ion. The pre-exponential factor in (18) provides the description of two oscillations within the range of which four nearest shells of ions in PrAl, are placed. This factor determines the magnitude of the &:,, coefficient of the CEF Hamiltonian (8), and the overall splitting of the CEF levels is controlled mainly by the screening parameter. Though the form of y (18) is quite general, because it allows the imitation of some of the nonlinear screening effects, still we cannot regard it as final due to the possible ambiguity caused by the large number of adjustable parameters. The parameters of K have been determined from the comparison of measured and calculated doubledifferential neutron cross-sections for PrAl,. An additional convenience of the RM,-type compound consists in the fact that all the crystallographical sites for the ions of the same type are equivalent. So all the parameters (Z,, p, q, s, R,, R’, R”) for the ions of the same type are identical and the space distribution of the crystal potential of the PrAl, compound is described by two sets of parameters. The coefficients A,, of the CEF
Hamiltonian (3) (see appendix) are expressed as derivatives of the potentials Vi. The matrix elements of the operators O,, for the wavefunctions of the Pr3+ ion (J = 4) split ground multiplet can easily by found using well known quantum-mechanical formulae [18,28]. The following values were taken for (r”) of the Pr3+ ion: (r2) = 1.086, (r4) = 2.822, (r6) = 15.726 au 1151. The best agreement between experimental and calculated results was achieved with the “shielding factors” a2 = 0.57, a4 = 0.84 and a6 = -0.64 (compare with the calculated values for the “free” Pr3+ ion [7]: a2 = 0.67, a4 = 0.03, a6 = -0.04). The mentioned possible ambiguity of the form of the potentials y (18) might cause certain variations of the values of the u,, factors. Still the considerable deviations of the a4 and a6 values from those for the “free” ion were necessary for any reasonable changes of the form of the V, potentials. The magnitude of the coefficients of the H&h (8) was the following: &o/aj = - 127 K, fi4,,/& = 17 K, fi6,,/vJ = 23.5 K and fi66/fi60 = 5. In fig. 2 the experimental [27] and the best theoretical double differential neutron cross sections for PrAl, are shown. The calculated cross sections correspond to the values of the parameters of the potentials V, from table 1. These potentials are shown in fig. 3 for comparison. The calculated level splittings and the wave functions for Pr3+ are shown in fig. 4a, b, together with the level scheme found in [27]. The theoretical cross sections utilized the results on the double scattering of cold neutrons on PrAl, samples [31]. The calculated cross sections are in satisfactory agreement with the experimental data except the range of energies near 11 meV where the calculated intensity is too low in comparison with the experimental one. It is interesting to note that the CPM revealed the high sensitivity of the CEF effects to the crystal structure parameters. The calculated level schemes in figs. 4a, b and the corresponding cross sections in fig. 2 for the temperatures 90 and 295 K were obtained with the assumption of the temperature independence of the potential parameters v but with account of the temperature expansion of the lattice parameters for PrAl, [27] (at T = 90
V. G. Orlov / Crystal potential model for the description of CEF effects
10 &
343
12
(meV)-
Fig. 2. Double differential neutron cross sections for PrAl 3: experimental data, 0, T = 90 K, 0, T= 295 K, the solid curves show the best theoretical results.
v 0.5
0.4
vi A13+ /
0.3
0.2
0.1
0
I
4s
I
A
6
5
r-0.1
-0.2 Fig. 3. Potentials V, (18) with parameters from the table 1. Distances from the Pr3+ Ion to the nearest ions in PrAI, are denoted by arrows.
V. G. Orlov / Crystal potential model for the description of CEF effects
344 Table 1 Parameters for PrAl,
of the potentials
Ion
V; (18) for the A13+ and Pr 3+ ions which were used for calculating
neutral
cross sections
Parameters
-7 Al Pr
+3 +3
P
s
4
1.2 2.6
2.3 6.9
1.91 2.07
K a=6.481 A, c=4.591 A and at T=295 K a = 6.511 A and c = 4.606 A). It follows from fig. 4 that the temperature expansion of the PrAl, lattice should decrease the energies of transitions between the CEF levels. This fact agrees with the temperature variations of the experimental cross sections in fig. 2 but needs more detailed experimental investigation. The most notable difference between the level schemes corresponding to the CPM (figs. 4a,b) and that found in ref. [27] (fig. 4c) consists in the magnitude of the IX-I, transition energy. The level scheme 4c was obtained [27] with arbitrary variations of all four parameters A,, of the Hamiltonian Hi&. It should be noted that the distance between the I, and I, levels could not be made in the CPM as large as in fig. 4c, even with considerable variations of the parameters of the potentials y. The form of the potential V, in fig. 3 corresponding to the Al 3+ ions is similar to that of the potential vLs for A13+ in fig. 1. The potential V, in fig. 3 for the Pr 3+ ions seems reasonable. It
meV
15.6
l-51 =
;;
the differential
=
10.9
0.2781’4>+0.961
10.8
0.70713>+0.7071-3>
l-4
_
7.5
0.70713>-0.7071-3>
r6
-
4.6
0
rl _
I T2>
r5,
=
14.8
r52
e
1o.5 10.3
l-3
R’
R”
(A)
(A)
1.733 1.661
1.437 2.570
2.581 3.918
reveals a deeper minimum and its repulsive part is placed at larger distances, in comparison with the v for A13+ ions. Similar tendencies were observed in the cases of the light nontransition metals and of the heavy d-metals [32].
Acknowledgements The author is deeply grateful to Dr. J. Jensen for valuable suggestions on the form of the crystalline electric field Hamiltonian and for discussion of the present work. The author thanks Professor A.R. Mackintosh and Drs. P.A. Alekseev and I.P. Sadikov for stimulating and fruitful discussions. The author acknowledges the hospitality of the NBI/NORDITA during his stay in Copenhagen.
Appendix We present expressions for the coefficients
r5,
=
15.6
0.813
l-3
-
14.3
0.70713>+0.7071-3>
1?4>+0.5821~2>
r52
_
9.3
0.582I-f4>-0.8131~2>
r4
_
7.2
r&
-
6.4
0.70713>-0.7071-3>
1+1>
r6
=
4.4
I?,
=
4.5
1+1>
lo>
r, -
0
r, _
0
IO>
a Fig. 4. The CEF level-splitting
b scheme for the ground
A,,
meV
meV
0.961/-+4>+0.2781+2>
R, 6)
C
state of Pr 3+ in PrAl,:
a) CPM, T = 90 K, b) CPM, T = 295 K,
C)
EPCM [27].
V. G. Orlov / Crystal potential model for the description of CEF effects
of the Hamiltonian A,, = ; c
(3):
(l/RJK(‘)
+(1/Rf)[3R;(2R;x+Rf,) + (RfJR?)
- lOR:( Rf,+6RfxRfy)
iI
+105 R;xR;y]yc4)
I [ x (
pq2) - 1 p) I Ri '
A,, = -eC [(l/Ri)v(‘)+
345
)
+(3/R;)[R;(R;-6Rfy)
+Y(~‘],
-R;Rf,(l2Rf-90R;,) + R;x(15R; - 140R;Y)] K(3)
A,, = -~c[(R~~/R4)~("+6(R~~/Ri) I
- (3/Rt0)[3R;(Rf-5R$)
x(R;-R;~)~~"
130RjR:,(R;-7R:,)
+(3/RT)(R;-6R;Rfx+5R:',) x (yc2)- (l/Ri) A,,=
vu’)]
,
+35 R;x(R'-9R;y)] (A-3)
-~cI(R~~/R4)(3R~~-R:,) x [ rc4; - (61~~)K(3) + (15/R;) x (y(2) - (l/Ri)
KC”)]
+( 3,'Rj)(R;/Rf,)
x[y'"- (3/Ri)(~(2)-(1/R,,~(1')]}, (A.4)
X[vlc4)-
x(vc2'- (l/Ri)y('))}.
(A-7)
In the formulae (A.l)-(A.7) cm) stand for the mth derivatives (m = 1, 2, . . . ,6) of the potential of the n th (n = 1, 2, . . .) ion coordination shell. One can get the coefficients A,, and A,, by replacing x by y or z in (A.6). The coefficients A 659 A,, and A,, can be written in an analogous way to (A.7). For the compound PrAl, the contributions of only four nearest shells of ions were taken into account. In writing down the Hamiltonian (3) account has been taken of the following properties of ion arrangement in the hexagonal structure:
(~/R~)K(~)
+( 15,‘R;)(~C2)-
(l,'Ri)~(l))], (A.51
~~~= - $C[(RP,/R~)J+~)+ 15(R;,/R;) x(
R;-
R;&(')+ 15(3R$4)-
10RfRfx
i
i
+7R;x)(R;x/R;)~(4)+(15/R~) x(R~-~~R~R~~+~~R;R;~-~~R~&(~~ -(45,'Rf')(R;-15R?R? +35R:R"IX -21Rfx)(vc2)-(l,R,&))], A,,=
-
gc{(
Rfx~fy/R:)p
The coefficients i. and A,,,,, of the Hamiltonian (6) can be expressed through the combinations of A nm (A.l)-(A.7) and powers of the f-electron radius r:
64.6)
A,=+A,, r2+(A41 -&A4,)r4 +f(A42+$A43)r2
+(R;x;R~)[R~(R;x+6R~y) -15RfxRf,]1/@)
+ & [15A,, + 6A,, + 8A,, +=(A,, +A,&?
G4.8)
V.G. Orlou / Crystal potential model for the description of CEF effects
346
i*,=A,,
+~A,,+&4,,+$4,&*
+~[A,,+~A,,-~(A,,+A,,)]~~, &,,
= &A,,
+ &
[ 15A,,
(A.9) -
+9(A,,
7A,,
-
4A,,
+A&*>
(A.10)
K60=~[A63+A66-A6,-~(A61+A62)1r
(A.ll) Al,, = t(&
-452).
(A.12)
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