- Email: [email protected]
Computation of principal values and directions of tensors in the spin Hamiltonian
JOLRNAL
OF MAGNETIC
RESONANCE
41, 195-199 (1980)
Computation of Principal Values and Directions of Tensors in the Spin Hamiltonian E. BULUGGIU Grrcppo
Nazionale
Struttura
della
Materia
AND A. VERA
de1 CNR,
Istituto
di Fisica
dell’Universitri,
Parma,
Italy
Received October 26, 1979; revised February 11, 1980 A method is presented for the evaluation of the parameters appearing in the spin Hamiltonian X = p6i.g.S + S.D.S + I-,A.S in the usual energy scheme in which gfiH > D > A. The tensors are supposed symmetric and not coaxial, and D is traceless. The method extends the well-known “three arbitrary planes” procedures to the case of a generic value of S taking into account the second-order effects due to the zero-field splitting term. An application to the evaluation of the magnetic parameters of the Cu(pyridine N-oxide)Cl,.H,O dimeric complex is given.
.A recurrent practical problem, in EPR experiments, of the parameters of the spin Hamiltonian X = @I-g+
+ s.D.S
is the accurate evaluation
+ I.A.S,
[II
where all tensors are assumed symmetric, generally noncoaxial, and D is traceless. Moreover, we consider the Zeeman term as the preponderant energy, so that a perturbation treatment is possible. Several procedures have been derived for determining the principal values and directions of the g tensor. These involve measurement of gerf = h~Ij3H for II covering a number of different directions on three arbitrary planes. Originally proposed for three orthogonal planes by Geusic and Brown (1) and in a modified form by Schonland (2), the procedure was generalized to the case of three not necessarily orthogonal planes in Ref. (2) and by Waller and Rogers (3). All these methods are applicable when no fine splitting is present and, hence, are strictly valid for the case S = M only. .4 similar procedure was applied by Lund and Vanngard (4) to the evaluation of the D and A tensors and, hence, to the case of arbitrary S values. So extended, the method appears undoubtedly advantageous. In fact, it allows one to deduce all the parameters of Eq. [l] without requiring either the knowledge a priori of the magnetic axes or specific orientations of the crystal. This is particularly useful for the case of noncoaxial tensors, where the directions of the magnetic axes can only approximately be estimated. Moreover, if many directions of the magnetic field are used, it allows one to obtain the tensors which give the best fit over the angular variation of the spectrum. 195 0022-2364/80/140195-05$02.00/O Copyright 0 1980 by Academic Press, Inc. All rights of reproduction in any form reserved.
1%
BULUGGIU
AND
VERA
The point is that, when applied to a case S f i/i, this method, as we will see, appears to be “intrinsically” limited to the first-order approximation of the perturbation treatment. This fact, which has a particular relevance for g, is not sufficiently clarified in Ref. (4). Thus, too large values for the g tensor and, as a consequence, for all the tensors of Eq. [l] may result, principally because one neglects second-order zero-field effects. This is particularly the case, obviously, when a spectrometer of sufficiently high frequency is not available; see also the example of Table 1. Therefore, it appears convenient to point out a procedure, valid for any value of S, which primarily reconciles the use of a “three-plane” method with a more correct evaluation of g and which, in principle, allows a higher-order treatment of all the parameters of Eq. [ 11. With this in mind, we limit the second-order perturbation treatment of Eq. [l] to the zero-field splitting term, according to the usual energy scheme g@Y > D > A. Then, for the allowed transitions in a strong magnetic field, we can write (5) hv = E,,, +-
- EM,-LM, = g,,PH 1
f3p11*
k&H
- 3&f
IW - y--
+ A,&,
(3Ms” - 3Ms + 1)
- ID1 1*(4S* + 4s - 1) + y
(2s* + 2s
1) 9 PI I
where g& = &if-h, g:,D,ff
[31
= kg*D*g*h,
141
[51
g2,ffA& = Ii-g-AZ-g-h, 1D1j2 =
ii-g-D*-g-h ~.g’ h
10~1~ = 2TrD*
- 4
- ‘;:;‘gh;;)*
WI
,
ii -g .D**g .h + (6.g.D.g.h)* ii -g* .h
0;.g2.h)*
’
171
and h is the unit vector along I-I. Equations [3], [4], and [5] are the basic formulas for the determination of the g, D, and A tensors, respectively. Following, for example, Refs. (1) and (4), by fitting the gEff, g2,,Deff, and g2,ffAZ,ff values measured for a number of directions of II on the plane, say (xy), one obtains the elements xx, yy, and xy of the g*, gDg, and gAZg tensors. Repeating this procedure for the (yz) and (xz) planes, all the missing elements can be obtained. It is possible in this way to deduce the g, D, and A principal values and directions. Note that gEff, gzffDeff, gzffAEff are considered as measured “locally,” that is, for each single direction of the magnetic field. The difficulty is that this appears rigorously possible for the trivial case S = ?4 only, where zero-field splitting and consequent second-order effects are absent. In a more general case, from Eq. [2] we can write
n The
8mm
3 cm
Wavelength
EPR
spectra
are described
2.0718 2.0654 2.0653
2.0675
8
1 2 3
2.1583 2.0606 2.0682 2.0675
gz
1 2 3 4
Step
by the spin
2.0822 2.0764 2.0764
2.0951
2.1946 2.0897 2.0957 2.0950
Hamiltonian
2.3268 2.3205 2.3206
2.3046
2.4121 2.2966 2.3054 2.3046
COMPUTATIONOFTHEMAGNETICPARAMETERSOFTHECV+-CU COMPLEX
0.0812 0.0811 0.0811
0.0781
0.0813 0.0777 0.0781 0.0781
D.Z,
ITERATIVE
1
X = p8.g.S
BY THE
TABLE
+ 5.D.S
0.0221 0.0218 0.0218
0.0232
0.0263 0.0231 0.0233 0.0232
D,,
I+ PAIR METHOD”
with
S = 1. All the
-0.1033 -0.1029 -0.1029
-0.1013
-0.1076 -0.1008 -0.1014 -0.1013
Dz,
IN CU(PYRIDINE
D values
Z
/ Y
X
Z
t Y
X
38 121 81
54 140 75
x’
60 39 68
53 50 62
y’
angles
are in cm-‘.
Resulting
N-OXIDE)CI,.H,O
(deg)
110 103 24
122 97 32
z’
198
BULUGGIU
(2s - 1)
hv geff
=
pH
-
Tgeffp2R2
AND
(25 - l)(D,12
VERA
+ 4M2
+ smaller terms, I
VI
where
f: gj=
MF-I
i W&.,s-, MS=-Sfl 2S(2Z + 1)
’
and Hi$+M.-l is the resonance field relative
to the electronic (MS, MI * MS - 1, MI). In the same approximation we have Den = 2gemP M--, i y$,,
W - &)H~,&M~-,/S(~S~
transitions
- 1X21 + 1)
191
s-
I-
and I
Aeff = -3gerrP 1 My-I
s C
M1H3s:ls,M,-,/2SZ(Z + 1)(2Z + 1).
[lOI
MS=-s-1
These are the operative relations which allow us to measure the g,,, Deff, and Aeff values from the experiment. As can be seen, for a generic value of S, these parameters can be deduced from a single spectrum only if one disregards the second-order part of Eq. [8], which requires more information about the g and D tensors than the available one. In this approximation, the use of Eqs. [8] to [lo] is equivalent to assuming, for each direction of the magnetic field, the geff, Deff, and Aeff values which give the least-squares best fit of all the resonance fields by means of the first-order part of Eq. [2] only. In order to take into account the zero-field effects we suggest the following iterative procedure. As the initial step, for each assumed direction on the three chosen orthogonal planes, we deduce the g eff local value from the mean of the resonance fields, i.e., neglecting the second-order terms of Eq. [8], and the corresponding Deff and Aeff values of Eqs. [9] and [lo]. This allows us to make a first rough evaluation of the g, D, and A tensors. By use of these, we are now able to estimate approximately the second-order term of Eq.. [8] for the various assumed directions. Thus, we obtain more accurate values for the local geff, Deff, and Aeff parameters. Such new values are used to redetermine the g, D, and A tensors, and so on up to convergence. As an application of this procedure, we have deduced the magnetic parameters of the dimeric Cu(II)(pyridine N-oxide)Cl,*H,O complex, where the two copper ions per unit cell are antiferromagnetically coupled through two bridges consisting of oxygen of the pyridine N-oxide (6). EPR spectra at room temperature are characteristic of a partially populated excited state with S = 1. There is no evidence of hyperfme structure and the angular behavior of the spectrum shows remarkable misalignment between the g and D tensors. This last fact makes use of a three-plane method advantageous with.respect to alternative more approximate and tedious techniques of systematic searching of extreme zero-field splittings and g values. The direction of the magnetic field was varied at regular intervals (10“) on three orthogonal planes. Owing to the unavailability of complete struc-
EVALUATION
OF
SPIN-HAMILTONIAN
PARAMETERS
199
tural data, we have chosen the three planes casually. For the same reason we give re‘lative rather than absolute principal directions for the g and D tensors. The results, for the X-band and Q-band frequencies, are given in Table 1. In both cases the same directions of the magnetic field were taken. Initially, the values are not corrected, as mentioned, for second-order effects. In this limit the X band values are unacceptably larger than those obtained at Q band which are more typical for Cu w ions. Moreover, the latter values may be considered close to the “true” ones, as shown by the small second-order modifications during the following steps. As one can see, the iterative procedure improves substantially the agreement between the parameters calculated at the two frequencies. Further, it is rapidly convergent and requires a negligible increase in computation time. In this paper we have taken into account the second-order zero-field effects only, because these appear to be the most important sources of error. However, a similar treatment of all the parameters of Eq. [l] is also possible and, moreover, sulch a procedure appears utilizable for any higher order of the perturbation treatment. ACKNOWLEDGMENTS
The authors wish to thank Professor G. Dascola for encouragement V. Varacca for critically reading a preliminary draft of the manuscript.
and help and Professor
REFERENCES I. 2. 3. 4. 5. 6.
J. E. GEIJSIC AND L. C. BROWN, Phys. Rev. 112, 64 (1958). D. S. SHONLAND, Proc. Phys. Sot. London 73, 788 (1959). W. G. WALLER AND M. T. ROGERS, J. Magn. Reson. 9, 92 (1973). A. LUND AND T. V;~NNG.&RD,J. Chem. Phys. 42, 2979 (1965). M. IWASAKI, J. Map. Reson. 16, 417 (1974). G. F. KOKOSZKA, H. C. ALLEN. JR., AND G. GORDON, J. Chem. Phys. 3020 (1967).
46,
3013 (1967); 46,
OF MAGNETIC
RESONANCE
41, 195-199 (1980)
Computation of Principal Values and Directions of Tensors in the Spin Hamiltonian E. BULUGGIU Grrcppo
Nazionale
Struttura
della
Materia
AND A. VERA
de1 CNR,
Istituto
di Fisica
dell’Universitri,
Parma,
Italy
Received October 26, 1979; revised February 11, 1980 A method is presented for the evaluation of the parameters appearing in the spin Hamiltonian X = p6i.g.S + S.D.S + I-,A.S in the usual energy scheme in which gfiH > D > A. The tensors are supposed symmetric and not coaxial, and D is traceless. The method extends the well-known “three arbitrary planes” procedures to the case of a generic value of S taking into account the second-order effects due to the zero-field splitting term. An application to the evaluation of the magnetic parameters of the Cu(pyridine N-oxide)Cl,.H,O dimeric complex is given.
.A recurrent practical problem, in EPR experiments, of the parameters of the spin Hamiltonian X = @I-g+
+ s.D.S
is the accurate evaluation
+ I.A.S,
[II
where all tensors are assumed symmetric, generally noncoaxial, and D is traceless. Moreover, we consider the Zeeman term as the preponderant energy, so that a perturbation treatment is possible. Several procedures have been derived for determining the principal values and directions of the g tensor. These involve measurement of gerf = h~Ij3H for II covering a number of different directions on three arbitrary planes. Originally proposed for three orthogonal planes by Geusic and Brown (1) and in a modified form by Schonland (2), the procedure was generalized to the case of three not necessarily orthogonal planes in Ref. (2) and by Waller and Rogers (3). All these methods are applicable when no fine splitting is present and, hence, are strictly valid for the case S = M only. .4 similar procedure was applied by Lund and Vanngard (4) to the evaluation of the D and A tensors and, hence, to the case of arbitrary S values. So extended, the method appears undoubtedly advantageous. In fact, it allows one to deduce all the parameters of Eq. [l] without requiring either the knowledge a priori of the magnetic axes or specific orientations of the crystal. This is particularly useful for the case of noncoaxial tensors, where the directions of the magnetic axes can only approximately be estimated. Moreover, if many directions of the magnetic field are used, it allows one to obtain the tensors which give the best fit over the angular variation of the spectrum. 195 0022-2364/80/140195-05$02.00/O Copyright 0 1980 by Academic Press, Inc. All rights of reproduction in any form reserved.
1%
BULUGGIU
AND
VERA
The point is that, when applied to a case S f i/i, this method, as we will see, appears to be “intrinsically” limited to the first-order approximation of the perturbation treatment. This fact, which has a particular relevance for g, is not sufficiently clarified in Ref. (4). Thus, too large values for the g tensor and, as a consequence, for all the tensors of Eq. [l] may result, principally because one neglects second-order zero-field effects. This is particularly the case, obviously, when a spectrometer of sufficiently high frequency is not available; see also the example of Table 1. Therefore, it appears convenient to point out a procedure, valid for any value of S, which primarily reconciles the use of a “three-plane” method with a more correct evaluation of g and which, in principle, allows a higher-order treatment of all the parameters of Eq. [ 11. With this in mind, we limit the second-order perturbation treatment of Eq. [l] to the zero-field splitting term, according to the usual energy scheme g@Y > D > A. Then, for the allowed transitions in a strong magnetic field, we can write (5) hv = E,,, +-
- EM,-LM, = g,,PH 1
f3p11*
k&H
- 3&f
IW - y--
+ A,&,
(3Ms” - 3Ms + 1)
- ID1 1*(4S* + 4s - 1) + y
(2s* + 2s
1) 9 PI I
where g& = &if-h, g:,D,ff
[31
= kg*D*g*h,
141
[51
g2,ffA& = Ii-g-AZ-g-h, 1D1j2 =
ii-g-D*-g-h ~.g’ h
10~1~ = 2TrD*
- 4
- ‘;:;‘gh;;)*
WI
,
ii -g .D**g .h + (6.g.D.g.h)* ii -g* .h
0;.g2.h)*
’
171
and h is the unit vector along I-I. Equations [3], [4], and [5] are the basic formulas for the determination of the g, D, and A tensors, respectively. Following, for example, Refs. (1) and (4), by fitting the gEff, g2,,Deff, and g2,ffAZ,ff values measured for a number of directions of II on the plane, say (xy), one obtains the elements xx, yy, and xy of the g*, gDg, and gAZg tensors. Repeating this procedure for the (yz) and (xz) planes, all the missing elements can be obtained. It is possible in this way to deduce the g, D, and A principal values and directions. Note that gEff, gzffDeff, gzffAEff are considered as measured “locally,” that is, for each single direction of the magnetic field. The difficulty is that this appears rigorously possible for the trivial case S = ?4 only, where zero-field splitting and consequent second-order effects are absent. In a more general case, from Eq. [2] we can write
n The
8mm
3 cm
Wavelength
EPR
spectra
are described
2.0718 2.0654 2.0653
2.0675
8
1 2 3
2.1583 2.0606 2.0682 2.0675
gz
1 2 3 4
Step
by the spin
2.0822 2.0764 2.0764
2.0951
2.1946 2.0897 2.0957 2.0950
Hamiltonian
2.3268 2.3205 2.3206
2.3046
2.4121 2.2966 2.3054 2.3046
COMPUTATIONOFTHEMAGNETICPARAMETERSOFTHECV+-CU COMPLEX
0.0812 0.0811 0.0811
0.0781
0.0813 0.0777 0.0781 0.0781
D.Z,
ITERATIVE
1
X = p8.g.S
BY THE
TABLE
+ 5.D.S
0.0221 0.0218 0.0218
0.0232
0.0263 0.0231 0.0233 0.0232
D,,
I+ PAIR METHOD”
with
S = 1. All the
-0.1033 -0.1029 -0.1029
-0.1013
-0.1076 -0.1008 -0.1014 -0.1013
Dz,
IN CU(PYRIDINE
D values
Z
/ Y
X
Z
t Y
X
38 121 81
54 140 75
x’
60 39 68
53 50 62
y’
angles
are in cm-‘.
Resulting
N-OXIDE)CI,.H,O
(deg)
110 103 24
122 97 32
z’
198
BULUGGIU
(2s - 1)
hv geff
=
pH
-
Tgeffp2R2
AND
(25 - l)(D,12
VERA
+ 4M2
+ smaller terms, I
VI
where
f: gj=
MF-I
i W&.,s-, MS=-Sfl 2S(2Z + 1)
’
and Hi$+M.-l is the resonance field relative
to the electronic (MS, MI * MS - 1, MI). In the same approximation we have Den = 2gemP M--, i y$,,
W - &)H~,&M~-,/S(~S~
transitions
- 1X21 + 1)
191
s-
I-
and I
Aeff = -3gerrP 1 My-I
s C
M1H3s:ls,M,-,/2SZ(Z + 1)(2Z + 1).
[lOI
MS=-s-1
These are the operative relations which allow us to measure the g,,, Deff, and Aeff values from the experiment. As can be seen, for a generic value of S, these parameters can be deduced from a single spectrum only if one disregards the second-order part of Eq. [8], which requires more information about the g and D tensors than the available one. In this approximation, the use of Eqs. [8] to [lo] is equivalent to assuming, for each direction of the magnetic field, the geff, Deff, and Aeff values which give the least-squares best fit of all the resonance fields by means of the first-order part of Eq. [2] only. In order to take into account the zero-field effects we suggest the following iterative procedure. As the initial step, for each assumed direction on the three chosen orthogonal planes, we deduce the g eff local value from the mean of the resonance fields, i.e., neglecting the second-order terms of Eq. [8], and the corresponding Deff and Aeff values of Eqs. [9] and [lo]. This allows us to make a first rough evaluation of the g, D, and A tensors. By use of these, we are now able to estimate approximately the second-order term of Eq.. [8] for the various assumed directions. Thus, we obtain more accurate values for the local geff, Deff, and Aeff parameters. Such new values are used to redetermine the g, D, and A tensors, and so on up to convergence. As an application of this procedure, we have deduced the magnetic parameters of the dimeric Cu(II)(pyridine N-oxide)Cl,*H,O complex, where the two copper ions per unit cell are antiferromagnetically coupled through two bridges consisting of oxygen of the pyridine N-oxide (6). EPR spectra at room temperature are characteristic of a partially populated excited state with S = 1. There is no evidence of hyperfme structure and the angular behavior of the spectrum shows remarkable misalignment between the g and D tensors. This last fact makes use of a three-plane method advantageous with.respect to alternative more approximate and tedious techniques of systematic searching of extreme zero-field splittings and g values. The direction of the magnetic field was varied at regular intervals (10“) on three orthogonal planes. Owing to the unavailability of complete struc-
EVALUATION
OF
SPIN-HAMILTONIAN
PARAMETERS
199
tural data, we have chosen the three planes casually. For the same reason we give re‘lative rather than absolute principal directions for the g and D tensors. The results, for the X-band and Q-band frequencies, are given in Table 1. In both cases the same directions of the magnetic field were taken. Initially, the values are not corrected, as mentioned, for second-order effects. In this limit the X band values are unacceptably larger than those obtained at Q band which are more typical for Cu w ions. Moreover, the latter values may be considered close to the “true” ones, as shown by the small second-order modifications during the following steps. As one can see, the iterative procedure improves substantially the agreement between the parameters calculated at the two frequencies. Further, it is rapidly convergent and requires a negligible increase in computation time. In this paper we have taken into account the second-order zero-field effects only, because these appear to be the most important sources of error. However, a similar treatment of all the parameters of Eq. [l] is also possible and, moreover, sulch a procedure appears utilizable for any higher order of the perturbation treatment. ACKNOWLEDGMENTS
The authors wish to thank Professor G. Dascola for encouragement V. Varacca for critically reading a preliminary draft of the manuscript.
and help and Professor
REFERENCES I. 2. 3. 4. 5. 6.
J. E. GEIJSIC AND L. C. BROWN, Phys. Rev. 112, 64 (1958). D. S. SHONLAND, Proc. Phys. Sot. London 73, 788 (1959). W. G. WALLER AND M. T. ROGERS, J. Magn. Reson. 9, 92 (1973). A. LUND AND T. V;~NNG.&RD,J. Chem. Phys. 42, 2979 (1965). M. IWASAKI, J. Map. Reson. 16, 417 (1974). G. F. KOKOSZKA, H. C. ALLEN. JR., AND G. GORDON, J. Chem. Phys. 3020 (1967).
46,
3013 (1967); 46,