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The optical spectrum and ground-state splitting of Mn2+ and Fe3+ lons in the crystal field of cubic symmetry
JOURNALOF MOLECULARSPECTROSCOPY12, 319-346 (1964)
The Optical Spectrum and Ground-State Splitting of Mn*’ and Fe3+ Ions in the Crystal Field of Cubic Symmetry W. Low AND Department
of Physics,
G.
ROSENGARTEN
The Hebreul Vnirtersity,
Jerusalem,
Israel
We ha,ve calculated the energy levels and ground-state splitting I’&, for Mn”+ and Fez+ in a cubic field. These calculations consist of t,he diagonalization of the various rj matrices of the complete (d5) configuration including cubic crystal field and the spin-orbit coupling. The parameters are chosen to fit the optical spectra of Mn’+ in crystals of MnF? and MnClz and hydrates as well as the spectra of Fe3+ in hydrat,es. It is found that the spectra ran be fitted with an rms error of a few hundred cm-‘. The eigenfunctions of these levels are obtained. The observed ground-stat)e splitting for Mn2+ and Fe3+ in hydrates can be accounted for if the value of the spin-rjrbit coupling in t,he ergstal is larger than that of the free ion. Similarly the calculated g factor deviat,es from the experimentally found value. Possible reasons for the discrepancies are given.
During the last few years crystal field theory has been used to great advant.age in interpreting the magnetic behavior (I ) and the optical spectra (2) of pammagnetic ions in crystals. The theory uses a conventional central field Hamiltonian and adds to it a perturbing potential, expanded in spherical harmonies, called the crystal field potential. This potential is supposed to act only on the elect,rons in the unfilled shell. The effect of the crystal field potential is to split the degeneracy of each ionic level into a number of Stark levels. The number of t’hese levels is dependent on the point symmetry at the site of the paramagnetic ion. The separation of these levels depends on the electrostat#ic parameters B, (‘, and on the crystal field strength parameters (called Dq in case of cubic symmetry). It is found esperimentally that the separation of various Stark splittings in the irou group elements are of the same magnitude as the term separations of the free ion. Diagonalizations of t’he matsrices rather than perturbation theory are, therefore, necessary in order to correlate the theory with experiments. The matrices in the stroug field case have been calculated (3, 4) for the cubic field symmetry. Recent data have shown that in some cases fine st~ructure of lines can be resolved. This additional st.ructure has been at,tributed t.o perturbation by t.he 319
320
LOW AND
ROSENGARTEN
spin-orbit coupling & It is, therefore, necessary to calculate matrices which include the spin-orbit coupling. Schonfeld (5, 6) has constructed the matrices for the cubic field and spin-orbit coupling for the two cases of d2 and d3. She has shown the correlation with the experimental data can be improved if the spinorbit coupling parameter is taken into account. We report an extension of these calculations to the case d5.’ While this work has been completed for some time, Gabriel, Johnston, and Powell (8) have published an important paper on the ground-state splitting of Mn (III) (d5) ions in a cubic field. (This paper will be referred to as G.J.P.) They calculate this splitting and part of the optical spectrum for the (d5)6Ss,z configuration as a function of the Racah parameters B and C, Dq, <, and the spin-spin parameters MO and Mt . Our method of calculation differs somewhat from theirs and sheds some additional light on the subject of the initial splitting of the ground state of the Mnzf as well as of the Fe3+ ions. It will be shown in particular that good correlation with optical spectra can be obtained without introducing additional covalency parameters. OUTLINE
OF THEORY
AND
METHOD
OF CALCULATION
We shall outline briefly the basis of our calculation. The Appendix gives the matrix elements used in this calculation and the method of computation. The matrices as well as the transformation matrices will be published elsewhere. The starting point is the assumption that the conventional atomic Hamiltonian can be used and that the crystal field perturbation does not affect the spherical symmetrical part of the atomic Hamiltonian. Other assumptions are that the interaction between different configurations is negligible, and that the charge transfer which usually occurs in the ultraviolet, does not appreciably influence the relative separation of the energy levels. In calculating the matrix elements it is conventional to use two extreme representations: the strong field and the weak field limits. Both methods have advantages and disadvantages. The strong field representation classifies the d electrons into de and d-y orbitals. The crystal field part is diagonal in these matrices and the electrostatic interaction off-diagonal. It has the advantages that one can in principle allow the de and dy orbitals to have different radial extension and different amounts of covalent bonding. This method has been used by Stout (9) [originally suggested by Koide and Pryce (IO)]. Stout claims that by introducing a new parameter, called the covalency parameter, better agreement between theory and experiment can be obtained. This method permits the introduction of an anisotropic spin-orbit coupling parameter. However it suffers from the disadvantage that it does not utilize the electrostatic matrices as well as the spin-orbit coupling matrices already available in the literature. 1The results of calculations Conference (7).
for d6 have been presented
at the IInd Quantum
Electronics
OPTICAL
SPECTRUM
AND
GROUND-STATE
SPLITTING
321
The weak field limit, however, takes advantage of the existing electrostatic and spin-orbit matrices. It uses further the existing computer programs for the diagonalization of the matrices and the possibility of obtaining the appropriate eigenvectors. The matrices contain, therefore, 5 parameters B, C, which are the linear cornbination of the Slater integrals , $ the spin-orbit coupling in the crystal, Q the Trees’ correction which is of importance in correlating the optical spectra, and finally Dq a measure of the cubic crystal field strength. We have neglected in this calculation the spin-spin interaction which would introduce two additional parameters. The spin-spin interaction affects only very slightly the position of the various excited levels. It affects somewhat more the initial splitting of the ground state but not very significantly as shown by G.J.P. The matrices are classified according to the irreducible representation of the double cubic group. The matrix I?* is the largest. The dimensions are for the doublets 1’8:25 X 25, rs:12 X 12, J?,: 13 X 13 and for the quartets J?,: 16 X 16, r’,: 8 X 8, I’,: 8 X 8. These were diagonalized on the Weizac computer in the following manner. We chose a set of parameters for the 5 variables, B, C, 4, LY,and Dq which by inspection will fit reasonably to the experimental data. The parameters B, C, [, and (Y cannot be taken as the free ion value since these parameters differ in various crystals (6). The free ion values which are the upper limit of these parameters have been calculated by Shadmi ( 11) . They are in units of cm-’ for AIn’+: B = 910, C = 3270, cr = 76, t = 320. For Fe3+: B = 1100, C = 3750, a = 90, E = 420. These values differ from those taken by Gabriel et al. who use Watson’s wave function ( 12). Their values for Mn’+: B = 900, C = 3300, CY= 0, F = 376, and for Fe3+:B = 1100, (’ = 400, (Y = 0, 4 = 440, all in cn-I. The difference in these values is significant since the initial splitting 3a is a funct#ion of all the five parameters. After obtaining a first, set of energy levels from our initial choice of parameters, we apply a “least square” technique (denoted in the tables as L-S) to find improved parameters which will give the best overall fit to the experimental spectrum. The least square procedure is meaningless unless the number of parameters is small compared with the number of “distinct” experimentally observed transit,ions. In the cases discussed below, the experimental data are not sufficient and it was found convenient to fix two parameters (Yand & The Trees’ correction was arbitrarily fixed at the free ion value. The matrices were diagonalized for a nunber of values of [. The optical spectrum itself is not very sensitive to the value of E but of course the initial splitting 3a is a sensitive function of $. We shall briefly review the main aspects where our calculations differ from those of G.J.P. 1. We use the same Hamiltonian as the authors. The largest matrix which is obt’ained is 42 X 42. G.J.P. used a convergent perturbation technique carried to
322
LOW AND
ROSENGARTEN
sixth order. It had to be carried to this order to find the magnitude of the groundstate splitting. We diagonalized the 42 X 42 matrix and obtained the eigenvalues. The results of the magnitude of the splitting of these two procedures differ significantly. Presumably the perturbation technique does not converge sufficiently rapidly. The ground-state splitting I’-I’~ depends on a sensitive manner on the admixture from higher excited states, We have been unable to obtain the eigenvectors of the lYslevels because of the limited high speed memory of the computer. We obtained, however, the eigenvectors for the I6 and r7 levels. We assume that the admixture of the rs levels and the I’6 and p7 levels coming from the same rL is not very different. This seems to be true in the case of the Ps and r, levels where the admixture is nearly the same, and is reasonable since the spin-orbit coupling is only a small perturbation in these calculations. 2. G.J.P. obtain the electrostatic parameters by trying to fit the optical spectrum to the transition rl-r3 which is nearly independent of the crystal field strength. The value 1OB + 5C is taken from the experimentally observed optical spectrum of the 1’1-1’3 transition. They proceed t’o calculate the initial splitting in a number of tables as a function of C/B, Dq, and 4. We have tried to get the best ovedljt to the optical spectrum for each individual crystal host by choosing a set of values for B, C, and Dq which will give the least square deviation from the experimental values. Obviously the transitions of higher wave numbers are more sensitive to the accurate choice of these parameters. The diagonalization of these matrices was made with one variable parameter .$ in order to fit the ground-state splitting. 3. We use five parameters including the Trees’ correction CLIt turns out that the Trees’ correction is important in reducing the overall error in the parameters B and C. On the other hand we do not use the spin-spin correction which G. J. P. include in some of bheir calculations. 4. We obtain in our calculation the admixture of the wave function, and in particular the admixture of the ground state by the P and G levels. We are therefore, able to calculate the g factor. 5. We extend these calculations to the case of Fe3+ which G. J. P. have not treated. COMPARISON
OF EXPERIMENTAL
RESULTS
WITH
THE
CALCULATION
The experimental data on X-state ions are rather scarce. The ground-state splitting of Nlnzf and Fe3+ in cubic fields has been measured in a number of crystal systems (I, 13). However, no reliable optical spectra for these ions has been reported in cubic point symmetries. Spectra in crystals of lower symmetries have been measured. For example Stout (9) has measured the spectrum of MnFn , Pappalardo (14) the spectrum of MnCls , and Stout has calculated the spectrum of RlnC& (15). Schliifer (16) has reported data on iron hydrates
(,I’TICAL
RPECTRURl
.43-I 1 (:ROI-SD-STATE
SI’IJTTISC;
323
[Fe (H20)i+] and this spectrum has also been studied by Rabinovitch and Stockmeyer (17) as well as by Pappalardo (18). Weak optical transitions have been reported in XlgO, some of which may be caused by Fe3+ impurities (19). The identification of the observed t,ransitions and the calculations of the levels have been made with these addit,ional assumptions. (a) The point symmetry of the ground and excited st’ates are cubic. (bj The levels are not assumed to be shifted by vibratiors, i.e., the barycentzer of all levels is not influenced by \-ibrations. If vibrations do influence the spectrum n-e assume t,hat. all levels are shifted equally. ( c 1 These crystals are assumed to haTee cubic point symmet,ry. The fact is t’hat (except, for MgO) they have lower symmetry. We have treat’ed the data as if the peak of the wide lines which are usually observed coincides with t,he cent,er of gravity of the lines in the cubic field case. ((1) In the absence of our knowledge of selection rules we have treated the absorption line as if they represent t’ransition to the r6 and 1?7levels, rather than to the bary-center of the spin-orbit split levels. This is done for convenience’s sake. Using these levels we were able to make a least square calculation (with smaller matrices) and find the improved parameters. The error caused by this assumption may be of t,he order of ,t and probably much smaller. As will be seen in Dhe det.ailed discussion of t,he experimental data the rn1.L; deviation is not large and the agreement between the theory and the experiment, as far as the optical absorption spectra is concerned, is better than expected. However, the calctilated ground-state split’ting is in all cases investigated, smaller than that found by experiment. We present a number of t,ypical results in Tables which summarize our calculaGons. We shall briefly comment on each of these tables. Table I In this table the position of the energy levels is computed with and without the spin-orbit coupling taken into account. We calculate the spectrum of Mn III using the electrostatic parameters of the free ions and the cubic field strength Dg - i40 cm-‘. This is a typical value found to fit the spectra of MnCls and MnF2 . We calculate these levels for .$ = 240 c~n-~ and t = 0. It is seen that t,he difference is indeed small, indicating that the levels are not admixed appreciably by means of the spin-orbit coupling. The largest shifts are found for the I’~ levels and t’hese do not amount to more than 150 cm-l or about &‘2. This table also gives the calculated levels for the two cases of Dy = f740 cni-‘, i.e., for the octahedral and body-centered-cubic symmetry. Watanabe !RO) had originally suggested t’hat, the ground splitting can only be perturbed by even powers of Dq. G. J. I’. have pointed out that t’he two cases of positive and negative Dy should give rise to different sets of energy levels. This is borne out by the results presented here. The difference is negligible for t,he r3 levels bllt is several hundred cn? for the I‘4 and Ts levels.
Assignment of Level
19% 32H il6%
4 r3 5 D
r5
r5 54D
r 35 4G
r5 S4G
32F
r4 60% 54~ + 39% 4P
r1 %
-1
5=0
”
32276.0
31315.3
30670.7
26970.0
25298.3
21170.8
10.369
-1 Value in cm
.
30818.1
F7
32244.6 32267.5
r7 r6
31142.1
30554.1
r6
r7
26961.5
25249.3
25327.1
r6
25347.0
25216.3
32267.6
32246.6
31471.1
30469.0
30765.0
26961.1
26956.0
21124.5
26955.8
-1
1
10.368
21202.4
-
21181.0
10.369
value in cm
5 a 240 cm
Dq = -740 cm?
21133.5
-
-1 Value in cm
r7
r6
r7
l-6
r7
l-7
-1
5 = 240 cm-l Assignment of Level
LJq= +740 cm-l
Parameters B = 910, C = 3270, cz = 76, 5 9 240, Dq = 740 in units of cm
Comparison of term values for Eq = + 740 cm
-1 and with the spin-orbit and Dq = - 740 cm -1 coupling parameters 5 = 0 and 5 = 240 cm .
TABLE I
z z
id
g
E!
z z
5 e
42409.8
r5 34% 52G + 20%
r
55F
2
48394.2
47098.9
46358.7
r1 52G
F
46403.4
r2 36% 51 + 24x 52F
r5 34
45809.7
T4 56?>2HI + 20% 32F + 159: 2H 3 II
44566.7
41041
r5 68 X52D + 21% 12D
r4 34 F
40825.5
!-/, 467 32F + 22X 5'F + 18"/, 32H3
2P 3
39247.3
r1 52I
35592.0
34614.3
r4 66% 521 + 16% 32F
r5 282 5211 + 26% 32F + 31% 52I II
34466.5
33955.8
54G
+ 35% 32F
r4 60 %34P + 39
r2 55 75% 7
6
?7
r6
r7
'6
r7
-6
r 7
r
?7
r7
r6
r6
r7
r6
'6
r7
7
48254.6
46989.1
47111 485553.8
47219.4 46997.4
46404.6
46380.6 46380.3 46384.7
45906.2
44501.6
44615.4 45697.6
44619.8
42548.3
41001.6
40726.7
39230.6
35558.8
44431.5
42293.7
41059.4
4OY59.4
39230.7
35645.5
34502.3
34401.8
34515.6 34733.0
34523.5
33975.4
34457.4
33975.1
326
LOW AND
ROSENGARTEN
OPTICAL
SPECTRUM
Ah-D GROUND-STATE
HPLITTlNG
327
Table II This table gives the calculated value and the experiment,al data of the energy levels for t,he crystal of MnClz . The first column gives the designation of the rL level and the degree of admixture. In the second column it shows the calculated center of gravity using a set of initial parameters. Column t’hree gives the experimental results as measured by Pappalardo (,14) and column four compares t’hese results. In column five is listed the calculated positiou of the Ils or r7 levels. These were used for a least square fit to the experimental results. The next two columns list the position of these levels using the improved parameters and the deviation between the observed and calculated results (0 - C’). A least square fit calculation became desirable when the high-frequency transitions of 40,650 and 42,370 cm-‘, respectively, were assigned to the quartet levels. These levels deviate by 1357 and 436 cm-‘, respectively, far more t8han the rms deviation of the other levels. Wit,h the new improved parameters the overall agreement is better but the deviation for the lower frequency transition is greater. If t,hese two high-frequency transitions are assigned t’o doublet levels, the agreement8 is improved considerably. Column eight gives Stout’s calculated value (lj), using two adjustable parameters: Dq and a covalency parameter. It is seen from column nine t,hat#t,he average deviation of Stout’s calculated levels is considerably larger than in our calculation. We have not given here Pappalardo’s calculation since his calculated level deviates in some cases a thousand wave numbers. We should like to mention t,hat Pappalardo’s assignment of “J’ t,o t,he 36,.500-cnl-’ t,ransition is not correct. Table III ‘The experimental results of RInF,! as measured by Stout (9:) are compared with t’he theoretical calculation. The data on RlnFz are more accurate than those of RlnCl? . On the other hand the point symmetry is orthorhombic, with a point, group of D?h . The data are taken from Table VII of Stout’s paper. The average deviation of Stout’s calculation is about 410 en?. Our average deviation is about, 1’5 -. ~~~~~~~ and if the level of 41,400 is omitted the agreement is even better. The stat)ernent’ by Stout that) “no such agreement is possible by lowering t,hr c+?ctrostatica interaction energy by the same factor for all d orbitals or by fit,ting Itacah coefficients to t#he crystal energy levels” is obviously not correct,. By keeping these parameters including the Dy parameter as adjustable variables and taking int)o account the spin-orbit, coupling c*onsiderably better agreement can be found. For the sake of completeness we present in Table III (bj all the energy levels including the rs levels. It’ is hoped that these levels will be useful in the interpretation of spectra in cubic crystals when such data become available. We have used in this calculation a value of 320 en-’ for the spin-orbit coupling parameter. This is probably an upper limit. The ground-st,ate splitting in t,his case is found to be 10 X lop4 cm-’ and the wavefunction of the ground state is given by !N.91 ‘I&5 Sr, + 0.07 74’9r4 + 0.02 5%4c:r4 .
l-626550
ill2
26750
26638
54Dr5
r7 28254
-430
l-6 22151
23825
I. I
- 11.4
r7 18533
r7
24255
-121
- 65
- 11.4
54CF3
22000
18500
0
Cdcll-
lated r or r7 Peve1
All values in cm-
Difference of o-c
22121
.
Fxperimental value
763 * SO
54w5
18565
+ 38% 34w4
61% 54@4
11.4
-
Calculated center of gravity usine initial values
740
240
240
5k
Name of Level
ns
76
3082 k 200
3150
76
758 + 73
In units of cm
-1
calcu-
L
26845
24266
22146
18219
- 237
lated r6 o= r7 after L-S fit
Improved Parameters
700
Initial Parameters
CALCULATED AND EXPBBIMP,NTA~ VALUES 0~ k*+
TABLE II 2
-
9.5
- 441
- 146
281
237
Difference of o-c
320
76
3270
910
27800
24000
22600
17900
0
Stout's calculated center of gravity E = 0.13
Free ion Parameters
IN WC1
,..
-1050
- 175
- 600
- 600
Difference o-c
g z
G
kz
T1
OPTICAL SPECTRUM
8 00 N
AND GROUND-STATE
8 2 m
329
SPLITTING
8
*x
52 Fp2 + 22% 5
42111
Experimental value
Difference of o-c
Calculated r err Peve 1 7
Difference of o-c
stout’s calculated center of gravity E = 0.13
Difference o-c
calculated r6 and ~7 levels using the using the so-called improved parameters, 40650, and 42370 cm-l which deviate are assigned to doublet levels the Finally, these results should be his calculated values and experimental
Calculated “6orr7 after L-S fit
Column 4 gives the difference between colum 3 and 2. Column 5 gives the initial set of parameters. Column 6 gives the calculated 7 and r levels which were obtained using a least square fit which included6the le
3x
calculated center of gravity using initial values
$
3
kG
0
s 3
-285
(28120 (28370
25452 2ki.530
54m 3
54Dr5
+ 48
25500
25438 (25300 (25190
-193
- 30
54Grl
23500
-110
19.57
23530
19440
0
54Gr5
-19.57
o-c
Difference
19550
of Level
752 k 18
750
Experimental Value.?
320
320
59%5 4Gr4 + 40%34Pr4
5k
Name
2
76
Calculated center of gravity using initial values
+
3158 " 64
801
Improved Paramtefs all values in cm
76
3150
820
Initial Parameters
rb
r6
'6
r7
r7
- 14
23514
2x7711
- 25
+ 62
- 57
19497
25438
-110
o-c
Difference
110
Calculated r6 or r7 levels after L-S fit
320
76
3270
910
Free Ion Paramters
COM'ARISON OF CALCULATED AND OBSERVED SPECTRA OF MF2
TABLE III (a)
l'1h50
25480
20 -1405
+
+
25190
55
- 180
o-c
Difference
23680
19440
0
Stout's calculation (parameter Dq chosen so as to fit the first 2 levels
332
LOWAND
ROSENGARTEN
OPTICAL
SPECTRUM
AND GROUND-STATE
SPLITTING
333
TABLE III (b) COMPARISON OF OPTICAL SPECTRA OF MF2
WITH CALCULATED VALUES
-1 Parameters and levels all in cm Crystal Parameters B = 820,
C = 3150,
Q! = 76,
%%rl + 0.07 x4pr4 + 0.03%4G
59%54Gr4
r4
+ 40
x4pr4
r7
Calculated value
Exp. value in
-19.574
Initial splitting unknown
-19.573
Deviation
wF2 + 19.6
r8
r
19510
7
r8
I
r8
II
T
54Gr5
Jhq = 750.
Irreducible representation
Name
99.9
s = 320,
19440
-
23500
- 30
110
19569 19613
6
23443
r7 ra I
23491 23580
r6
ra
19519
II
23587 25438
r8
25443
r
25454
25300
r 1 r76 8
25455
25500
r
28386
28120
- 48
4 Dl5 5
81
r r
28403
6
8
_ r7
- 160 II
28614 28777
28370
LOW AND
334
ROSENGARTEN
TABLE III (Continued)
Natne
Calculated Irreducible value representation
4 39% DI-5+ 22% 2I 5 IIl5
54DT3
r7
28699
r8
29096
r7
30099
ra
30114
; r6 57% Qr*
+ 40% Z3pr2
2 3 4 61% 'Ir4 + 19% HIII-4+ 17% 2m-
r7 '6 [ r8
59% 34Pr4 + 34%54W4
ra I r7
ra II ,r6 39%
'I r + 26% 5 15
r8 [ r7
sZIr3
rt3
521r1
'6 [ r8
2 38% 3FT4 +23%5
r8 [ '6 rs
Brp. value in
Deviation
mF2
30230
+ 115
33060
+
30132 31866 32375 32616 32958 32965
61
33024 33062 33393 33472 35410 37236 38164 38406 38473 38793 38885
39000
+
76
OPTICAL
SPECTRUM
AND
GROUND-STATE
SPLITTING
335
TABLE III (Continued)
Narw
Irreducible representation
Calculated value
Fxp. value in
Deviation
*F2
39727 39952 41019
i
‘f’
^.
-4
r6 r8 II 7. 61 7
41640 41757 41841 41924 42309
_%
43109 43379 43542
:g
2
J
43967
FT + 27% 52172 + 26% 32Fr2 2
A_,: ! 5
r8 I --7
_ + 24% 52F'4
[
r
44414 44419
i6
44466
!8 II
44568
i8
45175
r
45235
6
-_ -8
1 -. 7 (I‘,)
41400
393
LOW AND
336
TABLE III Irreducible representation
Nank?
48% 5 ‘m5
+ 31% 32Fr5
r7 [
27% 5’ Gr4+23%3
2H I4l-
r8 [
63% 5 ‘Fr2 41%
2 3 HIf4
+ 25% 32Fr2 + 38% 5
s2sr1 3
2D’5
32Gr4
+ 34%
2n4
S211?5
r6
r7 r6
[ 48% 3 2Hr5
Q3
r8
r8
Calculated value
48112 48123 48524 48743 49577 49476 49780 49996 50112
r6
50140
r8
59415
.?7
59481
0-8)
59757
(r8)
63886
r
65020
8
r7 r8
32Grl
(Continued)
r7
r6 32Gr5
ROSENGARTEN
r6
65134 66391 66414
Exp.
value =F2
in
Deviation
OPTICAL
SPECTRUM
AND
GROUND-STATE
337
SPLITTING
TABLE III (Continued)
ZPF
3
Deviation
mF2
77293
r8
il
Exp. value in
Calculated Value
Irreducible
representation
77313
T6 C-,1 2 DF 15
r8
85821
TABLE IV
COMPARISON OF CRYSTAL FIELD PARAMETER AND GROUND STATE SPLITTING OF b2+ IN VARIOUS CRYSTAtS
Parameter
ml2
&IF2
&h(6H20)2+ (Interpolated values) 780
B
801
?r 21
758
C
3158
+ 64
3082
cx
76
76
76
5
320
320
320
D4
752
Initial splitting
763
rt 18
4 70
3130
+ 200
755
+ 50
1 x 10 -3
1 x 10 -3
1 x 10 -3
Measured splitting
2-3
All
values
in
units
of
cm
x 10 -3
-1
Table IV This table summarizes the results of three crystal systems. The results on hydrates have been fitted to the experimental data of Holmes and McClure (21 j. The values are interpolated between those found from a number of diagonalizations on MnClz and MnF2 . The ground-state splitting of the hydrates is
338
LOW AND ROSENGARTEN TABLE V ENERGY LEVELS AND GROUND STATE SPLITTING OF Fe3f
Initial Parameters B = 730, C = 3150, a = 90, c = 420, Dq = 1350 Free ion Parameter B = 1100, C = 3750, a = 90, = = 420. -1 In Units of cm
Name of Level
rl 99.75% 6S + 0.19% 4P + 0.04% 5G
Calculated center of gravify in cm-
- 41.331
Fxprrimental Value in cm-1
0
r4 55% 54G + 41% 34P
13127
r5 23% 52111 + 22% 32H
17128
r5 41% 526 + 24% 34F + 22% 54D
18176
18500 (18200)
r
4G
24653
(24500)
i- 4G 35
24779
r2 44% 521 + 43% 32F
25286
y4 38% 521 + 17% 32HII + 15%
26295
r5 4J%54D i-35%54G
26749
r5 25%3 2F + 24%521,
27390
r 4D 35
28701
(29000)
?-427 yo2G+ 20%52F + 20%32F
32358
(31500)
r5 2P
33814
15
52D + 22% 52G
I4 40%34P +35%34F
35049 35698
r5
50%52D + 39% 2I 5 I
38309
12500 - 1,600
27500
OPTICAL SPECTRUM
AND GROUXIX3TATE
SPLITTISG
TABLE V (Continued)
Name of Level
Calculated center of gravity in cm-l
71 44% 526 + 37% 52I
38594
r4 39% 32HI -!27% 32F
42359
r5 52F
42539
T4 48% 34F + 21% 54G
43343
r 2 25F
43346
r5 43% 52F + 33% 52G
43621
r. 2 15s
46571
4F -5 3
47961
!-‘30% 32HI + 21% 52F
49216
r5 37%32F + 27% 5'G
49966
Y5 36/, c 2 III + 29% 32H
52843
r2 45% 32F + 26%52F + 25%321
53385
T4 41% 32HII + 33%521
53899
!- 2D 53
60265
2G ?4 3
63606
r 2G 53
67746
v 2 -1 3 G
68908
r 2P 43
74045
I- 2D 51
87803
Experimental value in cm-1
(42000)
Calculated Ground State Splitting 16 x 10-3 Experimental Ground State Splitting
N 40 x 10-3
Values in brackets are weak peaks in Schlafer's graph as interpreted by Pappalardo (1958).
340
LOW AND
ROSENGARTEN
TABLE VI COMPARISONOF MEASUREDAND CALCULATEDENERGY LEVELS OF Fe3+ IN VARIOUS SALTS
-1 All Levels in Units of cm
Fe(H20)63+ (1) 12500
Fe(H20)63+
Calculated
(2) 12600
Fe3+ in Beryl
Fe3+ in MgO
(3) 13170
(4) 12100
12300
(15200) 18000 - 19000
18200
18200
24599
24500
24694
18000 - 23600
- 25500 (?)
24820 27500
-4 )3al35x 10
-4 35 x 10
26500
26800
27300
28740
-4 16 x 10
-4 45 x 10
-4 61.5 x 10
(1) Schlafer,H. L., 1955, Z. Phys. Chem. Neue Folge 4, 116. (2) Holmes, 0. G., McClure D. S., 1957, Chem. Phys. 26, 1688. (3) Dvir, M., Low, W., 1960, Phys. Rev. llg, 587. (4) Low, W., RosengartenG., This paper. taken frontthe results on Tutton Salts and fluosilicate containing manganese (l,l%. Table V As mentioned before, reliable experimental data on Fe3+ are scarce. We have used the stronger peaks on the absorption spectrum of iron hydrate in Schlafer’s paper (16). Table V gives the results of the calculated values. Since only a few levels were measured, no least square calculation was performed. Pappalardo (18) detects some additional weak lines in Schlafer’s graph. These lines seem to
99.88
6SF1 +
in MgO
6Srl + 0.09
-1 cm
(3al = 61.5 x 10-3
0.19
-1 cm
4pr4 +
(3a( - 35 - 45 x LO-3
99.75
Experimental value in hydrates
Admixture
1350 16 x 1O-3
1350
3 x 1o-3
420
90
90
a
300
3250
3150
C
F
750
730
4pr4 +
-1
0.02
22 x
4Gr4 +
lo-3
2150
440
90
4000
1100
O-04
4Gr 4
-3 29 x 10
2150
440
0
4000
1100
Fe3+ AS A FUNCTION OF ELECTROSTATIC PARAMETERS
B
3a
%
OF
All values in cm
COMPARISON OF GROUND STATE SPLITTING
TABLE VII
342
LOW AND
ROSENGARTEN
fit reasonably well to doublet levels. Similarly Dvir and Low’s data on beryl (a!?) can be fitted with a slightly smaller B and larger D9 about 1500 cm-‘. The electrostatic parameters B and C are reduced considerably from those of the free ion value probably because of covalent bonding. The ground-state splitting is 160 X lo4 cm-’ and the wave function is given by 99.75 % 56s I’1 + 0.19 % 4P r4 + 0.04 % 4G I?4. Table VI This table summarizes the measured and calculated energy levels of FeSf for a number of salts. The calculated values are those which we fitted to the iron hydrate. Also included are a number of weak lines found in MgO which presumably are due to iron. It is clear that the fit is reasonably good but that the calculated ground-state splitting is smaller by a factor of 24 than that found by experiment . Table VII This table summarizes some of the calculated ground-state splitting as a function of [, Dq, and the electrostatic parameters. The first two columns show the variation of this splitting as a function of t. This splitting is a strong function of the spin-orbit coupling and varies approximately as t4-15. A change of a factor of 2 in the amount of 4P I’4 admixture changes the initial splitting by a factor of 5. Columns three and four list the results of our calculations using the electrostatic parameters and Dq of G.J.P. for two values, a! = 90 and (Y = 0 cm-‘. Two things are noteworthy. First the splitting is a function of o( but unfortunately its inclusion decreases the splitting. Second the initial splitting of 29 X 10m3 differs considerably from that calculated by G.J.P. who find a value of 53.6 X lo-” cm. DISCUSSION We have carried the crystal field of the iron group elements in cubic field symmetry as far as possible. Within this framework the following conclusions may be inferred: (a) The optical spectrum of &In*+ and Fe3+ can be fitted using five parameters of which only one is a crystal field parameter. Given the assumptions listed in the Introduction, the agreement between experimental and theoretical values is very satisfactory. There is no need to include additional parameters such as a covalency parameter; the freely adjustable parameters B, C, and Dq are sufficient to fit the spectrum. The lack of reliable experimental data is at present the handicap in the detailed interpretation of the spectra. (b) the situation is not so satisfactory as far as the ground-state splitting is concerned. The calculated value is found to be too small by a factor of two to four when the free ion spin-orbit coupling, & , value is used. It is possible to fit the optical spectrum and the initial ground-state splitting by assuming a spinorbit coupling which is 20-30 % larger than C;o.
OPTICAL
SPECTRI7M A?;D GEWUND-STATE
SPLITTING
3-u
‘l’ht calculated wavefunctions permit us to obtain the ‘Ltheoretical” g factors. These deviate from the experimentally found g factors of Xn”+ and even more ’ ‘+. Fidone and Stevens (~3) have pointed out that covalent. for the case of Ee bonding may affect the g-fact,or calculations. In some cases this may result, in a g factor larger t,han the g factor of the free electron. Apparently covalent bonding lllust influenc*e the admixture of the wavefunction and, therefore, also the grounds:tat’e splitt’ing. (;a.hriel rf nl. have discussed in detail the shortcomings of calculations which usr t,he naive crystal field theory. They suggest that the starting Hamiltonian may I)e incomplete. The terms omitted are those connected with spin-orbit, spill-other orbit, and orbit-orbit interactions. Two different cubic field parametc~s /)q, one for the spin-doublet and another for the spin-quartet states may flavr to ttc included; i.e., the crystal field experienced by the excited doublet st,at,cs may be different. In our calculations, we have not found any systematic deviations of t’he doublet levels which may suggest a significantly different value for I)(/ for t’hese levels. Another possibility mentioned by these authors is to assume different spin-orbit parameters for doublet-doublet, quartet-quartet, dollt)lrt-clual.tet, and quartet-sextet matrix elements because of different degrees in covalency for these doublet and quartet levels. This is an ad hoc assumption which needs an experimental confirmation. We are not so certain that this may lead to a significant change in the calculated splitting. IVe should like to suggest a few additional possibilities: (1) One may accept that, the spiii-orbit coupling in salbs in the case of S-state ions is really larger thalr the frecx ion value. This would mean that the manganese ion behaves diffrrtallt,ly than other iron group elements in salts. There is some indication from neut,roii tlifl’rnction data that the form factors of manganese and iron are diffrrclnt, from nickel (24, 2.5). The radial distribution of the S-state ion in its ground state may 1~ somewhat different than so far assumed theoretically (2). The point xymmct,ry of the Hamiltonian deviates from the cubic field for some of the oxcit8cd s:tat,es. The excited states may he of lower symmetry because of coupled vibrat,ions. These may cause admistllres to the ground states changing the perrtlnt,aye ot’ the 41’ wave function. (13) The Hamiltonian we have used does not, take ilrt80 account the electron transfer spectrum. This spectrum affects the relativcl transition probabilities of variolls optical transiGons. The electron transfer sp(>rtrurn may possibly rffert, the init.ial splitting and t.hc g factor, sincck hoth t,hcsf> clualltit its arc sensitive functions of small changes in the composition of the adniisturcs. 111coilclusion, one can remark that crystal field calculations are relatively succc4111 in csplaining the position of the energy levels t,o within a few hundred rm ‘. IIowever, this model is not po\yclrful enough to explain the finer details of Ihc
s~,c~ct~wnn.
LOW AND
344
ROSENGARTEN
ACKNOWLEDGMENT The authors are grateful to Professor G. Racah for many stimulating discussions and for considerable help in constructing the crystal field matrices. This work was supported in part by the Cambridge Research Laboratories, Bedford, O.A.R. through the European Office U.S.A.F. APPENIIIS
We shall give a brief outline of the method of obtaining the matrix elements. The cubic field potential for the d” electron system is given by licub cc [KO + (9&Y*
(YI” +
C)l
(1)
The potential can be considered as a component of an irreducible tensorial operator of order 4 (26). As mentioned in the text of this paper, it is convenient to work in the weak field scheme, i.e., in the vSLJl?j scheme. Here v is the seniority number, r the irreducible representation of the cubic group, and j one of its rows. According to the Wigner-Eckart theorem, we can factor the mat,rix element: (uSLJri
/ Vcub /
v’SL’J’I’~~) = (vSLJ
/i Cl4 11ZJ’SL’J’)..~ ;I, 1
$
I’
tl)
.
(2)
The first factor is the reduced matrix element which depends on the specific configuration, but not on the point symmetry. The second factor depends on the symmetry. This factor can be simplified to
Let us consider the reduced matrix elements. These are given for the cl” configurations (vSLJ
11 U4 I/ z,‘SL’J’)
= (-1)
(2J’ + l(]“” X W (LJL’J’;
S-L--I,‘PJ’ n[(2L + 1) (2L’ + 1) (25 + 1)
S4) c (-1)“” 21” L”Sl,
X W (2L2L’; L”4)
(8‘ USI, (/ d”-‘(v”S”L”) (d’-l
(u”S”L”)
dSL)
(4),
dSL’/} d” &ST,‘)
where W are the Racah coefficients and (I 1) are the fractional parentage coefficients. We can write the matrix element of the cubic potential in the VSLJM scheme as (vSLJM
/ Vcub 1v’SL’J’M’)
= (vSLJ
/I C4 11v’SL’J’)
I_\
OPTICAL
SPECTKI TM .4X 1) G BOUND-STATE
315
SPLITTIXG
where
j( .1’:11’, 10 1./N ‘, + f&p’
1iJ’:l;r’,
44 1 JM)
+
(,r’M’,
4 -
4 1 .JM)]
(fi)
( ./‘.I/‘, 10 1./Al ) and ( .J’;lJ’, 11 1 .J:lJ ) are Wigner coefficients. III order t,o c*alculat,e the ./‘clcnlents, it is convenient t#o oht,ain the transformation ( .J.l/ / #Jr,). These elenlents are given b> whrrr
Here> ( .131 i Jr’,) are the C~O~U~~IIS (for a given .J and I’j) of the transformation Iuat,ris. It. c:an also he shown t,hat they are t,he eigenvect.ors of the matrix (JJI
I’,.,,,, i .J’dJ’) and havct the cigcnvaluc
J J’ 4
.I’ I, r (
r
1)
. These 1 numbers
I
have beon
by Schijnfeld f 5). electrostatic mat8riws of the free ions ilt t#he vSLJM scheme have been .cal(wlattd for the d4 and d” configuration by Eacah, the spin-orbit matrices by Alurakawa (27) and by Racah and Shadmi (28, 11). It should be not,ed that the nlatrix elements have been calculated in the vSL./I sc*hcil~. Jt is not) dificult to obtain t,he eigenvectors in t)he more “physical WhflllP” raSI,I’,,I’., . ( I’,, represents the representation in the absence of the spirl-orbit wupling.) The progranl for the computer was arranged to take (*are of this. calclllated The
f I%-.Imw, “l’aritmagnet ic Resonance in Solids.” *%csdemic Press, New York, IMU. ! I).,C:. ~%C'I,I.KE, “Advances in Solid State Physics,” Vol. 9, p. 399. Arsdemic Press, Sew l.ork, 1950. 5. \-. 'rANARE 4N1) s. SII(;.~N(I,./. I'h!/n.Soc.Jnpetl9.i66 (1954). 4. Y. TANABE ANI) H. KAMIKI.I+ .f.f'h?/x.SW. Jupur~ 13, 39-L(l!l58). b. C;. scr+ijNFEI,I), M. SC. thesis, Jerusalem, 1961. h’. u’. I,ow.. “Quantum E;lectronics I,” p. 410. Columbia ITniv. Press, Sew York, 1960. 7. W. I,ow, “Qmtnturn LClect,ronic;i 11,” p. 138. (‘olumbiz+ Univ. Press, Sew York, 1961. 8. 11. .J. 1). POWELL, J.H. GBKIEI, AXI) I).F. JIJ~~NSIWN,Phys.ZZev. Letters6,145 (1960); .J. I{. ( IAHILIEI,, I). F. JCIHWIYIN AND M. J. 11).POWELL, I'ror.Ho!y.Sot. 264, 503 (1961). 8. J.~~.~~1~111:~1~,J.f’hern.f-‘h~/s.31,’iODCl%!)). 10. $. KOIDE AKDM. Ii. I,. J’RI-~E, Phil. ,lIng. 1813, 607 (1958). Il. Y. Str.4~~1.thesis, Jerusalem, 1961. 12. J<, XI. WATSON, M.T.T. Report No. 12 11!)59) (unpublished). 19. ‘1’. I’. P. HALL, W. HAYES ANTI F. t. Ill. WILI.IA>TS. I’m-. Phys. Sot. (London) 78, 883 (la51 1.
14. I<. I'AJ~I~.~I..\IzI)o, J. ('hew. Whys. 31, 1050 11959). 15. .J. Xl. STOI.I’, ./. (Ihem. I’hys. 33, 303 (1960).
346 16. 17. 18. 19. 20. 21. 22. 2s. 24. 25.
LOW AND ROSENGARTEN
W. L. SCHLKFER,2. Phys. Chem. Neue Folge 4,116 (1955). E. RABINOVITCH AND W. H. STOCKMEYER,J. _4m. Chem. Sot. 64,335 (1942). R. PAPPALARDO, NUOVOcinaento 8, 955 (1958). W. Low ANDM. WEGER, Phys. Rev. 118,1119 (1960). H. WATANABE, Progr. Theoret. Phys. Japan 18,405 (1957). 0. G. HOLMES AND D. S. MCCLURE, J. Chem. Phys. 26, 1686 (1957). M. DVIR AND W. Low, Phys. Rev. 119,1587 (1960). J. FIDONE ANU K. W. H. STEVENS, Proc. Phys. Sot. (London) 73,116 (1959). J. M. HASTINGS, M. ELLIOTT AND L. M. CORLISS, Phys. Rev. 116,13 (1959). R. NATHANS, S. J. PICHART AND W. A. ALPERIN, J. Phys. Sac. Japan 17. Supplement B-111 7 (1962). 26. V. FANO AND G. RACAH, “Irreducible Tensorial Sets.” Academic Press, New York, 1959. 07. K. MCRAKAWA, J. Phys. Sac. Japan 10,919 (1955). 28, G. RACAH, Phys. Rev. 62,438 (1942); 63,367 (1943); 76,1352 (1949).
The Optical Spectrum and Ground-State Splitting of Mn*’ and Fe3+ Ions in the Crystal Field of Cubic Symmetry W. Low AND Department
of Physics,
G.
ROSENGARTEN
The Hebreul Vnirtersity,
Jerusalem,
Israel
We ha,ve calculated the energy levels and ground-state splitting I’&, for Mn”+ and Fez+ in a cubic field. These calculations consist of t,he diagonalization of the various rj matrices of the complete (d5) configuration including cubic crystal field and the spin-orbit coupling. The parameters are chosen to fit the optical spectra of Mn’+ in crystals of MnF? and MnClz and hydrates as well as the spectra of Fe3+ in hydrat,es. It is found that the spectra ran be fitted with an rms error of a few hundred cm-‘. The eigenfunctions of these levels are obtained. The observed ground-stat)e splitting for Mn2+ and Fe3+ in hydrates can be accounted for if the value of the spin-rjrbit coupling in t,he ergstal is larger than that of the free ion. Similarly the calculated g factor deviat,es from the experimentally found value. Possible reasons for the discrepancies are given.
During the last few years crystal field theory has been used to great advant.age in interpreting the magnetic behavior (I ) and the optical spectra (2) of pammagnetic ions in crystals. The theory uses a conventional central field Hamiltonian and adds to it a perturbing potential, expanded in spherical harmonies, called the crystal field potential. This potential is supposed to act only on the elect,rons in the unfilled shell. The effect of the crystal field potential is to split the degeneracy of each ionic level into a number of Stark levels. The number of t’hese levels is dependent on the point symmetry at the site of the paramagnetic ion. The separation of these levels depends on the electrostat#ic parameters B, (‘, and on the crystal field strength parameters (called Dq in case of cubic symmetry). It is found esperimentally that the separation of various Stark splittings in the irou group elements are of the same magnitude as the term separations of the free ion. Diagonalizations of t’he matsrices rather than perturbation theory are, therefore, necessary in order to correlate the theory with experiments. The matrices in the stroug field case have been calculated (3, 4) for the cubic field symmetry. Recent data have shown that in some cases fine st~ructure of lines can be resolved. This additional st.ructure has been at,tributed t.o perturbation by t.he 319
320
LOW AND
ROSENGARTEN
spin-orbit coupling & It is, therefore, necessary to calculate matrices which include the spin-orbit coupling. Schonfeld (5, 6) has constructed the matrices for the cubic field and spin-orbit coupling for the two cases of d2 and d3. She has shown the correlation with the experimental data can be improved if the spinorbit coupling parameter is taken into account. We report an extension of these calculations to the case d5.’ While this work has been completed for some time, Gabriel, Johnston, and Powell (8) have published an important paper on the ground-state splitting of Mn (III) (d5) ions in a cubic field. (This paper will be referred to as G.J.P.) They calculate this splitting and part of the optical spectrum for the (d5)6Ss,z configuration as a function of the Racah parameters B and C, Dq, <, and the spin-spin parameters MO and Mt . Our method of calculation differs somewhat from theirs and sheds some additional light on the subject of the initial splitting of the ground state of the Mnzf as well as of the Fe3+ ions. It will be shown in particular that good correlation with optical spectra can be obtained without introducing additional covalency parameters. OUTLINE
OF THEORY
AND
METHOD
OF CALCULATION
We shall outline briefly the basis of our calculation. The Appendix gives the matrix elements used in this calculation and the method of computation. The matrices as well as the transformation matrices will be published elsewhere. The starting point is the assumption that the conventional atomic Hamiltonian can be used and that the crystal field perturbation does not affect the spherical symmetrical part of the atomic Hamiltonian. Other assumptions are that the interaction between different configurations is negligible, and that the charge transfer which usually occurs in the ultraviolet, does not appreciably influence the relative separation of the energy levels. In calculating the matrix elements it is conventional to use two extreme representations: the strong field and the weak field limits. Both methods have advantages and disadvantages. The strong field representation classifies the d electrons into de and d-y orbitals. The crystal field part is diagonal in these matrices and the electrostatic interaction off-diagonal. It has the advantages that one can in principle allow the de and dy orbitals to have different radial extension and different amounts of covalent bonding. This method has been used by Stout (9) [originally suggested by Koide and Pryce (IO)]. Stout claims that by introducing a new parameter, called the covalency parameter, better agreement between theory and experiment can be obtained. This method permits the introduction of an anisotropic spin-orbit coupling parameter. However it suffers from the disadvantage that it does not utilize the electrostatic matrices as well as the spin-orbit coupling matrices already available in the literature. 1The results of calculations Conference (7).
for d6 have been presented
at the IInd Quantum
Electronics
OPTICAL
SPECTRUM
AND
GROUND-STATE
SPLITTING
321
The weak field limit, however, takes advantage of the existing electrostatic and spin-orbit matrices. It uses further the existing computer programs for the diagonalization of the matrices and the possibility of obtaining the appropriate eigenvectors. The matrices contain, therefore, 5 parameters B, C, which are the linear cornbination of the Slater integrals , $ the spin-orbit coupling in the crystal, Q the Trees’ correction which is of importance in correlating the optical spectra, and finally Dq a measure of the cubic crystal field strength. We have neglected in this calculation the spin-spin interaction which would introduce two additional parameters. The spin-spin interaction affects only very slightly the position of the various excited levels. It affects somewhat more the initial splitting of the ground state but not very significantly as shown by G.J.P. The matrices are classified according to the irreducible representation of the double cubic group. The matrix I?* is the largest. The dimensions are for the doublets 1’8:25 X 25, rs:12 X 12, J?,: 13 X 13 and for the quartets J?,: 16 X 16, r’,: 8 X 8, I’,: 8 X 8. These were diagonalized on the Weizac computer in the following manner. We chose a set of parameters for the 5 variables, B, C, 4, LY,and Dq which by inspection will fit reasonably to the experimental data. The parameters B, C, [, and (Y cannot be taken as the free ion value since these parameters differ in various crystals (6). The free ion values which are the upper limit of these parameters have been calculated by Shadmi ( 11) . They are in units of cm-’ for AIn’+: B = 910, C = 3270, cr = 76, t = 320. For Fe3+: B = 1100, C = 3750, a = 90, E = 420. These values differ from those taken by Gabriel et al. who use Watson’s wave function ( 12). Their values for Mn’+: B = 900, C = 3300, CY= 0, F = 376, and for Fe3+:B = 1100, (’ = 400, (Y = 0, 4 = 440, all in cn-I. The difference in these values is significant since the initial splitting 3a is a funct#ion of all the five parameters. After obtaining a first, set of energy levels from our initial choice of parameters, we apply a “least square” technique (denoted in the tables as L-S) to find improved parameters which will give the best overall fit to the experimental spectrum. The least square procedure is meaningless unless the number of parameters is small compared with the number of “distinct” experimentally observed transit,ions. In the cases discussed below, the experimental data are not sufficient and it was found convenient to fix two parameters (Yand & The Trees’ correction was arbitrarily fixed at the free ion value. The matrices were diagonalized for a nunber of values of [. The optical spectrum itself is not very sensitive to the value of E but of course the initial splitting 3a is a sensitive function of $. We shall briefly review the main aspects where our calculations differ from those of G.J.P. 1. We use the same Hamiltonian as the authors. The largest matrix which is obt’ained is 42 X 42. G.J.P. used a convergent perturbation technique carried to
322
LOW AND
ROSENGARTEN
sixth order. It had to be carried to this order to find the magnitude of the groundstate splitting. We diagonalized the 42 X 42 matrix and obtained the eigenvalues. The results of the magnitude of the splitting of these two procedures differ significantly. Presumably the perturbation technique does not converge sufficiently rapidly. The ground-state splitting I’-I’~ depends on a sensitive manner on the admixture from higher excited states, We have been unable to obtain the eigenvectors of the lYslevels because of the limited high speed memory of the computer. We obtained, however, the eigenvectors for the I6 and r7 levels. We assume that the admixture of the rs levels and the I’6 and p7 levels coming from the same rL is not very different. This seems to be true in the case of the Ps and r, levels where the admixture is nearly the same, and is reasonable since the spin-orbit coupling is only a small perturbation in these calculations. 2. G.J.P. obtain the electrostatic parameters by trying to fit the optical spectrum to the transition rl-r3 which is nearly independent of the crystal field strength. The value 1OB + 5C is taken from the experimentally observed optical spectrum of the 1’1-1’3 transition. They proceed t’o calculate the initial splitting in a number of tables as a function of C/B, Dq, and 4. We have tried to get the best ovedljt to the optical spectrum for each individual crystal host by choosing a set of values for B, C, and Dq which will give the least square deviation from the experimental values. Obviously the transitions of higher wave numbers are more sensitive to the accurate choice of these parameters. The diagonalization of these matrices was made with one variable parameter .$ in order to fit the ground-state splitting. 3. We use five parameters including the Trees’ correction CLIt turns out that the Trees’ correction is important in reducing the overall error in the parameters B and C. On the other hand we do not use the spin-spin correction which G. J. P. include in some of bheir calculations. 4. We obtain in our calculation the admixture of the wave function, and in particular the admixture of the ground state by the P and G levels. We are therefore, able to calculate the g factor. 5. We extend these calculations to the case of Fe3+ which G. J. P. have not treated. COMPARISON
OF EXPERIMENTAL
RESULTS
WITH
THE
CALCULATION
The experimental data on X-state ions are rather scarce. The ground-state splitting of Nlnzf and Fe3+ in cubic fields has been measured in a number of crystal systems (I, 13). However, no reliable optical spectra for these ions has been reported in cubic point symmetries. Spectra in crystals of lower symmetries have been measured. For example Stout (9) has measured the spectrum of MnFn , Pappalardo (14) the spectrum of MnCls , and Stout has calculated the spectrum of RlnC& (15). Schliifer (16) has reported data on iron hydrates
(,I’TICAL
RPECTRURl
.43-I 1 (:ROI-SD-STATE
SI’IJTTISC;
323
[Fe (H20)i+] and this spectrum has also been studied by Rabinovitch and Stockmeyer (17) as well as by Pappalardo (18). Weak optical transitions have been reported in XlgO, some of which may be caused by Fe3+ impurities (19). The identification of the observed t,ransitions and the calculations of the levels have been made with these addit,ional assumptions. (a) The point symmetry of the ground and excited st’ates are cubic. (bj The levels are not assumed to be shifted by vibratiors, i.e., the barycentzer of all levels is not influenced by \-ibrations. If vibrations do influence the spectrum n-e assume t,hat. all levels are shifted equally. ( c 1 These crystals are assumed to haTee cubic point symmet,ry. The fact is t’hat (except, for MgO) they have lower symmetry. We have treat’ed the data as if the peak of the wide lines which are usually observed coincides with t,he cent,er of gravity of the lines in the cubic field case. ((1) In the absence of our knowledge of selection rules we have treated the absorption line as if they represent t’ransition to the r6 and 1?7levels, rather than to the bary-center of the spin-orbit split levels. This is done for convenience’s sake. Using these levels we were able to make a least square calculation (with smaller matrices) and find the improved parameters. The error caused by this assumption may be of t,he order of ,t and probably much smaller. As will be seen in Dhe det.ailed discussion of t,he experimental data the rn1.L; deviation is not large and the agreement between the theory and the experiment, as far as the optical absorption spectra is concerned, is better than expected. However, the calctilated ground-state split’ting is in all cases investigated, smaller than that found by experiment. We present a number of t,ypical results in Tables which summarize our calculaGons. We shall briefly comment on each of these tables. Table I In this table the position of the energy levels is computed with and without the spin-orbit coupling taken into account. We calculate the spectrum of Mn III using the electrostatic parameters of the free ions and the cubic field strength Dg - i40 cm-‘. This is a typical value found to fit the spectra of MnCls and MnF2 . We calculate these levels for .$ = 240 c~n-~ and t = 0. It is seen that t,he difference is indeed small, indicating that the levels are not admixed appreciably by means of the spin-orbit coupling. The largest shifts are found for the I’~ levels and t’hese do not amount to more than 150 cm-l or about &‘2. This table also gives the calculated levels for the two cases of Dy = f740 cni-‘, i.e., for the octahedral and body-centered-cubic symmetry. Watanabe !RO) had originally suggested t’hat, the ground splitting can only be perturbed by even powers of Dq. G. J. I’. have pointed out that t’he two cases of positive and negative Dy should give rise to different sets of energy levels. This is borne out by the results presented here. The difference is negligible for t,he r3 levels bllt is several hundred cn? for the I‘4 and Ts levels.
Assignment of Level
19% 32H il6%
4 r3 5 D
r5
r5 54D
r 35 4G
r5 S4G
32F
r4 60% 54~ + 39% 4P
r1 %
-1
5=0
”
32276.0
31315.3
30670.7
26970.0
25298.3
21170.8
10.369
-1 Value in cm
.
30818.1
F7
32244.6 32267.5
r7 r6
31142.1
30554.1
r6
r7
26961.5
25249.3
25327.1
r6
25347.0
25216.3
32267.6
32246.6
31471.1
30469.0
30765.0
26961.1
26956.0
21124.5
26955.8
-1
1
10.368
21202.4
-
21181.0
10.369
value in cm
5 a 240 cm
Dq = -740 cm?
21133.5
-
-1 Value in cm
r7
r6
r7
l-6
r7
l-7
-1
5 = 240 cm-l Assignment of Level
LJq= +740 cm-l
Parameters B = 910, C = 3270, cz = 76, 5 9 240, Dq = 740 in units of cm
Comparison of term values for Eq = + 740 cm
-1 and with the spin-orbit and Dq = - 740 cm -1 coupling parameters 5 = 0 and 5 = 240 cm .
TABLE I
z z
id
g
E!
z z
5 e
42409.8
r5 34% 52G + 20%
r
55F
2
48394.2
47098.9
46358.7
r1 52G
F
46403.4
r2 36% 51 + 24x 52F
r5 34
45809.7
T4 56?>2HI + 20% 32F + 159: 2H 3 II
44566.7
41041
r5 68 X52D + 21% 12D
r4 34 F
40825.5
!-/, 467 32F + 22X 5'F + 18"/, 32H3
2P 3
39247.3
r1 52I
35592.0
34614.3
r4 66% 521 + 16% 32F
r5 282 5211 + 26% 32F + 31% 52I II
34466.5
33955.8
54G
+ 35% 32F
r4 60 %34P + 39
r2 55 75% 7
6
?7
r6
r7
'6
r7
-6
r 7
r
?7
r7
r6
r6
r7
r6
'6
r7
7
48254.6
46989.1
47111 485553.8
47219.4 46997.4
46404.6
46380.6 46380.3 46384.7
45906.2
44501.6
44615.4 45697.6
44619.8
42548.3
41001.6
40726.7
39230.6
35558.8
44431.5
42293.7
41059.4
4OY59.4
39230.7
35645.5
34502.3
34401.8
34515.6 34733.0
34523.5
33975.4
34457.4
33975.1
326
LOW AND
ROSENGARTEN
OPTICAL
SPECTRUM
Ah-D GROUND-STATE
HPLITTlNG
327
Table II This table gives the calculated value and the experiment,al data of the energy levels for t,he crystal of MnClz . The first column gives the designation of the rL level and the degree of admixture. In the second column it shows the calculated center of gravity using a set of initial parameters. Column t’hree gives the experimental results as measured by Pappalardo (,14) and column four compares t’hese results. In column five is listed the calculated positiou of the Ils or r7 levels. These were used for a least square fit to the experimental results. The next two columns list the position of these levels using the improved parameters and the deviation between the observed and calculated results (0 - C’). A least square fit calculation became desirable when the high-frequency transitions of 40,650 and 42,370 cm-‘, respectively, were assigned to the quartet levels. These levels deviate by 1357 and 436 cm-‘, respectively, far more t8han the rms deviation of the other levels. Wit,h the new improved parameters the overall agreement is better but the deviation for the lower frequency transition is greater. If t,hese two high-frequency transitions are assigned t’o doublet levels, the agreement8 is improved considerably. Column eight gives Stout’s calculated value (lj), using two adjustable parameters: Dq and a covalency parameter. It is seen from column nine t,hat#t,he average deviation of Stout’s calculated levels is considerably larger than in our calculation. We have not given here Pappalardo’s calculation since his calculated level deviates in some cases a thousand wave numbers. We should like to mention t,hat Pappalardo’s assignment of “J’ t,o t,he 36,.500-cnl-’ t,ransition is not correct. Table III ‘The experimental results of RInF,! as measured by Stout (9:) are compared with t’he theoretical calculation. The data on RlnFz are more accurate than those of RlnCl? . On the other hand the point symmetry is orthorhombic, with a point, group of D?h . The data are taken from Table VII of Stout’s paper. The average deviation of Stout’s calculation is about 410 en?. Our average deviation is about, 1’5 -. ~~~~~~~ and if the level of 41,400 is omitted the agreement is even better. The stat)ernent’ by Stout that) “no such agreement is possible by lowering t,hr c+?ctrostatica interaction energy by the same factor for all d orbitals or by fit,ting Itacah coefficients to t#he crystal energy levels” is obviously not correct,. By keeping these parameters including the Dy parameter as adjustable variables and taking int)o account the spin-orbit, coupling c*onsiderably better agreement can be found. For the sake of completeness we present in Table III (bj all the energy levels including the rs levels. It’ is hoped that these levels will be useful in the interpretation of spectra in cubic crystals when such data become available. We have used in this calculation a value of 320 en-’ for the spin-orbit coupling parameter. This is probably an upper limit. The ground-st,ate splitting in t,his case is found to be 10 X lop4 cm-’ and the wavefunction of the ground state is given by !N.91 ‘I&5 Sr, + 0.07 74’9r4 + 0.02 5%4c:r4 .
l-626550
ill2
26750
26638
54Dr5
r7 28254
-430
l-6 22151
23825
I. I
- 11.4
r7 18533
r7
24255
-121
- 65
- 11.4
54CF3
22000
18500
0
Cdcll-
lated r or r7 Peve1
All values in cm-
Difference of o-c
22121
.
Fxperimental value
763 * SO
54w5
18565
+ 38% 34w4
61% 54@4
11.4
-
Calculated center of gravity usine initial values
740
240
240
5k
Name of Level
ns
76
3082 k 200
3150
76
758 + 73
In units of cm
-1
calcu-
L
26845
24266
22146
18219
- 237
lated r6 o= r7 after L-S fit
Improved Parameters
700
Initial Parameters
CALCULATED AND EXPBBIMP,NTA~ VALUES 0~ k*+
TABLE II 2
-
9.5
- 441
- 146
281
237
Difference of o-c
320
76
3270
910
27800
24000
22600
17900
0
Stout's calculated center of gravity E = 0.13
Free ion Parameters
IN WC1
,..
-1050
- 175
- 600
- 600
Difference o-c
g z
G
kz
T1
OPTICAL SPECTRUM
8 00 N
AND GROUND-STATE
8 2 m
329
SPLITTING
8
*x
52 Fp2 + 22% 5
42111
Experimental value
Difference of o-c
Calculated r err Peve 1 7
Difference of o-c
stout’s calculated center of gravity E = 0.13
Difference o-c
calculated r6 and ~7 levels using the using the so-called improved parameters, 40650, and 42370 cm-l which deviate are assigned to doublet levels the Finally, these results should be his calculated values and experimental
Calculated “6orr7 after L-S fit
Column 4 gives the difference between colum 3 and 2. Column 5 gives the initial set of parameters. Column 6 gives the calculated 7 and r levels which were obtained using a least square fit which included6the le
3x
calculated center of gravity using initial values
$
3
kG
0
s 3
-285
(28120 (28370
25452 2ki.530
54m 3
54Dr5
+ 48
25500
25438 (25300 (25190
-193
- 30
54Grl
23500
-110
19.57
23530
19440
0
54Gr5
-19.57
o-c
Difference
19550
of Level
752 k 18
750
Experimental Value.?
320
320
59%5 4Gr4 + 40%34Pr4
5k
Name
2
76
Calculated center of gravity using initial values
+
3158 " 64
801
Improved Paramtefs all values in cm
76
3150
820
Initial Parameters
rb
r6
'6
r7
r7
- 14
23514
2x7711
- 25
+ 62
- 57
19497
25438
-110
o-c
Difference
110
Calculated r6 or r7 levels after L-S fit
320
76
3270
910
Free Ion Paramters
COM'ARISON OF CALCULATED AND OBSERVED SPECTRA OF MF2
TABLE III (a)
l'1h50
25480
20 -1405
+
+
25190
55
- 180
o-c
Difference
23680
19440
0
Stout's calculation (parameter Dq chosen so as to fit the first 2 levels
332
LOWAND
ROSENGARTEN
OPTICAL
SPECTRUM
AND GROUND-STATE
SPLITTING
333
TABLE III (b) COMPARISON OF OPTICAL SPECTRA OF MF2
WITH CALCULATED VALUES
-1 Parameters and levels all in cm Crystal Parameters B = 820,
C = 3150,
Q! = 76,
%%rl + 0.07 x4pr4 + 0.03%4G
59%54Gr4
r4
+ 40
x4pr4
r7
Calculated value
Exp. value in
-19.574
Initial splitting unknown
-19.573
Deviation
wF2 + 19.6
r8
r
19510
7
r8
I
r8
II
T
54Gr5
Jhq = 750.
Irreducible representation
Name
99.9
s = 320,
19440
-
23500
- 30
110
19569 19613
6
23443
r7 ra I
23491 23580
r6
ra
19519
II
23587 25438
r8
25443
r
25454
25300
r 1 r76 8
25455
25500
r
28386
28120
- 48
4 Dl5 5
81
r r
28403
6
8
_ r7
- 160 II
28614 28777
28370
LOW AND
334
ROSENGARTEN
TABLE III (Continued)
Natne
Calculated Irreducible value representation
4 39% DI-5+ 22% 2I 5 IIl5
54DT3
r7
28699
r8
29096
r7
30099
ra
30114
; r6 57% Qr*
+ 40% Z3pr2
2 3 4 61% 'Ir4 + 19% HIII-4+ 17% 2m-
r7 '6 [ r8
59% 34Pr4 + 34%54W4
ra I r7
ra II ,r6 39%
'I r + 26% 5 15
r8 [ r7
sZIr3
rt3
521r1
'6 [ r8
2 38% 3FT4 +23%5
r8 [ '6 rs
Brp. value in
Deviation
mF2
30230
+ 115
33060
+
30132 31866 32375 32616 32958 32965
61
33024 33062 33393 33472 35410 37236 38164 38406 38473 38793 38885
39000
+
76
OPTICAL
SPECTRUM
AND
GROUND-STATE
SPLITTING
335
TABLE III (Continued)
Narw
Irreducible representation
Calculated value
Fxp. value in
Deviation
*F2
39727 39952 41019
i
‘f’
^.
-4
r6 r8 II 7. 61 7
41640 41757 41841 41924 42309
_%
43109 43379 43542
:g
2
J
43967
FT + 27% 52172 + 26% 32Fr2 2
A_,: ! 5
r8 I --7
_ + 24% 52F'4
[
r
44414 44419
i6
44466
!8 II
44568
i8
45175
r
45235
6
-_ -8
1 -. 7 (I‘,)
41400
393
LOW AND
336
TABLE III Irreducible representation
Nank?
48% 5 ‘m5
+ 31% 32Fr5
r7 [
27% 5’ Gr4+23%3
2H I4l-
r8 [
63% 5 ‘Fr2 41%
2 3 HIf4
+ 25% 32Fr2 + 38% 5
s2sr1 3
2D’5
32Gr4
+ 34%
2n4
S211?5
r6
r7 r6
[ 48% 3 2Hr5
Q3
r8
r8
Calculated value
48112 48123 48524 48743 49577 49476 49780 49996 50112
r6
50140
r8
59415
.?7
59481
0-8)
59757
(r8)
63886
r
65020
8
r7 r8
32Grl
(Continued)
r7
r6 32Gr5
ROSENGARTEN
r6
65134 66391 66414
Exp.
value =F2
in
Deviation
OPTICAL
SPECTRUM
AND
GROUND-STATE
337
SPLITTING
TABLE III (Continued)
ZPF
3
Deviation
mF2
77293
r8
il
Exp. value in
Calculated Value
Irreducible
representation
77313
T6 C-,1 2 DF 15
r8
85821
TABLE IV
COMPARISON OF CRYSTAL FIELD PARAMETER AND GROUND STATE SPLITTING OF b2+ IN VARIOUS CRYSTAtS
Parameter
ml2
&IF2
&h(6H20)2+ (Interpolated values) 780
B
801
?r 21
758
C
3158
+ 64
3082
cx
76
76
76
5
320
320
320
D4
752
Initial splitting
763
rt 18
4 70
3130
+ 200
755
+ 50
1 x 10 -3
1 x 10 -3
1 x 10 -3
Measured splitting
2-3
All
values
in
units
of
cm
x 10 -3
-1
Table IV This table summarizes the results of three crystal systems. The results on hydrates have been fitted to the experimental data of Holmes and McClure (21 j. The values are interpolated between those found from a number of diagonalizations on MnClz and MnF2 . The ground-state splitting of the hydrates is
338
LOW AND ROSENGARTEN TABLE V ENERGY LEVELS AND GROUND STATE SPLITTING OF Fe3f
Initial Parameters B = 730, C = 3150, a = 90, c = 420, Dq = 1350 Free ion Parameter B = 1100, C = 3750, a = 90, = = 420. -1 In Units of cm
Name of Level
rl 99.75% 6S + 0.19% 4P + 0.04% 5G
Calculated center of gravify in cm-
- 41.331
Fxprrimental Value in cm-1
0
r4 55% 54G + 41% 34P
13127
r5 23% 52111 + 22% 32H
17128
r5 41% 526 + 24% 34F + 22% 54D
18176
18500 (18200)
r
4G
24653
(24500)
i- 4G 35
24779
r2 44% 521 + 43% 32F
25286
y4 38% 521 + 17% 32HII + 15%
26295
r5 4J%54D i-35%54G
26749
r5 25%3 2F + 24%521,
27390
r 4D 35
28701
(29000)
?-427 yo2G+ 20%52F + 20%32F
32358
(31500)
r5 2P
33814
15
52D + 22% 52G
I4 40%34P +35%34F
35049 35698
r5
50%52D + 39% 2I 5 I
38309
12500 - 1,600
27500
OPTICAL SPECTRUM
AND GROUXIX3TATE
SPLITTISG
TABLE V (Continued)
Name of Level
Calculated center of gravity in cm-l
71 44% 526 + 37% 52I
38594
r4 39% 32HI -!27% 32F
42359
r5 52F
42539
T4 48% 34F + 21% 54G
43343
r 2 25F
43346
r5 43% 52F + 33% 52G
43621
r. 2 15s
46571
4F -5 3
47961
!-‘30% 32HI + 21% 52F
49216
r5 37%32F + 27% 5'G
49966
Y5 36/, c 2 III + 29% 32H
52843
r2 45% 32F + 26%52F + 25%321
53385
T4 41% 32HII + 33%521
53899
!- 2D 53
60265
2G ?4 3
63606
r 2G 53
67746
v 2 -1 3 G
68908
r 2P 43
74045
I- 2D 51
87803
Experimental value in cm-1
(42000)
Calculated Ground State Splitting 16 x 10-3 Experimental Ground State Splitting
N 40 x 10-3
Values in brackets are weak peaks in Schlafer's graph as interpreted by Pappalardo (1958).
340
LOW AND
ROSENGARTEN
TABLE VI COMPARISONOF MEASUREDAND CALCULATEDENERGY LEVELS OF Fe3+ IN VARIOUS SALTS
-1 All Levels in Units of cm
Fe(H20)63+ (1) 12500
Fe(H20)63+
Calculated
(2) 12600
Fe3+ in Beryl
Fe3+ in MgO
(3) 13170
(4) 12100
12300
(15200) 18000 - 19000
18200
18200
24599
24500
24694
18000 - 23600
- 25500 (?)
24820 27500
-4 )3al35x 10
-4 35 x 10
26500
26800
27300
28740
-4 16 x 10
-4 45 x 10
-4 61.5 x 10
(1) Schlafer,H. L., 1955, Z. Phys. Chem. Neue Folge 4, 116. (2) Holmes, 0. G., McClure D. S., 1957, Chem. Phys. 26, 1688. (3) Dvir, M., Low, W., 1960, Phys. Rev. llg, 587. (4) Low, W., RosengartenG., This paper. taken frontthe results on Tutton Salts and fluosilicate containing manganese (l,l%. Table V As mentioned before, reliable experimental data on Fe3+ are scarce. We have used the stronger peaks on the absorption spectrum of iron hydrate in Schlafer’s paper (16). Table V gives the results of the calculated values. Since only a few levels were measured, no least square calculation was performed. Pappalardo (18) detects some additional weak lines in Schlafer’s graph. These lines seem to
99.88
6SF1 +
in MgO
6Srl + 0.09
-1 cm
(3al = 61.5 x 10-3
0.19
-1 cm
4pr4 +
(3a( - 35 - 45 x LO-3
99.75
Experimental value in hydrates
Admixture
1350 16 x 1O-3
1350
3 x 1o-3
420
90
90
a
300
3250
3150
C
F
750
730
4pr4 +
-1
0.02
22 x
4Gr4 +
lo-3
2150
440
90
4000
1100
O-04
4Gr 4
-3 29 x 10
2150
440
0
4000
1100
Fe3+ AS A FUNCTION OF ELECTROSTATIC PARAMETERS
B
3a
%
OF
All values in cm
COMPARISON OF GROUND STATE SPLITTING
TABLE VII
342
LOW AND
ROSENGARTEN
fit reasonably well to doublet levels. Similarly Dvir and Low’s data on beryl (a!?) can be fitted with a slightly smaller B and larger D9 about 1500 cm-‘. The electrostatic parameters B and C are reduced considerably from those of the free ion value probably because of covalent bonding. The ground-state splitting is 160 X lo4 cm-’ and the wave function is given by 99.75 % 56s I’1 + 0.19 % 4P r4 + 0.04 % 4G I?4. Table VI This table summarizes the measured and calculated energy levels of FeSf for a number of salts. The calculated values are those which we fitted to the iron hydrate. Also included are a number of weak lines found in MgO which presumably are due to iron. It is clear that the fit is reasonably good but that the calculated ground-state splitting is smaller by a factor of 24 than that found by experiment . Table VII This table summarizes some of the calculated ground-state splitting as a function of [, Dq, and the electrostatic parameters. The first two columns show the variation of this splitting as a function of t. This splitting is a strong function of the spin-orbit coupling and varies approximately as t4-15. A change of a factor of 2 in the amount of 4P I’4 admixture changes the initial splitting by a factor of 5. Columns three and four list the results of our calculations using the electrostatic parameters and Dq of G.J.P. for two values, a! = 90 and (Y = 0 cm-‘. Two things are noteworthy. First the splitting is a function of o( but unfortunately its inclusion decreases the splitting. Second the initial splitting of 29 X 10m3 differs considerably from that calculated by G.J.P. who find a value of 53.6 X lo-” cm. DISCUSSION We have carried the crystal field of the iron group elements in cubic field symmetry as far as possible. Within this framework the following conclusions may be inferred: (a) The optical spectrum of &In*+ and Fe3+ can be fitted using five parameters of which only one is a crystal field parameter. Given the assumptions listed in the Introduction, the agreement between experimental and theoretical values is very satisfactory. There is no need to include additional parameters such as a covalency parameter; the freely adjustable parameters B, C, and Dq are sufficient to fit the spectrum. The lack of reliable experimental data is at present the handicap in the detailed interpretation of the spectra. (b) the situation is not so satisfactory as far as the ground-state splitting is concerned. The calculated value is found to be too small by a factor of two to four when the free ion spin-orbit coupling, & , value is used. It is possible to fit the optical spectrum and the initial ground-state splitting by assuming a spinorbit coupling which is 20-30 % larger than C;o.
OPTICAL
SPECTRI7M A?;D GEWUND-STATE
SPLITTING
3-u
‘l’ht calculated wavefunctions permit us to obtain the ‘Ltheoretical” g factors. These deviate from the experimentally found g factors of Xn”+ and even more ’ ‘+. Fidone and Stevens (~3) have pointed out that covalent. for the case of Ee bonding may affect the g-fact,or calculations. In some cases this may result, in a g factor larger t,han the g factor of the free electron. Apparently covalent bonding lllust influenc*e the admixture of the wavefunction and, therefore, also the grounds:tat’e splitt’ing. (;a.hriel rf nl. have discussed in detail the shortcomings of calculations which usr t,he naive crystal field theory. They suggest that the starting Hamiltonian may I)e incomplete. The terms omitted are those connected with spin-orbit, spill-other orbit, and orbit-orbit interactions. Two different cubic field parametc~s /)q, one for the spin-doublet and another for the spin-quartet states may flavr to ttc included; i.e., the crystal field experienced by the excited doublet st,at,cs may be different. In our calculations, we have not found any systematic deviations of t’he doublet levels which may suggest a significantly different value for I)(/ for t’hese levels. Another possibility mentioned by these authors is to assume different spin-orbit parameters for doublet-doublet, quartet-quartet, dollt)lrt-clual.tet, and quartet-sextet matrix elements because of different degrees in covalency for these doublet and quartet levels. This is an ad hoc assumption which needs an experimental confirmation. We are not so certain that this may lead to a significant change in the calculated splitting. IVe should like to suggest a few additional possibilities: (1) One may accept that, the spiii-orbit coupling in salbs in the case of S-state ions is really larger thalr the frecx ion value. This would mean that the manganese ion behaves diffrrtallt,ly than other iron group elements in salts. There is some indication from neut,roii tlifl’rnction data that the form factors of manganese and iron are diffrrclnt, from nickel (24, 2.5). The radial distribution of the S-state ion in its ground state may 1~ somewhat different than so far assumed theoretically (2). The point xymmct,ry of the Hamiltonian deviates from the cubic field for some of the oxcit8cd s:tat,es. The excited states may he of lower symmetry because of coupled vibrat,ions. These may cause admistllres to the ground states changing the perrtlnt,aye ot’ the 41’ wave function. (13) The Hamiltonian we have used does not, take ilrt80 account the electron transfer spectrum. This spectrum affects the relativcl transition probabilities of variolls optical transiGons. The electron transfer sp(>rtrurn may possibly rffert, the init.ial splitting and t.hc g factor, sincck hoth t,hcsf> clualltit its arc sensitive functions of small changes in the composition of the adniisturcs. 111coilclusion, one can remark that crystal field calculations are relatively succc4111 in csplaining the position of the energy levels t,o within a few hundred rm ‘. IIowever, this model is not po\yclrful enough to explain the finer details of Ihc
s~,c~ct~wnn.
LOW AND
344
ROSENGARTEN
ACKNOWLEDGMENT The authors are grateful to Professor G. Racah for many stimulating discussions and for considerable help in constructing the crystal field matrices. This work was supported in part by the Cambridge Research Laboratories, Bedford, O.A.R. through the European Office U.S.A.F. APPENIIIS
We shall give a brief outline of the method of obtaining the matrix elements. The cubic field potential for the d” electron system is given by licub cc [KO + (9&Y*
(YI” +
C)l
(1)
The potential can be considered as a component of an irreducible tensorial operator of order 4 (26). As mentioned in the text of this paper, it is convenient to work in the weak field scheme, i.e., in the vSLJl?j scheme. Here v is the seniority number, r the irreducible representation of the cubic group, and j one of its rows. According to the Wigner-Eckart theorem, we can factor the mat,rix element: (uSLJri
/ Vcub /
v’SL’J’I’~~) = (vSLJ
/i Cl4 11ZJ’SL’J’)..~ ;I, 1
$
I’
tl)
.
(2)
The first factor is the reduced matrix element which depends on the specific configuration, but not on the point symmetry. The second factor depends on the symmetry. This factor can be simplified to
Let us consider the reduced matrix elements. These are given for the cl” configurations (vSLJ
11 U4 I/ z,‘SL’J’)
= (-1)
(2J’ + l(]“” X W (LJL’J’;
S-L--I,‘PJ’ n[(2L + 1) (2L’ + 1) (25 + 1)
S4) c (-1)“” 21” L”Sl,
X W (2L2L’; L”4)
(8‘ USI, (/ d”-‘(v”S”L”) (d’-l
(u”S”L”)
dSL)
(4),
dSL’/} d” &ST,‘)
where W are the Racah coefficients and (I 1) are the fractional parentage coefficients. We can write the matrix element of the cubic potential in the VSLJM scheme as (vSLJM
/ Vcub 1v’SL’J’M’)
= (vSLJ
/I C4 11v’SL’J’)
I_\
OPTICAL
SPECTKI TM .4X 1) G BOUND-STATE
315
SPLITTIXG
where
j( .1’:11’, 10 1./N ‘, + f&p’
1iJ’:l;r’,
44 1 JM)
+
(,r’M’,
4 -
4 1 .JM)]
(fi)
( ./‘.I/‘, 10 1./Al ) and ( .J’;lJ’, 11 1 .J:lJ ) are Wigner coefficients. III order t,o c*alculat,e the ./‘clcnlents, it is convenient t#o oht,ain the transformation ( .J.l/ / #Jr,). These elenlents are given b> whrrr
Here> ( .131 i Jr’,) are the C~O~U~~IIS (for a given .J and I’j) of the transformation Iuat,ris. It. c:an also he shown t,hat they are t,he eigenvect.ors of the matrix (JJI
I’,.,,,, i .J’dJ’) and havct the cigcnvaluc
J J’ 4
.I’ I, r (
r
1)
. These 1 numbers
I
have beon
by Schijnfeld f 5). electrostatic mat8riws of the free ions ilt t#he vSLJM scheme have been .cal(wlattd for the d4 and d” configuration by Eacah, the spin-orbit matrices by Alurakawa (27) and by Racah and Shadmi (28, 11). It should be not,ed that the nlatrix elements have been calculated in the vSL./I sc*hcil~. Jt is not) dificult to obtain t,he eigenvectors in t)he more “physical WhflllP” raSI,I’,,I’., . ( I’,, represents the representation in the absence of the spirl-orbit wupling.) The progranl for the computer was arranged to take (*are of this. calclllated The
f I%-.Imw, “l’aritmagnet ic Resonance in Solids.” *%csdemic Press, New York, IMU. ! I).,C:. ~%C'I,I.KE, “Advances in Solid State Physics,” Vol. 9, p. 399. Arsdemic Press, Sew l.ork, 1950. 5. \-. 'rANARE 4N1) s. SII(;.~N(I,./. I'h!/n.Soc.Jnpetl9.i66 (1954). 4. Y. TANABE ANI) H. KAMIKI.I+ .f.f'h?/x.SW. Jupur~ 13, 39-L(l!l58). b. C;. scr+ijNFEI,I), M. SC. thesis, Jerusalem, 1961. h’. u’. I,ow.. “Quantum E;lectronics I,” p. 410. Columbia ITniv. Press, Sew York, 1960. 7. W. I,ow, “Qmtnturn LClect,ronic;i 11,” p. 138. (‘olumbiz+ Univ. Press, Sew York, 1961. 8. 11. .J. 1). POWELL, J.H. GBKIEI, AXI) I).F. JIJ~~NSIWN,Phys.ZZev. Letters6,145 (1960); .J. I{. ( IAHILIEI,, I). F. JCIHWIYIN AND M. J. 11).POWELL, I'ror.Ho!y.Sot. 264, 503 (1961). 8. J.~~.~~1~111:~1~,J.f’hern.f-‘h~/s.31,’iODCl%!)). 10. $. KOIDE AKDM. Ii. I,. J’RI-~E, Phil. ,lIng. 1813, 607 (1958). Il. Y. Str.4~~1.thesis, Jerusalem, 1961. 12. J<, XI. WATSON, M.T.T. Report No. 12 11!)59) (unpublished). 19. ‘1’. I’. P. HALL, W. HAYES ANTI F. t. Ill. WILI.IA>TS. I’m-. Phys. Sot. (London) 78, 883 (la51 1.
14. I<. I'AJ~I~.~I..\IzI)o, J. ('hew. Whys. 31, 1050 11959). 15. .J. Xl. STOI.I’, ./. (Ihem. I’hys. 33, 303 (1960).
346 16. 17. 18. 19. 20. 21. 22. 2s. 24. 25.
LOW AND ROSENGARTEN
W. L. SCHLKFER,2. Phys. Chem. Neue Folge 4,116 (1955). E. RABINOVITCH AND W. H. STOCKMEYER,J. _4m. Chem. Sot. 64,335 (1942). R. PAPPALARDO, NUOVOcinaento 8, 955 (1958). W. Low ANDM. WEGER, Phys. Rev. 118,1119 (1960). H. WATANABE, Progr. Theoret. Phys. Japan 18,405 (1957). 0. G. HOLMES AND D. S. MCCLURE, J. Chem. Phys. 26, 1686 (1957). M. DVIR AND W. Low, Phys. Rev. 119,1587 (1960). J. FIDONE ANU K. W. H. STEVENS, Proc. Phys. Sot. (London) 73,116 (1959). J. M. HASTINGS, M. ELLIOTT AND L. M. CORLISS, Phys. Rev. 116,13 (1959). R. NATHANS, S. J. PICHART AND W. A. ALPERIN, J. Phys. Sac. Japan 17. Supplement B-111 7 (1962). 26. V. FANO AND G. RACAH, “Irreducible Tensorial Sets.” Academic Press, New York, 1959. 07. K. MCRAKAWA, J. Phys. Sac. Japan 10,919 (1955). 28, G. RACAH, Phys. Rev. 62,438 (1942); 63,367 (1943); 76,1352 (1949).