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Electron paramagnetic resonance of LiNbO3: Fe3+
Solid State Communications,
Vol. 11, pp. 15-19, 1972. Pergamon Press.
Printed in Great Britain
ELECTRON PARAMAGNETIC RESONANCE OF LiNbO3: Fe 3* F. Mehran and B.A. Scott IBM Thomas ]. Watson Research Center, Yorktown Heights, New York 10598
(Received 16 February 1972 by E. Burstein)
We have investigated the X-band electron paramagnetic resonance of stoichiometric and congruent LiNbO 3 crystals grown from melts doped with ~ 500 weight ppm Fe203. The analysis of the observed Fe 3÷ spectrum for a general direction of magnetic field is complicated due to comparable magnitudes of the Zeeman and trigonal field interactions. From the observed lines with the magnetic field parallel to the c-axis and for a temperature range of 40°C < T < 300°C the three parameters in the Hamiltonian are determined to be: g
:
1.995
O
=
3B~2 =
= (1.61
+__ 0.005
x
(0.168
-
5.0
×
1 0 - ~ T ) c m -~
10 -3
-
1.4
x
10-6T)cm -'
These parameters were found to be independent of crystal stoichiomerry.
IN TI-HS paper we report the X-band electron paramagnetic resonance of LiNbO3: Fe 3÷. Although there have been many studies of impurity states in LiNbO 3 by EPR, 1-3 the analysis of the Fe impurity is especially of importance because this impurity appears to be responsible for optical refractive index damage and phase holographic storage in LiNbO 3 as wet1 as other electrooptic crystals. 4-~ The analysis of the LiNbO3: Fe 3+ spectrum is also of interest due to complications arising from comparable magnitudes of trigonal field and Zeeman interactions, so that the usual perturbation theory approach cannot be applied.
Holographic storage tests showed ~ 40% diffraction efficiency in a 3 mm thick section of congruent LiNbO 3 containing 450 ppm Fe203. This crystal was used for the EPR measurements. The stoichiometric LiNbO 3 crystal studied by EPR contained 300ppm Fe2Os. The EPR spectrum of Fe 3+ (3d5,6S~/2)in LiNbO3 can be understood on the basis of C3v site symmetry in the R3c LiNbO~ structure. The Hamiltonian can be written: ~ J( = /ggn.-$'
+ D
( 3s) S~ -
~-~
1 B4o(35S, 950 2 2835) 12 4 Sz +
Single crystals used in the present investigation were grown by the Czochralski technique in an 02 temperature. Both the congruent (Li/Nb) = 0.95 and stoichiometric (Li/Nb) = 1 crystal were grown, the latter from Li 20 rich melts. Fe2 03 was added to the melts in concentrations which would yield 300-500 wt. ppm in the crystals.
where D - 3B °.
15
(1)
16
ELECTRON PARAMAGNETIC RESONANCE
In equation (I) the three terms are the Zeeman, the second and the fourth order crystal field interactions respectively and the z-axis is chosen along the c-axis of the crystal. In the present case of Fe 3~doped LiNbO3, the first two terms in the Hamiltonian, the Zeeman and the second order interactions, are comparable in magnitude whereas the fourth order interaction given by the third term is much smaller. The secular equation should, therefore, be solved exactly for the first two terms and the third term may then be treated as a perturbation. The sixth degree secular equation cannot be solved exactly for a general direction of magnetic field. However, for the two special directions of the magnetic field parallel and perpendicular to the trigonal axis, the secular equation may be exactly solved (see Appendix) with the following results:
E ± ~1 = _+ ~11 3 g H -
HI1 c-axis
E + 3
58 D
+_ 3 BgH - 2 D
5
5
E ± 5 = ± 5_13gH +ZOo 2 2 3
I
of the expectation values of the fourth order term for the field parallel to the c-axis is very simple, and if we include this term equation (2) becomes modified in the following way:
E ±~=1
±.21g13H - 38 D -
10B~
[4(a)]
E ±_3 = .,_._3g13H _ _2 O + 15B ° [4(b)] 2 2 3 E ± 5 = ± 5g13H +10 D 2 2 3
5 B e [4(c)]
The allowed transition energies from equation (4) are:
+_3_<..>± 5_ h~ = 13gH ± 4D-T- 20B ° 2 2
[5(a)]
±3<..>±1 2 2
hw = BgH ± 2D +25B °
[5(b)]
hv = BgH
[5(c)]
[2(a)]
[2(b)] 1
1
2
2
_<->-_
[2(c)] Figures 1 and 2 show the energy levels and the observed X-band microwave transitions corresponding to equations (3-5).
E ±1= ± l g / ~ / _ Q(cos0+ x/3 sin0) [3(a)] 2
HA- c-axis
Vol. 11, No. 1
E ± 3 : :klgl~H_Q(cosO_~/3sinO ) [3(b)]
±5: where 8 D a + 4(g~H)2 T- ~g~ 4 HD J],/2 Q = I 2~-
In equations (2) and (3) the subscripts refer to the quantum numbers of the Sz operator in the
[3(c)I
The effects of the fourth order term on the energy levels for the direction of the magnetic field perpendicular to the c-axis have not been calculated since equation (5) is more than sufficient for determining the three parameters in the Hamiltonian. Equation (2) was then used to check for the consistency of the results. Using equation [5(c)] for the M = -~-I t o + ± 2 transition, we determine the g-factor to be g = 1.995 ± 0.005. The parameters D and B ° were then determined from equations [5(a-b)] and were found to be temperature dependent. Figures 3 and g show the temperature variations of D and B°~ respectively. These variations can be approximated by the following relationships (Y in °C):
Vol. 11, No. 1
E L E C T R O N PARAMAGNETIC RESONANCE
300:
225
22'5L
200
2.5C-
1.75
2.25-
1.50
2.00-
17
I
M=5/2/M=-5/2~.
I.~
M=5/2
1.75-
1.00
i
1.50
0.75 05O
1.00~-
= °75 'L,,! ; e50
.....
025
'E ODO
-°25 F
M='3/2
,
F,,-o.50p-~. : g (XX~.
~-o~[ ~ ' ~
M=I/2
~ ~
-025 f -(150 -075 i
-2.012
"x,~= -I/2
- 1.50~ -1.75L -2.00[0
'z~5o
' ~
' 6~o
' ~oo
' ,oooo
MAGNETIC FIELD (GAUSS)
FIG. 1. Energy l e v e l s and o b s e r v e d X-band microwave transitions for .H I[ c-axis at T = 40°C.
D = (0.168 - 5.0 × l O - S T ) c m - ' , B°
= (1.61 × I 0 - 3 -
1.4 x l O - ~ T ) c m - ' .
Within experimental error, t h e s e parameters were found to be independent of crystal stoichiometry. It is of interest to note that the linear relationship between D and T is similar to r e s u l t s that were obtained in the E P R of Gd 3+ and F e ~ in B a T i 0 3 , 8 which s u g g e s t e d that D(T) cc P~(T), where Ps (7") is the s p o n t a n e o u s polarization. In the c a s e of BaTiO 3, D is related to the tetragonal field splitting and D --, 0 a s Ps --* 0 at the ferroe l e c t r i c - t o - p a r a e l e c t r i c transition temperature T c = 130°C. On the other hand, the symmetry of LiNbO 39 above T c = 1200°C is trigonal R 3 c , so that D must remain finite. Extrapolation of the data of Fig. 3 to high temperatures yields
-Z7~.
r
2oo0
i
,~o
I
6doo
sooo
,owo
MAONET,C FI~_D.
FIG. 2. Energy levels and o b s e r v e d X-band microwave transition for .H i c-axis at T = 40°C.
D = 0.11cm -~ at the Curie point in LiNbO 3. The temperature variation of B ° may be attributed to the c h a n g e s in the c r y s t a l field potential due to the polarization effects arising from the lack of local charge compensation, l° Acknowledgements - We wish to thank K. Nichols for expert experimental a s s i s t a n c e in the growth of the c r y s t a l s used in this study and B.L. Olson, R. J o h n s o n and G. P i g e y for chemical a n a l y s i s results. We are also grateful to Dr. K.B. Pennington for the holographic s t o r a g e measurements and to Drs. R.S. Title and G. Burns for valuable comments on the manuscript.
ELECTRON P A R ~ I A G N E T I C RESONANCE
18
Vol. 11, No. 1
t.55
m~ F
0.641
1.50
0.163I 1.45
0,161 i-
'~ 1.40:
0.160 '~ 0159 Q
0.158 QI57
1.50
0.156 0.155
1.25
0.154 0.153 0.152L
1.20
,50
Ioo
150 200
250
300
TEMPERATURE('C)
o
FIG. 3. The temperature variation of the second order field parameter D.
50
tO0 15o zoo zso 3oo
TEMPERATUR ("C) E
FIG. 4. The temperature variation of the fourth order field parameter B °.
REFERENCES 1.
BURNS G., O'KANE D.F. and T I T L E R.S. Phys. Lett. 23, 56 (1966); Phys. Rev. 167, 314 (1968).
2.
TAKEDA T., WATANABE A. and SUGIHARA K., Phys. Lett. 27A, 114 (1968).
3.
DANNER J.C., RANON U. and STAMIRES D.N., Chem. Phys. Lett. 2, 605 (1968).
4.
AMODEI J . J . , P H I L L I P S W. and STAEBLER D.L., IEEE J. Quant. Elec. QE7, 63 (1971).
5.
AMODEI J . J . , STAEBLER D.L. and STEPHENS A.W., Appl. Phys. Lett. 18, 507 (1971).
6.
PETERSON G.E., GLASS A.M. and NEGRAN T.J., Appl. Phys. Lett. 19, 130 (1971).
7.
See, for example, ABRAGAM A. and BLEANEY B., Electron Paramagnetic Resonance of Transition
Ions, Clarendon P r e s s , Oxford, (1970). 8.
RIMAI L. and DEMARS G.A., Phys. Rev. 127, 702 (1962).
9.
ABRAHAMS S.C., LEVINSTEIN H.J. and REDDY J.M., J. Phys. Chem. Solids 27, 1019 (1966).
10.
RIMAI L., DEUTSCH T. and SILVERMAN B.D., Phys. Rev. 133, Al123 (1964). Nous avons ~tudi~ la resonance paramagnetique en bande X de cristaux de LiNbO_ sto~chiom~triques et congruents dop6s au fer. L e s cristaux sont tires a partir de bains dopes avec ",~ S00ppm de F e 2 0 3. L ' a n a l y s e du spectre de 1'ion Fe 3-~ pour une direction quelconque du champ magnetique e s t compliqu~e car le champ Zeeman et le champ trigonal sont de grandeurs comparables. En etudiant les tales obtenues en champ magnetique parall~le ~ 1'axe C dans la gamme de temperatures 40°C < T < 300°C, nous obtenons les valeurs des trois param~tres de l'Hamiltonien de spin: q D
= 1.995 -- 0 . 0 0 5 = 3B ° = (0.168 -
B° =
( 1 . 6 1 × 10 - ~ -
5.0 x 10 - ~ T ) c m - , 1.4 × 10 -~ T ) c m - '
Ces param~tres ne dependent pas de la sto~chiom~trie du cristal.
Vol. II, No. 1
ELECTRON PAR.~IAGNET/C RESONANCE
19
APPENDIX F o r a g e n e r a l d i r e c t i o n o f t h e m a g n e t i c f i e l d t h e s e c u l a r e q u a t i o n for t h e H a m i l t o n i a n ( n e g l e c t i n g t h e fourth o r d e r term' i s :
+
i
!>
o
o
I-3_>
I-~>
o
o
o
<~[ 2
o
o
o
o
V
~gH+
_
(At)
~5 H ~ ~ 10 -f D-E
w h e r e H+ = H z ± i H v , a n d t h e s t a t e s h a v e b e e n l a b e l e d by t h e q u a n t u m n u m b e r s of t h e S z o p e r a t o r w h i c h are ' g o o d ' in t h e a b s e n c e of t h e m a g n e t i c f i e l d . For the field along the c-axis; H+ = H_ = 0 and the secular equation is a product of six linear equations with
t h e s o l u t i o n s g i v e n in e q u a t i o n (2). F o r t h e f i e l d p e r p e n d i c u l a r to t h e c - a x i s we c h o o s e t h e d i r e c t i o n o f t h e m a g n e t i c f i e l d a s t h e z - a x i s a n d t h e c - a x i s a s t h e x - a x i s . T h e H a m i l t o n i a n ( n e g l e c t i n g t h e f o u r t h o r d e r time) i s then:
and the secular equation is:
-
VTD
<_--3,2
0
-~-3'/2D
<_5,
0
0
0
_ ~/3.oflz 3 + 1 D-E
0
2
0
0
0
0
0
O
v'10D
0
-5F~.~Hz - 5 D - E 2~
<- 11 2
0
<31
0
0
0
0
0
~-
3
v'10 D
2
0
- 18gH, 4 D-E 2
(A3)
3 v'2 D
4
3~/2
= 0
"2"-
~3~ H
+ ~D-E
where the states are l a b e l e d by the eigen values of Sz in the absence of the crystal field. The secular equation (A3) is thus a product of two cubic equations and m a y be solved analytically with the solutions given in equation (3).
Vol. 11, pp. 15-19, 1972. Pergamon Press.
Printed in Great Britain
ELECTRON PARAMAGNETIC RESONANCE OF LiNbO3: Fe 3* F. Mehran and B.A. Scott IBM Thomas ]. Watson Research Center, Yorktown Heights, New York 10598
(Received 16 February 1972 by E. Burstein)
We have investigated the X-band electron paramagnetic resonance of stoichiometric and congruent LiNbO 3 crystals grown from melts doped with ~ 500 weight ppm Fe203. The analysis of the observed Fe 3÷ spectrum for a general direction of magnetic field is complicated due to comparable magnitudes of the Zeeman and trigonal field interactions. From the observed lines with the magnetic field parallel to the c-axis and for a temperature range of 40°C < T < 300°C the three parameters in the Hamiltonian are determined to be: g
:
1.995
O
=
3B~2 =
= (1.61
+__ 0.005
x
(0.168
-
5.0
×
1 0 - ~ T ) c m -~
10 -3
-
1.4
x
10-6T)cm -'
These parameters were found to be independent of crystal stoichiomerry.
IN TI-HS paper we report the X-band electron paramagnetic resonance of LiNbO3: Fe 3÷. Although there have been many studies of impurity states in LiNbO 3 by EPR, 1-3 the analysis of the Fe impurity is especially of importance because this impurity appears to be responsible for optical refractive index damage and phase holographic storage in LiNbO 3 as wet1 as other electrooptic crystals. 4-~ The analysis of the LiNbO3: Fe 3+ spectrum is also of interest due to complications arising from comparable magnitudes of trigonal field and Zeeman interactions, so that the usual perturbation theory approach cannot be applied.
Holographic storage tests showed ~ 40% diffraction efficiency in a 3 mm thick section of congruent LiNbO 3 containing 450 ppm Fe203. This crystal was used for the EPR measurements. The stoichiometric LiNbO 3 crystal studied by EPR contained 300ppm Fe2Os. The EPR spectrum of Fe 3+ (3d5,6S~/2)in LiNbO3 can be understood on the basis of C3v site symmetry in the R3c LiNbO~ structure. The Hamiltonian can be written: ~ J( = /ggn.-$'
+ D
( 3s) S~ -
~-~
1 B4o(35S, 950 2 2835) 12 4 Sz +
Single crystals used in the present investigation were grown by the Czochralski technique in an 02 temperature. Both the congruent (Li/Nb) = 0.95 and stoichiometric (Li/Nb) = 1 crystal were grown, the latter from Li 20 rich melts. Fe2 03 was added to the melts in concentrations which would yield 300-500 wt. ppm in the crystals.
where D - 3B °.
15
(1)
16
ELECTRON PARAMAGNETIC RESONANCE
In equation (I) the three terms are the Zeeman, the second and the fourth order crystal field interactions respectively and the z-axis is chosen along the c-axis of the crystal. In the present case of Fe 3~doped LiNbO3, the first two terms in the Hamiltonian, the Zeeman and the second order interactions, are comparable in magnitude whereas the fourth order interaction given by the third term is much smaller. The secular equation should, therefore, be solved exactly for the first two terms and the third term may then be treated as a perturbation. The sixth degree secular equation cannot be solved exactly for a general direction of magnetic field. However, for the two special directions of the magnetic field parallel and perpendicular to the trigonal axis, the secular equation may be exactly solved (see Appendix) with the following results:
E ± ~1 = _+ ~11 3 g H -
HI1 c-axis
E + 3
58 D
+_ 3 BgH - 2 D
5
5
E ± 5 = ± 5_13gH +ZOo 2 2 3
I
of the expectation values of the fourth order term for the field parallel to the c-axis is very simple, and if we include this term equation (2) becomes modified in the following way:
E ±~=1
±.21g13H - 38 D -
10B~
[4(a)]
E ±_3 = .,_._3g13H _ _2 O + 15B ° [4(b)] 2 2 3 E ± 5 = ± 5g13H +10 D 2 2 3
5 B e [4(c)]
The allowed transition energies from equation (4) are:
+_3_<..>± 5_ h~ = 13gH ± 4D-T- 20B ° 2 2
[5(a)]
±3<..>±1 2 2
hw = BgH ± 2D +25B °
[5(b)]
hv = BgH
[5(c)]
[2(a)]
[2(b)] 1
1
2
2
_<->-_
[2(c)] Figures 1 and 2 show the energy levels and the observed X-band microwave transitions corresponding to equations (3-5).
E ±1= ± l g / ~ / _ Q(cos0+ x/3 sin0) [3(a)] 2
HA- c-axis
Vol. 11, No. 1
E ± 3 : :klgl~H_Q(cosO_~/3sinO ) [3(b)]
±5: where 8 D a + 4(g~H)2 T- ~g~ 4 HD J],/2 Q = I 2~-
In equations (2) and (3) the subscripts refer to the quantum numbers of the Sz operator in the
[3(c)I
The effects of the fourth order term on the energy levels for the direction of the magnetic field perpendicular to the c-axis have not been calculated since equation (5) is more than sufficient for determining the three parameters in the Hamiltonian. Equation (2) was then used to check for the consistency of the results. Using equation [5(c)] for the M = -~-I t o + ± 2 transition, we determine the g-factor to be g = 1.995 ± 0.005. The parameters D and B ° were then determined from equations [5(a-b)] and were found to be temperature dependent. Figures 3 and g show the temperature variations of D and B°~ respectively. These variations can be approximated by the following relationships (Y in °C):
Vol. 11, No. 1
E L E C T R O N PARAMAGNETIC RESONANCE
300:
225
22'5L
200
2.5C-
1.75
2.25-
1.50
2.00-
17
I
M=5/2/M=-5/2~.
I.~
M=5/2
1.75-
1.00
i
1.50
0.75 05O
1.00~-
= °75 'L,,! ; e50
.....
025
'E ODO
-°25 F
M='3/2
,
F,,-o.50p-~. : g (XX~.
~-o~[ ~ ' ~
M=I/2
~ ~
-025 f -(150 -075 i
-2.012
"x,~= -I/2
- 1.50~ -1.75L -2.00[0
'z~5o
' ~
' 6~o
' ~oo
' ,oooo
MAGNETIC FIELD (GAUSS)
FIG. 1. Energy l e v e l s and o b s e r v e d X-band microwave transitions for .H I[ c-axis at T = 40°C.
D = (0.168 - 5.0 × l O - S T ) c m - ' , B°
= (1.61 × I 0 - 3 -
1.4 x l O - ~ T ) c m - ' .
Within experimental error, t h e s e parameters were found to be independent of crystal stoichiometry. It is of interest to note that the linear relationship between D and T is similar to r e s u l t s that were obtained in the E P R of Gd 3+ and F e ~ in B a T i 0 3 , 8 which s u g g e s t e d that D(T) cc P~(T), where Ps (7") is the s p o n t a n e o u s polarization. In the c a s e of BaTiO 3, D is related to the tetragonal field splitting and D --, 0 a s Ps --* 0 at the ferroe l e c t r i c - t o - p a r a e l e c t r i c transition temperature T c = 130°C. On the other hand, the symmetry of LiNbO 39 above T c = 1200°C is trigonal R 3 c , so that D must remain finite. Extrapolation of the data of Fig. 3 to high temperatures yields
-Z7~.
r
2oo0
i
,~o
I
6doo
sooo
,owo
MAONET,C FI~_D.
FIG. 2. Energy levels and o b s e r v e d X-band microwave transition for .H i c-axis at T = 40°C.
D = 0.11cm -~ at the Curie point in LiNbO 3. The temperature variation of B ° may be attributed to the c h a n g e s in the c r y s t a l field potential due to the polarization effects arising from the lack of local charge compensation, l° Acknowledgements - We wish to thank K. Nichols for expert experimental a s s i s t a n c e in the growth of the c r y s t a l s used in this study and B.L. Olson, R. J o h n s o n and G. P i g e y for chemical a n a l y s i s results. We are also grateful to Dr. K.B. Pennington for the holographic s t o r a g e measurements and to Drs. R.S. Title and G. Burns for valuable comments on the manuscript.
ELECTRON P A R ~ I A G N E T I C RESONANCE
18
Vol. 11, No. 1
t.55
m~ F
0.641
1.50
0.163I 1.45
0,161 i-
'~ 1.40:
0.160 '~ 0159 Q
0.158 QI57
1.50
0.156 0.155
1.25
0.154 0.153 0.152L
1.20
,50
Ioo
150 200
250
300
TEMPERATURE('C)
o
FIG. 3. The temperature variation of the second order field parameter D.
50
tO0 15o zoo zso 3oo
TEMPERATUR ("C) E
FIG. 4. The temperature variation of the fourth order field parameter B °.
REFERENCES 1.
BURNS G., O'KANE D.F. and T I T L E R.S. Phys. Lett. 23, 56 (1966); Phys. Rev. 167, 314 (1968).
2.
TAKEDA T., WATANABE A. and SUGIHARA K., Phys. Lett. 27A, 114 (1968).
3.
DANNER J.C., RANON U. and STAMIRES D.N., Chem. Phys. Lett. 2, 605 (1968).
4.
AMODEI J . J . , P H I L L I P S W. and STAEBLER D.L., IEEE J. Quant. Elec. QE7, 63 (1971).
5.
AMODEI J . J . , STAEBLER D.L. and STEPHENS A.W., Appl. Phys. Lett. 18, 507 (1971).
6.
PETERSON G.E., GLASS A.M. and NEGRAN T.J., Appl. Phys. Lett. 19, 130 (1971).
7.
See, for example, ABRAGAM A. and BLEANEY B., Electron Paramagnetic Resonance of Transition
Ions, Clarendon P r e s s , Oxford, (1970). 8.
RIMAI L. and DEMARS G.A., Phys. Rev. 127, 702 (1962).
9.
ABRAHAMS S.C., LEVINSTEIN H.J. and REDDY J.M., J. Phys. Chem. Solids 27, 1019 (1966).
10.
RIMAI L., DEUTSCH T. and SILVERMAN B.D., Phys. Rev. 133, Al123 (1964). Nous avons ~tudi~ la resonance paramagnetique en bande X de cristaux de LiNbO_ sto~chiom~triques et congruents dop6s au fer. L e s cristaux sont tires a partir de bains dopes avec ",~ S00ppm de F e 2 0 3. L ' a n a l y s e du spectre de 1'ion Fe 3-~ pour une direction quelconque du champ magnetique e s t compliqu~e car le champ Zeeman et le champ trigonal sont de grandeurs comparables. En etudiant les tales obtenues en champ magnetique parall~le ~ 1'axe C dans la gamme de temperatures 40°C < T < 300°C, nous obtenons les valeurs des trois param~tres de l'Hamiltonien de spin: q D
= 1.995 -- 0 . 0 0 5 = 3B ° = (0.168 -
B° =
( 1 . 6 1 × 10 - ~ -
5.0 x 10 - ~ T ) c m - , 1.4 × 10 -~ T ) c m - '
Ces param~tres ne dependent pas de la sto~chiom~trie du cristal.
Vol. II, No. 1
ELECTRON PAR.~IAGNET/C RESONANCE
19
APPENDIX F o r a g e n e r a l d i r e c t i o n o f t h e m a g n e t i c f i e l d t h e s e c u l a r e q u a t i o n for t h e H a m i l t o n i a n ( n e g l e c t i n g t h e fourth o r d e r term' i s :
+
i
!>
o
o
I-3_>
I-~>
o
o
o
<~[ 2
o
o
o
o
V
~gH+
_
(At)
~5 H ~ ~ 10 -f D-E
w h e r e H+ = H z ± i H v , a n d t h e s t a t e s h a v e b e e n l a b e l e d by t h e q u a n t u m n u m b e r s of t h e S z o p e r a t o r w h i c h are ' g o o d ' in t h e a b s e n c e of t h e m a g n e t i c f i e l d . For the field along the c-axis; H+ = H_ = 0 and the secular equation is a product of six linear equations with
t h e s o l u t i o n s g i v e n in e q u a t i o n (2). F o r t h e f i e l d p e r p e n d i c u l a r to t h e c - a x i s we c h o o s e t h e d i r e c t i o n o f t h e m a g n e t i c f i e l d a s t h e z - a x i s a n d t h e c - a x i s a s t h e x - a x i s . T h e H a m i l t o n i a n ( n e g l e c t i n g t h e f o u r t h o r d e r time) i s then:
and the secular equation is:
-
VTD
<_--3,2
0
-~-3'/2D
<_5,
0
0
0
_ ~/3.oflz 3 + 1 D-E
0
2
0
0
0
0
0
O
v'10D
0
-5F~.~Hz - 5 D - E 2~
<- 11 2
0
<31
0
0
0
0
0
~-
3
v'10 D
2
0
- 18gH, 4 D-E 2
(A3)
3 v'2 D
4
3~/2
= 0
"2"-
~3~ H
+ ~D-E
where the states are l a b e l e d by the eigen values of Sz in the absence of the crystal field. The secular equation (A3) is thus a product of two cubic equations and m a y be solved analytically with the solutions given in equation (3).