- Email: [email protected]
Transverse zeeman effect for ions in uniaxial crystals with and without mirror plane symmetry
J. Phys. Chem. Solids Vol. 48, No. 6, pp. 491499, Printed in Great Britain..
1987 0
0022-3697/87 1987 Pergamon
$3.00 + 0.00 Journals Ltd.
TRANSVERSE ZEEMAN EFFECT FOR IONS IN UNIAXIAL CRYSTALS WITH AND WITHOUT MIRROR PLANE SYMMETRY C. DEW. VAN SICLEN Idaho National Engineering Laboratory, Idaho Falls, ID 83415, U.S.A.? and Department of Physics and Astronomy, The Johns Hopkins University, Baltimore, MD 21218, U.S.A. (Received 16 July 1986; accepted 2 October 1986) Abstract-Intra-configuration electric dipole transitions may occur for an ion in a crystal at a site that is not a center of inversion. When an external magnetic field is directed perpendicularly to the principal crystal symmetry axis c, intensity variations in the optical spectrum due to transitions between Zeeman states may appear as the crystal is rotated about c or as the direction of polarization of the incident light is changed with respect to the Zeeman field direction. The electric dipole selection rules giving rise to these intensity fluctuations are derived by consideration of Eamey’s mirror plane symmetry, and are shown to be identical to those obtained by other means. Keywords: Zeeman effect, crystal field, uniaxial symmetry, electric dipole transitions, intensity alternations.
1. INTRODUCTION The crystalline environment of an ion at a site of uniaxial symmetry may permit intra-configuration electric dipole transitions that appear as sharp lines
in the absorption spectrum. This is due to the admixing of opposite-parity states by the odd-parity part of the crystal field. The addition of an external magnetic field H directed perpendicularly to the crystal’s principal symmetry axis c will then split the absorption lines into their Zeeman components. These Zeeman lines may exhibit intensity fluctuations as the crystal is rotated about its axis. The transverse Zeeman effect has received considerable experimental and theoretical attention. (See the papers by Judd and Runciman [l] and Van Siclen [2] for references and brief discussions of the work contained therein.) To describe the influence of the magnetic field direction on absorption intensity for the case of rare-earth ions in ethylsulphate crystals at C,, symmetry sites, Murao et al. [3] constructed an anti-unitary operator that commutes with the total Hamiltonian for all directions of the magnetic field in the basal plane. Its eigenvalues label the Zeeman states, and allow their classification into types A and B according to whether the quantum number for the level is unaffected or not as the crystal is rotated in the magnetic field. Earney [4] derived a general form for the anti-unitary operator and used it to obtain selection rules for electric dipole transitions for several uniaxial point groups with an even-fold rotation or even-fold rotation-inversion axis. Kambara et al. [5] showed the type to be related to the irreducible representations of C,, that the states belong to at the special magnetic field directions for
which C,, is a symmetry of the system. (The horizontal reflection plane implied by the subscript h contains the c axis and is perpendicular to the field H, as is illustrated in Fig. 1. Here and in what follows, the notation of Koster et al. [6] is used.) If the representations are preserved when the Zeeman field is rotated from the C;, symmetry axis to the C$, symmetry axis, the level is type A; if they are not, the level is type B. Judd and Runciman [l] simplified and extended this group theoretical treatment in their analysis of intraconfiguration electric dipole transitions for ions at sites of symmetries corresponding to all tetragonal, hexagonal and trigonal point groups not possessing a center of inversion. They concluded that, for uniaxial crystals in the limit of small H, the only
t Present address. P.C.S.
48,6A
497
Fig. I. The two twofold symmetry axes corresponding to the point groups Cg and CA, which describe the reduced symmetry of the crystalline environment when the direction of the applied Zeeman field H coincides with one or the other axis. The two axes are separated by 30” in the hexagonal case pictured here and lie in the basal plane perpendicular to the principal crystal symmetry axis c (directed out of the page); the tetragonal and trigonal symmetries have Cl,, symmetry axes separated by 45” and 60”, respectively. A mirror plane containing the c axis and perpendicular to the C& axis is depicted by the dashed line.
498
C. DEW. VAN SICLEN
occurrence of intensity maxima and minima as functions of the polarization of the incident light and the angle of rotation of the crystal with respect to the static magnetic field. The assignment of type (as discussed in section 1) to the states involved in a transition is then primarily to simplify the analysis. The odd-parity part V, of the crystal field potential transforms like the identity representation F, of the symmetry group describing that potential. F, of this group necessarily occurs in the reduction of an odd-parity irreducible representation of the appropriate higher symmetry group that contains C;, and Cg,, as subgroups. In this way, the representations of C;, and C&, that correspond to V, may be found. Judd and Runciman [l] showed that when they are different, electric dipole transitions between doublets of the same type will display wide fluctuations in intensity as the crystal is rotated, but when they are the same, the fluctuations will occur for transitions between doublets of different type. By expanding V, as a sum of terms proportional to the modified 2. MIRROR PLANE SYMMETRY spherical harmonics C,(k), Van Siclen [2] showed these relationships to be a consequence of the non-zero For low-lying states, parity is a reasonably good terms (k, q) that contribute to V,: those symmetries quantum number even for ions at sites that are not for which q is a half-integer multiple of d (the centers of inversion. The energy-level structure is then multiplicity of the principal rotation axis; d = 4, 6 determined simply by the even-parity part V, of the and 3 for tetragonal, hexagonal and trigonal symcrystal-field Hamiltonian V, effectively increasing the metries, respectively) may exhibit the periodic insymmetry of the situation. For example, the tetragotensity alternations only for same-type transitions, nal groups Dq, C,, and D, share the same even-parity terms V, (and hence energy-level structure) with D4,,. while those symmetries for which q is an integer This higher symmetry possesses two twofold axes of multiple of d may exhibit the alternations only for symmetry (corresponding to the groups C;, and C&) different-type transitions. The purely type A or type B behavior of a level is not preserved under configuthat lie in the plane perpendicular to the fourfold principal symmetry axis and which make an angle of ration mixing in the first case, but is preserved in the second. The intensity fluctuations anticipated for type n/4 with each other. The ion’s environment retains A-type A or type B-type B electric dipole transitions the C;, symmetry when a magnetic field is directed as the crystal is rotated are therefore due to tranalong the C;, twofold axis, and so permits a classifisitions between the different-parity, different-type cation of the Zeeman energy levels by the irreducible representations of C;, . A similar classification of the parts of the two levels involved. When the transverse Zeeman field H lies along one energy levels by irreducible representations of C’& can be made when the crystal is rotated about c by of the twofold symmetry axes, V, transforms like the irreducible representation F; or F; of the group C,, an angle x/4 to bring the C$, symmetry axis into According to the character table for CZhr a function coincidence with the direction of the field. transforming like FF is unaffected by reflection Electric dipole transitions within a configuration through a plane perpendicular to the CZh symmetry result from the action of an effective operator V,E*r, where V, is the odd-parity part of the crystal field axis while a function transforming like F; undergoes responsible for admixing opposite-parity states and E a change in sign. If V, transforms like F;, then, the is the polarization vector of the incident light. A crystal symmetry must contain a mirror plane that is transition is allowed or disallowed according to perpendicular to H. Judd and Runciman worked through in some detail the example of an ion with an whether the direct product of the irreducible repodd number of electrons at a site of DZdsymmetry to resentations corresponding to the initial and final illustrate the transverse Zeeman effect; it also serves states contains the irreducible representation correto illustrate Earney’s mirror plane symmetry. They sponding to this effective dipole operator. When these considered the case for which the incident light is r belong to the group C;, or C$,,, meaning that both the dipole induced by the incident light and the polarized along the direction of H, which produces an effective operator V,Ey transforming like F: crystallographic x and y axes make special directions with the Zeeman field, the allowed transitions have (r; x F,) of C&. For vanishing intensity, Eamey requires that this operator connect two states that their maximum intensity and the forbidden trantransform under reflection through the mirror plane sitions their minimum (zero) intensity. It is consequently a straightforward matter to predict the with the same sign-that is, two states that transform transverse Zeeman patterns to show intensity alternations (in which each component vanishes as the crystal is rotated about c) are those corresponding to site symmetries D,, Clv and D2+ Analogous variations can take place for D,, C,, and D,, symmetries when H is sufficiently large to mix neighboring crystal-field states. Earney discovered an additional condition on the intensities of electric dipole spectral lines that arises when the symmetry of the ion’s environment contains a mirror plane that is perpendicular to the applied Zeeman field. In particular, he found that transitions between two states that transform under reflection through the mirror plane with the same sign are forbidden if the incident light is polarized along the direction of the field. The purpose of the present paper is to rederive Earney’s result and explore other consequences of the mirror plane symmetry by casting the analysis in group theoretical terms.
Transverse Zeeman effect for ions in uniaxial crystals
under the operation a, of the group C, with the same sign. One would thus expect the transitions r; + r; and r; + r; to be forbidden, as indeed Judd and Runciman have found for the field H parallel to the C$‘,,axis in their D,, example. Additional selection rules follow from the generalization of Earney’s condition. If the crystal symmetry contains a mirror plane that is perpendicular to H (V, transforms like r; of CZh), electric dipole transitions between two states that transform under reflection through the mirror plane with the same sign (two states that transform like the same irreducible representation of C,,) are allowed or forbidden according to whether the pola~~tion vector E is perpendicular (transforms like r; of C,) or parallel (transforms like r; of Czh) to the direction of the applied field H. Similarly, transitions between two states that transform under reflection with different signs (that transform like different irreducible representations of C,,) are forbidden or allowed according to whether E is perpendicular or parallel to H. The converse of these relationships holds when the crystal symmetry does not contain a mirror plane that is perpendicular to H (V, transforms like r; of CZh). In either case, intensity fluctuations with a periodicity of n may be evident in the absorption spectrum when the polarization vector of the incident light is rotated in a plane containing the C, symmetry axis while holding the crystal fixed with respect to the Zeeman field. Fluctuations with nonvanishing intensity minima may occur for rotations E in any other plane containing the ion. The intensity alternations having a periodicity of 2x/d as the crystal is rotated about c (with the direction of H along a C2,,symmetry axis and E fixed) are also predicted by these selection rules. It is interesting to note that the intensity-sensitive electric dipole transitions between levels of the same type occur for ions at site symmetries that are characterized by the presence of a mirror plane perpendicular to one of the twofold axes and the absence of a similar mirror plane perpendicular to the other. Conversely, intensity-sensitive transitions between levels of different type occur for ions at site symmetries not containing a mirror plane, or containing two such planes perpendicular to the C& and C& symmetry axes. The tetragonal crystal field symmetries D4, C,, and D,, raised to D4,, by exclusion of the odd part V, from V possess zero, two and one mirror planes, respectively; similarly, the hexagonal symmetries D,, C,, and D3,, raised to D6,, possess zero, two and one mirror planes. The mirror planes for D,, and D,, contain the e axis and are perpendicular to the C& symmetry axis (see Fig. 1). The trigonal symmetries D, and C,, are raised to D,, and have zero and three mirror planes perpendicular to the three C, axes (the last trigonal group, C,, is raised to Csi which does not contain a twofold symmetry axis). For the tetragonal
499
groups C, and S, and the hexagonal groups C6 and CSh that are raised in symmetry to C.+,, and C,,, respectively, x‘ and y’ axes not coincident with the crystallographic axes can be chosen so that the leading terms of the even-parity part of V are identical to the corresponding ones in D,,,, and D,,. The odd-parity part V, will not transform like a single irreducible representation of Cz,, since the axis of this group is fixed by an independent requirement for the form of V,, but the intensity fluctuations described above may again appear. (See Van Siclen [Z] for a detailed analysis of the Zeeman patterns for the case of an ion with an even number of electrons at a C&, sy~etry site.) 3. CONCLUSION A plane that contains the principal crystal symmetry axis c and is perpendicular to a C,, symmetry
axis is a mirror plane if the odd-parity part V, of the crystal field is unaffected by reflection through it; it is not a mirror plane if V, undergoes a change in sign. When the direction of the applied magnetic field H coincides with that C,, symmetry axis, initial and final states of an electric dipole transition may or may not be reflected through the plane with a change in sign as well, which, because a matrix element must certainly be invariant, quickly give dipole selection rules based solely on the behavior of the states and effective operator under reflection. It is easy to see, then, that a generalization of Earney’s mirror plane condition [4] reproduces the results of Judd and Runciman [I] found by considering the transformation properties of the states and effective dipole operator under the operations of the group Czh. They note that their analysis is only slightly changed when the polarization vector E of the incident light is taken to be perpendicular to H; the present paper includes that case explicitly as well as the case of E parallel to H and derives the additional intensity periodicity arising from a rotation of the polarization E in a plane containing the CZh symmetry axis. Acknowledgement-1 thank Professor Brian R. Judd for helpful conversations. This work was supported in part by the United States Department of Energy.
REFERENCES 1. Judd B. R. and Runciman W. A., Proc. R. Sot. Land.
A352, 91 (1976). 2. Van Siclen C. Dew., J. Phys. C: Solid St. Phys. 14,4611 (I98I). 3. Murao T., Haas W. J., Syme R. W. G., Spedding F. H. and Good Jr R. H.. J. them. Phvs. 47. I572 f19671. 4. Barney J. J., J. Phyi. C: Solid Si. Phyi. 2, 45? (1969). 5. Kambara T., Haas W. J., Spedding F. H. and Good Jr R. H., J. &em. Phys. 56,4475 (1972). 6. Koster G. F., Dimmock J. O., wheeler R. G. and Statz H., Properties oy the Thirty-two Point Groups. M.I.T. Press, Cambridge (I 963).
1987 0
0022-3697/87 1987 Pergamon
$3.00 + 0.00 Journals Ltd.
TRANSVERSE ZEEMAN EFFECT FOR IONS IN UNIAXIAL CRYSTALS WITH AND WITHOUT MIRROR PLANE SYMMETRY C. DEW. VAN SICLEN Idaho National Engineering Laboratory, Idaho Falls, ID 83415, U.S.A.? and Department of Physics and Astronomy, The Johns Hopkins University, Baltimore, MD 21218, U.S.A. (Received 16 July 1986; accepted 2 October 1986) Abstract-Intra-configuration electric dipole transitions may occur for an ion in a crystal at a site that is not a center of inversion. When an external magnetic field is directed perpendicularly to the principal crystal symmetry axis c, intensity variations in the optical spectrum due to transitions between Zeeman states may appear as the crystal is rotated about c or as the direction of polarization of the incident light is changed with respect to the Zeeman field direction. The electric dipole selection rules giving rise to these intensity fluctuations are derived by consideration of Eamey’s mirror plane symmetry, and are shown to be identical to those obtained by other means. Keywords: Zeeman effect, crystal field, uniaxial symmetry, electric dipole transitions, intensity alternations.
1. INTRODUCTION The crystalline environment of an ion at a site of uniaxial symmetry may permit intra-configuration electric dipole transitions that appear as sharp lines
in the absorption spectrum. This is due to the admixing of opposite-parity states by the odd-parity part of the crystal field. The addition of an external magnetic field H directed perpendicularly to the crystal’s principal symmetry axis c will then split the absorption lines into their Zeeman components. These Zeeman lines may exhibit intensity fluctuations as the crystal is rotated about its axis. The transverse Zeeman effect has received considerable experimental and theoretical attention. (See the papers by Judd and Runciman [l] and Van Siclen [2] for references and brief discussions of the work contained therein.) To describe the influence of the magnetic field direction on absorption intensity for the case of rare-earth ions in ethylsulphate crystals at C,, symmetry sites, Murao et al. [3] constructed an anti-unitary operator that commutes with the total Hamiltonian for all directions of the magnetic field in the basal plane. Its eigenvalues label the Zeeman states, and allow their classification into types A and B according to whether the quantum number for the level is unaffected or not as the crystal is rotated in the magnetic field. Earney [4] derived a general form for the anti-unitary operator and used it to obtain selection rules for electric dipole transitions for several uniaxial point groups with an even-fold rotation or even-fold rotation-inversion axis. Kambara et al. [5] showed the type to be related to the irreducible representations of C,, that the states belong to at the special magnetic field directions for
which C,, is a symmetry of the system. (The horizontal reflection plane implied by the subscript h contains the c axis and is perpendicular to the field H, as is illustrated in Fig. 1. Here and in what follows, the notation of Koster et al. [6] is used.) If the representations are preserved when the Zeeman field is rotated from the C;, symmetry axis to the C$, symmetry axis, the level is type A; if they are not, the level is type B. Judd and Runciman [l] simplified and extended this group theoretical treatment in their analysis of intraconfiguration electric dipole transitions for ions at sites of symmetries corresponding to all tetragonal, hexagonal and trigonal point groups not possessing a center of inversion. They concluded that, for uniaxial crystals in the limit of small H, the only
t Present address. P.C.S.
48,6A
497
Fig. I. The two twofold symmetry axes corresponding to the point groups Cg and CA, which describe the reduced symmetry of the crystalline environment when the direction of the applied Zeeman field H coincides with one or the other axis. The two axes are separated by 30” in the hexagonal case pictured here and lie in the basal plane perpendicular to the principal crystal symmetry axis c (directed out of the page); the tetragonal and trigonal symmetries have Cl,, symmetry axes separated by 45” and 60”, respectively. A mirror plane containing the c axis and perpendicular to the C& axis is depicted by the dashed line.
498
C. DEW. VAN SICLEN
occurrence of intensity maxima and minima as functions of the polarization of the incident light and the angle of rotation of the crystal with respect to the static magnetic field. The assignment of type (as discussed in section 1) to the states involved in a transition is then primarily to simplify the analysis. The odd-parity part V, of the crystal field potential transforms like the identity representation F, of the symmetry group describing that potential. F, of this group necessarily occurs in the reduction of an odd-parity irreducible representation of the appropriate higher symmetry group that contains C;, and Cg,, as subgroups. In this way, the representations of C;, and C&, that correspond to V, may be found. Judd and Runciman [l] showed that when they are different, electric dipole transitions between doublets of the same type will display wide fluctuations in intensity as the crystal is rotated, but when they are the same, the fluctuations will occur for transitions between doublets of different type. By expanding V, as a sum of terms proportional to the modified 2. MIRROR PLANE SYMMETRY spherical harmonics C,(k), Van Siclen [2] showed these relationships to be a consequence of the non-zero For low-lying states, parity is a reasonably good terms (k, q) that contribute to V,: those symmetries quantum number even for ions at sites that are not for which q is a half-integer multiple of d (the centers of inversion. The energy-level structure is then multiplicity of the principal rotation axis; d = 4, 6 determined simply by the even-parity part V, of the and 3 for tetragonal, hexagonal and trigonal symcrystal-field Hamiltonian V, effectively increasing the metries, respectively) may exhibit the periodic insymmetry of the situation. For example, the tetragotensity alternations only for same-type transitions, nal groups Dq, C,, and D, share the same even-parity terms V, (and hence energy-level structure) with D4,,. while those symmetries for which q is an integer This higher symmetry possesses two twofold axes of multiple of d may exhibit the alternations only for symmetry (corresponding to the groups C;, and C&) different-type transitions. The purely type A or type B behavior of a level is not preserved under configuthat lie in the plane perpendicular to the fourfold principal symmetry axis and which make an angle of ration mixing in the first case, but is preserved in the second. The intensity fluctuations anticipated for type n/4 with each other. The ion’s environment retains A-type A or type B-type B electric dipole transitions the C;, symmetry when a magnetic field is directed as the crystal is rotated are therefore due to tranalong the C;, twofold axis, and so permits a classifisitions between the different-parity, different-type cation of the Zeeman energy levels by the irreducible representations of C;, . A similar classification of the parts of the two levels involved. When the transverse Zeeman field H lies along one energy levels by irreducible representations of C’& can be made when the crystal is rotated about c by of the twofold symmetry axes, V, transforms like the irreducible representation F; or F; of the group C,, an angle x/4 to bring the C$, symmetry axis into According to the character table for CZhr a function coincidence with the direction of the field. transforming like FF is unaffected by reflection Electric dipole transitions within a configuration through a plane perpendicular to the CZh symmetry result from the action of an effective operator V,E*r, where V, is the odd-parity part of the crystal field axis while a function transforming like F; undergoes responsible for admixing opposite-parity states and E a change in sign. If V, transforms like F;, then, the is the polarization vector of the incident light. A crystal symmetry must contain a mirror plane that is transition is allowed or disallowed according to perpendicular to H. Judd and Runciman worked through in some detail the example of an ion with an whether the direct product of the irreducible repodd number of electrons at a site of DZdsymmetry to resentations corresponding to the initial and final illustrate the transverse Zeeman effect; it also serves states contains the irreducible representation correto illustrate Earney’s mirror plane symmetry. They sponding to this effective dipole operator. When these considered the case for which the incident light is r belong to the group C;, or C$,,, meaning that both the dipole induced by the incident light and the polarized along the direction of H, which produces an effective operator V,Ey transforming like F: crystallographic x and y axes make special directions with the Zeeman field, the allowed transitions have (r; x F,) of C&. For vanishing intensity, Eamey requires that this operator connect two states that their maximum intensity and the forbidden trantransform under reflection through the mirror plane sitions their minimum (zero) intensity. It is consequently a straightforward matter to predict the with the same sign-that is, two states that transform transverse Zeeman patterns to show intensity alternations (in which each component vanishes as the crystal is rotated about c) are those corresponding to site symmetries D,, Clv and D2+ Analogous variations can take place for D,, C,, and D,, symmetries when H is sufficiently large to mix neighboring crystal-field states. Earney discovered an additional condition on the intensities of electric dipole spectral lines that arises when the symmetry of the ion’s environment contains a mirror plane that is perpendicular to the applied Zeeman field. In particular, he found that transitions between two states that transform under reflection through the mirror plane with the same sign are forbidden if the incident light is polarized along the direction of the field. The purpose of the present paper is to rederive Earney’s result and explore other consequences of the mirror plane symmetry by casting the analysis in group theoretical terms.
Transverse Zeeman effect for ions in uniaxial crystals
under the operation a, of the group C, with the same sign. One would thus expect the transitions r; + r; and r; + r; to be forbidden, as indeed Judd and Runciman have found for the field H parallel to the C$‘,,axis in their D,, example. Additional selection rules follow from the generalization of Earney’s condition. If the crystal symmetry contains a mirror plane that is perpendicular to H (V, transforms like r; of CZh), electric dipole transitions between two states that transform under reflection through the mirror plane with the same sign (two states that transform like the same irreducible representation of C,,) are allowed or forbidden according to whether the pola~~tion vector E is perpendicular (transforms like r; of C,) or parallel (transforms like r; of Czh) to the direction of the applied field H. Similarly, transitions between two states that transform under reflection with different signs (that transform like different irreducible representations of C,,) are forbidden or allowed according to whether E is perpendicular or parallel to H. The converse of these relationships holds when the crystal symmetry does not contain a mirror plane that is perpendicular to H (V, transforms like r; of CZh). In either case, intensity fluctuations with a periodicity of n may be evident in the absorption spectrum when the polarization vector of the incident light is rotated in a plane containing the C, symmetry axis while holding the crystal fixed with respect to the Zeeman field. Fluctuations with nonvanishing intensity minima may occur for rotations E in any other plane containing the ion. The intensity alternations having a periodicity of 2x/d as the crystal is rotated about c (with the direction of H along a C2,,symmetry axis and E fixed) are also predicted by these selection rules. It is interesting to note that the intensity-sensitive electric dipole transitions between levels of the same type occur for ions at site symmetries that are characterized by the presence of a mirror plane perpendicular to one of the twofold axes and the absence of a similar mirror plane perpendicular to the other. Conversely, intensity-sensitive transitions between levels of different type occur for ions at site symmetries not containing a mirror plane, or containing two such planes perpendicular to the C& and C& symmetry axes. The tetragonal crystal field symmetries D4, C,, and D,, raised to D4,, by exclusion of the odd part V, from V possess zero, two and one mirror planes, respectively; similarly, the hexagonal symmetries D,, C,, and D3,, raised to D6,, possess zero, two and one mirror planes. The mirror planes for D,, and D,, contain the e axis and are perpendicular to the C& symmetry axis (see Fig. 1). The trigonal symmetries D, and C,, are raised to D,, and have zero and three mirror planes perpendicular to the three C, axes (the last trigonal group, C,, is raised to Csi which does not contain a twofold symmetry axis). For the tetragonal
499
groups C, and S, and the hexagonal groups C6 and CSh that are raised in symmetry to C.+,, and C,,, respectively, x‘ and y’ axes not coincident with the crystallographic axes can be chosen so that the leading terms of the even-parity part of V are identical to the corresponding ones in D,,,, and D,,. The odd-parity part V, will not transform like a single irreducible representation of Cz,, since the axis of this group is fixed by an independent requirement for the form of V,, but the intensity fluctuations described above may again appear. (See Van Siclen [Z] for a detailed analysis of the Zeeman patterns for the case of an ion with an even number of electrons at a C&, sy~etry site.) 3. CONCLUSION A plane that contains the principal crystal symmetry axis c and is perpendicular to a C,, symmetry
axis is a mirror plane if the odd-parity part V, of the crystal field is unaffected by reflection through it; it is not a mirror plane if V, undergoes a change in sign. When the direction of the applied magnetic field H coincides with that C,, symmetry axis, initial and final states of an electric dipole transition may or may not be reflected through the plane with a change in sign as well, which, because a matrix element must certainly be invariant, quickly give dipole selection rules based solely on the behavior of the states and effective operator under reflection. It is easy to see, then, that a generalization of Earney’s mirror plane condition [4] reproduces the results of Judd and Runciman [I] found by considering the transformation properties of the states and effective dipole operator under the operations of the group Czh. They note that their analysis is only slightly changed when the polarization vector E of the incident light is taken to be perpendicular to H; the present paper includes that case explicitly as well as the case of E parallel to H and derives the additional intensity periodicity arising from a rotation of the polarization E in a plane containing the CZh symmetry axis. Acknowledgement-1 thank Professor Brian R. Judd for helpful conversations. This work was supported in part by the United States Department of Energy.
REFERENCES 1. Judd B. R. and Runciman W. A., Proc. R. Sot. Land.
A352, 91 (1976). 2. Van Siclen C. Dew., J. Phys. C: Solid St. Phys. 14,4611 (I98I). 3. Murao T., Haas W. J., Syme R. W. G., Spedding F. H. and Good Jr R. H.. J. them. Phvs. 47. I572 f19671. 4. Barney J. J., J. Phyi. C: Solid Si. Phyi. 2, 45? (1969). 5. Kambara T., Haas W. J., Spedding F. H. and Good Jr R. H., J. &em. Phys. 56,4475 (1972). 6. Koster G. F., Dimmock J. O., wheeler R. G. and Statz H., Properties oy the Thirty-two Point Groups. M.I.T. Press, Cambridge (I 963).