A Appendix: The Magnetic Space-Group and Its Applications
S. Sugano
2471-2849 Ageya, Nagano, 380-0888 Japan
Abstract. In order to help the readers understand magneto-optics from the group theoretical view-point, we introduce the magnetic-space group, a non-unitary group, citing a typical antiferromagnetic crystal, MnF2. The co-representation of a non• unitary group is discussed on the basis of the Wigner test. Finally, applications of the magnetic space-group to spectroscopy for magnetically ordered crystals are discussed, again using the example of MnF2 antiferromagnets.
A.l Non-Unitary Group
In paramagnetic crystals, the time-average of microscopic magnetic moments is zero, so that these crystals are invariant to time-reversal which changes the sign of the magnetic moment. On the other hand, in ferromagnets, an• tiferromagnets, and ferrimagnets, the time-averages of microscopic magnetic moments are non-zero, and they are not invariant to time-reversal. For ex• ample, let us consider the crystal structure of an antiferromagnet, MnF2, as depicted in Fig. A.l. It is invariant under the following symmetry operations
Fig. A.l. Distribution of microscopic magnetic moments in the antiferromagnetic MnF2 crystal. Black circles indicate Mn2+ ions, white ones F- and arrows magnetic moments 320 S. Sugano involving minimum translations; {EIO}, {C2IO}, {C2xlr}, {C2ylr}, {IIO}, {ahiO}. {avxlr}, {avylr} (A.l) Here, the notations of symmetry operations are those normally used for space• group [1]. A non-primitive translation, r, is the translation from point I to point II in Fig. A.l. However, the crystal is not invariant under the symmetry operations {C4, C41lr }, {C2x, C2yiO}, {S4, S4 1lr }, {adx+y, adx-yiO}, which, together with the operations in (A.l), make the paramagnetic MnF2 crystal (rutile• type) invariant as the time-average of microscopic magnetic moments are vanishing. It is easily seen that the antiferromagnetic MnF2 crystal is invari• ant to these operations only when they are combined with the time-reversal operator K: {C4,C41 Ir}K, {C2x+y,C2x-yiO}K, {S4,S4 1 Ir}K, {adx+y,adx-yiO}K, (A.2) For convenience, we explain some symmetry operations appearing in (A.l) and (A.2) as follows:
: rotations around the z-axis by 1r and 1r /2, respectively,
: rotations by 1r around the axes defined by the vectors x + y and x - y, respectively,
: C2 2 /, C2xl, C2yl, respectively (inversion center is the origin), : c;;li, : c2 x+yl, c2 x-yl, respectively.
The symmetry operations in (A.l) are unitary operators but those in (A.2) are anti-unitary. The number of these operators are the same. An aggregate of the unitary operators forms a unitary group G. An aggregate of both the unitary and anti-unitary operators forms a non-unitary group, a magnetic space group Gm for MnF2 , as given by Gm =G+aoG, ao = {C4Ir}K. (A.3) As seen in this example, symmetry of a magnetic crystal with a distri• bution of non-vanishing microscopic magnetic moments belongs to a non• unitary group not containing the time-reversal operator as an element. In A Appendix: The Magnetic Space-Group and Its Applications 321 other words, the operator a0 in (A.3) is not K itself. In general, this kind of non-unitary group is called a magnetic group. The number of magnetic space-groups is known to be 1191.
A.2 Co-Representation
Let us assume that the Hamiltonian for a magnetic crystal is invariant under symmetry operations of a magnetic space-group Gm. If an anti-unitary op• erator, a2, acts on one of the orthogonal eigenfunctions, '1/Jv (v = 1, 2, · · ·, n) of the state with n-fold degeneracy, one obtains n a2'1/Jv = L '¢1LDILv(a2) · (A.4) IL
Operating with a 1 on ( 4) from the left, one obtains n a1a2'1/Jv = L 'l/J>..D>..IL(al)DJL,v(a2)*, (A.5) >..,JL which yields the relation, D(a1a2) = D(al)D(a2)*. Similarly, it is easy to derive the following relations:
D(u1)D(u2) = D(u1 u2), D(u)D(a) = D(ua), D(a)D(u)* = D(au), D(al)D(a2)* = D(a1 a2), (A.6) which indicate that the matrix D(a, u) is not the representation of a group in the usual sense. Matrices satisfying relations (A.6) are called the co• representations of a non-unitary group. Since the co-representation is derived by use of (A.4), D is irreducible. From the first relation of (A.6), it is clear that D(u) is a representation of the unitary subgroup, G. In what follows, we assume that D(u) has an irreducible form as a representation of the unitary subgroup, and that the irreducible representation, Ll( u), of the smallest dimension l is located at the upper left corner of matrix, D( u). This can be done without loss of generality. Now let '1/Jv (v = 1,2,· · ·,l) be the basic functions of representation Ll(u). It is straightforward to prove that ao'l/Jv (v = 1, 2, · · ·, l) with anti-unitary operator a0 given in (3) are also the bases of the representation Ll(a01ua0 )* of the unitary subgroup G. Namely, one may prove the relation,
l uao'l/Jv = Lao'¢1LL11Lv(a01uao)*. (v = 1,2,· · ·,l) (A.7)
In the simplest case of ao = K, equation (A. 7) tells us that K '1/Jv are the bases of Ll( u)*. In what follows we shall use the abbreviation, Ll'(u) = Ll(a01ua0 )*. 322 S. Sugano
Since we have assumed that an irreducible representation of the smallest dimension in D(u) is l-dimensional, the l--dimensional representation Ll' ( u) of the unitary subgroup should be irreducible. Note that Ll( u) is also irreducible. According to Wigner (2], one may have only the following simple relations among D(u), Ll(u) and Ll'(u);
Case 1: The dimension of D(u) is l, and Ll(u) and Ll'(u) are equivalent, i.e. Ll'(u) = u-1 Ll(u)U where U is a unitary matrix of dimension land satisfies the relation, UU* = Ll(a~). The degeneracy of the Ll state is not changed by adding a0 G to G. The irreducible co-representations of a non-unitary group Gm are given as D(u) = Ll(u), (A.8)
Case II: The dimension of D(u) is 2l, and Ll(u) and Ll'(u) are equivalent. U satisfies the relation, UU* = -Ll(a~). The degeneracy of the Ll state is doubled by adding a0G to G but the new state may be specfied by Ll. The irreducible co-representations of Gm are given as
Ll(u) 0 ) _ ( 0 Ll(aa01)U) D(u) = ( 0 Ll(u) ' D(a)- -Ll(aa(}l)U 0 ~A.9)
Case III: The dimension of D(u) is 2l, and Ll(u) and Ll'(u) are not equiva• lent. The Ll and Ll' (Ll-:/= Ll') states of G both having the same degeneracy l becomes degenerate by adding a 0 G to G. The irreducible co-representations of Gm are given as
( Ll(u) 0 ) ( 0 Ll(aao)) (A.lO) D(u) = 0 Ll'(u) ' D(a) = Ll(a01a)* 0 ·
A.3 Wigner Test
Wigner has found a very convenient test to judge which case, I, II, or III, Lli belongs to, when Gm and Lli are given. This test is called the Wigner test. In what follows, the test will be given without stating the proof [1,2], although it may be derived by using the relations hitherto used in the previous section. Let xi be the character of irreducible representation Ll i. Introducing vo satisfying ao = Kvo, the Wigner test may be summarized as follows, for the case of discussing single-valued representations of a unitary subgroup as encountered later in the exciton problems;
LXi(vo uvo u) = h (case I) u = -h (case II) =0 (case III), (A.ll) A Appendix: The Magnetic Space-Group and Its Applications 323 where his the number of elements in G. In order to obtain the Wigner test for a non-unitary space group, we. assume a 0 = K{.Bolbo} in (A.2). As in the case of a unitary-space group, we consider a non-unitary group of k denoted as c:;.. The unitary operators u of this group can be expressed as {EI.Rn}{.Bib}. The minimum translation vectors b0 and b are zero or non-primitive translations associated with .Bo and ,8, respectively. Considering the time reversal, Kk = -k, one obtains for c:;. the relations, (A.12)
.Bk=k+Kq. (A.13) Keeping in mind the relation,
L\i({EI.Rn}) = L\i({EIO})exp(ik · Rn), (A.14) the Wigner test in (11) may be re-expressed as follows:
L xi[{.Bolbo}{.Bib}{.Bolbo}{.Bib}] = m (case I) {3 =-m (case II) =0 (case III) (A.15) where m is the number ofrotational operators in {.Bib} of the unitary space• group of k.
A.4 Applications to Spectroscopy
A.4.1 Exciton Absorption For simplicity let us assume that a unit cell contains a single magnetic ion and the state of the magnetic ion at Rm being excited is given as ¢' (r- Rm). We further assume that this excited state has no degeneracy. It is easy to show that the function,
.,P/.(r) = N-1/ 2 L exp( -ik · Rm)¢'(r- Rm), (A.16) m is a Bloch function which can be the eigenfunction of the excited state of a crystal. N is the number of unit cells in the crystal. The excited state described by (A.16) is called a Frenkel exciton. In general, a unit cell contains p-magnetic ions at Rni (i = 1, 2, · · ·, p) and the excited state ¢'(r- Rni) has g-fold degeneracy associated with Tt irreducible representation of the site symmetry group in a unitary group G: the excited state may better be denoted as ¢'(r-Rni :Tnt) bt = 1, 2, ···,g), but the suffices Tnt will be omitted provided no confusion occurs. 324 S. Sugano
In the case of p--magnetic ions in a unit cell, the wave-function of Frenkel exciton is given by p 1/J~,r-r(r) = L 1/J~i(r)Ci,r-r
(A.17) The coefficients, ci,T-y (i = 1' 2, ... 'p)' are determined so that 1/Jk' ,r-y (r) is the 1 base of irreducible representation r of factor group Gk/Tk. Here Gk is the unitary subgroup of k. The translation group of k, Tk, is defined to be an aggregate of {EIRk} whose Rk satisfy the relation, k · Rk = 27rm (m: integers). For example, in the case of MnF2 crystal, Tx is the aggregate of {Ein1t1 + 2n2t2 + n3t3} where n1, n2, n3 are integers: the X point is at the surface of the Brillouin zone as depicted in Fig. 2.12. In this case, the symmetry operations of G x /Tx are 4 operations in ( A.1) plus 4 similar ones with the translations to which t 2 is added. If k is inside the Brillouin zone, Tk is identical to T. A symmetry operation in factor group Gk/Tk will be denoted as {alt}k. It is clear that p {ait}k1jJ~i = L 1/J~iDji( {ait}k), (i = 1, 2, .. ·,p) (A.18) j where Dji( {a it }k) is non-vanishing only when { alt}k transforms the i-th ion into the j-th ion. By using (A.17) and (A.18), one obtains {ait}k1/J~,r-y(r) = L 1/J~,r'-r'(r)DT'-y',r-r( {alt}k), (A.19) r'r' where D( {a it }k) and D' ( {alt }k) are connected by a similarity transformation, D( {alt}k) = c-1D'( {alt}k)C. (A.20) Equation (A.19) shows that 1/J~,r.., (r) is the 1 base of irreducible represen• tation r of factor group Gk/Tk. The similarity transformation (A.20) leads us to (A.21) r which indicates that, if one calculates the right side of (A.21), one can know what kind of irreducible representation r is needed to specify the Frenkel exciton of interest. As an example, let us consider the excitation at the k = 0 point. From (A.21) one obtains
2:x =0 (otherwise) , (A.22) A Appendix: The Magnetic Space-Group and Its Applications 325 where Ft is the irreducible representation of the site-symmetry group at R,.i to which ¢'(r-R,.i: Fnt) belongs. The irreducible representation Ft is given as Ft = F; X Fe where Fg and Fe are the irreducible representations of the site symmetry group for the ground and the excited state of the magnetic ion, respectively. The complex-conjugate symbol of F9 comes from the fact that the motion of a hole is just the time-reversed motion of the corresponding electron. In the case of the C2h site symmetry group as found for magnetic MnF2 , Table A.l shows that Ft = A9 for the magnetic dipole excitation with the polarization parallel to the z axis, and Ft = B9 for that with the x, y polarization. By use of (A.22) and Table A.l, we obtain Table A.2. Since the unitary factor group of k = 0, ar /T, of the antiferromagnetic MnF2 crystal consists of the symmetry operations in (A.l), it is isomorphous to point-group D2h and denoted as D~~. The character table for this group is given in Table A.3. By comparing the results in Table A.2 with characters in Table A.3, we can derive the irreducible representations of Frenkel exciton r for given Ft by use of (A.22): Ft = A9 : r=rt, rt Ft = B9 : r=rt, rt. (A.23) Denoting the z, x, y- polarized magnetic dipoles as Mz, Mx, My, and inspect• ing the characters in Table A.3, one may conclude that the following excitons are created by the polarized dipoles; rt exciton : forbidden, F:f exciton : for Mz , r;t exciton : for Mx, rt exciton: for My. (A.24) Table A.l. Character table for the C2h double group: R is the rotation by 211". The upper {lower) sign corresponds to g(u). 8(8, M.) is a spin function Irred. represent. E R c2 C2R I IR CTh CThR Bases A 9 ,u (rt) 1 1 1 1 ±1 ±1 ±1 ±1 (g: Lz), (u: z) B 9 ,u (rf) 1 1 -1 -1 ±1 ±1 =F1 =F1 (g: Lx, Ly), (u: x, y) E:;/2,g,u (ra±) 1 -1 -i ±1 ±1 ±i =Fi g : 8{1/2, -1/2) Eit.2,fJ.,U (rl) 1 -1 -i ±1 =F1 =Fi ±i g : 8{1/2, + 1/2) Note that the F's appearing in (A.23) are the irreducible representation of the unitary sub-group G of the magnetic space-group Gm. In order to examine how these representations are related to the co-representation, we use the Wigner test in (A.15). Assuming {,Bolbo} = {C4Ir} as given in (A.3), we conclude that r 1+ , r.+. 2 . Case I r.3+ , r+. 4 • Caselli. (A.25) 326 S. Sugano Table A.2. Table of L:r x Ft =A9 Ft = B 9 {EIO} 2 2 {C2IO} 2 -2 {110} 2 2 {o-hiO} 2 -2 Table A.3. The character table for D~~ group of k = 0 r± r.± r.± r± 1 2 3 4 {EIO} 1 1 1 1 {C2IO} 1 1 -1 -1 {C2xlr} 1 -1 1 -1 {C2ylr} 1 -1 -1 1 {JIO} ±1 ±1 ±1 ±1 {o-hiO} ±1 ±1 =t=1 =t=1 {uvxlr} ±1 =t=1 ±1 =t=1 {o-vylr} ±1 =t=1 =t=1 ±1 This result shows that r{ and F:f are not degenerate, but r;t and rt are degenerate. The energy difference between the r{ and F:f excitons coming from the A9 single-ion excitation is called Davydov-splitting. The splitting, however, is not observable, as the creation of the r{ exciton is forbidden by magnetic dipole of any polarization as shown in (A.24). A similar derivation of the r excitons at specific k points for antiferro• magnetic MnF2 has been done and the results are summarized in Table A.4. A.4.2 Two-Magnon Absorption It is well known that a magnon deriving from spin-deviation of magnetic ions (Ft = B 9 ) is created by the magnetic dipole with the x andy polarization. From the group-theoretical point of view, the treatment of a magnon excita• tion is quite the same as that of the Frenkel exciton, although their physical properties are quite different. In this section, we are going to apply the group-theoretical results obtained above to the two-magnon absorption that has been discussed in detail in Sect. 2.3.2. First of all, we remark that the k vectors of the two magnons are A Appendix: The Magnetic Space-Group and Its Applications 327 Table A.4. r excitons at specific k points for antiferromagnetic MnF2 • F's con• nected by+ are degenerate. Refer to Fig. 2.12 for the k points k points Ft = A 9 Ft=B9 r rt, r: r:, r 4+ X x1 x2 z z1 z2 R R+1 R+2 M M:{,Mi MJ,Mt A A1 A2 opposite to each other, as the k vector of light is almost zero. The absorption strength of the two-magnon absorption, a(E), is given as a(E) oc L j1r(kW6(E- Ek - E-k). (A.26) k Here j1r(k)j2 is a factor giving the selection rule for the excitation of two magnons with k and -k, and Ek the energy of a magnon with k. Since the unitary sub-group for antiferromagnetic MnF2 contains inversion {IIO}, we have Ek = E-k· If j1r(k)j2 is a slowly varying function of k, equation (A.26) may be approximated by the product of j1r(k)j2 and the density of states of two-magnons, I:k 6(E- Ek- E-k)· The density of states of one magnon has been determined by neutron experiments and is given in Fig. 2.12. Now we are in a position to perform a group theoretical examination of 11r(k)j2. It is known that two-magnon absorption is induced by an electric dipole. Table A.3 tells us that the electric dipole with the z polarization is Specified by r2- and the one With X and y polarizations by r3- and r 4-. Since a magnon is created by the single-ion excitation of Ft = B 9 , Table A.4 shows that we have to deal with the X2 magnon at the X point. The use of the character table for factor group G x /Tx indicates that the characters of X2 x X2 are 4 only for {EIO}, {Eit2}, {ah!O}, and {ah!t2}: they are vanishing otherwise. If this result is compared with Table A.3, one obtains (A.27) which leads us to the conclusion that two-magnon absorption is forbidden at the X point for the electric dipole with the z polarization ( r 2-), but allowed for the one with the x andy polarization (r3-, F4). Two-magnon absorption is forbidden at the R and M points for the electric dipole with any polariza• tion. This is because a magnon at these points has a definite parity as shown in Table A.4: no irreducible representation of odd parity appears in reducing the product Rt x Rt. 328 S. Sugano Problem 1: Derive the character table of factor group G x /Tx for antiferro• magnetic MnF 2. Problem 2: Argue a selection rule for the two-magnon absorption at the Z point in antiferromagnetic MnF2 [1]. References 1. T.lnui, Y.Tanabe, Y.Onodera: Group Theory and Its Applications in Physics, 2nd edn., (Springer-Verlag, Berlin, Heidelberg, New York 1996) 2. E. P. Wigner: Group Theory (Academic Press 1959) Index 3d-4f interaction 53 bound magnetic polarons 181 4f-4f interaction 56 bound state 64, 65,69 4d-transition metal ions 164 branching ratio 22 5d-transition metal ions 164 breakdown of the k = 0 selection rule 70, 72 absorption coefficient 142, 155, 167 Brillouin function 1 70 absorption edge 162 Brillouin zone 50, 324 absorption spectra of Cr20a 109 bulk linear susceptibility x(l) 125 absorption spectra of CrXa 148 absorption spectra of ruby 109 Cso(TDAE)., 246 absorption spectra of YIG 159 c-tensor, irreducible tensor operator activation energy 14 118,119 activator 16 C jN ratio 295 afterglow 21 calculated interference spectra 130 Ah0a:Ti3+ 27 calculated SH spectra 129 all solid-state laser 24 carrier levels 298 alloy potential fluctuation 181 carrier trapping 20 a-spectrum 9 CD 278 amplifier noise 300 CD (compact disk) 273 anisotropy 97 Cdl-xMn.,Te 219 anisotropy field 80 Ce-based compounds 172 anomalous metal 78 center aperture detection 311 antiferromagnetic domains 107 character table 325, 326 antiferromagnetism charge transfer band 5, 258 - frustrated 220 charge transfer excitation 76 - type-Ill 220 charge transfer transition 138, 149, antisymmetric exchange interaction 160 53,70 chirality vector 57 anti-unitary operator 320 chromium trihalides 148 APF 188,191 circularly polarized light 114, 141, astigmatic aberration method 278 152,161 asymmetric quantum wells 203 Clebsch-Gordan coefficient 118 closure approximation 120 BaClF:Sm2+ 31 Co ferrite 303 BaFBr:Eu2+ 30 Co1.5[Fe(CN)6 )· 6H20 260 basic functions symmetry adapted to compensation temperature 253 Ca 111 complex nature of the matrix elements Bi3+ -substituted garnets 155 117 hi-substituted garnet material 303 concentration quenching 18 bifurcation 202 configuration-coordinate model 12 Bloch function 323 conventional trigonal field 120 blue-violet laser diode 294 cooperative energy transfer 20 BMP 182 cooperative excitation 40, 53 bound hole pairs 100 cooperative luminescence 19 330 Index cooperative optical absorption 19 effective electric dipole moment 121 cooperative optical transition 18 effective mobility edge 187 cooperative transition 56 electric and magnetic dipole transitions co-representation 321 108 Coulomb repulsion 76 electric diplole transition moment 40 CrBr3 138 electric dipole 327 criterion as to the reality of the matrix electric dipole mechanism 151 elements 118 electric dipole moment 41, 47, 116, cross relaxation 18, 128 327 crystal and magnetic structure of electric second order susceptibility Cr203 111 128 crystal field parameter 8, 163 electron paramagnetic resonance 234 crystal field splitting 3 elliptically polarized light 142 crystal field theory 2 ellipticity 142 crystal field transitions 148, 157 EMP 191 CsFeCh · 2H20 60 energy levels of d3 in a cubic field 110 exchange coupling 308 Davydov-splitting 54, 326 exchange integral 76, 127 Debye- Waller factor 11 exchange interaction 170, 180 degree of delocalization 173 exchange-type interaction 128 Dieke diagram 3 excitation transfer 127 dielectric permeability tensor 140 exciton 47, 54, 64, 69, 70 dielectric susceptibility tensor 145 exciton absorption 323 differential push-pull 289 exciton diffusion 187 diffracted X-ray topography 108 exciton dispersion 65 digital radiography 22 exciton-magnon absorption (magnon diluted magnetic semiconductor 180, sideband) 47 211 experimental phase-difference spectra diode laser 24 131 dipole-dipole interaction 248 expression of xe at one site 117 direct exchange interaction 248 expression of xm at one site 117 direct process 12 extinction coefficients 142 disk noise 300 displacement vector T 116 F center 30 dissipative effect in local-field correction F+ center 30 132 factor group 324 dodecahedral sites 154 Faraday and Kerr effects 137 domain wall displacement detection Faraday and Kerr rotations 147 312 Faraday ellipticity 143 double excitation 38 Faraday rotation 137, 138, 142, 143, double group 325 164 Drude 93 Faraday rotator 138 DVD (digital versatile disk) 273 [Fe(Cp*)2]"+[TCNE]"- 246 11 DyCr03 70 Fe L5[Crm(CN)6 ]·7.5H20 256,261 (Fe.,Mnl-xh.5[Cr(CN)6 ]· 7.5H20 256 effect in multi-layer films 291 ferrimagnetic material 155, 282 effect in thick magnetic films 291 ferromagnetic exchange interaction effect of boundary conditions 128 250 effective carrier number 93 field-cooled magnetization (FCM;) 258 Index 331 fluorescence line-narrowing 33 K2CrCl4 52 FMP 184,195 Ko.4Co1.3[Fe(CN) 6 ]· SH20 257,260 focusing servo 279 Kerr effect 288 forced electric dipole transition 5 Kerr ellipticity 145, 290 four-level system 22 Kerr enhancement 292 Franck-Condon principle 14 Kerr rotation 137, 139, 145,288 free-magnetic polaron 181 Kerr rotation spectrum 148, 153 Frenkel exciton 324 kinetic energy 184 Fresnel relations 143 kinetic exchange mechanism 248 Frohlich 180 Knight shift 102 front aperture detection 305 Kubo formula 145 fundamental light 107 La2-xSrxCU04 77 Gat-xMnxAs 231 lanthanide 2 GaAs:Mn 233 Larmor precession 239 GGG 164 laser noise 300 Goodenough-Kanamori rule 248 lifetime broadening 12 green laser diode 294 ligand 2 ligand field theory 2, 148 Hartree approximation 7 light intensity modulation 284 high frequency peak 121 linearly polarized light 114, 137, 141 high-Tc superconductors 77, 134 liquid phase epitaxy (LPE) method high-density magneto-optical recording 156,158,164 294 local field Bloc 125 hole localization 190 local field correction 124 homogeneous broadening 12 local field correction factor LFCF 124 Huang-Rhys factor 10 localized magnetic polarons 186 Hund's rule 248 Lorentz field 126 low frequency peak 121 i-tensor, irreducible tensor operator 118 macroscopic theory of the nonlinear improved rear aperture detection 309 susceptibilities 112 in-gap state 103 magnetic circular dichroism (MCD) inhomogeneous and homogeneous 142 solution 112 magnetic dipole moment 116 inhomogeneous broadening 12, 132 magnetic dipoles 325 initial permeability 245 magnetic disks 314 interaction between orbital angular magnetic domain 139 momentum 127 magnetic domain soliton 60 internal transition 149 magnetic field modulation 286 intra-atomic d-d* transition of Mn2+ magnetic garnets 138, 154 ion 221 magnetic permeability tensor 139 ion implantation 229 magnetic phase diagram ionic linear susceptibility Xion 125 - Cdt-xMnx Te 220 Ir4+ -substituted magnetic garnets magnetic polaron 179, 187 164 magnetic potential 185 iron group 6 magnetic potential well 181, 184, 202 irreducible representation 322 magnetic second order susceptibility isolated Mn spin 191 128 332 Index magnetic semiconductors 172 molecular field theory 253 magnetic space group 116, 319 molecule-based magnets 246 magnetic symmetry 116 Mott insulator 44, 76 magnetic triple-layer film 308 multielectron energy level 151, 161, magnetic-spectroscopy 183 168 magnetically induced superresolution multilayer materials 1 72 304 multiplet terms of Cr3+ in cubic field magneto-optic effects 110 - Faraday effect 236 - figure of merit 237 Neel temperature 107 - Kerr effect 239 Nli{).4Co~~ 3 [Fen(CN) 6 ] · 5H20 257 - magnetic circular dichroism 216 near field recording 312 magneto-optic waveguide 235 negative magnetization 246, 257, 263 magneto-optical device 154 neutron diffraction topography 108 magneto-optical effect 155, 172, 288 Ni{l5 [Crm(CN)6 ] 250 magneto-optical figure of merit 155 (Ni~1 Mn~1-x)1.5[Crm(CN) 6 ] 250, 255, magneto-photoluminescence spectra 256 195 NiO 44 magnon 40,46 NMR 97,102 magnon Raman scattering 46 noise 300 magnon sideband 47, 55,65 non-diagonal exchange integral 40 mask 304 non-diagonal exchange interaction 40 mastering 272 non-primitive translation 320 Maxwell equations 139 non-unitary group 320 MD DATA2 287 nonlinear Hamiltonian 185 mean-field approximation 185 nonlinear second-order susceptibilities metals 172 107 nonlinear source term 125 metamagnetic transition 61,69 nonradiative decay 15 micro-crystals 207 nonreciprocal property 138 microchip laser 24 microscopic mechanism of SHG 108 (02)2 38 microscopic theory of the nonlinear octahedral ligand field 110 susceptibilities 116 octahedral sites 154, 157 MiniDisc (MD) 286 one-magnon Raman scattering 46 mirror reflection ad 116 one-magnon Raman spectra 46 mirror-symmetry relation 14 one-magnon scattering 84 mixed ferro-ferrimagnetism 250, 256, Onsager relations 140 263 optical conductivity 93 Mn~~5[Crm(CN) 6 ] 250,255 optical diffraction 275 [Mn(Crm(ox)3 )t 246 optical diffraction limit 276 Mnncun(pbaOH)(H20)3 246 optical isolator 137, 138 MnF2 41,47,53,65,319 opticallimit 275 MnSe 223 optical pickup 278 MnTe 220 MO disks 139 p-d exchange interaction 228, 230, 234 MO materials 172 p-d hybridization 212,231 mode-locking 26 Pauli principle 248 molecular field 112, 151, 161, 169 perovskite structures 134 Index 333 persistent spectral hole-burning 21, second harmonic generation, SHG 107 31 self-trapped 180 perturbation calculation of xm and xe sensitized fluorescence 16 119 sensitizer 16 phase-change disk 274,314 SHG solid state green laser 295 phonon polaron 180 short wavelength light source 294 phonon Raman scattering 83 shot noise 300 phonon-sideband 11 u-spectrum 9 photo-induced magnetic pole inversion single-frequency laser 24 245,263 site-symmetry group 325 photo-induced magnetization 183 Slater transition state 149, 150 photo-magnetism 245,268 Sm-doped ZnS microcrystals 33 photochromism 20 solid immersion lens 313 photoionization 20 photoluminescence excitation spectrum solid-state laser 22 187 source term 112 photon-avalanche up-conversion 19 sp-d exchange interaction 212, 239 photon-gated hole-burning 31 space inversion 111 photostimulable phosphor 30 spin dynamics 207 1r-spectrum 9 spin frustration 57 polarized light microscopy 108 spin gap 78, 97, 101 polarizing beam splitter 289 spin ladder 79, 88, 90 potential exchange mechanism 248 spin wave 80,81 Prussian blue 245,246 spin-flip Raman scattering 182 Prussian blue families 246 spin-lattice relaxation time 103 pure dephasing 12 spin-orbit coupling constant 149, 152, 160 Q-switching 24 spin-orbit interaction 112, 151, 152, 160,168 Racah parameter 8 spontaneous symmetry breaking 202, RACAH software 152, 160 203 radiographic imaging 21,30 sputtering 274 Raman luminescence 56 square lattice 87 Raman tensor 84 Sr14-x-yCax YyCu24041 90 rare-earth 2 SrAl204:Eu2+, Dya+ 29 rare-earth transition metal alloy 282 SrTiOa:FeH, Mo6+ 20 real field, c type 115 stamping 272 real field, i type 115 rear aperture detection 305 Stark level 3 reduced matrix elements 118 steady-state photoluminescence 188 refractive index 113 Stokes' shift 14 resistivity 94, 98 strong field scheme 6 resonance energy transfer 16 sublattice magnetization 256 resonant SHG 107 superconducting quantum interference Rh4+ -substituted magnetic garnets device 183 164 superexchange mechanism 248 room-temperature hole-burning 31 symmetric quantum wells 198 rotation C2x 116 symmetry of antiferromagnetic Cr2 Oa ruby 9, 16, 109 111 334 Index symmetry of the magnetic and electric two-photon absorption 135 susceptibilities 113 symmetry of the paramagnetic Cr20a unitary group 320 109 unitary operator 320 system noise 300 unitary subgroup 321, 324 up-conversion 19 Tanabe-Sugano diagram 9 USCF-XaSW method 149,167 TbFeCo 282 tensor, c tensor 115 V{C6 He)2 246 tensor, i tensor 115 Verdet constant 137 tetrahedral sites 154,157 V(TCNE)., · yCH2Ch 246 theoretical phase-difference spectra VX2 {X=Cl, Br, and I) 57 131 thermal noise 300 wave equation 112 thermo-magnetic recording 280 weak field scheme 3 third harmonic generation 135 Wigner 3 · j and 6 · j symbols 152 third-order optical responses 135 Wigner test 322 three-spot method 279 Wigner-Eckart theorem 118, 152, 169, time reversal 112, 115,320 170 time-invariant contribution {i type) Wigner-Racah calculus 152, 160 109 write compensation 284 time-noninvariant contribution (c type) 109 Y 202S:Eua+ 3, 5 total noise 300 YAG:Nda+ 23 tracking servo 279 YAG:Yba+ 25 transfer integral 76 Ybo.9Gdo.1P04 56 transient photoluminescence spec- YbCrOa 53 troscopy 196 YIG 154 transmission 141 YV04:Nda+ 24 triangular lattice antiferromagnets 57 trigonal field Virig 119 z2 vortex 57 twisting crystalline fields 120 Zeeman energy in the molecular field two-exciton absorption 52 119 two-magnon absorption 40, 326 Zeeman splitting 214 two-magnon Raman Hamiltonian 87 zero-phonon line 11 two-magnon Raman scattering 46 Znl_.,Co., Te 227, 229 two-magnon scattering 84, 86,88 Znl-xCr.,Te 227 two-phonon Raman scattering 12, 23 Znl-xNi., Te 228 Springer Series in Solid-State Sciences Editors: M. Cardona P. Fulde K. von Klitzing H.-J. Queisser Principles of Magnetic Resonance 24 Physics in High Magnetics Fields 3rd Edition By C. P. Slichter Editors: S. Chikazumi and N. Miura 2 Introduction to Solid-State Theory 25 Fundamental Physics of Amorphous By 0. Madelung Semiconductors Editor: F. Yonezawa 3 Dynamical Scattering of X-Rays in 26 Elastic Media with Microstructure I Crystals By Z. G. Pinsker One-Dimensional Models By I. A. Kunin 4 Inelastic Electron Tunneling Spectroscopy 27 Superconductivity of Transition Metals Editor: T. Wolfram Their Alloys and Compounds 5 Fundamentals of Crystal Growth I By S. V. Vonsovsky, Yu. A. lzyumov. Macroscopic Equilibrium and Transport and E. Z. Kurmaev Concepts 28 The Structure and Properties of Matter By F. E. Rosenberger Editor: T. Matsubara 6 Magnetic Flux Structures in 29 Electron Correlation and Magnetism in Superconductors By R. P. Huebener Narrow-Band Systems Editor: T. Moriya 7 Green's Functions in Quantum Physics 30 Statistical Physics I Equilibrium 2nd Edition Statistical Mechanics 2nd Edition By E. N. Economou By M. Toda, R. Kubo, N. Saito 8 Solitons and Condensed Matter Physics 31 Statistical Physics II Nonequilibrium Editors: A. R. Bishop and T. Schneider Statistical Mechanics 2nd Edition 9 Photoferroelectrics By V. M. Fridkin By R. Kubo, M. Toda, N. Hashitsume I 0 Phonon Dispersion Relations in 32 Quantum Theory of Magnetism Insulators By H. Bilz and W. Kress 2nd Edition By R. M. White II Electron Transport in Compound 33 Mixed Crystals By A. I. Kitaigorodsky Semiconductors By B. R. Nag 34 Phonons: Theory and Experiments I 12 The Physics of Elementary Excitations Lattice Dynamics and Models By S. Nakajima, Y. Toyozawa, and R. Abe of Interatomic Forces By P. Brliesch 13 The Physics of Selenium and Tellurium 35 Point Defects in Semiconductors II Editors: E. Gerlach and P. Grosse Experimental Aspects Lannoo 14 Magnetic Bubble Technology 2nd Edition By J. Bourgoin and M. By A. H. Eschenfelder 36 Modern Crystallography III 15 Modern Crystallography I Crystal Growth Fundamentals of Crystals By A. A. Chernov Symmetry, and Methods of Structural 37 Modern Chrystallography IV Crystallography Physical Properties of Crystals 2nd Edition Editor: L. A. Shuvalov By B. K. Vainshtein 38 Physics of Intercalation Compounds 16 Organic Molecular Crystals Editors: L. Pietronero and E. Tosatti Their Electronic States By E. A. Silinsh 39 Anderson Localization 17 The Theory of Magnetism I Editors: Y. Nagaoka and H. Fukuyama Statics and Dynamics 40 Semiconductor Physics An Introduction By D. C. Mattis 6th Edition By K. Seeger I 8 Relaxation of Elementary Excitations 41 The LMTO Method Editors: R. Kubo and E. Hanamura Muffin-Tin Orbitnls and Electronic Structure 19 Solitons Mathematical Methods By H. L. Skriver for Physicists 42 Crystal Optics with Spatial Dispersion, By. G. Eilenberger and Excitons 2nd Edition 20 Theory of Nonlinear Lattices By V. M. Agranovich and V. L. Ginzburg 2nd Edition By M. Toda 43 Structure Analysis of Point Defects in Solids 21 Modern Crystallography II An Introduction to Multiple Magnetic Resonance Structure of Crystals 2nd Edition Spectroscopy By B. K. Vainshtein, V. L. lndenbom, By J.-M. Spaeth, J. R. Niklas. and R. H. Bartram and V. M. Fridkin 44 Elastic Media with Microstructure II 22 Point Defects in Semiconductors I Three-Dimensional Models By I. A. Kunin Theoretical Aspects 45 Electronic Properties of Doped Semiconductors By M. Lannoo and J. Bourgoin By B. I. Shklovskii and A. L. Efros 23 Physics in One Dimension 46 Topological Disorder in Condensed Matter Editors: J. Bernasconi and T. Schneider Editors: F. Yonezawa and T. Ninomiya Springer Series in Solid-State Sciences Editors: M. Cardona P. Fulde K. von Klitzing H.-J. Queisser 47 Statics and Dynamics of Nonlinear Systems 68 Phonon Scattering in Condensed Matter V Editors: G. Benedek, H. Bilz, and R. Zeyher Editors: A. C. Anderson and J.P. Wolfe 48 Magnetic Phase Transitions 69 Nonlinearity in Condensed Matter Editors: M. Ausloos and R. J. Elliott Editors: A. R. Bishop, D. K. Campbell, 49 Organic Molecular Aggregates P. Kumar, and S. E. Trullinger Electronic Excitation and Inieraction Processes 70 From Hamiltonians to Phase Diagrams Editors: P. Reineker, H. Haken, and H. C. Wolf The Electronic and Statistical-Mechanical Theory 50 Multiple Diffraction of X-Rays in Crystals of sp-Bonded Metals and Alloys By J. Hafner By Shih-Lin Chang 71 High Magnetic Fields in Semiconductor Physics 51 Phonon Scattering in Condensed Matter Editor: G. Landwehr Editors: W. Eisenmenger, K. LaBmann, 72 One-Dimensional Conductors and S. Diittinger By S. Kagoshima, H. Nagasawa, and T. Sambongi 52 Superconductivity in Magnetic and Exotic 73 Quantum Solid-State Physics Materials Editors: T. Matsubara and A. Kotani Editors: S. V. Vonsovsky and M. I. Katsnelson 53 Two-Dimensional Systems, Heterostructures, 74 Quantum Monte Carlo Methods in Equilibrium and Superlattices and Nonequilibrium Systems Editor: M. Suzuki Editors: G. Bauer, F. Kuchar, and H. Heinrich 75 Electronic Structure and Optical Properties of 54 Magnetic Excitations and Fluctuations Semiconductors 2nd Edition Editors: S. W. Lovesey, U. Balucani, F. Borsa, By M. L. Cohen and J. R. Chelikowsky and V. Tognetti 76 Electronic Properties of Conjugated Polymers 55 The Theory of Magnetism II Thermodynamics Editors: H. Kuzmany. M. Mehring, and S. Roth and Statistical Mechanics By D. C. Mattis 77 Fermi Surface Effects 56 Spin Fluctuations in Itinerant Electron Editors: J. Kondo and A. Yoshimori Magnetism By T. Moriya 78 Group Theory and Its Applications in Physics 57 Polycrystalline Semiconductors 2nd Edition Physical Properties and Applications By T. Inui, Y. Tanabe, andY. Onodera Editor: G. Harbeke 79 Elementary Excitations in Quantum Fluids 58 The Recursion Method and Its Applications Editors: K. Ohbayashi and M. Watabe Editors: D. G. Petti for and D. L. Weaire 80 Monte Carlo Simulation in Statistical Physics 59 Dynamical Processes and Ordering on Solid An Introduction 3rd Edition Surfaces Editors: A. Yoshimori and By K. Binder and D. W. Heermann M. Tsukada 81 Core-Level Spectroscopy in Condensed Systems 60 Excitonic Processes in Solids Editors: J. Kanamori and A. Kotani By M. Veta, H. Kanzaki, K. Kobayashi, 82 Photoelectron Spectroscopy Y. Toyozawa. and E. Hanamura Principle and Applications 2nd Edition 61 Localization, Interaction, and Transport By S. Hiifner Phenomena Editors: B. Kramer, G. Bergmann, 83 Physics and Technology of Submicron and Y. Bruynseraede Structures 62 Theory of Heavy Fermions and Valence Editors: H. Heinrich, G. Bauer, and F. Kuchar Fluctuations Editors: T. Kasuya and T. Saso 84 Beyond the Crystalline State An Emerging 63 Electronic Properties of Perspective By G. Venkataraman. D. Sahoo. Polymers and Related Compounds and V. Balakrishnan Editors: H. Kuzmany, M. Mehring, and S. Roth 85 The Quantum Hall Effects 64 Symmetries in Physics Group Theory Fractional and Integral 2nd Edition Applied to Physical Problems 2nd Edition By T. Chakraborty and P. Pietiltiinen By W. Ludwig and C. Falter 86 The Quantum Statistics of Dynamic Processes 65 Phonons: Theory and Experiments II By E. Fick and G. Sauermann Experiments and Interpretation of 87 High Magnetic Fields in Semiconductor Experimental Results By P. Briiesch Physics II 66 Phonons: Theory and Experiments III Transport and Optics Editor: G. Landwehr Phenomena Related to Phonons 88 Organic Superconductors 2nd Edition By P. Briiesch By T. lshiguro, K. Yamaji. and G. Saito 67 Two-Dimensional Systems: Physics 89 Strong Correlation and Superconductivity and New Devices Editors: H. Fukuyama, S. Maekawa, Editors: G. Bauer, F. Kuchar, and H. Heinrich and A. P. Malozemoff Springer Series in Solid-State Sciences Editors: M. Cardona P. Fulde K. von Klitzing H.-J. Queisser Managing Editor: H. K.V. Latsch 90 Earlier and Recent Aspects I 09 Transport Phenomena in Mesoscopic of Superconductivity Systems Editors: H. Fukuyama and T. Ando Editors: J. G. Bednarz and K. A. Muller I 10 Superlattices and Other Heterostructures 91 Electronic Properties of Conjugated Symmetry and Optical Phenomena 2nd Edition Polymers III Basic Models and Applications By E. L. Ivchenko and G. E. Pikus Editors: H. Kuzmany, M. Mehring, and S. Roth I I I Low-Dimensional Electronic Systems 92 Physics and Engineering Applications of New Concepts Magnetism Editors: Y. Ishikawa and N. Miura Editors: G. Bauer, F. Kuchar, and H. Heinrich 93 Quasicrystals Editors: T. Fujiwara and T. Ogawa 112 Phonon Scattering in Condensed Matter VII 94 Electronic Conduction in Oxides Editors: M. Meissner and R. 0. Pohl By N. Tsuda, K. Nasu, A. Yanase, and K.Siratori 113 Electronic Properties 95 Electronic Materials of High-Tc Superconductors A New Era in Materials Science Editors: H. Kuzmany, M. Mehring, and J. Fink Editors: J. R. Chelikowsky and A. Franciosi 114 Interatomic Potential and Structural Stability 96 Electron Liquids 2nd Edition By A. Isihara Editors: K. Terakura and H. Akai 97 Localization and Confinement of Electrons in Semiconductors I I 5 Ultrafast Spectroscopy of Semiconductors Editors: F. Kuchar, H. Heinrich, and G. Bauer and Semiconductor Nanostructures 2nd Edition By J. Shah 98 Magnetism and the Electronic Structure of Crystals By Y.A. Gubanov. A.!. Liechtenstein, 116 Electron Spectrum of Gapless Semiconductors and A. V. Postnikov By J. M. Tsidilkovski 99 Electronic Properties of High-Tc 117 Electronic Properties of Fullerenes Superconductors and Related Compounds Editors: H. Kuzmany, J. Fink, M. Mehring, Editors: H. Kuzmany, M. Mehring, and J. Fink and S. Roth I 00 Electron Correlations in Molecules I 18 Correlation Effects and Solids 3rd Edition By P. Fulde in Low-Dimensional Electron Systems I 0 I High Magnetic Fields in Semiconductor Editors: A. Okiji and N. Kawakami Physics I II Quantum Hall Effect, Transport 119 Spectroscopy of Mott Insulators and Optics By G. Landwehr and Correlated Metals I 02 Conjugated Conducting Polymers Editors: A. Fujimori andY. Tokura Editor: H. Kiess 120 Optical Properties of I II-V Semiconductors I 03 Molecular Dynamics Simulations The Influence of Multi-Valley Band Structures Editor: F. Y onezawa By H. Kalt I 04 Products of Random Matrices 121 Elementary Processes in Excitations in Statistical Physics By A. Crisanti, and Reactions on Solid Surfaces G. Paladin, and A. Yulpiani Editors: A. Okiji, H. Kasai. and K. Makoshi I 05 Self-Trapped Excitons 122 Theory of Magnetism 2nd Edition By K. S. Song and R. T. Williams By K. Yosida I 06 Physics of High-Temperature 123 Quantum Kinetics in Transport and Optics Superconductors of Semiconductors Editors: S. Maekawa and M. Sato By H. Haug and A.-P. Jauho I 07 Electronic Properties of Polymers Orientation and Dimensionality 124 Relaxations of Excited States and Photo• of Conjugated Systems Editors: H. Kuzmany, Induced Structural Phase Transitions M. Mehring. and S. Roth Editor: K. Nasu I 08 Site Symmetry in Crystals 125 Physics and Chemistry Theory and Applications 2nd Edition of Transition-Metal Oxides By R. A. Evarestov andY. P. Smirnov Editors: H. Fukuyama and N. Nagao sa