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Weak fine structure components in ESR spectra of nonoriented solids
JOURN.1LOF MOLECULhRSPECTROSCOPY %, 149-153 (1967)
Weak
Fine Structure Components in ESR Spectra of Nonoriented Solids E. E. GENSER Shell Development
Company,
Enaeryville,
California
94608
The weak transitions observed in the ESR powder pattern of a dilute chromic alum (S = S$) are shown to be accounted for by perturbat,ion theory. The transition fields are calculated under the assumptions of an isotropic q tensor and axial symmetry in the strong field limit,. The physical interpretation of the results are discussed along with the limitations of the perturbation met,hod employed. INTRODUCTION
The analysis of powder patterns in the electron spin resonance (ESR) of ions or molecules with total spin, S 2 1 may be a somewhat) complicated problem (l-4). The complications result from the fact t’hat the fine structure arising from the Stark split’ting of the levels in zero magnetic field introduces a large anisotropy in the spectrum. The secular equation is no less than cubic for t,hese cases and cannot’ be factored for an arbitrary angle. Some of the difficulties may be circumvented by an approximate solution. The close resemblance to the problem of nuclear quadrupole interaction in diatomic molecules in molecular beam experiments (5), and also to the analysis of powder patsterns in nuclear quadrupole resonance (6) is to be noted. The pert’urbation methods employed in these in st,ances are primarily concerned with t’he allowed transitions (AAl, = f l), for which perturbation theory in the first, order is often adequate-occasionally supplemented by second order theory. However, the greater sensitivity of ESR, combined with the feasibility of reducing inhomogeneous line broadening by using magnetically dilute solid solutions, raises the possibility of observing the transitions. In this forbidden ( /Aar,~ > 1) as well as the allowed (,Ad~s = fl) paper, the second-order theory is ext,ended to cover these weak transitions for :I sp&fic case where the\- have been observed. THEORTWe assume
metry. The
at the outset,
approximate
spin x
an isotropic Hamiltonian
= g@I*S
g tensor
and the presence
of axial sym-
is (7’):
+ D[Sz” 149
(>$)X(S
+ l)],
(1:)
150
GENSER
where the axis of quantization, 2, is t’he principal symmetry axis in the species; S is the spin vector with components S, , Sr , Sz ; p is the Bohr magneton; g and D are scalar constants. We are interested in the case in which the magnetic (Zeeman) splitting of the levels is greater than the zero field splitting. The latter is treated as a perturbation on the former and the perturbation theory is carried to second order. It is convenient t#o transform Eq. (1) from the principal axis system (XYZ) into the laboratory frame (zyx) with H = HZ . The effect of this transformation is to make the Zeeman term diagonal (8). The Hamiltonian operator is now X = g@HS, + { ($$D)S2(3 co2 0 - 1) + D sin 0 cos O[S+(S, + 45) + US,
-
MI1 + MD)
sin2 e[S+2 + K2]},
(2)
where 0 is the angle between the principal symmetry axis 2, and H, and where t’he spin operators have their usual significance, e.g., S, ) M) = M 1Al). The symbol 1Al) denot,es an eigenstate of the Zeeman energy. To proceed with the perturbat’ion theory, we rewrit.e Eq. (2) : x
= X0 + X’,
(3)
where x0 = g/3HS, is the unperturbed Hamiltonian and x’, the t,erm in the curly brackets in Eq. (S), is t’he perturbation which is carried to second order using standard nondegenerate perturbat’ion theory. To give a specific illustration, the result for the case with S = Fi in the state \ 31,) 3 13,;) is E’s/2 = 3 g@H +
Similar
(9;‘S)D(3
cos2e -
1) +
3D2 ‘;;ie
0
sin2e + “;;$“.
(4) 0
results
are obtained for the other perturbed magnetic substates. Here splitting, which is equal. to the microwave quantum, 6. We first compute all of the transition energies E:p2 -+ E:,2 B3/2---f I&;2 , etc. These will give some equations for cos e as a function of H, D, 6, and g. Since all possible values of cost occur in a powdered sample then the absorption intensity per unit field strength takes the form (2) g/3Hois the unperturbed
g
-
T
B [H, e(H)]
1’ co;;(H)
1,
where B[H, e(H)] is t,he appropriate transition probability. Resonance absorptions are then expected at the fields which produce singularities in ] d cos e(H)/ CEH1 for a given 6, and D and for a finite value of B[H, e(H)]. We assume, for the sake of simplicity, a vanishing line width which implies a delta function for t’he absorption line. The level separations lead to a quartic equation in CONe(H) which may be factored into two quadratics. Considering the real roots of these, there occur
ESR PPECTRA OF NONORIENTISl)
SOI,II)S
151
TABLE I ‘I’IIEOREWAL EXPRESSIONS FOR THE TRANSITION ENERGIES BETKEEN THE PEKTURAEI) SUBSTATES OF S = 3/2 Transition
/ l/2’,
+ 13/2’;)
6 +
D
1 -3/2’)
-
1 -l/Y!
6 -
D
/ -1/2’i
+ j l/2’) 26 -
! -3/2’,
-
j l/Y)
/ -3/m
--) I 3/2’1
3 -
0
.___ D
6
2
2
‘6 + D
86
6
1
3
o-4
--
.w -
-
D
D2 6
for S = B.J, two distinct singularities for each of t’he transitions. The t’hcoretical expressions are given in Table I along with the designation of the transition in terms of the perturbed states. A specific illustrations is the dilute chromic nhmr: Cr3f in (iSH,)
hl(SOe)z.ll’
H@,
in which the chromic ion is surrounded by six water molecules in an :ipprosimately octahedral configuration with :Lsmall trigonal distortion along a direction through two opposite faces of the octahedron. A powder spectrum of the mnterial, obtained by Yan Reijen at 6 = 0.3164 cm+ is shown in Fig. 1 (9). Using the parameters obtained from an earlier single crystal st,udy of this material (IO), viz., g = 1.97 and D = 0.049 cm-l, the values of H1 and Hz have been computed from t)he expressions in Table I and are compared in Table II with t’he e.uperimental results from Fig. 1. These values are in quite good agreement with experiment in view of the error involved in illt,erpol~tion. Only one of t,he predicted transitions in the region studied is not observed. Ko account is taken here of t#he kansition probabilities B[iY, e(H)]. The experimental spectrum shows character of the that enough mixing of states occurs to relax the “forbidden” j AM, j > 1 transitions. One interest~ing feature emerges here with regard to the transitions labeled HI and Hz . The former occur when cos 6 = 0, i.e., the principal symmet~rg axis lies along the field direction and the mixing of states by the zero field splitting is, in part, suppressed. For the fields labeled Hz , cos2 B = +:;, and this is precisely
GENSER
152
bv = 0.3164
lO_Fold
cm-’
Magnification
3000 Gauss
ESR Spectrum
of the
Powder
FIG. 1. Electron spin resonance of CP
in (NH4)Al(SO&12H20
TABLE II COMPARISON OF THE THEORETICAL ANDEXPERIMENTAL FIELDS (GAUSS) FOR WHICHTHE DESIGNATED TRANSITIONSOCCURIN CHROMICALUM WITH d = 0.3164 cm-r, g = 1.97, ANDD = &(@I cm-i Transition
I 1/n 1-3/2’) 1-l/2’) 1-l/2’) I -3/2’) I -3/2’)
Br Theoretical HI ExperimentalaHz Theoretical Hs Experimental”
+ 13/2’) + 1-l/2’) + [ l/2’) --+ 13/2’) + I l/2’)
3972 2879 3378 1956 1423
3933 2917 3366 1957 -
1766 2293 3.556 7434 8506
1734 2366 3533 -
-+ I 3/2’)
1126
1149
1166
1073
BFigure 1 obtained from Reference 9.
the angle at which the first-order splitting vanishes. Since the residual anisotropy arises from smaller, higher-order terms, the net effect in both of these cases is to increase the “density” of transitions at these angles. As a final comment, it should be emphasized that while the perturbation theory leads to good agreement with experiment for the dilute chromic alum, in more complicated situat,ions such as in the weak field limit, or in the presence of lessthan-axial symmetry, one must generally resort to computer aided numerical calculations (3, 4, 9, li, 12). ACKNOWLEDGMENT The author wishes to thank Dr. L. L. van Reijen for his kind permission to reprint the spectrum of the chromic alum obtained from his thesis. RECEIVED: January 25, 1967
ESR
SPECTRA
OF NONOl~I~NT~D
SOLIDS
153
REFEEENCES 1. 2. 3. 4. 6. 6. ?. 8. 9. 10. il. 12.
S. SINGER, J. Chem. Phys. 23, 379 (1955). S. DE GRQOT AND J. H. VAN DER WAALS, illol. Phya. 3, 190 (1960). K~TTIS AND 13. LEFEBVRE, J. Chem. Phys. 39, 393 (1963). WASSERMAN, L. C. SNYDER, AND W. A. YAGIR, J. Chem. Phys. 41, 1763 (1964). T. FELII AND W. E. LAMB, Phys. Rev. 67, 15 (1945). BLOEMBERBEN, Report of t,he Conference on Defects in Crystalline Solids, 1954, p. 1. The Physical Society, London, 1955. B. BLEANEY AND K. W. H. STEVENS, Repfs. Prop. Phys. 16, 108 (1953j. B. BLEANEP, Phil. Mug. 42, 441 (1951). L. L. VAN REIJEN, t,hesis, Eindhoven, The NetheriaIlds, 1964. C. A, W~~ITNER, ft. T. WEIDNER, J. S. NSIANQ, AND P. R. WEISS, Phys. Rev. 74, 1478 ( 1948). J. D. SWA~EN AND H. M. GLADNEY, IBM J. Res. Develop. 6,515 (1964). Ii. LEFEBYRE; AND J. MARUANI, J. Chem. Phgs. 42, 1480 (1965).
L. M. P. E. B. N.
Weak
Fine Structure Components in ESR Spectra of Nonoriented Solids E. E. GENSER Shell Development
Company,
Enaeryville,
California
94608
The weak transitions observed in the ESR powder pattern of a dilute chromic alum (S = S$) are shown to be accounted for by perturbat,ion theory. The transition fields are calculated under the assumptions of an isotropic q tensor and axial symmetry in the strong field limit,. The physical interpretation of the results are discussed along with the limitations of the perturbation met,hod employed. INTRODUCTION
The analysis of powder patterns in the electron spin resonance (ESR) of ions or molecules with total spin, S 2 1 may be a somewhat) complicated problem (l-4). The complications result from the fact t’hat the fine structure arising from the Stark split’ting of the levels in zero magnetic field introduces a large anisotropy in the spectrum. The secular equation is no less than cubic for t,hese cases and cannot’ be factored for an arbitrary angle. Some of the difficulties may be circumvented by an approximate solution. The close resemblance to the problem of nuclear quadrupole interaction in diatomic molecules in molecular beam experiments (5), and also to the analysis of powder patsterns in nuclear quadrupole resonance (6) is to be noted. The pert’urbation methods employed in these in st,ances are primarily concerned with t’he allowed transitions (AAl, = f l), for which perturbation theory in the first, order is often adequate-occasionally supplemented by second order theory. However, the greater sensitivity of ESR, combined with the feasibility of reducing inhomogeneous line broadening by using magnetically dilute solid solutions, raises the possibility of observing the transitions. In this forbidden ( /Aar,~ > 1) as well as the allowed (,Ad~s = fl) paper, the second-order theory is ext,ended to cover these weak transitions for :I sp&fic case where the\- have been observed. THEORTWe assume
metry. The
at the outset,
approximate
spin x
an isotropic Hamiltonian
= g@I*S
g tensor
and the presence
of axial sym-
is (7’):
+ D[Sz” 149
(>$)X(S
+ l)],
(1:)
150
GENSER
where the axis of quantization, 2, is t’he principal symmetry axis in the species; S is the spin vector with components S, , Sr , Sz ; p is the Bohr magneton; g and D are scalar constants. We are interested in the case in which the magnetic (Zeeman) splitting of the levels is greater than the zero field splitting. The latter is treated as a perturbation on the former and the perturbation theory is carried to second order. It is convenient t#o transform Eq. (1) from the principal axis system (XYZ) into the laboratory frame (zyx) with H = HZ . The effect of this transformation is to make the Zeeman term diagonal (8). The Hamiltonian operator is now X = g@HS, + { ($$D)S2(3 co2 0 - 1) + D sin 0 cos O[S+(S, + 45) + US,
-
MI1 + MD)
sin2 e[S+2 + K2]},
(2)
where 0 is the angle between the principal symmetry axis 2, and H, and where t’he spin operators have their usual significance, e.g., S, ) M) = M 1Al). The symbol 1Al) denot,es an eigenstate of the Zeeman energy. To proceed with the perturbat’ion theory, we rewrit.e Eq. (2) : x
= X0 + X’,
(3)
where x0 = g/3HS, is the unperturbed Hamiltonian and x’, the t,erm in the curly brackets in Eq. (S), is t’he perturbation which is carried to second order using standard nondegenerate perturbat’ion theory. To give a specific illustration, the result for the case with S = Fi in the state \ 31,) 3 13,;) is E’s/2 = 3 g@H +
Similar
(9;‘S)D(3
cos2e -
1) +
3D2 ‘;;ie
0
sin2e + “;;$“.
(4) 0
results
are obtained for the other perturbed magnetic substates. Here splitting, which is equal. to the microwave quantum, 6. We first compute all of the transition energies E:p2 -+ E:,2 B3/2---f I&;2 , etc. These will give some equations for cos e as a function of H, D, 6, and g. Since all possible values of cost occur in a powdered sample then the absorption intensity per unit field strength takes the form (2) g/3Hois the unperturbed
g
-
T
B [H, e(H)]
1’ co;;(H)
1,
where B[H, e(H)] is t,he appropriate transition probability. Resonance absorptions are then expected at the fields which produce singularities in ] d cos e(H)/ CEH1 for a given 6, and D and for a finite value of B[H, e(H)]. We assume, for the sake of simplicity, a vanishing line width which implies a delta function for t’he absorption line. The level separations lead to a quartic equation in CONe(H) which may be factored into two quadratics. Considering the real roots of these, there occur
ESR PPECTRA OF NONORIENTISl)
SOI,II)S
151
TABLE I ‘I’IIEOREWAL EXPRESSIONS FOR THE TRANSITION ENERGIES BETKEEN THE PEKTURAEI) SUBSTATES OF S = 3/2 Transition
/ l/2’,
+ 13/2’;)
6 +
D
1 -3/2’)
-
1 -l/Y!
6 -
D
/ -1/2’i
+ j l/2’) 26 -
! -3/2’,
-
j l/Y)
/ -3/m
--) I 3/2’1
3 -
0
.___ D
6
2
2
‘6 + D
86
6
1
3
o-4
--
.w -
-
D
D2 6
for S = B.J, two distinct singularities for each of t’he transitions. The t’hcoretical expressions are given in Table I along with the designation of the transition in terms of the perturbed states. A specific illustrations is the dilute chromic nhmr: Cr3f in (iSH,)
hl(SOe)z.ll’
H@,
in which the chromic ion is surrounded by six water molecules in an :ipprosimately octahedral configuration with :Lsmall trigonal distortion along a direction through two opposite faces of the octahedron. A powder spectrum of the mnterial, obtained by Yan Reijen at 6 = 0.3164 cm+ is shown in Fig. 1 (9). Using the parameters obtained from an earlier single crystal st,udy of this material (IO), viz., g = 1.97 and D = 0.049 cm-l, the values of H1 and Hz have been computed from t)he expressions in Table I and are compared in Table II with t’he e.uperimental results from Fig. 1. These values are in quite good agreement with experiment in view of the error involved in illt,erpol~tion. Only one of t,he predicted transitions in the region studied is not observed. Ko account is taken here of t#he kansition probabilities B[iY, e(H)]. The experimental spectrum shows character of the that enough mixing of states occurs to relax the “forbidden” j AM, j > 1 transitions. One interest~ing feature emerges here with regard to the transitions labeled HI and Hz . The former occur when cos 6 = 0, i.e., the principal symmet~rg axis lies along the field direction and the mixing of states by the zero field splitting is, in part, suppressed. For the fields labeled Hz , cos2 B = +:;, and this is precisely
GENSER
152
bv = 0.3164
lO_Fold
cm-’
Magnification
3000 Gauss
ESR Spectrum
of the
Powder
FIG. 1. Electron spin resonance of CP
in (NH4)Al(SO&12H20
TABLE II COMPARISON OF THE THEORETICAL ANDEXPERIMENTAL FIELDS (GAUSS) FOR WHICHTHE DESIGNATED TRANSITIONSOCCURIN CHROMICALUM WITH d = 0.3164 cm-r, g = 1.97, ANDD = &(@I cm-i Transition
I 1/n 1-3/2’) 1-l/2’) 1-l/2’) I -3/2’) I -3/2’)
Br Theoretical HI ExperimentalaHz Theoretical Hs Experimental”
+ 13/2’) + 1-l/2’) + [ l/2’) --+ 13/2’) + I l/2’)
3972 2879 3378 1956 1423
3933 2917 3366 1957 -
1766 2293 3.556 7434 8506
1734 2366 3533 -
-+ I 3/2’)
1126
1149
1166
1073
BFigure 1 obtained from Reference 9.
the angle at which the first-order splitting vanishes. Since the residual anisotropy arises from smaller, higher-order terms, the net effect in both of these cases is to increase the “density” of transitions at these angles. As a final comment, it should be emphasized that while the perturbation theory leads to good agreement with experiment for the dilute chromic alum, in more complicated situat,ions such as in the weak field limit, or in the presence of lessthan-axial symmetry, one must generally resort to computer aided numerical calculations (3, 4, 9, li, 12). ACKNOWLEDGMENT The author wishes to thank Dr. L. L. van Reijen for his kind permission to reprint the spectrum of the chromic alum obtained from his thesis. RECEIVED: January 25, 1967
ESR
SPECTRA
OF NONOl~I~NT~D
SOLIDS
153
REFEEENCES 1. 2. 3. 4. 6. 6. ?. 8. 9. 10. il. 12.
S. SINGER, J. Chem. Phys. 23, 379 (1955). S. DE GRQOT AND J. H. VAN DER WAALS, illol. Phya. 3, 190 (1960). K~TTIS AND 13. LEFEBVRE, J. Chem. Phys. 39, 393 (1963). WASSERMAN, L. C. SNYDER, AND W. A. YAGIR, J. Chem. Phys. 41, 1763 (1964). T. FELII AND W. E. LAMB, Phys. Rev. 67, 15 (1945). BLOEMBERBEN, Report of t,he Conference on Defects in Crystalline Solids, 1954, p. 1. The Physical Society, London, 1955. B. BLEANEY AND K. W. H. STEVENS, Repfs. Prop. Phys. 16, 108 (1953j. B. BLEANEP, Phil. Mug. 42, 441 (1951). L. L. VAN REIJEN, t,hesis, Eindhoven, The NetheriaIlds, 1964. C. A, W~~ITNER, ft. T. WEIDNER, J. S. NSIANQ, AND P. R. WEISS, Phys. Rev. 74, 1478 ( 1948). J. D. SWA~EN AND H. M. GLADNEY, IBM J. Res. Develop. 6,515 (1964). Ii. LEFEBYRE; AND J. MARUANI, J. Chem. Phgs. 42, 1480 (1965).
L. M. P. E. B. N.