Ref. p. 411 1 Magnetic susceptibilities: introduction 1 Part One Magnetic susceptibilitjr of coordination and organon metallic transition metal compounds 1.1 Introduction 1.1.1 List of symbols 4 b, c CA1 lattice parameters A admixture parameter of F and P terms B magnetic induction B, C [cm-‘, Oersted, Oe] electronic repulsion parameters (RACAH parameters) BJ( 4 Brillouin function c concentration (g % = g of substance in 100 g of solution; M = molar, mM = millimolar) C Curie constant C, [emu K mol-I] Curie constant per mole CP axial ligand field parameter axis normal to crystal a axis and rotation axis b in the (010) plane of monoclinic C’ crystal D, E [cm- ‘1 zero-field splitting parameters of the spin Hamiltonian ~=D[CSZ2-:S(S+1)]+E(SxZ-Sy2)+g~BS.H as a consequence of axially symmetric and rhombic contributions to the crystal field 4 cubic ligand field parameter e electronic charge eg, tzs subshells of d electrons in an octahedral field g spectroscopic splitting factor 8111g 1 spectroscopic splitting factor parallel and perpendicular to the principal magnetic axis (in uniaxial crystals) spectroscopic splitting factor parallel to the crystallographic axes a, b, c (in ortho- ga, gb, gc rhombic crystals) degeneracy number &I H [Gauss, G] magnetic field strength HII, HI [Gauss, Gl magnetic field strength parallel and perpendicular to the unique crystallographic axis (in uniaxial crystals) H effective magnetic field strength eff internal magnetic field strength Hint AH [kJ mol-‘1 heat capacity change 2 Hamiltonian J exchange energy (values quoted as J/k in [K], J/he in [cm-‘] or J/T in multiples of k) J, Jij, JIZ exchange integral J total angular momentum k Stevens’ orbital reduction factor k Boltzmann constant K II) KI [emu/molecule] principal molecular susceptibilities parallel and perpendicular to the principal magnetic axis (for uniaxial molecules) principal molecular susceptibilities parallel to the principal magnetic axes KI, K2, K3 [emu/molecule] L ligand L orbital angular momentum Kiinig /Kiinig 2 1 Magnetische Suszeptibilitlten: Einleitung [Lit. S.41 M [Gausse mu,lcm3 magnetization (per unit volume) = Gauss, G] M,(=a) [Gauss emu!g] magnetization per gram (specific magnetization) Mm(=a,) magnetization per mole [Gauss emu~mol] Mo(=ao) [Gauss emu,@J specific weak (parasitic) magnetization as resulting from M = Mo+xgH M, [Gauss] saturation magnetization (per unit volume) M,,( = a,) [Gauss emu!g] saturation magnetization per gram M,,(=a,J saturation magnetization per mole [Gauss emu,‘mol] M,,( = TV) spontaneous magnetization (in the appropriate units) Me [Gauss pg] magnetization in Bohr magnetons squared electron mass ;I molecular weight N Avogadro number NI number of atoms per unit volume Nz [emq!mol] TIP per mole (temperature-independent paramagnetic susceptibility) CPBI effective magnetic moment per molecule (in Bohr magnetons) Pm CPBI principal effective magnetic moments (for orthorhombic, monoclinic and triclinic (PA > hJ2~ h~3 crystals; in Bohr magnetons) s spin angular momentum AS [J K- ’ mol-‘1 entropy change T CKI temperature (in Kelvin) TIP temperature independent paramagnetism T, critical temperature, in particular at a phase transition eB subshells of n electrons in octahedral field tzg. V molecular volume axial held splitting in units of the spin-orbit coupling constant (c= A/).) a TIP per molecule direction cosines al, a2, a3: PI. B2 etc. B Cde,el obtuse angle between a and c axes in the monoclinic system A [cm-‘] orbital energy separation due to an axial (i.e. tetragonal or trigonal) field Ao=lODq [cm-‘] octahedral orbital energy separation spin-orbit coupling constant (parameter) for a single d electron end [cm-‘1 0 C%l angle between second magnetic axis and (001) plane in monoclinic crystals Q, CKI paramagnetic Curie temperature (= Weiss constant) WI principal paramagnetic Curie temperatures (= Weiss constants) for orthorhombic, (Qph 3 (Qph 7 to,)3 monoclinic and triclinic crystals Qc CKI ferromagnetic Curie temperature 0,~ WI NCel temperature K [emu’cm3=cm3/cm3] volume susceptibility E. [cm-‘] spin-orbit coupling constant (parameter) for the ground state Pascal constitutive corrections AP Bohr magneton IJB frequency 1’ I7 [cm-‘] mean spin-pairing energy e Ce’cm7 density u [Gauss ems/g] = M, magnetic moment per gram (=specific magnetization) u, [Gauss emu’mol] magnetic moment per mole (= magnetization per mole) =M, u. [Gauss emu/g] = M. specific weak (parasitic) magnetization as obtained from a=oo+xB H KSnig /Kiinig Ref. p. 411 1 Magnetic susceptibilities: introduction 3 ns [Gauss emu/g] = M,, saturation magnetization per gram Q,, [Gauss emu/mol] saturation magnetization per mole = Mm ~nlsp-- K, spontaneous magnetization per mole xe [emu/g= cm3/sl magnetic susceptibility per gram (specific susceptibility) x,, [emu/m01= cm3/mol] magnetic susceptibility per mole (molar magnetic susceptibility) XA [emu/g-atom magnetic susceptibility per gram-atom =cm3/g-atom] XII3X l principal molar magnetic susceptibility parallel and perpendicular to the principal [emu/m01= cm3/mol] magnetic axis (for tetragonal, trigonal and hexagonal crystals) principal molar magnetic susceptibility parallel to the principal magnetic axes Xl, x2, x3 [emu/mol= cm3/mol] for orthorhombic, monoclinic and triclinic crystals xa, Xb, Xc principal molar magnetic susceptibility parallel to crystal axes a, b, c [emu/m01= cm3/mol] ti Cdegl angle between first magnetic axis and c axis in monoclinic crystals 1.1.2 General remarks The first part of the present volume complements and extends the collection of magnetic susceptibility data for coordination and organometallic transition metal compounds which was presented in the volumes 11/2, II/& II/10 and II/11 of the Landolt-Bornstein, New Series [137,138,139,140]. The literature covered here adds directly onto that included in the previous volume Cl403 and extends through the end of 1974. It is of interest to note that the magnetic data compiled in this volume equal to 68.7% of the total amount of data in the volume. The corresponding number for volume II/IO which covers the years 1969 and 1970 is about 76%. This compar- ison indicates a stabilization of researcha ctivities in the general area of the magnetism of transition metal com- pounds. Although a large number of susceptibility data refers again to a very limited range of temperature or to room temperature only, the number of more detailed studies is increasing where susceptibilities down to 4.2 K or even below 4.2 K were measured.S tudies of principal magnetic susceptibilities were also reported in the period coveredh ere, although rather limited in number. Following our earlier practice, thesed ata are included as special inserts into the tables. The list of compounds included here is as comprehensive as in the previous volumes of the series.N ot in- cluded are magnetic data for perovskites, spinels, garnets and hexagonal ferrites having certain special types of crystal structure, since these are collected in full detail elsewhere [141,142,143,144,145]. For more details on contents and arrangement of the tables, the reader is referred to section 1.1.4b elow. The following sections serve a dual purpose: (i) to introduce, define and briefly discuss all those quantities and concepts which directly occur in the tables and (ii) to provide a short introduction to the literature on mag- netism of transition metal compounds. For additional information on the theoretical and/or experimental aspectso f the method, the literature listed in section 1.1.8.6o f the referencesm ay be useful. It should be noted that apart from the general referenceso n magnetism, viz. section 1.1.8.1,t hese sections list only literature which appearedi n the years coveredb y this volume. Therefore, the reader may wish sometimest o consult the additional referencesi n volumes 11/2,1 1/8,I I/10 and II/11 of this series.A n annual review covering magnetism studies on transition metal compounds is also available [147,148]. 1.1.3 Theoretical basis 1.1.3.1 Definitions and fundamental relations The magnetic field strength inside a substance differs from the field strength of the applied field H (measuredi n the electromagnetic cgs system of units in vacua) according to B=(l+47clc)H (1) Here, B is the magnetic induction and M M=lcH (2) is the magnetization or the magnetic moment per unit volume. In general, B, H and M are vector quantities. In addition, ic, the volume susceptibility, is a scalar (units emu/cm3= cm3/cm3).E vidently, rc is a measure of K6nig / KSnig Ref. p. 411 1 Magnetic susceptibilities: introduction 3 ns [Gauss emu/g] = M,, saturation magnetization per gram Q,, [Gauss emu/mol] saturation magnetization per mole = Mm ~nlsp-- K, spontaneous magnetization per mole xe [emu/g= cm3/sl magnetic susceptibility per gram (specific susceptibility) x,, [emu/m01= cm3/mol] magnetic susceptibility per mole (molar magnetic susceptibility) XA [emu/g-atom magnetic susceptibility per gram-atom =cm3/g-atom] XII3X l principal molar magnetic susceptibility parallel and perpendicular to the principal [emu/m01= cm3/mol] magnetic axis (for tetragonal, trigonal and hexagonal crystals) principal molar magnetic susceptibility parallel to the principal magnetic axes Xl, x2, x3 [emu/mol= cm3/mol] for orthorhombic, monoclinic and triclinic crystals xa, Xb, Xc principal molar magnetic susceptibility parallel to crystal axes a, b, c [emu/m01= cm3/mol] ti Cdegl angle between first magnetic axis and c axis in monoclinic crystals 1.1.2 General remarks The first part of the present volume complements and extends the collection of magnetic susceptibility data for coordination and organometallic transition metal compounds which was presented in the volumes 11/2, II/& II/10 and II/11 of the Landolt-Bornstein, New Series [137,138,139,140]. The literature covered here adds directly onto that included in the previous volume Cl403 and extends through the end of 1974. It is of interest to note that the magnetic data compiled in this volume equal to 68.7% of the total amount of data in the volume. The corresponding number for volume II/IO which covers the years 1969 and 1970 is about 76%. This compar- ison indicates a stabilization of researcha ctivities in the general area of the magnetism of transition metal com- pounds. Although a large number of susceptibility data refers again to a very limited range of temperature or to room temperature only, the number of more detailed studies is increasing where susceptibilities down to 4.2 K or even below 4.2 K were measured.S tudies of principal magnetic susceptibilities were also reported in the period coveredh ere, although rather limited in number. Following our earlier practice, thesed ata are included as special inserts into the tables. The list of compounds included here is as comprehensive as in the previous volumes of the series.N ot in- cluded are magnetic data for perovskites, spinels, garnets and hexagonal ferrites having certain special types of crystal structure, since these are collected in full detail elsewhere [141,142,143,144,145]. For more details on contents and arrangement of the tables, the reader is referred to section 1.1.4b elow. The following sections serve a dual purpose: (i) to introduce, define and briefly discuss all those quantities and concepts which directly occur in the tables and (ii) to provide a short introduction to the literature on mag- netism of transition metal compounds. For additional information on the theoretical and/or experimental aspectso f the method, the literature listed in section 1.1.8.6o f the referencesm ay be useful. It should be noted that apart from the general referenceso n magnetism, viz. section 1.1.8.1,t hese sections list only literature which appearedi n the years coveredb y this volume. Therefore, the reader may wish sometimest o consult the additional referencesi n volumes 11/2,1 1/8,I I/10 and II/11 of this series.A n annual review covering magnetism studies on transition metal compounds is also available [147,148]. 1.1.3 Theoretical basis 1.1.3.1 Definitions and fundamental relations The magnetic field strength inside a substance differs from the field strength of the applied field H (measuredi n the electromagnetic cgs system of units in vacua) according to B=(l+47clc)H (1) Here, B is the magnetic induction and M M=lcH (2) is the magnetization or the magnetic moment per unit volume. In general, B, H and M are vector quantities. In addition, ic, the volume susceptibility, is a scalar (units emu/cm3= cm3/cm3).E vidently, rc is a measure of K6nig / KSnig 4 1 Magnet&he SuszeptibilitZten: Einleitung [Lit. S. 41 the easeo f magnetic polarization of the substance.T he quantity that is actually measuredi s the sus’ceptihilit~ per grant ls (specific susceptibility) defined by Xg= K/P (3) and therefrom the wsrrptihiliry per tnole x,, is derived where Xm=X&w=K~ (4) Here, e is the density, (M) the molecular weight in glmol, and V the molecular volume. The atomic and ionic susceptibilities ;IA and yAn are defined similar to Eq. (4) by multiplying xr for a particular atom or ion by the atomic or ionic weight. respectively. If, in Eq. (2), x8 or I,, is used instead of K, we obtain the specific nmgrwfizn- rim M, (magnetic moment per gram) or the nlngwtizotior~ per mole M, (magnetic moment per mole), respec- tively. Thus it is M,=x,H (5) and M,=%mH. (6) It should be noted that the magnetization M, and M, is sometimesd enoted as Q and Q,,,,r espectively. Most experimental results in the literature are expressedi n terms of the quantities )I~ or lrn. The system of units employed almost exclusively in the field of magnetism is therefore the electromagnetic cgs system. Thus the unit of xs is 10e6 emu,‘g (10e6 cm3/g) and that of ,Y,,i s 10e6 emu/mol (10m6c m3/mol). Similarly, the unit usually employed for M, and M, is 10m3G auss. emu/g and 10m3G auss. emu/mol, respectively. If a rationalized system of units would be chosen by defining B= H+M, the experimental results would have to be multiplied by 47~T. his is the casef or SI units which are consequently little used in magnetic studies [30, 31, 321. For the molar magnetic susceptibility, the conversion from SI to cgs units is determined by XmI cgsl=g XmC SII. (7) For several other important quantities, SI units and conversion factors are listed in Table 1 below. Note that, in the cgs system, the Bohr magneton (cJ section 1.1.3.3)i s pLg=0.92731x lo-*‘erg Gauss-‘, whereas in SI units pa=O.92731 x 1O-23A m* molecule- 1. With these values, the effective magnetic moment pm follows as pm = 2.8279( xmT )1’2 pB in cgs units and pn,= 7.9774x lo* (xmT )‘/* pa in SI units. Both numerical values for pm are identical, however. Table 1. SI units and conversion factors in magnetism *) Unit Conversion factor Quantity electro- SI JJ magnetic Cwl cgs H Oersted Am-’ 103/4X B Gauss Wb m-*=Tesla 10-4 K scalar scalar 4n XE cm3 g- ’ m3 kg-1 47go3 cm3 mol-1 m3 mol- 1 47qlO" Xm 1 Pm PA PO In anisotropic substances,e .g. a non-cubic crystal, the magnetization M depends on the direction and on the magnitude of the magnetic field strength H and Eq. (2) has to be replaced by M=(K)H (8) where (K) is a symmetric tensor of rank 2. In component notation, Eq. (8) is (9) *) Note that in this volume 1 emu= 1 cm3i s usedt hroughout KSnig /Kiinig Ref. p. 411 1 Magnetic susceptibilities: introduction 5 In Eq. (9), the vector components of M and H refer to an orthogonal coordinate system fixed in the crystal. Similar to K, the quantities xs, x,,, etc. for single crystals are, in general, second order tensors. If powder samples with random orientation of the individual polycrystals are considered, only the average susceptibility Xm,a”=:(X1+Xz+X3) (10) is measured where xi, i=l, 2, 3, are the principal molar susceptibilities, ct section 1.1.3.4. Substancesw ith a negative magnetic susceptibility, i.e. xs<O, are called diamagnetic and are considered in section 1.1.3.2.S ubstancesh aving a positive susceptibility, viz. xs>O, are called paramagnetic and these will be discussedi n section 1.1.3.3. 1.1.3.2 Diamagnetism and diamagnetic corrections The effect of a magnetic field on the motion of electrons is to cause a precession of each electronic orbit about the direction of H. The atom to which the electrons belong thus acquires a magnetic moment proportional in magnitude but opposite in direction to the field (Larmor precession theorem). The resulting susceptibility per gram-atom is then given by xi= -$ F (rT>. Here, e and m refer to the charge and mass of the electron, c is the velocity of light, N the Avogadro number and (r-2) the expectation value (mean value) of the square distance of the electron from the nucleus. According to Eq. (ll), diamagnetic susceptibilities should be independent of temperature which is observed in practice to a good approximation. Since all substancesc ontain electrons, diamagnetism is an inherent property of all atomic and molecular systems. An important property of diamagnetism is the law of additivity of atomic diamagnetic susceptibilities. According to Pascal [16, 331, the molar diamagnetic susceptibility of a pure compound is approximately de- termined by Xrn=C &XAi+C nj&j. (12) 1 j Here, ni is the number of atoms of kind i, XAi is their susceptibility per gram-atom and the sum extends over all atoms in the molecule. In addition, nj is the number of certain structural elementso f kind j, l,j their contribu- tion to the susceptibility, and the sum is over all these contributions. The quantities XAj and l,j are called the Pascal constants and the constitutive corrections, respectively. The most frequently used constants XA and constitutive corrections & are listed in Table 7. In general, the diamagnetic susceptibility of an organic substance may be estimated on the basis of the tabulated values with an accuracy of a few percent. Even though the Pascal schemew as establishedp urely on an empirical basis,i t is justified by more recent calculations involving molecular orbital theory [34]. Critical discussions and reviews on the subject are available in the literature [35, 361 and illuminating examples for the calculation of susceptibilities within the Pascal schemea re included in various textbooks [16,37,38]. In coordination compounds, the total observed paramagnetic susceptibility x,, may be written as x,,,=(xR& + N U)+ 1 ni ~2:. (13) I In Eq. (13), the quantity Na is the temperature-independent paramagnetism which will be considered in section 1.1.3.3b elow, whereas the last term is the diamagnetic contribution of all constituents in the molecule. We will be interested, in particular, in the susceptibility xr$, of the paramagnetic metal ion within the com- pound. Frequently, only the so-called “corrected” molar susceptibility XL is determined whereby x:.=~r&+Nu. (14) This value may be obtained by subtracting, from the observedv alue of I,,,, the susceptibilities of all diamagnetic groups in the molecule including the diamagnetic contribution for the paramagnetic metal ion, &=&--C ni#~. (15) Since, in general, c ni J& is considerably smaller than x.,,, this quantity is usually estimated on the basis of Eq. (12), particularly if organic type ligands are involved. However, in all complicated organic moieties, it is more accurate to measure 1 ni ~2: directly where possible. If inorganic anionic type ligands are involved, the determination, from experimental data, of ionic dia- magnetic susceptibilities is based on the assumption Xm= Xcation + Xanion . (16) Without going into details of the allocation of the total x,,, between cation and anion [16, 39, 401, we present in Table 6 a consistent set of diamagnetic susceptibilities for simple ions according to Klemm [41]. Kihig / Kiinig Ref. p. 411 1 Magnetic susceptibilities: introduction 5 In Eq. (9), the vector components of M and H refer to an orthogonal coordinate system fixed in the crystal. Similar to K, the quantities xs, x,,, etc. for single crystals are, in general, second order tensors. If powder samples with random orientation of the individual polycrystals are considered, only the average susceptibility Xm,a”=:(X1+Xz+X3) (10) is measured where xi, i=l, 2, 3, are the principal molar susceptibilities, ct section 1.1.3.4. Substancesw ith a negative magnetic susceptibility, i.e. xs<O, are called diamagnetic and are considered in section 1.1.3.2.S ubstancesh aving a positive susceptibility, viz. xs>O, are called paramagnetic and these will be discussedi n section 1.1.3.3. 1.1.3.2 Diamagnetism and diamagnetic corrections The effect of a magnetic field on the motion of electrons is to cause a precession of each electronic orbit about the direction of H. The atom to which the electrons belong thus acquires a magnetic moment proportional in magnitude but opposite in direction to the field (Larmor precession theorem). The resulting susceptibility per gram-atom is then given by xi= -$ F (rT>. Here, e and m refer to the charge and mass of the electron, c is the velocity of light, N the Avogadro number and (r-2) the expectation value (mean value) of the square distance of the electron from the nucleus. According to Eq. (ll), diamagnetic susceptibilities should be independent of temperature which is observed in practice to a good approximation. Since all substancesc ontain electrons, diamagnetism is an inherent property of all atomic and molecular systems. An important property of diamagnetism is the law of additivity of atomic diamagnetic susceptibilities. According to Pascal [16, 331, the molar diamagnetic susceptibility of a pure compound is approximately de- termined by Xrn=C &XAi+C nj&j. (12) 1 j Here, ni is the number of atoms of kind i, XAi is their susceptibility per gram-atom and the sum extends over all atoms in the molecule. In addition, nj is the number of certain structural elementso f kind j, l,j their contribu- tion to the susceptibility, and the sum is over all these contributions. The quantities XAj and l,j are called the Pascal constants and the constitutive corrections, respectively. The most frequently used constants XA and constitutive corrections & are listed in Table 7. In general, the diamagnetic susceptibility of an organic substance may be estimated on the basis of the tabulated values with an accuracy of a few percent. Even though the Pascal schemew as establishedp urely on an empirical basis,i t is justified by more recent calculations involving molecular orbital theory [34]. Critical discussions and reviews on the subject are available in the literature [35, 361 and illuminating examples for the calculation of susceptibilities within the Pascal schemea re included in various textbooks [16,37,38]. In coordination compounds, the total observed paramagnetic susceptibility x,, may be written as x,,,=(xR& + N U)+ 1 ni ~2:. (13) I In Eq. (13), the quantity Na is the temperature-independent paramagnetism which will be considered in section 1.1.3.3b elow, whereas the last term is the diamagnetic contribution of all constituents in the molecule. We will be interested, in particular, in the susceptibility xr$, of the paramagnetic metal ion within the com- pound. Frequently, only the so-called “corrected” molar susceptibility XL is determined whereby x:.=~r&+Nu. (14) This value may be obtained by subtracting, from the observedv alue of I,,,, the susceptibilities of all diamagnetic groups in the molecule including the diamagnetic contribution for the paramagnetic metal ion, &=&--C ni#~. (15) Since, in general, c ni J& is considerably smaller than x.,,, this quantity is usually estimated on the basis of Eq. (12), particularly if organic type ligands are involved. However, in all complicated organic moieties, it is more accurate to measure 1 ni ~2: directly where possible. If inorganic anionic type ligands are involved, the determination, from experimental data, of ionic dia- magnetic susceptibilities is based on the assumption Xm= Xcation + Xanion . (16) Without going into details of the allocation of the total x,,, between cation and anion [16, 39, 401, we present in Table 6 a consistent set of diamagnetic susceptibilities for simple ions according to Klemm [41]. Kihig / Kiinig 6 1 Magnetische Suszeptibilitlten: Einleitung [Lit. S. 41 1.1.3.3 Paramagnetic susceptibility of polycrystalline substances Molecular paramagnetism is a characteristic property of unpaired electron systems.I n most coordination compounds of the transition metals, and some organometallic compounds as well, paramagnetic behaviour is encountered due to the incompletely filled 3d, 4d or Sd electron shell. It should be noted that simple para- magnetism will be found only if there is sufficient magnetic dilution. This is the case if, due to the presenceo f large organic ligands, the paramagnetic centers are well separated,t hus avoiding cooperative interactions of the ferro- and antiferromagnetic type. In an external magnetic field, the ground state of a free atom or ion with angular momentum J is split into 25 + 1 equally spacede nergy levels. It may be shown [42,43] that the magnetization for this systemi s determined by M=NvgJPeBdX) (17) where denotes the Brillouin function BJ(x) s,(x)=+ctgh [v]-&C&h [$I, Here, Nv is the number of atoms per unit volume and x=gJp&I/kT. For x91, i.e. very large values of H and low temperatures,E q. (17) predicts the snturarion mngnetization as M,=NvgJkt. (19) On the other hand, if x 4 1, Eq. (17) gives the Curie law M_NJ(J+l)g%; NP: Cm (20) H 3kT =jkT=?-’ In Eq. (20) g is the spectroscopic splitting factor, N the Avogadro number, k the Boltzmann constant, and C, the Curie constant per mole. The quantity pm (often denoted by pen) is called the effective magnetic moment, pm=gCJ(J+1)l”*~rs (21) and ue (also abbreviated by /I or B.M.) is the Bohr magneton pa=-$&=O.927314~ 10m2’ erg Gauss-‘. The Curie law Eq. (20) is found to be followed by many magnetically dilute substanceso ther than free atoms or ions. There is, in addition, a second-order contribution to the paramagnetic susceptibility, the so-called tem- perature-independent paramagnetism Na (also abbreviated by TIP) which arises from states separated from the ground state by an energy Eg k ‘1:T he molar susceptibility corrected for diamagnetism,i .e. the paramagnetic susceptibility of the transition metal ion under study, follows as The quantity Na is of particular importance for the understanding of molar susceptibilities up to -100 x lO-‘j emu!mol, e.g. in low-spin 3d” transition metal compounds [44]. In the chemical literature, the magnetic behaviour is often describedi n terms of the effectivem agnetic moment, pm.O n the bais of Eq. (20) it is pm= $ 1’2 (X;7-)“2pB. (24) ( ) If required, the temperature-independentp aramagnetism may be taken into account by using 1’2 [(XL-Na) 7’11’2pB (25) In Eq. (24) and Eq. (25),t he combination of universal constants is (3k/Nu$1’2 =2.828. The temperature dependenceo f the susceptibility for substancesw hich are nor magnetically dilute usually follows the Curie-Weiss low Gl NP: (26) “=T-O,= 3k(T-0,) where 9, is the Weiss constant or paramagnetic Curie temperature. Eq. (26) is applicable, in particular, to ferromagnetic (ok. @,>O) and antiferromagnetic (oiz. O,<O) substancesf ar above the Curie and Ntel tem- perature, respectively. Moreover, the majority of paramagnetic substancesa lso follows the Curie-Weiss rather than the Curie law at higher temperatures.T he reasonsf or this behaviour have been discussedi n section 1.1.3.3 of volume II/8 of this series [138]. If Eq. (26) is followed over an extended range of temperature, the effective magnetic moment is obtained therefrom as p,=2.828 [$,,(T- Op)]1’2pB. (27) KSnig / Kihig Ref. p. 411 1 Magnetic susceptibilities : introduction 7 In all other cases,t he effective magnetic moment becomesa function of temperature and is rather calculated on the basis of Eq. (24). The values of C, and 0, are usually determined graphically from the straight line which fits best to the experimental l/xk data plotted versus temperature (Curie-Weiss plot). The TIP contribution may be also ob- tained graphically if the molar magnetic susceptibility xrn is plotted as function of l/T and extrapolated for l/T-O. Often NGIi s incorporated into Eq. (27), its numerical value being estimated on the basis of ligand field theory [22,45]. For methods to determine Ncr empirically, cf: [46,47]. Finally, it should be noted that some authors still use the alternative form of the Curie-Weiss law G Xk=Tf@, (28) a practice which is not recommended( cJ:s ection 1.1.4.2). 1.1.3.4 Paramagnetic anisotropy The magnetic susceptibility (1) of single crystals of lower than cubic symmetry has, to very good approxima- tion, the character of a symmetric tensor of rank 2, cf: Eq. (9). If the six unique elementso f, say, the tensor (x,,,) are known, it is always possible to find a transformation to a new coordinate system in which (x,,,)i s diagonal, (Xm)c=c(XEl) (29) x10 0 (30) El)= 0 x2 0 i 0 0 x3 I where C is the transformation matrix. The diagonal elements xi, xZ, x3 are called the principal molar suscep- tibilities of the crystal and their directions are the principal magnetic axes. Thus the magnetic properties of a single crystal may be completely specified by giving the magnitudes and directions of the principal crystal susceptibilities. The magnetic anisotropies of the crystal are the differences (x1--&, (x1-x3) and &-x3). The orientation of the principal axes of the susceptibility tensor with respect to the crystal axes depends on the crystal system to which the considered single crystal belongs. (1) The triclinic system is characterized by the crystal axes a=!=b +=c and the respective angles a+/I+ y. Since there is no symmetry element other than a onefold axis of rotation or inversion, there is no definite relationship between the crystal axes and the magnetic axes, cf. Fig. 1. It follows that six angles between the crystal and magnetic axes have to be specified. (2) In the monoclinic system, a+ b =I=c and c(= y = 90” + j. The crystal possessesa single twofold axis usually denoted as the b-axis perpendicular to the (010) plane. In this case,t he principal magnetic axis corresponding to x3 is uniquely fixed along the b-axis, the other two lying in the (010) plane with x1>x2 being inclined at angles $ and 8 to the crystal c- and a-axis, respectively (cf: Fig. 2). The relation between the angles $, 0 and /I is /3=$+$+S. (31) b axis a axis Fig. 1. Orientation of crystalline magnetic tensors in Fig. 2. Orientation of crystalline magnetic tensors in triclinic system. monoclinic system. KSnig / KSnig Ref. p. 411 1 Magnetic susceptibilities : introduction 7 In all other cases,t he effective magnetic moment becomesa function of temperature and is rather calculated on the basis of Eq. (24). The values of C, and 0, are usually determined graphically from the straight line which fits best to the experimental l/xk data plotted versus temperature (Curie-Weiss plot). The TIP contribution may be also ob- tained graphically if the molar magnetic susceptibility xrn is plotted as function of l/T and extrapolated for l/T-O. Often NGIi s incorporated into Eq. (27), its numerical value being estimated on the basis of ligand field theory [22,45]. For methods to determine Ncr empirically, cf: [46,47]. Finally, it should be noted that some authors still use the alternative form of the Curie-Weiss law G Xk=Tf@, (28) a practice which is not recommended( cJ:s ection 1.1.4.2). 1.1.3.4 Paramagnetic anisotropy The magnetic susceptibility (1) of single crystals of lower than cubic symmetry has, to very good approxima- tion, the character of a symmetric tensor of rank 2, cf: Eq. (9). If the six unique elementso f, say, the tensor (x,,,) are known, it is always possible to find a transformation to a new coordinate system in which (x,,,)i s diagonal, (Xm)c=c(XEl) (29) x10 0 (30) El)= 0 x2 0 i 0 0 x3 I where C is the transformation matrix. The diagonal elements xi, xZ, x3 are called the principal molar suscep- tibilities of the crystal and their directions are the principal magnetic axes. Thus the magnetic properties of a single crystal may be completely specified by giving the magnitudes and directions of the principal crystal susceptibilities. The magnetic anisotropies of the crystal are the differences (x1--&, (x1-x3) and &-x3). The orientation of the principal axes of the susceptibility tensor with respect to the crystal axes depends on the crystal system to which the considered single crystal belongs. (1) The triclinic system is characterized by the crystal axes a=!=b +=c and the respective angles a+/I+ y. Since there is no symmetry element other than a onefold axis of rotation or inversion, there is no definite relationship between the crystal axes and the magnetic axes, cf. Fig. 1. It follows that six angles between the crystal and magnetic axes have to be specified. (2) In the monoclinic system, a+ b =I=c and c(= y = 90” + j. The crystal possessesa single twofold axis usually denoted as the b-axis perpendicular to the (010) plane. In this case,t he principal magnetic axis corresponding to x3 is uniquely fixed along the b-axis, the other two lying in the (010) plane with x1>x2 being inclined at angles $ and 8 to the crystal c- and a-axis, respectively (cf: Fig. 2). The relation between the angles $, 0 and /I is /3=$+$+S. (31) b axis a axis Fig. 1. Orientation of crystalline magnetic tensors in Fig. 2. Orientation of crystalline magnetic tensors in triclinic system. monoclinic system. KSnig / KSnig