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Quantitative spin determination by EPR spectroscopy
JOURNAL
OF MAGNETIC
49,425-430
RESONANCE
(1982)
uantitative Spin Determination
by EPR Spectroscopy
R. ZIMMERMANN Physikalisches
Institut
der Universitiit 8520 Erlangen,
Erlangen-Niimberg, West Germany
Giiickstrasse
IO,
Received January 12, 1982 The spin concentration of a sample can be determined by comparing the EPR spectrum of the sample with that of a sample of known spin concentration. In simple cases it is sufficient to compare the total areas of the integrated EPR signals with regard to its g-tensor dependence which is calculated. Peaks of the EPR spectrum can also be used for spin determination even in the case of partly overlapping lines.
EPR spectroscopy on Kramers doublet systems can be applied to the determination of the spin concentration in a sample. The method conventionally used is to compare the spectrum of a sample of unknown spin concentration with that of a sample of known spin concentration. The comparison can rest on the integrate intensity from isolated EPR lines or on the total intensities of EPR spectra as foun by double integration. For the comparison the functional dependence of the intensity n the g values of the Kramers doublet has to be known. Aasa and Vanngard (I) ave pointed out that in some publications (2-9) a wrong functional dependence has been used as a result of not observing that the g dependence is different for field-swept spectrometers and frequency-swept ones. Aasa and Vanngard (I) give an analytical expression for the intensity of an isolated EPR peak and report an approximation formula for the total intensity of an integrated EPR spectrum, In the present paper we want to point out that an exact expression for the total intensity can be derived. In discussing steps of the derivation we find an expression for the stick diagram which may be helpful for the determination of the intensity of an EPR peak in case the EPR peak is not well isolated. The Stick Diagram The magnetic properties of a Kramers doublet can be described by the eigenvalues of the g tensor, g,, gu, and g,. The application of an external magnetic field 1-I splits the doublet into two states separated by the energy where 0 is the Bohr magneton, g = [gz sin’ 8 cos’ Cp+ gz sin’ 9 sin2 p + gz cos2 9]‘/2
i23
is the g value belonging to that particular orientation of the external fiel are the polar angles of the external field. The lineshape of an EPR powder 3, 425 0022-2364/82/120425-06$02.00/o Copyright 0 1982 by Academic Press, Inc. All rights of reproduction in any form reserved.
426
R. ZIMMERMANN
spectrum P(H) is obtained by integrating the product of the transition probability @‘(a, ‘P) with the lineshape function L over all possible orientations. If HE = I?,J (&) is the resonance field for a particlar orientation of the external field we can write P(H) = J
The transition
probability
W{tJ, tp) = + {gf(g:
J2r W(S, (P)L(H - H,) sin bd&P.
s31
0 0
is proportional
to (IO, 1 I)
+ gz) sin2 3 cos2 P + g:(gz + gz) sin2 0 sin* P + ‘a?:
+ g:> cm2 191. f41
The lineshape depends on the particular problem. The integrated lineshape E may be Gaussian, Lorentzian, or a Gaussian distribution of Lorentzian lines. The integrated lineshape is subject to normalization on the frequency scale. For a Lorentzian line we have’
cc r/(h) (E- ,Q2+(r/2)2 dE=l’ s--m
L51
When observing the spectrum on the magnetic field scale we have to transform the integral Eq. [5] to the variable H. Since j- L(E)dE
we find that the integrated
= s L(H) g
dH,
lineshape function is normalized
to l/(pg),
i.e.,
= l/(&).
m L(H - HJdH
s -a,
f+u
For Iineshapes of very small width the lineshape function becomes a 6 function L(H - Hnz) -
( 1/PgPW
- HA
P3
In this case the powder spectrum degenerates to the stick diagram
SW)= s,’s,2m & W(I?, F’)6(H
- H,)
sin 8dzYdcP.
The integral in Eq. [9] can be expressed by elliptic integrals. To derive an expression we may make a transformation from the variable 8 to the variable The integration over H,,, is trivial due to the sifting property of the 6 f~nctiou. The remaining integral can be transformed to elliptic integrals. We find XH)
’ The
gjg5 + g%? = [(g; - g;)(g,2 - g2)]‘/2
iower
limit
of integration
can be taken
- Kg; - g:)(g:
to be -co,
since
Em + I’.
-
g2)1”2E
QUANTITATIVE
EPR
SPECTROSCOPY
FIG. 1. Theoretical EPR spectrum for a,Kramers doublet with g values g, = 2.015, g, = 3.78, g, = 4.275. The microwave frequency was 9.1067 GHz. Drawing above: The stick diagram of the spectrum (dashed line) and the integrated EPR spectrum for field-dependent Gaussian lineshape line). Drawing below: The EPR spectrum as obtained by deriving the integrated EPR spectrum respect to the external field H. (Note: The spectrum shown is a well-fitting simulation of the spectrum of the native MoFe protein from A. vine&&i (14, 1.5))
424
and EPR (fun with EPR
wherek2= (g* - g%g$ - g$/(g: - d.)(gi - g2),g = Em/W), and
Thereby F(a/2, k) and E(7r/2, k) are the complete elliptic integrals of the first and second kind, respectively (12). In the above expression for the stick dia a constant factor has been dropped for convenience. The elliptic integrals Eqs. [lo] are usually approximated by expansion series. It seems better, however, to use polynomial approximations which have been derived for digital computers (13). With expressions containing only six terms the values can be approached to better than 4 X 1Op5 (13). The expressions are in the Appendix and can be handled easily with a pocket calculator. A stick diagram of an EPR spectrum has been plotted with a dashed line in Pi IA. By folding the stick diagram with the lineshape function normalized on the field scale, LH(H - II,) = /3gL(H - H,), 81nl
428
R. ZIMMERMANN
we can return to the lineshape of the powder spectrum and obtain P(H) = Jrn S(H,)LH(H
- H,)dH,
0
~
1121
The solid line of Fig. 1A shows the simulation of the integrated EP assuming Gaussian lineshape. Figure 1B corresponds to the derivative line in Fig. IA and represents the EPR signal as normally observed. Since the stick diagram has an infinity at g = g,, the computer calculation of Eq. [la] creates some problems. The problems can be solved by observing that the infinity is logarithmic. Hence, the integral over a small region of the stick diagram around g = g, stays finite and it is this integral value which is needed for a computer calculation. Unfortunately the formula for the area of the stick diagram in of g = gY becomes rather lengthy. Thus it is questionable whether an EP should be simulated on a computer by using the stick diagram. On the if we want to simulate the peaks of the EPR spectrum at g = g, or g = g,, we are usually far enough from the logarithmic infinity at g = g, in order to have a good approximation to the spectrum at g = g, or g = g, when neglecting the infinity. Examples are the low-field and high-field peaks in the EPR spectrum of Fig. 1B. The Intensity
of Isolated Peaks
Aasa and Vanngard (I) have shown that isolated peaks of the EPR spectrum at g = g, and g = g, may be used for a determination of the spin concentration of a powder sample provided that the linewidth is smaller than the anisotropy of the g values. When approximating the stick diagram of Fig. 1A at g = field resonance) by a step function it can be shown that in the case of small the lineshape of the EPR peak at g = g, is of the form -L(H - E,//3g,) the area of the peak, A, is given by A = S%/PgJ,
[13a
which gives the simple expression
L&g;+ gz> A = 5 [(g,”- g$)(gf- g:)]“’ * Up to a constant factor (which is of no importance here) formula [ 13b] is identical to the one reported by Aasa and Vanngard (I). It is a good approximation to the area of the high-field peak in Fig. lB, since the stick diagram has a very small slope for external fields just below the step at HX = E,/@gx. In principle, formula [ 13b] can be also used for the area of the peak at = E,/@gZ when exchanging g, and g, in formula [ 13b]. This approximation however, not very satisfactory for the EPR peak at Hz in Fig. 1B, since the ~o~~~r~ slope of the stick diagram just above Hz is neglected. Due to the nonzero slope lineshape of the EPR peak becomes more complex. It can be seen from Fig. that in particular the solid line of the low peak does not return to the baseline. Hence, the area of the EPR peak is not given by S(E,/Pg,) as would be supposed by the result in Eq. [ 131. Which value is to be used? The answer can be found in Fig. IA. For an inter-
QUANTITATIVE
EPR SPECTROSCOPY
429
mediate region between g = g, and g = g, the solid line and the dashed Iine coincide. In this region the area of the EPR spectrum coincides with the area as found from the stick spectrum. Without large error we may assume that the stick spectrum and the folded spectrum (dashed and full line in Fig. lA, respectively) have the same values at the external field H,,,i,, which defines the minimum of the EP signal in Fig. 1 B. Hence, the area below the EPR peak at II,, as found by integration unto the limit H = Hmin, is given by A = S(Hmi,).
1141
This formula can be used in the more general case of partly overlapping EPR peaks and approaches Eq. [ 131 for completely separated EPR peaks. The EPR spectrum in Fig. 1 is a simulation of the EPR signal of the 0Fe protein of nitrogenase (I 4, 15). It has a resonance at g = 2 which may be f fied by impurities. In this case the resonance at H = Hz may be of particular importance for a determination of the spin concentration and the slightly refined metho have discussed is necessary for a correct determination of the spin concentration. The Double Integrated
Intensity
The total intensity I of an EPR spectrum can be obtained by integrating stick diagram Eqs. [ lOa] and [lob]. For g, G g,, =Gg, we obtain the result
the
where k2 = (g,” - g;)/(gz - g!J and cos 50 = gX/gz. The functions F(p, k) and E((P, k) are incomplete elliptic integrals of the first and second kind, respectively. Their values can be found in numerical tables or by approximation formulae (14). The factor in front of the bracket (a * . } in Eq. [ 151 has been chosen such that I = I for g, = g,, = g, = 1. Aasa and Vanngard (1) have found the approximation
I = ; w + gy’+ gz’)/W’ + ; (8, + g, + gJ/3, which despite of its simplicity gives astonishing good results. CONCLUSION
uantitative determination of spins by EPR spectroscopy is a very important tool in biology (14, 15). This is possible even without simulation of the spectrum. e method is based on the total area of integrated EPR spectrum as found by uble integration of the EPR signal. This method may be complicated by baseline problems and by overlapping EPR signals from other species or from impurities. In this case the evaluation of the area of peaks in the EPR spectrum is of particular importance. We have shown that this method can be used even if the EPR peak overlaps with the resonance at g = g, APPENDIX
Complete elliptic integrals of the first and second kind can be approximated the following expressions to better than 4 X lo-’ (I 3):
by
430
R. ZIMMERMANN
F
= [a, + alml + azm:] + [bo + b,m, t b,m:] In $
>
with m, = 1 - k2 and
a0 = 1.38629 44,
b,, = 0.5,
al = 0.11197 23,
b, = 0.12134 78,
a2 = 0.07252 96,
b, = 0.02887 29;
a, = 0.46301 51,
b, = 0.24527 27,
a2 = 0.10778 12,
b2 = 0.04124 96.
E with ml = 1 - k2 and
ACKNOWLEDGMENT The author wants to thank Professor E. Mtinck for many helpful discussions. The financial support of the Deutsche Forschungsgemeinschaft is appreciated. REFERENCES I. 2. 3. 4. 5. 6. 7. S. 9. IO. II. 12. 13. 14. 15.
R. AASA AND T. V~NNGARD, J. Magn. Reson. 19, 308 (1975). R. AASA AND T. V~NNGARD, J. Chem. Phys. 52, 1612 (1970). F. K. KNEUB~HL AND B. NATTERER, Helv. Phys. Acta 34, 710 (1961). T. V~NNGARD AND R. AASA, “Paramagnetic Resonance” (W. Low, Ed.), Vol. II, p. 509, Academic Press, New York, 1963. I. CHEN, M. ABKOWITH, AND J. H. SHARP, J. Chem. Phys. 50, 3416 (1969). L. D. ROLLMANN AND S. I. CHAN, J. Chem. Phys. 50, 3416 (1969). A. ISOMOTO, H. WATARI, AND M. KOTANI, J. Phys. Sot. Jpn. 29, 1571 (1971). A. ABRAGAM AND B. BLEANEY, “Electron Paramagnetic Resonance of Transition Ions,” p. 201, Oxford Univ. Press (Clarendon), Oxford, 1970. P. A. NARAYANA AND K. V. L. N. SASTY, J. Chem. Phys. 57, 1805 (1972). A. ABRAGAM AND B. BLEANEY, “Electron Paramagnetic Resonance in Transition Ions,” p. 100, Oxford Univ. Press (Clarendon), Oxford, 1970. R. AASA, B. G. MALMSTR~~M, P. SALTMAN, AND T. V~NNGARD, Biochem. Biophys. Acra ‘ET,203 (1963). P. F. BYRD AND M. D. FRIEDMAN, “Handbook of Elliptic Integrals for Engineers and Physicists,” Springer-Verlag, Berlin, 1954. M. ABRAMOWITZ AND I. A. STEGUN, “Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables,” 7th ed., Dover, New York, 1965. E. M~~NcK, H. RHODES, W. H. ORME-JOHNSON, L. D. DAVIS, W. J. BRILL, AND V. K. SNAW, Biochim. Biophys. Acta 400, 32 (1975). J. RAWLINGS, V. K. SHAH, J. R. CHISNEL, W. J. BRILL, R. ZIMMERMANN, E. M~NCK, AND W. H. ORME-JOHNSON, J. Biol. Chem. 253, 1001 (1978).
OF MAGNETIC
49,425-430
RESONANCE
(1982)
uantitative Spin Determination
by EPR Spectroscopy
R. ZIMMERMANN Physikalisches
Institut
der Universitiit 8520 Erlangen,
Erlangen-Niimberg, West Germany
Giiickstrasse
IO,
Received January 12, 1982 The spin concentration of a sample can be determined by comparing the EPR spectrum of the sample with that of a sample of known spin concentration. In simple cases it is sufficient to compare the total areas of the integrated EPR signals with regard to its g-tensor dependence which is calculated. Peaks of the EPR spectrum can also be used for spin determination even in the case of partly overlapping lines.
EPR spectroscopy on Kramers doublet systems can be applied to the determination of the spin concentration in a sample. The method conventionally used is to compare the spectrum of a sample of unknown spin concentration with that of a sample of known spin concentration. The comparison can rest on the integrate intensity from isolated EPR lines or on the total intensities of EPR spectra as foun by double integration. For the comparison the functional dependence of the intensity n the g values of the Kramers doublet has to be known. Aasa and Vanngard (I) ave pointed out that in some publications (2-9) a wrong functional dependence has been used as a result of not observing that the g dependence is different for field-swept spectrometers and frequency-swept ones. Aasa and Vanngard (I) give an analytical expression for the intensity of an isolated EPR peak and report an approximation formula for the total intensity of an integrated EPR spectrum, In the present paper we want to point out that an exact expression for the total intensity can be derived. In discussing steps of the derivation we find an expression for the stick diagram which may be helpful for the determination of the intensity of an EPR peak in case the EPR peak is not well isolated. The Stick Diagram The magnetic properties of a Kramers doublet can be described by the eigenvalues of the g tensor, g,, gu, and g,. The application of an external magnetic field 1-I splits the doublet into two states separated by the energy where 0 is the Bohr magneton, g = [gz sin’ 8 cos’ Cp+ gz sin’ 9 sin2 p + gz cos2 9]‘/2
i23
is the g value belonging to that particular orientation of the external fiel are the polar angles of the external field. The lineshape of an EPR powder 3, 425 0022-2364/82/120425-06$02.00/o Copyright 0 1982 by Academic Press, Inc. All rights of reproduction in any form reserved.
426
R. ZIMMERMANN
spectrum P(H) is obtained by integrating the product of the transition probability @‘(a, ‘P) with the lineshape function L over all possible orientations. If HE = I?,J (&) is the resonance field for a particlar orientation of the external field we can write P(H) = J
The transition
probability
W{tJ, tp) = + {gf(g:
J2r W(S, (P)L(H - H,) sin bd&P.
s31
0 0
is proportional
to (IO, 1 I)
+ gz) sin2 3 cos2 P + g:(gz + gz) sin2 0 sin* P + ‘a?:
+ g:> cm2 191. f41
The lineshape depends on the particular problem. The integrated lineshape E may be Gaussian, Lorentzian, or a Gaussian distribution of Lorentzian lines. The integrated lineshape is subject to normalization on the frequency scale. For a Lorentzian line we have’
cc r/(h) (E- ,Q2+(r/2)2 dE=l’ s--m
L51
When observing the spectrum on the magnetic field scale we have to transform the integral Eq. [5] to the variable H. Since j- L(E)dE
we find that the integrated
= s L(H) g
dH,
lineshape function is normalized
to l/(pg),
i.e.,
= l/(&).
m L(H - HJdH
s -a,
f+u
For Iineshapes of very small width the lineshape function becomes a 6 function L(H - Hnz) -
( 1/PgPW
- HA
P3
In this case the powder spectrum degenerates to the stick diagram
SW)= s,’s,2m & W(I?, F’)6(H
- H,)
sin 8dzYdcP.
The integral in Eq. [9] can be expressed by elliptic integrals. To derive an expression we may make a transformation from the variable 8 to the variable The integration over H,,, is trivial due to the sifting property of the 6 f~nctiou. The remaining integral can be transformed to elliptic integrals. We find XH)
’ The
gjg5 + g%? = [(g; - g;)(g,2 - g2)]‘/2
iower
limit
of integration
can be taken
- Kg; - g:)(g:
to be -co,
since
Em + I’.
-
g2)1”2E
QUANTITATIVE
EPR
SPECTROSCOPY
FIG. 1. Theoretical EPR spectrum for a,Kramers doublet with g values g, = 2.015, g, = 3.78, g, = 4.275. The microwave frequency was 9.1067 GHz. Drawing above: The stick diagram of the spectrum (dashed line) and the integrated EPR spectrum for field-dependent Gaussian lineshape line). Drawing below: The EPR spectrum as obtained by deriving the integrated EPR spectrum respect to the external field H. (Note: The spectrum shown is a well-fitting simulation of the spectrum of the native MoFe protein from A. vine&&i (14, 1.5))
424
and EPR (fun with EPR
wherek2= (g* - g%g$ - g$/(g: - d.)(gi - g2),g = Em/W), and
Thereby F(a/2, k) and E(7r/2, k) are the complete elliptic integrals of the first and second kind, respectively (12). In the above expression for the stick dia a constant factor has been dropped for convenience. The elliptic integrals Eqs. [lo] are usually approximated by expansion series. It seems better, however, to use polynomial approximations which have been derived for digital computers (13). With expressions containing only six terms the values can be approached to better than 4 X 1Op5 (13). The expressions are in the Appendix and can be handled easily with a pocket calculator. A stick diagram of an EPR spectrum has been plotted with a dashed line in Pi IA. By folding the stick diagram with the lineshape function normalized on the field scale, LH(H - II,) = /3gL(H - H,), 81nl
428
R. ZIMMERMANN
we can return to the lineshape of the powder spectrum and obtain P(H) = Jrn S(H,)LH(H
- H,)dH,
0
~
1121
The solid line of Fig. 1A shows the simulation of the integrated EP assuming Gaussian lineshape. Figure 1B corresponds to the derivative line in Fig. IA and represents the EPR signal as normally observed. Since the stick diagram has an infinity at g = g,, the computer calculation of Eq. [la] creates some problems. The problems can be solved by observing that the infinity is logarithmic. Hence, the integral over a small region of the stick diagram around g = g, stays finite and it is this integral value which is needed for a computer calculation. Unfortunately the formula for the area of the stick diagram in of g = gY becomes rather lengthy. Thus it is questionable whether an EP should be simulated on a computer by using the stick diagram. On the if we want to simulate the peaks of the EPR spectrum at g = g, or g = g,, we are usually far enough from the logarithmic infinity at g = g, in order to have a good approximation to the spectrum at g = g, or g = g, when neglecting the infinity. Examples are the low-field and high-field peaks in the EPR spectrum of Fig. 1B. The Intensity
of Isolated Peaks
Aasa and Vanngard (I) have shown that isolated peaks of the EPR spectrum at g = g, and g = g, may be used for a determination of the spin concentration of a powder sample provided that the linewidth is smaller than the anisotropy of the g values. When approximating the stick diagram of Fig. 1A at g = field resonance) by a step function it can be shown that in the case of small the lineshape of the EPR peak at g = g, is of the form -L(H - E,//3g,) the area of the peak, A, is given by A = S%/PgJ,
[13a
which gives the simple expression
L&g;+ gz> A = 5 [(g,”- g$)(gf- g:)]“’ * Up to a constant factor (which is of no importance here) formula [ 13b] is identical to the one reported by Aasa and Vanngard (I). It is a good approximation to the area of the high-field peak in Fig. lB, since the stick diagram has a very small slope for external fields just below the step at HX = E,/@gx. In principle, formula [ 13b] can be also used for the area of the peak at = E,/@gZ when exchanging g, and g, in formula [ 13b]. This approximation however, not very satisfactory for the EPR peak at Hz in Fig. 1B, since the ~o~~~r~ slope of the stick diagram just above Hz is neglected. Due to the nonzero slope lineshape of the EPR peak becomes more complex. It can be seen from Fig. that in particular the solid line of the low peak does not return to the baseline. Hence, the area of the EPR peak is not given by S(E,/Pg,) as would be supposed by the result in Eq. [ 131. Which value is to be used? The answer can be found in Fig. IA. For an inter-
QUANTITATIVE
EPR SPECTROSCOPY
429
mediate region between g = g, and g = g, the solid line and the dashed Iine coincide. In this region the area of the EPR spectrum coincides with the area as found from the stick spectrum. Without large error we may assume that the stick spectrum and the folded spectrum (dashed and full line in Fig. lA, respectively) have the same values at the external field H,,,i,, which defines the minimum of the EP signal in Fig. 1 B. Hence, the area below the EPR peak at II,, as found by integration unto the limit H = Hmin, is given by A = S(Hmi,).
1141
This formula can be used in the more general case of partly overlapping EPR peaks and approaches Eq. [ 131 for completely separated EPR peaks. The EPR spectrum in Fig. 1 is a simulation of the EPR signal of the 0Fe protein of nitrogenase (I 4, 15). It has a resonance at g = 2 which may be f fied by impurities. In this case the resonance at H = Hz may be of particular importance for a determination of the spin concentration and the slightly refined metho have discussed is necessary for a correct determination of the spin concentration. The Double Integrated
Intensity
The total intensity I of an EPR spectrum can be obtained by integrating stick diagram Eqs. [ lOa] and [lob]. For g, G g,, =Gg, we obtain the result
the
where k2 = (g,” - g;)/(gz - g!J and cos 50 = gX/gz. The functions F(p, k) and E((P, k) are incomplete elliptic integrals of the first and second kind, respectively. Their values can be found in numerical tables or by approximation formulae (14). The factor in front of the bracket (a * . } in Eq. [ 151 has been chosen such that I = I for g, = g,, = g, = 1. Aasa and Vanngard (1) have found the approximation
I = ; w + gy’+ gz’)/W’ + ; (8, + g, + gJ/3, which despite of its simplicity gives astonishing good results. CONCLUSION
uantitative determination of spins by EPR spectroscopy is a very important tool in biology (14, 15). This is possible even without simulation of the spectrum. e method is based on the total area of integrated EPR spectrum as found by uble integration of the EPR signal. This method may be complicated by baseline problems and by overlapping EPR signals from other species or from impurities. In this case the evaluation of the area of peaks in the EPR spectrum is of particular importance. We have shown that this method can be used even if the EPR peak overlaps with the resonance at g = g, APPENDIX
Complete elliptic integrals of the first and second kind can be approximated the following expressions to better than 4 X lo-’ (I 3):
by
430
R. ZIMMERMANN
F
= [a, + alml + azm:] + [bo + b,m, t b,m:] In $
>
with m, = 1 - k2 and
a0 = 1.38629 44,
b,, = 0.5,
al = 0.11197 23,
b, = 0.12134 78,
a2 = 0.07252 96,
b, = 0.02887 29;
a, = 0.46301 51,
b, = 0.24527 27,
a2 = 0.10778 12,
b2 = 0.04124 96.
E with ml = 1 - k2 and
ACKNOWLEDGMENT The author wants to thank Professor E. Mtinck for many helpful discussions. The financial support of the Deutsche Forschungsgemeinschaft is appreciated. REFERENCES I. 2. 3. 4. 5. 6. 7. S. 9. IO. II. 12. 13. 14. 15.
R. AASA AND T. V~NNGARD, J. Magn. Reson. 19, 308 (1975). R. AASA AND T. V~NNGARD, J. Chem. Phys. 52, 1612 (1970). F. K. KNEUB~HL AND B. NATTERER, Helv. Phys. Acta 34, 710 (1961). T. V~NNGARD AND R. AASA, “Paramagnetic Resonance” (W. Low, Ed.), Vol. II, p. 509, Academic Press, New York, 1963. I. CHEN, M. ABKOWITH, AND J. H. SHARP, J. Chem. Phys. 50, 3416 (1969). L. D. ROLLMANN AND S. I. CHAN, J. Chem. Phys. 50, 3416 (1969). A. ISOMOTO, H. WATARI, AND M. KOTANI, J. Phys. Sot. Jpn. 29, 1571 (1971). A. ABRAGAM AND B. BLEANEY, “Electron Paramagnetic Resonance of Transition Ions,” p. 201, Oxford Univ. Press (Clarendon), Oxford, 1970. P. A. NARAYANA AND K. V. L. N. SASTY, J. Chem. Phys. 57, 1805 (1972). A. ABRAGAM AND B. BLEANEY, “Electron Paramagnetic Resonance in Transition Ions,” p. 100, Oxford Univ. Press (Clarendon), Oxford, 1970. R. AASA, B. G. MALMSTR~~M, P. SALTMAN, AND T. V~NNGARD, Biochem. Biophys. Acra ‘ET,203 (1963). P. F. BYRD AND M. D. FRIEDMAN, “Handbook of Elliptic Integrals for Engineers and Physicists,” Springer-Verlag, Berlin, 1954. M. ABRAMOWITZ AND I. A. STEGUN, “Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables,” 7th ed., Dover, New York, 1965. E. M~~NcK, H. RHODES, W. H. ORME-JOHNSON, L. D. DAVIS, W. J. BRILL, AND V. K. SNAW, Biochim. Biophys. Acta 400, 32 (1975). J. RAWLINGS, V. K. SHAH, J. R. CHISNEL, W. J. BRILL, R. ZIMMERMANN, E. M~NCK, AND W. H. ORME-JOHNSON, J. Biol. Chem. 253, 1001 (1978).