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Note on the gyromagnetic ratio of Co++ and on the Jahn-teller effect in Fe++
Van Vlcck,
J. H.
Physica
1960
26
544-552
NOTE ON THE GYROMAGNETIC RATIO OF CO++ AND ON THE JAHN-TELLER EFFECT IN Fe++ J.
H. VAN
VLECK
Physics Department,
Harvard
University,
of pseudo-angular
momentum
U.S.A. *)
Synopsis By the method that in a triply
degenerate
the same commutation method value
thus
obtained
these materials Teller
the gyromagnetic
is 1.65, in better
do not have
effect,
is included; coefficients
agreement
cubic symmetry.
fluosilicate. however,
factor.
obeys This
with
Sucksmith’s
spin-only
experimental
value g = 2, even though application
is to the Jahnof this effect
on Fe++, .and at the same time this mechanism
At first sight, one would level
for a proportionality
The other
role in the Abragam-Pryce
as its lowest
of the fact
angular momentum
is given why Low finds no evidence
resonance measurements
plays an important of copper
the utilization
ratio of a Co++ ion in a cubic field. The
on CoSO4 and CoCls than is the
effect in Fe++. An explanation
in his magnetic
J-T
rules as for a t, state, except
is used to calculate
measurements
is meant
level in a cubic field, the orbital
is triply
theory
of the magnetic
expect
the Fe++ ion to show a large
degenerate
even
the linear terms in the distortional
when
susceptibility
spin-orbit
coordinates
interaction
have very
small
and so are ineffective.
Introduction. In a perfectly cubic crystalline or ligand field, certain orbital levels, called Ti or Tz in Mulliken’s terminology, or rq, rs in Rethe’s, are triply degenerate. In the present paper, we shall be concerned only with matrix elements which are internal to such a triplet, or in other words deal only with the projection on three particular substates. Then, except for a proportionality factor, the matrix elements of the so-projected orbital angular momentum vector whose components we denote by l’%, lU’, lz’ are the same as those of the angular momentum vector for an atomic p state. Consequently, if angular momenta are measured in multiples of h/2n, WC have 1,‘s + lg’2 + 129 = 2ys &‘I, - l,‘l,’ = iyl,‘, etc., This resemblance was first pointed out by Abragam appropriate values of y can be shown to be Dr5
FY,
Fr5
-#
+*
y=-1 *)
Temporarily
at the Instituut
Lorentz
der Rijksuniversiteit,
-
544
-
and Pryce
1). The
(1) Leiden,
Nederland.
GYROMAGNETIC
The value
-
RATIO
+ given
OF co++
for Fr4
AND
JAHN-TELLER
EFFECT
IN Fe++
545
appliez to Co++(d7 4F) only if there is strong
Russell-Sounders coupling; actually the crystalline field will mix in some Pr4, and so reduce (71. In the limiting case that the field is very large compared to the R-S interaction, the value of y is - 1. Corresponding corrections are absent for the ferrous configuration d6, as there is only one state of maximum multiplicity. When spin is included, the need of writing down explicit wave functions can be avoided for many purposes by utilizing the fact that the problem of finding the proper linear combinations is equivalent to compounding ot spin and orbital angular momenta in a fi state. The resulting states can be described by a quantum number J’ which assumes the values J’ = 1s -
11, s, s +
1
We may term J’ a pseudo-inner quantum number, the true angular momentum. The spin orbit-energy EJ~ = +yn [J’(J’
+
1) -
S(S +
as it does not measure is given by 1) -
21
(4
where L is the ordinary atomic multiplet constant, The levels in the absence of an applied magnetic field are 2J’ + 1 degenerate, but the degeneracy will be reduced by interaction with other states when J’ > 1. This procedure has been discovered and used by Griffith 2) to simplify the calculation of magnetic susceptibilities in a perfectly cubic field, and also independently by Mof f i t t and collaborators 3) in a study of the colors of transition metal fluorides. The reader is referred to their papers for explanation of the method in greater detail. The writer has also hit upon the same trick in connection with two other applications, which it is the purpose of the present paper to describe. Before doing so, it may be noted parenthetically that the method of pseudoangular momentum vectors, to coin a name for the procedure, is in some ways like that which we devised for treating molecular coupling problems 4). As Klein 5) and Casimir 6) have shown, the angular momentum matrices, when projected on axes fixed in the molecule rather than space, satisfy commutation relations with a reversed sign, corresponding to a value y = - 1 for the ratio y which we introduced above, but without the restriction to I = 1. The gyromagnetic ratio of Co ++. Our first application is to gyromagnetic ratios. The fact that y # 1 might at first seem to imply that there is something anomalous about the orbital gyromagnetic ratio in a r4 or rs state. This is not really the case; it is still unity. When spin is included, the magnetic moment and total angular momentum vectors are respectively proportional to the projections of 1’ + 2s and of 1’ + Son J’. Hence, by the usual formulas for projecting angular momenta such as are used in deriving the Land6 and hyperfine g-factors, the gyromagnetic ratio at T = 0, where only one state
546
J.
is inhabited
H. VAN
VLECK
is seen to be
w + ])I+ se + 1)l+
r[J’(J’ + 1) + 2 Y[J’(J’ + 1) + 2 -
go’ =
2[J’(J’+ 1) + S(S + 1) - 21 (3) [J’(J’ + 1) + S(S + 1) - 21.
This ratio is not to be confused with the spectroscopic gJ,ZYLR7’
fl)
+ 2 -
+])I + XT'U $1) + w 2J'U'+ 1)
SF
splitting
factor
+I) - 21
(4)
At temperatures which are not small compared to the multiplet constant 1, a correction should be made for the second order Zeeman effect, i.e. the influence of the elements of magnetic moment and angular momentum nondiagonal in J’. Unlike the ordinary case, the angular momentum has matrix elements non-diagonal in J’, as it is yl + S rather than 1 + S. Furthermore the Boltzmann factors for upper states of the triplet J’ = S - 1, S, S + 1 are not negligible. (Unlike true multiplets, these pseudo-multiplets are regular rather than inverted in the right half of the iron group, and so have their narrowest interval at the bottom rather than top). When the influence of non-diagonal elements and upper states is included, the expression for the gyromagnetic ratio becomes
Cd(2J’+ l)[gJ~2J’(J’+ 1)+ g’ = Cr{(2J+ where Ep, hJ, =
gJr
l)[gdJ,J’(
Wy)-l(gJr--
y)(gJT--
2)]} e--E”iltT (5)
J’+ 1)+2(~y)-~(gJ~-y)[hJ~-l)]}e-“J”“”
are defined in (2), (4) and
r[J’(J’ + 1) + 2 - .‘G + 1)l+ [J’(J’ + 1) + W + 1) - 4 . 2J’tJ’ + 1)
(6)
Eq. (5) is established by extending Griffith’s procedure to include the expectation values of angular momentum as well as magnetic moment. In the limit AjkT < I, the expression (5) for g’ reduces, as one would expect, to g,’
=
4S(S
+ 1) + 2Y2
2S(S
+
(7)
1) + 2Y2
This limit is scarcely one that can be realized practically without other corrections not included in our model, for at high temperatures the influence of other cubic Stark states must be considered. When a little computation is essayed with (5), it turns out that the gyromagnetic ratio is a rather insensitive function of temperature. The situation in this regard is different from that of the etfective Bohr magneton number, which has been discussed by Griffith2 and by Cossee 7)*) (with formulas supplied by Pryce). The limiting values given by (3) and (7) are co++ go’ =
1.625 (y = -
2) go’ =
1.71 (y = -
1); Fe++ga’
=
1.75
co++ g,’
1.625 (y = -
4) g,’
1.79 (y = -
1) ; Fe++ g,’ =
1.86
=
f) The writer is indebted
to Dr. Cossee
=
for seeing this manuscript
in advance
of publication.
GYROMAGNETICRATIOOFCO
as Co++ (d7 4Fr4)
++AND
TAHN-TELLEREFFECTIN
and Fe++ (d6 SOTS) have respectively
Fe++
547
S = 3/2, 2, and
for their states of lowest spin-orbit energy have respectively J’ = +, 1. At intermediate temperatures, the value of g’ in some cases falls slightly outside the limits given by go’, g,’ but still the variation with temperatire is slow. So, especially in view of the paucity of gyromagnetic measurements, it does not seem worth while to give the explicit numerical formulas for the coefficients in (5) for Co++ and Fe++. They can of course be computed from (4) and (6), or more readily by a rather obvious adaptation of formulas for the susceptibility given by Griffith and by Cossee, who, unlike Griffith, does not make the specialization y = - 1. The susceptibility formulas involve essentially the numerator of (5), but the corresponding denominators are readily constructed. The values which we compute for room temperatures from (5) are g’ =
1.65 (Co++)
g’ =
1.73 (Fe++)
In obtaining these values we have supposed that the spin-orbit constant is about 15% lower in the solid than the gas, as some reduction in the solid is now a well-known effect. Also we have taken the round value y = - 413 for Co++ as a compromise between the two extremes - 1 .O and - 1.5, of which the latter is probably the more correct. With this choice of y, the values of g’ are practically the same at 0 and 300”. Cossee finds the best fit of susceptibility of Co0 embedded in the cubic crystal MgO is obtained with y = - 1.28, but Low’s 8) spectroscopic measurements indicate y = - 1.42. A crude estimate of crystalline fields by Finkelstein and Van Vleck 9) gave y = - 1.38. The only cobaltous paramagnetic materials for which gyromagnetic measurements are available are CoSO4 and CoCls. Here Sucksmith’s 10) early measurements yield g’ =
1.57(CoSO4)
g’ =
1.45 (CoCls)
with the value for the chloride probably less reliable than for the sulphate. Exact agreement with experiment cannot be expected, since in the cobaltous salts used by Sucksmi t h, the paramagnetic ion is not in an exactly cubic field. However, Sucksmith’s measurements agree very much better with our values calculated on the basis of such a field than with the Spin-only value 2. For FeS04 S u c ksmi t h finds g’ = 1.89 as a mean of four measurements which, however, have considerable scatter, ranging from 1.75 to 2.09. In this case the spin-only hypothesis works a little better than a cubic field. It is not surprising that the spin-only approximation is more satisfactory in Fe++ than in Co++, for Fe++ has about half as large a spin-orbit constant, and so spin-orbit interaction is more easily suppressed by the non-cubic portion of the field. In this connection it should be noted that the suscepti-
548
J. H. VAN
VLECK
-_____
bilities calculated by Griffith, while somewhat too high for experiment in Fe++ are rather too low in Co ++. This fact indicates that actually the role of the orbital moment in raising the susceptibility above the “spin only” value is more important in Co ++ than in Fe++. Incidentally, Griffith calculates his effective Bohr magneton number 4.7 for Co++ on the assumption y = -1. Use of y =1.3 (cf. Cossee 7) raises this number to 5.0, in somewhat better agreement with experiment (4.8 - 5.1). The Jahn-Teller effect an Fe ++. We now turn to our other application of the method of pseudo-angular momentum - that to the Jahn-Teller effect rr), especially in ferrous cubic salts in octahedral fields. The essence of the Jahn-Teller effect is that in symmetrical fields (in our case cubic) the degenerate levels, except for Kramers doublets, are in general not stable with respect to small distortions in the symmetry. The resonance spectra of paramagnetic ions embedded in MgO (a cubic crystal) are admirably adapted to the study of the J-T effect. Excellent experimental data of this character have been provided by Low. He finds the JahnTeller effect conspicuous by its absence. In the case of the Ni++, Mnff, and Fe+++ ions, this is certainly to be expected, for their ground states are void of any orbital degeneracy in a cubic crystalline field of coordination number 6, and the observed spectroscopic splitting factors g are nearly 2. In the case of Co++, this factor is experimentally 4.28, in very close agreement with the value 4.33 yielded by Eq. (4) for an unperturbed 4FI’a state with RS coupling in a perfectly cubic field. The small discrepancy is further reduced by considering such refinements as perturbations by other states, and partial breakdown of RS coupling. On the other hand, with a strong J-T effect the crystalline potential becomes asymmetrical because of the resulting distortion, the orbital moment is quenched, and g would be nearly 2. The explanation of Low’s measurements on Co+-+ lies in the fact that under certain conditions spin-orbit interaction can stabilize the energy, or stated more expressively, virtually wipe out the J-T effect, as has already been indicated by 0 pi k and Pryce 12) and by Low 8). That this is indeed the case in Co++ can be readily seen by the method of pseudoangular momentum. As y and il are both negative, the lowest state in Co++ has J’ = 4, and so only the Kramers double degeneracy which cannot be lifted by the J-T effects. There are, to be sure’, matrix elements linear in the vibrational coordinates which join different eigenvalues of the spin-orbit interaction, but their effect will be only a minor, second order one if this interaction is sufficiently large. Low 8) also finds that in Fe++, the spectroscopic splitting factor can be computed on the assumption of a perfectly cubic field, without any J-T effect. Eq. (4) with y = - 1, J’ = 1, S = 2 gives g = 3.5 and he finds experimentally g = 3.43. It is not so readily seen that the J-T effect does not reduce g to nearly 2 in Fe++. The lowest state has J’ = 1, and the linear
Co++ AND
GYROMAGNETICRATIOOF
J-T
potential
can have matrix
elements
Fe++
549
manifold
that
TAHN-TELLEREFFECTIN
inside this triplet
lift the degeneracy. The explanation is that these matrix elements, though non-vanishing, are very small. This fact was mentioned in a paper 1s) which the writer presented at the Dublin symposium of the Faraday Society, and we now present the details of the proof. The salient reduction factor 1/lo which we derive has also been obtained independently in unpublished work by Pryce and in an article in course of publication by Low and Weger, who use detailed wave functions rather than the labor-saving trick of pseudo-angular momentum. First let us consider only the vibrational coordinate Qa which gives a displacement along one of the principal cubic axes, as this is the simplest to treat. We use notation for the Q’s to conform with a previous paper by the writer la), which should be consulted if the reader desires to know the detailed geometry of the normal coordinates which we use. We begin by considering a D state in a system of representation in which the spin is decoupled and the spin-orbit energy neglected. The displacement Qa reduces the cubic symmetry to tetragonal, splitting the ra doublet and decomposing the I’4 triplet into a single and twofold level. This state of affairs can be expressed as E(DI'3)
Wr5)
= Eo(Dr5)
= +
Eo(Dr3)
~(3~2~ -
f
~93,
2)Qs
(8) (W=O,f
1)
(9)
where c(, a are constants. The factor (3~212 - 2) is clearly equivalent to the familiar harmonic 31,s - Z(Z+ 1) for a + state I = 1. Consider now the 5Dr5 state of Fe++. If the spin-orbit energy is sufficiently large, it is better to use the J’, MJ’ than the ml, MS system of representation. In the J’, MJ’ system, the energy inclusive of spin-orbit interaction will be ot the form Wr5)
=
+
Eo(Dr5)
‘4J’[3MJj2
+
-
B+(J’(J’ + 1) - 2 - -S(S+ I)] + J’(J’ + !)lQ3
(Mr
= - J’, . . . . + J’).
(10)
As I and y are both negative, the lowest state has J’ = 1. The ratio Ap/a can be computed by the method of pseudo-angular momentum and the invariance of diagonal sums. The point is that the problem is exactly like that of converting matrix elements of an axially symmetric second order potential from a MS, ML to a J, k!J system of representation for a 5P state, a type of problem treated almost thirty years ago by Penney and Schlappr5) for the rare earths, and subsequently elaborated and extensively tabulated by Stevens 16). One thus finds
A1
1
a
10
-=----*
(11)
It is the factor 1/IO in the coefficient of Qa in (10) as compared with (9) which, in our opinion, explains why the J-T effect is inoperative in Low’s Physica 26
550
J.
H. VAN
VLECK
experiments
on Fe+++ in MgO. It implies a reduction of J-T by much more than a factor 10. In the first place, if we add a potential AQ3 to a vibrational energy +CQs2, the energy change &AZ/C is quadratic in ,A. Furthermore, and this is the most important point, there are many different energetically equivalent J-T displacements, as Opik and Pryce 12) show (two, i.e. displacement up or down, for our simplified model that includes only Q3). The zero point energy will carry one “over the top” from one position to another, and the decomposition effect arising from the last term of (10) is virtually eliminated. The situation may be compared to that of a hypothetical ammonia molecule in which the potential barrier is so small that the tunnel doubling effect is lost because the energy vibrations carry one way above the hill from one valley to another. It is for this reason, indeed, that no observable J-T effect is found in the rare earths despite the high degree of degeneracy sometimes encountered there, and our point is that in Fe++ the factor l/ 10 makes the J-T potential more nearly typical of the rare earth than iron group in order of magnitude. The first case in the transition series where the J-T effect was definitely established is in CuSiFc*6HzO and even here it disappears at low temperatures where the crystal is distorted statically rather than dynamically. Abragam and Pryce 17) show that the magnetic behavior of CuSiFe.6H20 at higher temperatures is comprehensible only with the aid of the J-T effect. In Cu++ the deepest state is 2Dr3 so one must use (8) rather than (9) or (10). It therefore remains to compare the constant A1 in Fe++ with that a in Cu++. If the atomic potentials are the same, and we have ideal R-S coupling, so that <~a>, are the same throughout the d shell in a given degree of ionization, then the constant a is the same for d, d6, and simply inverted in sign for da, dg. The same is also true of a. The ratio a/a can be shown 14) to be -
-4d6)
4dg)
44 = -__
44
=
ifI - $f2
where fl and f2 are the fractions of the decomposition 2a of 2Dr3 arising from the second and fourth order harmonics of the axial crystalline potential generated by the displacement Q3. (Note that fl + fz = 1, but one can conceivably have fl > 1) As presumably fz is appreciably smaller than fl, we see that probably Al/a is even smaller than l/10. Furthermore, the 2Dr3 level is void of an orbital magnetic moment except insofar as there are perturbations by other levels, and so there is no spin-orbit interaction to stabilize things and wash out the J-T effect. It is thus no longer a puzzle that the J-T effect is operative in CUSiF6*6H2O and still does not make itself felt in Low’s experiments. Our discussion has been incomplete in that so far we have considered only a particular type Q3 of possible J-T displacement. There are other
GYROMAGNETIC
vibrations
OF CO ++ AND
RATIO
JAHN-TELLER
EFFECT
Qs, Q4, Qs, Q6 which give rise to potentials
4,21Q2, b(L& + 4AQ4,
1/3a[lX2 -
V&
+
IN
Fe++
551 I.
of the structure
WQ5,
b&l,
+
L4Qe
in DI’, where lz, I,, I, are angular momentum matrices for a p state; the matrix elements in the J’, rn~, system of representation are correspondingly d3A J,[Jz’2
-
Jy'21Q2,(AJ,/a) Wz'Jy' + Jv'J~')Q~,
(&,/a)4 Jy’Jz’ + Jz'Jy')Q59@b/a) Wz'Jz' + JzrJzj)Q6 as the proportionality factor in passing from one to the other system of representation is the same for all harmonics of a given degree. The essential factor 1/lo in Fe++ thus still remains when one includes other types of displacements. The splitting effect of Qs on Dra is essentially similar to that of Qs. However, Q4, Q5, Qe do not contribute to the decomposition of Drs, and the J-T effect of CuSiFs must be attributed to the Qs, Qa type of displacement. It is perhaps of interest to examine the possibility of a J-T in resonance experiments on the first half of the iron series. Cr+f resembles Cu++, in having no orbital moment in first approximation, and so may have a J-T effect. For Ti+++(d 2Drs) and V +++(d23Fr4) the spin-orbit constant 3, is positive and their lowest states have respectively J’ = 312, J’ = 2. The corresponding values of the constant AJ’ in (10) are A 312
=
+a
One can also show that, neglecting
W2) 44
--
= -
As = &a differences
in screening
if1 + &fz.
In this connection it should be noted that the J-T effectiveness is perhaps best gauged by the overall-splitting generated by MJ, in (10) which is 3Jis AJ, rather than Ap, so the reduction effect may be less than in the case of Fe++ because of the large value of J’. It would therefore not be surprising if Ti+++ or V+++ should show appreciable J-T effects. Possibly this effect may be the explanation of why their magnetic susceptibilities cannot be described by a consistent static crystalline field theory 13). Received 4-4-60 REFERENCES
1) 4
Abragam, A., and Pryce, M., Proc. ray. Sm. A. PO5 (1951) 135. Griffith, J. S.,Trans. Far. Sm. 54 (1958) 1109.
3)
Moffitt, W., Goodman, G. L., Fred, M., and Wenstock, B., Molecular Physics 2 (1959) 109; cf. also Moffitt, W., and Thorson, W., Phys. Rev. l@#F (1957) 1251.
4) Van Vleck, J. H., Rev. Mod. Phys. 23 (1951) 213. 5) Klein, O., Zs. f. Physik 58 (1928) 730.
552
GYROMAGNETIC
RATIO
OF CO ++ AND JAHN-TELLER
EFFECT
IN
Fe++
6) Casimir, H. B. G., Rotation of a Rigid Body in Quantum Mechanics, Dissertation, Leiden, 1931. 7) Cossee, P., Molecular Physics, 3 (1960) 25. 8) Low, W., Phys. Rev. 101 (1956) 1827; 100 (1958) 256. 9) Finkelstein, R. and Van Vleck, J. H., J. Chem. Phys. 8 (1940) 797. footnote 21. 10) Sucksmith, W., Proc. roy. Sot. A 133 (1931) 179. 11) Jahn, H. A. and Teller, E., Proc. roy. Sot. 161 (1937)220; Jahn, H. A., ibid. 164 (1937) 117. 12) Opik, U., and Pryce, M. H. L., Proc. roy. Sot. A, e58 (1957) 425; 11. H. L., and Sack, R. A., Proc. ray, Sot. A. Longuet-Higgins, H. C., Opik, U., Pryce, i?44 (1958) 1. 13) Van Vleck, J. H., Far. Sot. Discussions 1958, p. 96. 14) Van Vleck, J. H., J. Chem. Phys., 7 (1939) 79. 15) Penney, W. G., and Schlapp, R., Phys. Rev. 41 (1932) 194; Van Vleck, J. h., ibid. 41 (1932) 208. 16) Stevens, K. W. H., Proc. Phys. Sot. Lon., 65A (1952) 209. 17) Abragam, .4. and Pryce, M. H. L., Proc. Phys. Sot. Lon., 63A (1949) 409.
J. H.
Physica
1960
26
544-552
NOTE ON THE GYROMAGNETIC RATIO OF CO++ AND ON THE JAHN-TELLER EFFECT IN Fe++ J.
H. VAN
VLECK
Physics Department,
Harvard
University,
of pseudo-angular
momentum
U.S.A. *)
Synopsis By the method that in a triply
degenerate
the same commutation method value
thus
obtained
these materials Teller
the gyromagnetic
is 1.65, in better
do not have
effect,
is included; coefficients
agreement
cubic symmetry.
fluosilicate. however,
factor.
obeys This
with
Sucksmith’s
spin-only
experimental
value g = 2, even though application
is to the Jahnof this effect
on Fe++, .and at the same time this mechanism
At first sight, one would level
for a proportionality
The other
role in the Abragam-Pryce
as its lowest
of the fact
angular momentum
is given why Low finds no evidence
resonance measurements
plays an important of copper
the utilization
ratio of a Co++ ion in a cubic field. The
on CoSO4 and CoCls than is the
effect in Fe++. An explanation
in his magnetic
J-T
rules as for a t, state, except
is used to calculate
measurements
is meant
level in a cubic field, the orbital
is triply
theory
of the magnetic
expect
the Fe++ ion to show a large
degenerate
even
the linear terms in the distortional
when
susceptibility
spin-orbit
coordinates
interaction
have very
small
and so are ineffective.
Introduction. In a perfectly cubic crystalline or ligand field, certain orbital levels, called Ti or Tz in Mulliken’s terminology, or rq, rs in Rethe’s, are triply degenerate. In the present paper, we shall be concerned only with matrix elements which are internal to such a triplet, or in other words deal only with the projection on three particular substates. Then, except for a proportionality factor, the matrix elements of the so-projected orbital angular momentum vector whose components we denote by l’%, lU’, lz’ are the same as those of the angular momentum vector for an atomic p state. Consequently, if angular momenta are measured in multiples of h/2n, WC have 1,‘s + lg’2 + 129 = 2ys &‘I, - l,‘l,’ = iyl,‘, etc., This resemblance was first pointed out by Abragam appropriate values of y can be shown to be Dr5
FY,
Fr5
-#
+*
y=-1 *)
Temporarily
at the Instituut
Lorentz
der Rijksuniversiteit,
-
544
-
and Pryce
1). The
(1) Leiden,
Nederland.
GYROMAGNETIC
The value
-
RATIO
+ given
OF co++
for Fr4
AND
JAHN-TELLER
EFFECT
IN Fe++
545
appliez to Co++(d7 4F) only if there is strong
Russell-Sounders coupling; actually the crystalline field will mix in some Pr4, and so reduce (71. In the limiting case that the field is very large compared to the R-S interaction, the value of y is - 1. Corresponding corrections are absent for the ferrous configuration d6, as there is only one state of maximum multiplicity. When spin is included, the need of writing down explicit wave functions can be avoided for many purposes by utilizing the fact that the problem of finding the proper linear combinations is equivalent to compounding ot spin and orbital angular momenta in a fi state. The resulting states can be described by a quantum number J’ which assumes the values J’ = 1s -
11, s, s +
1
We may term J’ a pseudo-inner quantum number, the true angular momentum. The spin orbit-energy EJ~ = +yn [J’(J’
+
1) -
S(S +
as it does not measure is given by 1) -
21
(4
where L is the ordinary atomic multiplet constant, The levels in the absence of an applied magnetic field are 2J’ + 1 degenerate, but the degeneracy will be reduced by interaction with other states when J’ > 1. This procedure has been discovered and used by Griffith 2) to simplify the calculation of magnetic susceptibilities in a perfectly cubic field, and also independently by Mof f i t t and collaborators 3) in a study of the colors of transition metal fluorides. The reader is referred to their papers for explanation of the method in greater detail. The writer has also hit upon the same trick in connection with two other applications, which it is the purpose of the present paper to describe. Before doing so, it may be noted parenthetically that the method of pseudoangular momentum vectors, to coin a name for the procedure, is in some ways like that which we devised for treating molecular coupling problems 4). As Klein 5) and Casimir 6) have shown, the angular momentum matrices, when projected on axes fixed in the molecule rather than space, satisfy commutation relations with a reversed sign, corresponding to a value y = - 1 for the ratio y which we introduced above, but without the restriction to I = 1. The gyromagnetic ratio of Co ++. Our first application is to gyromagnetic ratios. The fact that y # 1 might at first seem to imply that there is something anomalous about the orbital gyromagnetic ratio in a r4 or rs state. This is not really the case; it is still unity. When spin is included, the magnetic moment and total angular momentum vectors are respectively proportional to the projections of 1’ + 2s and of 1’ + Son J’. Hence, by the usual formulas for projecting angular momenta such as are used in deriving the Land6 and hyperfine g-factors, the gyromagnetic ratio at T = 0, where only one state
546
J.
is inhabited
H. VAN
VLECK
is seen to be
w + ])I+ se + 1)l+
r[J’(J’ + 1) + 2 Y[J’(J’ + 1) + 2 -
go’ =
2[J’(J’+ 1) + S(S + 1) - 21 (3) [J’(J’ + 1) + S(S + 1) - 21.
This ratio is not to be confused with the spectroscopic gJ,ZYLR7’
fl)
+ 2 -
+])I + XT'U $1) + w 2J'U'+ 1)
SF
splitting
factor
+I) - 21
(4)
At temperatures which are not small compared to the multiplet constant 1, a correction should be made for the second order Zeeman effect, i.e. the influence of the elements of magnetic moment and angular momentum nondiagonal in J’. Unlike the ordinary case, the angular momentum has matrix elements non-diagonal in J’, as it is yl + S rather than 1 + S. Furthermore the Boltzmann factors for upper states of the triplet J’ = S - 1, S, S + 1 are not negligible. (Unlike true multiplets, these pseudo-multiplets are regular rather than inverted in the right half of the iron group, and so have their narrowest interval at the bottom rather than top). When the influence of non-diagonal elements and upper states is included, the expression for the gyromagnetic ratio becomes
Cd(2J’+ l)[gJ~2J’(J’+ 1)+ g’ = Cr{(2J+ where Ep, hJ, =
gJr
l)[gdJ,J’(
Wy)-l(gJr--
y)(gJT--
2)]} e--E”iltT (5)
J’+ 1)+2(~y)-~(gJ~-y)[hJ~-l)]}e-“J”“”
are defined in (2), (4) and
r[J’(J’ + 1) + 2 - .‘G + 1)l+ [J’(J’ + 1) + W + 1) - 4 . 2J’tJ’ + 1)
(6)
Eq. (5) is established by extending Griffith’s procedure to include the expectation values of angular momentum as well as magnetic moment. In the limit AjkT < I, the expression (5) for g’ reduces, as one would expect, to g,’
=
4S(S
+ 1) + 2Y2
2S(S
+
(7)
1) + 2Y2
This limit is scarcely one that can be realized practically without other corrections not included in our model, for at high temperatures the influence of other cubic Stark states must be considered. When a little computation is essayed with (5), it turns out that the gyromagnetic ratio is a rather insensitive function of temperature. The situation in this regard is different from that of the etfective Bohr magneton number, which has been discussed by Griffith2 and by Cossee 7)*) (with formulas supplied by Pryce). The limiting values given by (3) and (7) are co++ go’ =
1.625 (y = -
2) go’ =
1.71 (y = -
1); Fe++ga’
=
1.75
co++ g,’
1.625 (y = -
4) g,’
1.79 (y = -
1) ; Fe++ g,’ =
1.86
=
f) The writer is indebted
to Dr. Cossee
=
for seeing this manuscript
in advance
of publication.
GYROMAGNETICRATIOOFCO
as Co++ (d7 4Fr4)
++AND
TAHN-TELLEREFFECTIN
and Fe++ (d6 SOTS) have respectively
Fe++
547
S = 3/2, 2, and
for their states of lowest spin-orbit energy have respectively J’ = +, 1. At intermediate temperatures, the value of g’ in some cases falls slightly outside the limits given by go’, g,’ but still the variation with temperatire is slow. So, especially in view of the paucity of gyromagnetic measurements, it does not seem worth while to give the explicit numerical formulas for the coefficients in (5) for Co++ and Fe++. They can of course be computed from (4) and (6), or more readily by a rather obvious adaptation of formulas for the susceptibility given by Griffith and by Cossee, who, unlike Griffith, does not make the specialization y = - 1. The susceptibility formulas involve essentially the numerator of (5), but the corresponding denominators are readily constructed. The values which we compute for room temperatures from (5) are g’ =
1.65 (Co++)
g’ =
1.73 (Fe++)
In obtaining these values we have supposed that the spin-orbit constant is about 15% lower in the solid than the gas, as some reduction in the solid is now a well-known effect. Also we have taken the round value y = - 413 for Co++ as a compromise between the two extremes - 1 .O and - 1.5, of which the latter is probably the more correct. With this choice of y, the values of g’ are practically the same at 0 and 300”. Cossee finds the best fit of susceptibility of Co0 embedded in the cubic crystal MgO is obtained with y = - 1.28, but Low’s 8) spectroscopic measurements indicate y = - 1.42. A crude estimate of crystalline fields by Finkelstein and Van Vleck 9) gave y = - 1.38. The only cobaltous paramagnetic materials for which gyromagnetic measurements are available are CoSO4 and CoCls. Here Sucksmith’s 10) early measurements yield g’ =
1.57(CoSO4)
g’ =
1.45 (CoCls)
with the value for the chloride probably less reliable than for the sulphate. Exact agreement with experiment cannot be expected, since in the cobaltous salts used by Sucksmi t h, the paramagnetic ion is not in an exactly cubic field. However, Sucksmith’s measurements agree very much better with our values calculated on the basis of such a field than with the Spin-only value 2. For FeS04 S u c ksmi t h finds g’ = 1.89 as a mean of four measurements which, however, have considerable scatter, ranging from 1.75 to 2.09. In this case the spin-only hypothesis works a little better than a cubic field. It is not surprising that the spin-only approximation is more satisfactory in Fe++ than in Co++, for Fe++ has about half as large a spin-orbit constant, and so spin-orbit interaction is more easily suppressed by the non-cubic portion of the field. In this connection it should be noted that the suscepti-
548
J. H. VAN
VLECK
-_____
bilities calculated by Griffith, while somewhat too high for experiment in Fe++ are rather too low in Co ++. This fact indicates that actually the role of the orbital moment in raising the susceptibility above the “spin only” value is more important in Co ++ than in Fe++. Incidentally, Griffith calculates his effective Bohr magneton number 4.7 for Co++ on the assumption y = -1. Use of y =1.3 (cf. Cossee 7) raises this number to 5.0, in somewhat better agreement with experiment (4.8 - 5.1). The Jahn-Teller effect an Fe ++. We now turn to our other application of the method of pseudo-angular momentum - that to the Jahn-Teller effect rr), especially in ferrous cubic salts in octahedral fields. The essence of the Jahn-Teller effect is that in symmetrical fields (in our case cubic) the degenerate levels, except for Kramers doublets, are in general not stable with respect to small distortions in the symmetry. The resonance spectra of paramagnetic ions embedded in MgO (a cubic crystal) are admirably adapted to the study of the J-T effect. Excellent experimental data of this character have been provided by Low. He finds the JahnTeller effect conspicuous by its absence. In the case of the Ni++, Mnff, and Fe+++ ions, this is certainly to be expected, for their ground states are void of any orbital degeneracy in a cubic crystalline field of coordination number 6, and the observed spectroscopic splitting factors g are nearly 2. In the case of Co++, this factor is experimentally 4.28, in very close agreement with the value 4.33 yielded by Eq. (4) for an unperturbed 4FI’a state with RS coupling in a perfectly cubic field. The small discrepancy is further reduced by considering such refinements as perturbations by other states, and partial breakdown of RS coupling. On the other hand, with a strong J-T effect the crystalline potential becomes asymmetrical because of the resulting distortion, the orbital moment is quenched, and g would be nearly 2. The explanation of Low’s measurements on Co+-+ lies in the fact that under certain conditions spin-orbit interaction can stabilize the energy, or stated more expressively, virtually wipe out the J-T effect, as has already been indicated by 0 pi k and Pryce 12) and by Low 8). That this is indeed the case in Co++ can be readily seen by the method of pseudoangular momentum. As y and il are both negative, the lowest state in Co++ has J’ = 4, and so only the Kramers double degeneracy which cannot be lifted by the J-T effects. There are, to be sure’, matrix elements linear in the vibrational coordinates which join different eigenvalues of the spin-orbit interaction, but their effect will be only a minor, second order one if this interaction is sufficiently large. Low 8) also finds that in Fe++, the spectroscopic splitting factor can be computed on the assumption of a perfectly cubic field, without any J-T effect. Eq. (4) with y = - 1, J’ = 1, S = 2 gives g = 3.5 and he finds experimentally g = 3.43. It is not so readily seen that the J-T effect does not reduce g to nearly 2 in Fe++. The lowest state has J’ = 1, and the linear
Co++ AND
GYROMAGNETICRATIOOF
J-T
potential
can have matrix
elements
Fe++
549
manifold
that
TAHN-TELLEREFFECTIN
inside this triplet
lift the degeneracy. The explanation is that these matrix elements, though non-vanishing, are very small. This fact was mentioned in a paper 1s) which the writer presented at the Dublin symposium of the Faraday Society, and we now present the details of the proof. The salient reduction factor 1/lo which we derive has also been obtained independently in unpublished work by Pryce and in an article in course of publication by Low and Weger, who use detailed wave functions rather than the labor-saving trick of pseudo-angular momentum. First let us consider only the vibrational coordinate Qa which gives a displacement along one of the principal cubic axes, as this is the simplest to treat. We use notation for the Q’s to conform with a previous paper by the writer la), which should be consulted if the reader desires to know the detailed geometry of the normal coordinates which we use. We begin by considering a D state in a system of representation in which the spin is decoupled and the spin-orbit energy neglected. The displacement Qa reduces the cubic symmetry to tetragonal, splitting the ra doublet and decomposing the I’4 triplet into a single and twofold level. This state of affairs can be expressed as E(DI'3)
Wr5)
= Eo(Dr5)
= +
Eo(Dr3)
~(3~2~ -
f
~93,
2)Qs
(8) (W=O,f
1)
(9)
where c(, a are constants. The factor (3~212 - 2) is clearly equivalent to the familiar harmonic 31,s - Z(Z+ 1) for a + state I = 1. Consider now the 5Dr5 state of Fe++. If the spin-orbit energy is sufficiently large, it is better to use the J’, MJ’ than the ml, MS system of representation. In the J’, MJ’ system, the energy inclusive of spin-orbit interaction will be ot the form Wr5)
=
+
Eo(Dr5)
‘4J’[3MJj2
+
-
B+(J’(J’ + 1) - 2 - -S(S+ I)] + J’(J’ + !)lQ3
(Mr
= - J’, . . . . + J’).
(10)
As I and y are both negative, the lowest state has J’ = 1. The ratio Ap/a can be computed by the method of pseudo-angular momentum and the invariance of diagonal sums. The point is that the problem is exactly like that of converting matrix elements of an axially symmetric second order potential from a MS, ML to a J, k!J system of representation for a 5P state, a type of problem treated almost thirty years ago by Penney and Schlappr5) for the rare earths, and subsequently elaborated and extensively tabulated by Stevens 16). One thus finds
A1
1
a
10
-=----*
(11)
It is the factor 1/IO in the coefficient of Qa in (10) as compared with (9) which, in our opinion, explains why the J-T effect is inoperative in Low’s Physica 26
550
J.
H. VAN
VLECK
experiments
on Fe+++ in MgO. It implies a reduction of J-T by much more than a factor 10. In the first place, if we add a potential AQ3 to a vibrational energy +CQs2, the energy change &AZ/C is quadratic in ,A. Furthermore, and this is the most important point, there are many different energetically equivalent J-T displacements, as Opik and Pryce 12) show (two, i.e. displacement up or down, for our simplified model that includes only Q3). The zero point energy will carry one “over the top” from one position to another, and the decomposition effect arising from the last term of (10) is virtually eliminated. The situation may be compared to that of a hypothetical ammonia molecule in which the potential barrier is so small that the tunnel doubling effect is lost because the energy vibrations carry one way above the hill from one valley to another. It is for this reason, indeed, that no observable J-T effect is found in the rare earths despite the high degree of degeneracy sometimes encountered there, and our point is that in Fe++ the factor l/ 10 makes the J-T potential more nearly typical of the rare earth than iron group in order of magnitude. The first case in the transition series where the J-T effect was definitely established is in CuSiFc*6HzO and even here it disappears at low temperatures where the crystal is distorted statically rather than dynamically. Abragam and Pryce 17) show that the magnetic behavior of CuSiFe.6H20 at higher temperatures is comprehensible only with the aid of the J-T effect. In Cu++ the deepest state is 2Dr3 so one must use (8) rather than (9) or (10). It therefore remains to compare the constant A1 in Fe++ with that a in Cu++. If the atomic potentials are the same, and we have ideal R-S coupling, so that <~a>,
-4d6)
4dg)
44 = -__
44
=
ifI - $f2
where fl and f2 are the fractions of the decomposition 2a of 2Dr3 arising from the second and fourth order harmonics of the axial crystalline potential generated by the displacement Q3. (Note that fl + fz = 1, but one can conceivably have fl > 1) As presumably fz is appreciably smaller than fl, we see that probably Al/a is even smaller than l/10. Furthermore, the 2Dr3 level is void of an orbital magnetic moment except insofar as there are perturbations by other levels, and so there is no spin-orbit interaction to stabilize things and wash out the J-T effect. It is thus no longer a puzzle that the J-T effect is operative in CUSiF6*6H2O and still does not make itself felt in Low’s experiments. Our discussion has been incomplete in that so far we have considered only a particular type Q3 of possible J-T displacement. There are other
GYROMAGNETIC
vibrations
OF CO ++ AND
RATIO
JAHN-TELLER
EFFECT
Qs, Q4, Qs, Q6 which give rise to potentials
4,21Q2, b(L& + 4AQ4,
1/3a[lX2 -
V&
+
IN
Fe++
551 I.
of the structure
WQ5,
b&l,
+
L4Qe
in DI’, where lz, I,, I, are angular momentum matrices for a p state; the matrix elements in the J’, rn~, system of representation are correspondingly d3A J,[Jz’2
-
Jy'21Q2,(AJ,/a) Wz'Jy' + Jv'J~')Q~,
(&,/a)4 Jy’Jz’ + Jz'Jy')Q59@b/a) Wz'Jz' + JzrJzj)Q6 as the proportionality factor in passing from one to the other system of representation is the same for all harmonics of a given degree. The essential factor 1/lo in Fe++ thus still remains when one includes other types of displacements. The splitting effect of Qs on Dra is essentially similar to that of Qs. However, Q4, Q5, Qe do not contribute to the decomposition of Drs, and the J-T effect of CuSiFs must be attributed to the Qs, Qa type of displacement. It is perhaps of interest to examine the possibility of a J-T in resonance experiments on the first half of the iron series. Cr+f resembles Cu++, in having no orbital moment in first approximation, and so may have a J-T effect. For Ti+++(d 2Drs) and V +++(d23Fr4) the spin-orbit constant 3, is positive and their lowest states have respectively J’ = 312, J’ = 2. The corresponding values of the constant AJ’ in (10) are A 312
=
+a
One can also show that, neglecting
W2) 44
--
= -
As = &a differences
in screening
if1 + &fz.
In this connection it should be noted that the J-T effectiveness is perhaps best gauged by the overall-splitting generated by MJ, in (10) which is 3Jis AJ, rather than Ap, so the reduction effect may be less than in the case of Fe++ because of the large value of J’. It would therefore not be surprising if Ti+++ or V+++ should show appreciable J-T effects. Possibly this effect may be the explanation of why their magnetic susceptibilities cannot be described by a consistent static crystalline field theory 13). Received 4-4-60 REFERENCES
1) 4
Abragam, A., and Pryce, M., Proc. ray. Sm. A. PO5 (1951) 135. Griffith, J. S.,Trans. Far. Sm. 54 (1958) 1109.
3)
Moffitt, W., Goodman, G. L., Fred, M., and Wenstock, B., Molecular Physics 2 (1959) 109; cf. also Moffitt, W., and Thorson, W., Phys. Rev. l@#F (1957) 1251.
4) Van Vleck, J. H., Rev. Mod. Phys. 23 (1951) 213. 5) Klein, O., Zs. f. Physik 58 (1928) 730.
552
GYROMAGNETIC
RATIO
OF CO ++ AND JAHN-TELLER
EFFECT
IN
Fe++
6) Casimir, H. B. G., Rotation of a Rigid Body in Quantum Mechanics, Dissertation, Leiden, 1931. 7) Cossee, P., Molecular Physics, 3 (1960) 25. 8) Low, W., Phys. Rev. 101 (1956) 1827; 100 (1958) 256. 9) Finkelstein, R. and Van Vleck, J. H., J. Chem. Phys. 8 (1940) 797. footnote 21. 10) Sucksmith, W., Proc. roy. Sot. A 133 (1931) 179. 11) Jahn, H. A. and Teller, E., Proc. roy. Sot. 161 (1937)220; Jahn, H. A., ibid. 164 (1937) 117. 12) Opik, U., and Pryce, M. H. L., Proc. roy. Sot. A, e58 (1957) 425; 11. H. L., and Sack, R. A., Proc. ray, Sot. A. Longuet-Higgins, H. C., Opik, U., Pryce, i?44 (1958) 1. 13) Van Vleck, J. H., Far. Sot. Discussions 1958, p. 96. 14) Van Vleck, J. H., J. Chem. Phys., 7 (1939) 79. 15) Penney, W. G., and Schlapp, R., Phys. Rev. 41 (1932) 194; Van Vleck, J. h., ibid. 41 (1932) 208. 16) Stevens, K. W. H., Proc. Phys. Sot. Lon., 65A (1952) 209. 17) Abragam, .4. and Pryce, M. H. L., Proc. Phys. Sot. Lon., 63A (1949) 409.