Magnetism

MAGNETISM Types of Magnetism There are four types of magnetism: (i) Diamagnetism (ii) Paramagnetism (iii) Ferromagnetism (iv) Antiferromagnetism Diamagnetism It is caused by paired, filled shell electrons. It arises from the motion of the electron. This effect is submerged by unpaired electron spins. A diamagnetic material experiences a force in a direction opposite to that of the magnetic field gradient. The magnitude of susceptibility is small and negative. It is approximately equal to –1 x 10-6 cgs units. This is independent of magnetic field, H and temperature, T. (e.g.) H2O, KCl, Organic ligands Paramagnetism It is caused by unpaired electron spin and orbital motion. The magnitude is positive and approximately equal to 10 100 x 10-6 cgs units. This is independent of magnetic field, H and varies inversely with T. (e.g.) [Cr(H2O)6]SO4, (NH4)2[Mn(H2O)6](SO4)2 Ferromagnetism Neighboring particles with unpaired spins interacting among themselves cause this. The susceptibility is positive and very large and approximately equal to 10-2 to 104 x 10-6 cgs units. They may be aligned parallel so that the material possesses an overall magnetic moment and is ferromagnetic. Above the Curie temperature, Tc, the gram susceptibility,g, varies inversely with T. But below Tc, g rises abruptly and dependent on H.

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(e.g.) metallic iron Antiferromagnetism Neighboring particles with unpaired spins interacting among themselves cause this. The unpaired electrons may be aligned in antiparallel fashion giving overall magnetic moment and antiferromagnetic behavior. The susceptibility is positive, but very small and approximately equal to 0.1 x 10-6 cgs units. Effects of Temperature Curie Law The law is stated as follows: “The magnetic susceptibility is inversely proportional to temperature.  = C/T where ‘C’ is the Curie constant. Many paramagnetic substances obey the simple Curie law, especially at high temperatures. Curie – Weiss Law This law is stated mathematically as follows:  = C/(T + ) where ‘’ is the Weiss constant. When -1 is plotted against T, we get the curves as sown in Figure 1. -1 Curie – Weiss law

Curie law

 T(K)

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The temperature dependence of various magnetic susceptibilities are schematically shown below:

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For all materials, the effect of increasing temperature is to increase the thermal energy possessed by ions and electrons. Therefore, there is a natural tendency for increasing structural disorder when the temperature increases. In the case of paramagnetic materials, applied magnetic field orders the spins of the molecules. But the thermal energy partially cancels this ordering effect. As soon as the magnetic field is removed, the orientation of the electron spins becomes disordered. Hence, for paramagnetic materials,  decreases with increasing temperature according to Curie or Curie – Weiss law fashion. In the case of ferro- and antiferromagnetic materials, spins are perfectly arranged as parallel or antiparallel. However, temperature introduces disorderliness in this arrangement. Hence, there is a rapid decrease in  with increasing temperature for ferromagnetic materials. For antiferromagnetic materials, this leads to a decrease in the degree of antiparallel ordering. Therefore,  increases.

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Zeeman Effect The splitting of levels by the external magnetic field is called Zeeman effect. This is of two types: 1. First order Zeeman effect 2. Second order Zeeman effect First Order Zeeman Effect The effects produced on the ground levels (i.e. splitting of the levels) by the first power of the magnetic field are called first order Zeeman effect. That is, the change in energy of the levels is proportional to H. Second Order Zeeman Effect The effects produced on the levels by the second order of magnetic field are called second order Zeeman effect. That is, the change in energy is proportional to H2. This is explained as follows: When the magnetic field acts on an ion, the electron distribution is distorted. Therefore, the ground state mixes with a small amount of some higher state. This mixing is proportional to the magnetic field, H. this magnetic field and the consequent mixing lowers the energy of all the components of the ground state by an amount proportional to H2. This is called second order Zeeman effect. This effect is illustrated for Cu2+. The ground state is 2E. That is, the ground state is a doublet. Both the components of this doublet are lowered by the quantity cH2: Cu2+ system

E

doublet No field

gH gH

H 1st order

cH2 Magnetic field 2nd order Magnetic field

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The lowering of energy by the magnetic field, that is, E/H is linear in H. Therefore the susceptibility remains independent of the magnetic field. In other words, when a non-degenerate level lies lowest, Zeeman effect cannot split it. Therefore, there can be no contribution to the magnetic susceptibility from the first – order Zeeman effect. However, if any paramagnetic susceptibility is present, it is due to the second – order Zeeman effect. This happens when kT is less than the energy separation between the ground level and the higher lying levels. The susceptibility arising in this way can be very large. Temperature Independent Paramagnetism (T.I.P) Thermal distribution between the ground and higher levels will occur if the separation between them is in the order of kT. However, if the level that mixes with the ground level lies much more than kT above the ground level, thermal distribution between the levels does not occur. Hence, the contribution made to the susceptibility is independent of temperature. Therefore it is called temperature independent Paramagnetism (T.I.P). In cupric ion, T.I.P is about 60 x 10-6 cgs unit and this adds to the molar susceptibility of about 1500 x 10-6 cgs units at room temperature. Co3+; t2g6, spin paired This ion should be diamagnetic because there are no unpaired electrons. Nevertheless, molar susceptibilities of about 100 x 10-6 cgs units are observed for this ion. It is due to T.I.P. Here, there is no first order splitting because all the spins are paired. However, the energy of the singlet ground state is lowered due to the second order magnetic field and similarly that of the higher singlet state.

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Magnetic Moment and Magnetic Susceptibility (eff. And ) Magnetic properties of complexes are discussed in terms of a quantity called the ‘magnetic moment’, eff. rather than that of the susceptibility, . Both are related as follows: eff. = (3k/N2)(aT) = 2.828(aT) where eff. Is the paramagnetic moment, ‘k’ is the Boltzmann constant and ‘N’ is the Avogadro number. If Curie’s law is obeyed, eff. should be independent of temperature. eff. = 2.828(AT) A = C/T (Curie’s law)  eff. = 2.828{(C/T) x T)} i.e. eff. = 2.828 x C

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The Magnetic Properties of Free Ions The magnetic properties of free ions depend on the following factors: 1. First order Zeeman effect 2. Second order Zeeman effect 3. States approximately equal to kT 4. States << kT 1.First order Zeeman effect Let us consider a system and let the degeneracy of its term be lifted by spin-orbit coupling. This leads to states. When a magnetic field is applied, the remaining degeneracy of the states may be lifted. Let us consider the ground state, J = 2, and let the other states lie very much above kT. The first order Zeeman effect splits the levels according to the relation, WJ = MJgH and the splitting is shown below: MJ 2 2gH 1 gH J = 2 0 0 -1 -gH -2 -2gH

No magnetic field

Magnetic field

The magnetic moment of a system, which consists of ground state only, is given by Equation (1): eff. = g[J(J+1)] ---------- (1) The above equation applies well to most of the lanthanide ions, where the ground state alone is considered in determining the 8

magnetic properties. In these ions, spin – orbit coupling is so large that states other than the ground states are thermally inaccessible. As far as the ‘f’ electrons are concerned, the ions are essentially free. The splitting factor, g, is a function of the amount of orbital and spin angular momenta which the state possesses. If a state specified by ‘J’ arises from a term specified by L and S, then S(S+1) – L(L+1) + J(J+1) g = 1+ 2J(J+1) For a system in which there is no orbital angular momentum, L=0. Therefore, J=S (because J=L+S; J=0+S; J=S). S(S+1) – 0 + S(S+1)  g = 1 + 2S(S+1) 2S(S+1) i.e.,

g = 1 + 2S(S+1)

or ,

g = 1+1 g = 2

In other words, when the magnetic behavior of an ion is due to the spin angular momentum alone, g = 2. Then from equation (1), we have, eff. = g[J(J+1)] = 2[S(S+1)] (because g = 2 and J = S) If the number of unpaired electrons in the ion is equal to ‘n’, then S = n/2.  eff. = 2[n/2(n/2+1)] = 2/2[n(n+2)] = [n(n+2)] --------- (2) 9

Equation (2) is called the spin – only formula for the magnetic moment because they correspond to the contribution from spin angular momentum alone. The spin only values for different ‘n’ values are given in Table 1. Table 1 Spin – only values n 2S+1 eff.s.o A 1 2 1.73 1250 2 3 2.83 3333 3 4 3.87 6250 4 5 4.90 10000 5 6 5.92 14600 6 7 6.93 20000 7 8 7.94 26250 When the spin-orbit coupling constant,  , is of the order of kT, more than one state is thermally accessible. That is, more than one level can be populated. Now the susceptibility of the system as a whole is due to the first order Zeeman effect contribution from each state. This contribution is proportional to its Boltzmann population. 2. Second order Zeeman effect When the state having J = 0 lies lowest and that with J = 1 lies >>kT above it, the susceptibility arises from the second order Zeeman effect alone.

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3. States approximately equal to kT When there are states separated from the ground state by energy of the order of kT (kT = 210 cm-1 at 300K), both first order and second order Zeeman effects from adjacent states contribute. Quenching of Orbital Angular Momentum by Ligand Fields When a free ion is converted into its complex, there is a loss in the orbital angular momentum. If the orbital angular momentum is to exist, two requirements must be satisfied. Requirement 1 When an orbital can be rotated by about an axis to give an identical and degenerate orbital, orbital angular momentum results. For example, in the free ion, dxz orbital can be rotated about the z-axis by 90o to give the dyz orbital and vice versa. Similarly, the dxy orbital can be rotated about the z-axis to give dx2-y2 orbital. Hence, orbital angular momentum results between these orbitals. However, in the presence of a cubic ligand field, the dxy and dx2-y2 orbitals are no longer degenerate and no orbital angular momentum results between them. That is, the ligand field quenches orbital angular momentum. Nevertheless, the ligand field does not quench all the orbital angular momentum because the dxz and dyz orbitals remain degenerate. Orbital angular momentum remains to some extent with the t2g orbital. The reason is that the rotation about the z-axis turns dxz into dyz and rotation about the x- or y-axis turns dxy into dxz or dyz respectively. However, no rotation can turn the dz2 orbital into dx2-y2 orbital because they differ in shape. Hence, there is no orbital angular momentum associated with the eg set.

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Requirement 2 A second requirement for the existence of orbital angular momentum due to orbital rotation is that there should not be an electron in the second orbital with the same spin as that in the commencing orbital. In the configurations t2g0, t2g3 and t2g6, it is not possible to make the required transformations of the d-orbitals because t2g0 is vacant; t2g3 and t2g6 have no vacant orbital. The configuration and the corresponding terms are given below: Configuration

Terms

t2g3eg0 t2g3eg1 t2g3eg2 t2g6eg0 t2g6eg1 t2g6eg2 t2g6eg3

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A2g Eg 6 A1g 1 A1g 2 Eg 3 A2g 2 Eg 5

All the above configurations do not have orbital angular momentum. The terms are either A or E. Thus orbital angular momentum is quenched for A and E terms. The configurations t2g1egm, t2g2egm, t2g4egm and t2g5egm lead to T terms. The orbital angular momentum is not completely quenched. That is, the orbital angular momentum remains for T terms. Spin-orbit coupling constant and magnetic moment If spin-orbit coupling is present, the observed magnetic moment will be different from the spin-only value.  and  The symbol  denotes spin-orbit coupling constant of the free ion. It is given by the expression,

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Ze2h2  =

82m2c2r3

 is the spin-orbit coupling constant of the complex ion.  and  are related as follows:  =   / 2S =/n where ‘n’ is the number of unpaired electrons. ‘+’ sign applies to less than half-filled shells and ‘-’ sign applies to more than half-filled shells. Magnitude of  or  The magnitude depends on the following factors: 1.The effective nuclear charge Zeff. When the atomic number increases, Zeff. increases and  or  increases. 2. Oxidation state. For a given metal ion, an increase in the positive oxidation state leads to a decrease in ‘r’. Zeff. increases and  increases. 3. Within a group. Going down the transition series (3d to 4d to 5d) along a group increases  or . (e.g. (Cr3+) = 275 cm-1, (Mo3+) = 820 cm-1, (W3+) = 1800 cm-1) 4. The effective orbit radius. In a complex, the effective orbit radius increases due to electron cloud expansion. Therefore,  increases. The value of  for a complex ion is 20 – 25% less than that for a free ion. In a weak octahedral crystal field, that is, in a high-spin complex, the number of unpaired electrons is the same as in the case of free ion. In such a case  will be +ve if the number

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of electrons is less than five and  will be –ve if the number of electrons is more than five. In a strong crystal field, that is, in a low-spin complex, the t2g orbitals comprise the shell. In a low-spin d4 complex, the arrangement will be t2g4 and more than half filled and  is negative. All the tetrahedral complexes are high-spin. Hence, the sign of  will be the same as that of the free ion. For 6A1g, we have an orbital singlet, and therefore,  becomes redundant (meaningless). Whenever L or S is zero, there is no spin-orbit interaction and  becomes redundant (meaningless). Spin-orbit coupling constants (in cm-1) for 3d transition series metal ions are given in Table 2. Magnetic moments and multiplet widths Case 1. Multiplet width >> kT. In this case, L and S couple very strongly. The two vectors do not remain distinct. The spin-orbit coupling constant  must be very high so that L and S vectors interact strongly. Now the lowest lying component (lowest lying J) alone is populated. The vectors L and S precess rapidly about the direction of the resultant vector J. It can be assumed that L and S also precess rapidly about J. Effectively, LS is precessing rapidly about J. Finally, LS is given approximately by J. J = g[J(J+1)] From the above equation, equations for L and S are derived as follows: When there is only orbital magnetic moment, S = 0 so that J = L, and hence g = 1.  L = [L(L+1)] For the spin-only magnetic moment, L = 0, so that J = S.  g = 2

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 S = 2[S(S+1)] = [4S(S+1)] = [2S(2S+2)] However, the resultant spin quantum number, S = ns = n() where ‘n’ is the number of unpaired electrons.  S = [n(n+2)] S is called the spin-only magnetic moment. eff. = [M3kT /N]  emu = [M3kT/N2]  = eh/4mc [ erg/gauss = Bohr magneton] eff. = 2.83(MT) B.M. Case 2. Multiplet width < kT When the multiplet width is quite small as compared to kT, all the J levels may be populated. Now, the coupling of L and S vectors is insignificant. Therefore, the two vectors remain distinct. Hence, the magnetic moment is due to the sum total of the spin and orbital effects. M = NL2/3kT + NS2/3kT = N/3kT(L2 + S2)  = [M3kT/N2] Substituting the value of M, we get, L+S = [L(L+1) + 4S(S+1)] Case 3. Multiplet width  kT When the multiplet width is comparable to kT, to calculate the total magnetic susceptibility, the magnetic

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susceptibility due to each ‘J’ level, along with the population of the ‘J’ level is to be considered. The three cases can be summarized as follows: Multiplet width (B.M.) Temperature dependence of M  Large g[J(J+1)] M  1/T compared to kT Small [L(L+1)+4S(S+1)] M  1/T compared to kT Comparable to Complicated function of Curie’s law not kT J and T obeyed

Van Vleck Equation The magnetic property of a paramagnetic substance originates from the permanent magnetic dipoles present in it. When a magnetic field is applied the following effects take place. (i) the magnetic dipoles tend to align themselves along the field direction against the opposing influence of temperature. As a result, the MJ levels of a particular J are symmetrically split into lower and upper levels. This splitting is known as the first order Zeeman effect. (ii) The orbitals get distorted via the introduction of some character of the excited state into the ground state. This is known as the second order Zeeman effect. Derivation In deriving the Van Vleck equation, two assumptions are made: (i) the paramagnetic susceptibility is independent of the applied magnetic field,H. 16

(ii)

the energy of the ith level of the atom or ion is a power series in H. That is,

Wi = W(0) + W(1)H + W(2)H2, Where Wi(0) is the energy of the ith level in absence of H, and Wi(1) and Wi(2) are the first order and second order Zeeman coefficients respectively. If a substance is devoid of permanent magnetic dipoles, there is no first order Zeeman interaction with H, which means Wi(1) will be zero. A magnetic dipole, when it interacts with the magnetic field, lowers its energy. The Van Vleck equation is: N[Wi2(1)/(kT) – 2Wi(2)]e-Wi(0)/(kT)] M = eWi(0)/(kT) The above equation can be segregated as follows: N[Wi(1)2/(kT)e-Wi(0)/(kT)] Mfirst order) = e-Wi(0)2/(kT) NWi(1)2/(kT) (since Wi(0)=0) = 2J+1 ------- (1) where 2J+1 is the multiplicity, i.e., degeneracy of the level concerned. -2NWi(2)e-Wi(2)e-Wi(0)/(kT) M(second order) =

e-Wi(0)/(kT) -2NWi(2)

=

since Wi(0) = 0 ------ (2) 2J+1

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The above equation can be expressed as: -2NWi(2) M(second order) = N = 2J+1 2 or

 =

Wi(2) --------------- (3) 2J+1

Temperature – dependence of Second Order Magnetic Susceptibility The first order Zeeman magnetic susceptibility equation(1) carries kT, whereas the second order Zeeman magnetic susceptibility equation(2) does not. Therefore, the magnetic susceptibility due to second order effect is usually independent of temperature. When the separation of the interacting levels is small and of the order of kT, the excited level is actually populated. Hence, only at very low temperatures, the second order magnetic susceptibility will be independent of temperature (because higher levels will not be populated). When the separation of the interacting levels is much greater than kT, the upper level is not populated and the second order contribution will be independent of temperature. Magnitude of Second Order Contribution The second order Zeeman effect is inversely proportional to the separation of the interacting levels. This separation is usually quite large, and so M due to this effect is rather small. It is usually one order of magnitude smaller than the first order contribution. The second order contribution is very important in a complex with no spin magnetic moment. If the separation is large, mixing is small and second order contribution will be less. The overall sign of M(second order) is positive.

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Relation of Second Order Magnetic Susceptibility to Curie Law The second order M is not inversely proportional to ‘T’. It is a constant term. The magnetic moment derived from the second order effect will be a function of T. Consider the relation,M = N22/(3kT). When ‘T’ decreases, M should decrease. But it is a constant. Therefore, to maintain M a constant  should decrease. Constant N is added to the magnetic susceptibility as a correction term. (This constant should not be used as a variable parameter.)  M = N22/(3kT) + N Effect of Magnetic Field on Second Order Magnetic Susceptibility The second order magnetic susceptibility, M depends on the magnetic field, H. When H = 0, the second order effect vanishes. Hence, second order magnetic susceptibility is an induced effect like diamagnetism. However, unlike diamagnetism, it makes a positive contribution to M(first order). Its magnitude is of the same order as that of diamagnetism. Other Names for Second-Order Magnetic Susceptibility The other names for second-order magnetic susceptibility are: (i) Temperature – independent Paramagnetism (T.I.P) (ii) Van Vleck Paramagnetism (iii) Residual Paramagnetism Ground State Diamagnetic (S=0) and Induced Paramagnetism When the ground state is diamagnetic (S=0), even a mixing with the excited state terms via spin – orbit coupling cannot generate paramagnetic susceptibility. However, paramagnetism

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can be induced via mixing of the ground state and the excited state, which is capable of making an orbital contribution, under the influence of a magnetic field. This is the case with lowspin (t2g6; diamagnetic) cobalt(III) complexes and d0 systems like KmnO4 and K2CrO4. The ground state in such instances is spin and orbital singlet, and the excited state is several times kT above the ground state. The T.I.P for an octahedral cobalt(III) complex can be calculated using the formula, 2

eh

24

4.085

T.I.P =

N( ) = 3 2mc D D 1 1 where ‘D’ is the energy of the A1g T1g transition and is obtained experimentally. The ratio of the experimental T.I.P. and the theoretical T.I.P. gives k2, where ‘k’ is the orbital reduction factor. A ‘k’ value in the range 0.5 – 0.9 indicates the covalent nature, and a ‘k’ value of ‘1’, the ionic nature of a complex. A high – spin d5 system (Fe3+ and Mn2+) has the 6A1g ground state term in an octahedral field. This term is an orbital singlet and has no orbital angular momentum. Such a system has no excited term with the spin multiplicity six. Therefore, there is no mixing and has no second order Zeeman effect. Spin – Orbit Coupling on A, E and T terms There is no orbital rotation in the A and E terms. That is, one orbital cannot be converted into another by any symmetry operation. Hence, there is no orbital magnetic moment in the A and E terms. But the T term has orbital rotation. That is, dxz orbital can be converted into dyz orbital by rotation through 90o about the z-axis. Hence, orbital magnetic moment is present in the T terms. However, the orbital magnetic moment maybe brought into the A and E ground state terms from the excited T terms via spin-orbit coupling.

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Magnetic Moment and 10Dq The magnetic moment and 10Dq obtained from the electronic spectrum can be quantitatively connected as follows: g = 2 for spin only system. For 3A2g term, g = 2 - 8/(10Dq) = 2[1-4/(10Dq)] eff. = 2[1-4/(10Dq)][S(S+1)] = [1-4/(10Dq)]s (because s = 2[S(S+1)]) This equation quantitatively connects the magnetic moment and 10Dq obtained from the electronic spectrum. Depending the sign of , the eff. values may be higher or lower than the s values. Sign of :  is +ve for less than half full shells and  is –ve for more than half full shells. Example: d7 tetrahedral cobalt(II) has ground state 4A2 and d8 octahedral nickel(II) has ground state 3A2g. These have more than half full shells. Therefore,  is –ve and therefore eff.>s. d3 chromium(III) octahedral has ground state 4A2g and  is positive because it is less than half full. Therefore, eff.<s. For a d5 octahedral system, the Russell – Saunders term in the absence of a magnetic field is 6S and the corresponding spectroscopic term is 6A1g. It is an orbital singlet term and hence, has no orbital angular momentum associated with it. In addition, there is no other term having the same multiplicity as 6 A1g. Therefore, interaction via the spin-orbit coupling or via the second order Zeeman effect is not possible. Therefore, the observed magnetic moment of the system is due to the s only. For a low-spin octahedral d6 system [cobalt(III)] , the ground state term is 1A1g and the complex is diamagnetic. But there is interaction with the excited state terms, 1T1g and 1T2g

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due to the second order Zeeman effect. These excited state terms have orbital contribution. Hence, a positive molar susceptibility arises to the extent of +100 x 10-6 cgs units. But no spin-orbit coupling can occur since the ground state term and the excited state terms are diamagnetic. ( All are singlet terms; 2S+1 =1; S = 0, that is completely paired and hence diamagnetic.). The second order Zeeman susceptibility is given by 8N2/(10Dq) Octahedral high-spin d4, low-spin d7 and d9 configurations have E ground state terms. The spin-orbit interaction of an E term with a higher energy T2 term having the same multiplicity as that of the E term, gives the ‘g’ value as follows: g = 2[1-2/(10Dq)] eff. = [1-2/(10Dq)]s For the tetrahedral complexes of vanadium(IV),  is +ve, so that eff.<s. For the octahedral complexes of copper(II),  is – ve, so that eff.>s Spin-orbit coupling on T terms T terms retain orbital angular momentum. Therefore, the actual magnetic moment will exceed s. Example: Hexaaquotitanium(III) sulphate - d1 system. 2 Eg 2

D

 2 2

T2g

J= 0 -/2 J=3/2 g

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In order to calculate the L,S and J values, correspondence between the p and t2g orbitals must be arrived at. It can be shown that the orbital angular momentum integrals of the members of the ‘p’ orbital set are equal in magnitude but opposite in sign to those of the corresponding members of the t2g set. Therefore, l = 1 for the T terms and there is a negative correspondence between the ‘p’ and t2g orbital sets. For the T terms, the t2g is less than half full. Therefore J with higher value will be the lower energy state. Considering the 2T2g term, S = ; L= 1; J = L+S = 1+ = 3/2 J can have values L+S , L+S-1, L+S-2 -------- , L-S and in this case 3/2 and . The lower level will have J = 3/2. Lande Interval Rule The energy difference between two successive J levels is given by the product of  and the larger of the two J values. In this case larger value of J = 3/2. Therefore, the energy difference = 3/2. Importance of this rule In most substances, the magnetic properties originate from the ground state ‘J’level. But sometimes the first, or even the second excited state may be significantly populated. In such cases, we should know the energy separating the excited and the ground state. For the above titanium(III) complex, d1 system,  = +155cm-1 and the energy separation between the two J levels is equal to 3/2 = 3/2 x 155 = 232.5cm-1. The value of kT at 300K is equal to 0.69504cm-1 x 300 = 208.5cm-1 because kT at 1K = 0.69504cm-1. Thus the energy separation between the two J levels is in the range of kT. g(t2g) = -1 + 3[J(J+1) + S(S+1) – L(L+1)]/[2J(J+1)] For J = 3/2, g = -1+3[3/2(3/2+1)+(+1)–1(1+1)]/[2 x 3/2(3/2+1)] = -1+1 = 0 The –ve sign is introduced to show the negative correspondence between the t2g set and ‘p’ orbitals and ‘3’ is introduced because the degeneracy is ‘3’. 23

Similarly it can be shown that g = 2 for J =  state. Since g = 0 for the ground state, magnetic property will not arise from this state. Therefore, M will arise from the second order Zeeman effect. Then the magnetic property will be independent of temperature. When the temperature is increased, the higher level, J = H, will be populated. For this level, g0. Hence, this level has first order Zeeman effect. Therefore, M is dependent on temperature. The above figure is also applicable to a tetrahedral 3d9 copper(II) complex. This also has 2T2 ground term. But for copper(II),  = -830 cm-1 and J is –ve. Therefore, J =  level becomes the ground state and g  0 for this level. Hence, this level shows first order Zeeman effect and M is temperature dependent. As the temperature decreases, this level will be populated more so that  approaches S = 1.73 B.M., since g = 2. When the temperature increases, J = 3/2 level will be populated. This level ( g= 0) experiences second order Zeeman effect. Hence, the susceptibility will be due to the spin – only contribution and the second order Zeeman effect. 3d2 configuration (3T1g term) For the 3T1g term, L = 1 because there is correspondence between p orbital set and t2g set, though –ve, S = 1 and J is found out as follows: J = L + S = 1+1 =2; L – S = 1-1 = 0.  J = 2,1,0. Because of the –ve correspondence between the t2g and p orbitals, the ground state for the d2 system will be J = 2 and the first excited state will be J = 1 and so on. The energy gap between the successive levels is given by the Lande interval rule, that is, the product of  and the larger of the two J values. Thus, the difference between the J = 2 and J = 1 states is 2A, where A = (1.5 – c2)/(1 + c2) and ‘c’ is the mixing coefficient. c = (6Dq + E)/(4Dq)

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g is obtained as a complicated function in A. (e.g.) Octahedral vanadium(III) complex. The ground state term is 3T1g. For V(III),  = +105 cm-1. Using this value, the Van Vleck equation predicts  = 2.7 B.M. at 300 K. This should decrease with the fall in temperature till  = 0.62 B.M. at 0 K. Actually, ammonium vanadate(III) alum records  = 2.7 B.M. at 300 K as well as at 80 K. That is,  is temperature independent. The reason is that this does compound does not possess a truly octahedral geometry but belongs to a low symmetry. This low symmetry destroys the orbital angular momentum associated with the T term. If this splitting by the low symmetry is larger than that due to the spin – orbit coupling, then  will tend to approach s and its temperature dependence will not be significant. (e.g. 2) K3[Mn(CN)6] ; low-spin ; d4 system The ground state term is 3T1g.  - 3.50 B.M. at 300 K and 3.31 B.M. at 80 K. The magnetic moment rapidly decreases to zero at 0 K. Thus the magnetic moment is temperature dependent. This shows that the orbital angular momentum is not destroyed. (Oh symmetry). The higher levels are also populated and hence the higher levels also contribute to magnetic moment. The extent of population depends on temperature. As the temperature is lowered, the population also decreases. Hence, contribution to magnetic moment also decreases. Electron Delocalisation and Consequent Effects on Magnetic Properties There is always some overlap between the metal and ligand orbitals. This leads to electron delocalisation and this must be considered to get an agreement with the experimental results.In such cases, the orbital angular momentum is reduced by a factor ‘k’, called the delocalisation factor. When k = 1, there is no delocalisation. In a covalent complex, it is less than unity. For A2g term, 25

g = 2[1 – 4k2/10Dq] eff. = S[1 – 4k2/10Dq] M (second order) = 8k2N2/10Dq For Eg term, g = 2[1-2k2/10Dq] eff. = S[1 – 2k2/10Dq] M(second order) = 4k2N2/10Dq The total magnetic susceptibility is given by the sum of the first and second order effects. M(A2g) = M(spin–only)[1 – 8k2/10Dq] + 8k2N2/10Dq M(Eg) = M(spin-only)[1 – 4k2/10Dq] + 4k2N2/10Dq Examples: 1. [Ni(H2O)6]SO4 M = 4340 x 10-6 cgs units at 300 K From electronic spectrum, 10Dq = 8900 cm-1 corresponding to the transition 3 3 A2g T2g For free ion  = -315 cm-1 Second order Zeeman coefficient, 8N2/10Dq = 235 x 10-6 cgs/mol (N2 = 0.261 cm-1 erg/gauss2 mol) M(A2g) = M(spin-only)[1 – 8k2/10Dq] + 8k2N2/10Dq 4340 x 10-6 = [3333(1+8 x 315k2/8900) + 235k2] x 10-6 k = 0.93 The value of ‘k’ indicates that the degree of delocalisation of the ‘d’ electrons of the metal ion is small. 2. [Cu(H2O)6]SO4. K2SO4 The ground state term for Cu(II) is 2Eg M = 1520 x 10-6 cgs units at 300K

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10Dq = 12000 cm-1 corresponding to the transition, 2 2 Eg T2g For free ion,  = -830 cm-1 M(Eg) = M(spin-only)[1 – 4k2/10Dq] + 4k2N2/10Dq 1520 x 10-6 = 1250{1+(4k2 x 830)/12000} + (4k2 + 0.261/12000) x 106]x10-6 k = 0.69 This shows that the delocalisation of the metal ‘d’ electrons is significant.

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Anomalous Magnetic Moments

When the magnetic moment for a metal ion falls outside the range of predicted value based on the spin angular and orbital angular momenta of electrons, it is called anomalous value. Discrete molecular species (Magnetically dilute) Here, there is no secondary magnetic interaction between the neighboring molecules. That is, there is no ferromagnetic or antiferromagnetic interaction in these systems. That is, the systems are magnetically dilute. (e.g.) Nickel(II) complexes In the octahedral symmetry, the ground state term is 3A2g and it acquires some orbital contribution from the 3T2g excited term due to spin-orbit coupling. The overall magnetic moment is dependent on 10Dq and  as follows: eff. = S(1-/10Dq) where  = 4 for the A term and 2 for the E term. Reasons for anomalous magnetic behavior 1. 2. 3. 4. 5.

Equilibrium between two spin states. Magnetically non-equivalent sites in the unit cell. Solute-solvent interaction. Solute-solute interaction. Configurational equilibrium.

1. Equilibrium between two spin states There may be equilibrium between high-spin and low-spin states in certain cases. For example, in the octahedral geometry, when the crystal field strength is in the region of the critical 10Dq, a spin state equilibrium may arise for the d4, d5, d6 and d7 configurations. But it is not so in the case of 3d8 nickel(II) octahedral complexes because the 3A2g ground state 28

term and 1Eg excited state terms are almost parallel at all crystal field strengths. But if there is a tetragonal distortion of an octahedral geometry, the spin-state of nickel(II) complex may change. This distortion will take place if the complex is a mixed ligand complex like Nia4b2. The ligands are present in axial positions. x2-y2

b1g b1g eg

2 a1g 1 b2g

3d

a1g

xy z2

b2g t2g eg eg

xz,yz

Tetragonal Distortion of Octahedral Complex If the ligands a and b have similar crystal field strengths, 1 and 2 will be small.  will be approximately equal to 3.0 B.M. But if ‘b’ is a very weak ligand compared to the ligand ‘a’, 1 or 2 may be higher than the pairing energy, P (z – out distortion). Therefore pairing takes place resulting in diamagnetism. Hence, when 1 or 2 is in the range of pairing energy, P, there will be a spin state equilibrium in a tetragonal nickel(II) complex. 29

(e.g.) Dichlorotetrakis(N,N-diethylthiourea)nickel(II) This complex is spin-paired below 194 K and  = 0. It becomes partially paramagnetic when the temperature is increased. Nickel(II) has a tetragonal field and the magnetic property is decided only by the thermal population of the two spin-states. The equilibrium is represented as follows: Singlet(low-spin) triplet(high-spin) The equilibrium constant, [triplet] [high-spin] K= = [singlet] [low-spin] ‘K’ is calculated from the knowledge of the mole fractions, Nlow-spin and Nhigh-spin The relation between M and mole fraction is given by the following expressions: M(expected) = Nlow-spinM(low-spin) + Nhigh-spinM(high-spin) (Nlow-spin + Nhigh-spin = 1) The above expression can be expressed in terms of magnetic moment as follows: expected2- low-spin2 K= high-spin2-expected2 Difference between affixed mixture of two spin states and spinstate equilibrium Anomalous magnetic moment can also arise from a fixed mixture of two spin-states(say S=0 and S=1). Now a linear Curie-Weiss plot will still be obtained for the S = 1 state. But if the spin-state equilibrium is temperature dependent, the composition of the mixture will change with a change in temperature. Now the Curie-Weiss plot will be non-linear. Thus, form the nature of the Curie-Weiss plot, it is possible to distinguish between (i) a fixed mixture of two spin – states and (ii) a spin-state equilibrium between two spin-states.

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In an octahedral crystal field, the spin-state equilibrium is possible for the following configurations: d4

t2g3eg1 – 5Eg; t2g4- 3T1g Cr2+; Mn3+

d5

t2g3eg2 – 6A1g; t2g5 – 2T2g Mn2+; Fe3+

d6 t2g4eg2 – 5T2g; t2g6 – 1A1g Fe2+; Co3+ d7 t2g5eg2 – 4T1g; t2g6eg1 – 2Eg Co2+ In each of the above configurations, around the cross-over region, the energies of the two spin states differ by  kT. Therefore, the relative populations of the two states vary with temperature. Plot of K Vs. 1/T K is the equilibrium constant between the two spin-states. If the plot of lnK versus 1/T for a complex is a straight line, it means that the modification of the crystal lattice does not influence the magnetic property. If the plot of lnK versus 1/T for a complex is not a straight line, it means that the modification of the crystal lattice influences the magnetic property in addition to the spin-state equilibrium. 2.Magnetically non-equivalent site in the unit cell In a unit cell, two situations may arise. (i). The metal ions may have the same coordination number, the same set of ligands but different geometries, and (ii). The metal ions may have different coordination numbers and hence different geometries. Example for situation 1.

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The complex, dibromobis(benzylphosphine)nickel(II) is green colored and exhibits an anomalous magnetic moment of 2.7 B.M. The X-ray crystallographic study of this compound has revealed that the unit cell has three nickel(II) complexes – one square planar and two tetrahedral. Considering that square of the magnetic moment is additive and also the mole fractions of the different complexes, we get, 2 = 0.33 x 2Ni2+(square planar) + 2 x 0.33 x 2Ni2+(Td) [Square planar : Tetrahedral = 1:2 Total number of moles = 3 Mole fraction of square planar = 1/3 = 0.33 Mole fraction of Td = 2/3 = 2 x 0.33] 2.72 = 0.33 x 0 + 2 x 0.33 x 2Ni2+(Td) Ni2+(Td) = 3.3 B.M. Example for situation 2. A complex has a magnetic moment of 2.58 B.M. The unit cell of this compound was found to contain one 4coordinate square planar nickel(II) complex and two 6coordinate pseudooctahedral nickel(II) complexes. 3. Solute – solvent interaction Anomalous magnetic moment may also arise when a particular complex interacts with a coordinating solvent. Thus many square planar diamagnetic nickel(II) complexes become partially paramagnetic due to an equilibrium of the following type: Solvent + Square planar complex(dia) = pseudooctahedral Complex(para) 32

4.Solute – solute interaction When two or more molecules of a complex are associated, coordination number of the metal – ion increases. This interaction changes the spin-state of the metal ion. (e.g.) Bis(N-methylsalicylaldiminato)nickel(II) R O

H N

C

Ni C H

N

O R

This is diamagnetic in the solid state. But in soluion, this shows anomalous magnetic moment in the range 1.9 – 2,3 B.M. This value depends on the nature of the non-coordinating solvent. This behavior is not due to planar tetrahedral equilibrium because the complex has zero dipole moment in benzene or dioxan. When the concentration of the complex is increased in these solvents, the m.wt and magnetic moment also increase, indicating the presence of solute – solute interaction. When a substituent is introduced in the ortho position, the complex is no longer paramagnetic. That is, the solute – solute interaction does not take place due to steric hindrance. 5.Configurational equilibrium Octahedral square planar equilibrium also gives rise to anomalous magnetic moment.

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Antiferromagnetism and Ferromagnetism (Magnetically concentrated system) When the neighboring magnetic centers are close enough, direct or indirect overlap of orbitals takes place. Now the magnetic exchange interaction takes place. This interaction affects the magnetic property of the complex. Almost all paramagnetic compounds are involved in exchange interaction to a certain extent. This interaction is dominant only at a very low temperature. When the exchange interaction energy is greater than kT, both ferromagnetism and Antiferromagnetism jointly operate. Because of the exchange interaction, the magnetic moment may be more or less than that of a complex not involved in exchange interaction. Ferromagnetic interaction increases the magnetic moment of a complex while antiferromagnetic interaction decreases the magnetic moment. Antiferromagnet The neighboring magnetic centers are opposed to each other ( ). Ferromagnet The neighboring magnetic centers are aligned parallel ( ) Ferromagnetism is rarely encountered but much more useful than Antiferromagnetism. Types of Antiferromagnetism There are two types, viz., intramolecular and intermolecular Intramolecular Antiferromagnetism In this type, the interacting paramagnetic centers are present within the same molecule, which may be dimeric or polymeric. (e.g.) Copper(II) acetate monohydrate dimmer, Cu2(CH3COO)4.2H2O

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The room temperature magnetic moment per copper(II) ion is 1.4 B.M. But if there were no interactions,  > S = 1.73 B.M. That is, the magnetic moment is decreased due to antiferromagnetic exchange interaction. K4[Ru2Ocl10] ; K4[Cl5Ru – O – RuCl5] This complex is diamagnetic due to the coupling of neighboring spins. A monometallic low-spin Ru(IV), d4 complex is expected to exhibit two unpaired spins. Intermolecular antiferromagnetism A weak intermolecular antiferromagnetism occurs in many transition metal complexes. This effect may be found out by measuring the magnetic susceptibility in solution and in solid state. The effect will differ. Antiferromagnetic exchange pathways The exchange may take place by any one of the following two methods: 1. Direct metal – metal interaction via overlap of suitable metal orbitals. 2. Super exchange arising from the transfer of paramagnetic spin density from one metal ion through the orbital overlap of the diamagnetic bridging atoms to an adjacent metal ion. Direct metal-metal interaction If the complex has a suitable structure, the metal orbitals carrying the unpaired electrons may overlap. Copper(II) acetate monohydrate is a dimmer both in the solid state and in solution. In this complex, the two copper atoms are 2.64 Å apart. In metallic copper, the Cu-Cu distance is 2.65 Å. This structure allows a lateral overlap of the dx2-y2 orbitals of each of the two copper(II) ions, generating a  bond. The ‘d’

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orbital electron distribution and the energy order of the ‘d’ orbitals in this complex are given below: dxy( ) < dxz( ) < dyz( )
But if the acetate groups are substituted, Cu-Cu distance is longer ( 2.69 Å) than that in the metallic copper (2.65 Å) and hence direct interaction may not be possible. Mn2(CO)10 is diamagnetic. The oxidation state of manganese in this complex is zero. An overlap of the two hybrid orbitals of the two adjacent Mn(0) containing the unpaired electron leads to diamagnetism. Superexchange The super exchange phenomenon usually involves three atoms. In most coordination complexes, the shell of the ligands surrounding the metal ion is such that a long metalmetal distance is usually maintained. This prevents direct interaction between the metal ions. The system is similar to magnetically dilute system. But in a magnetically concentrated system, the metal-metal distance will be usually short and direct interaction will take place or the bridging ligand atoms successfully transmit the interaction between the magnetic dipoles. In superexchange, suitable partially filled metal orbitals overlap with a filled ligand orbital. The overlap may be

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 or -type. In some cases both types of overlap may take place.

-overlap In this case each metal ion, M, has one unpaired electron in its dz2 orbital. The filled ligand orbital is pz. Linear overlap, that is, -overlap takes place. -overlap The unpaired electron will be present in the dxz orbital of each metal ion. The filled ligand orbital will be px. Lateral overlap, that is, -overlap takes place. Antiferromagnetic binuclear complexes S =  system ; copper(II) The magnetic exchange interaction in copper(II) acetates may depend on two factors, namely,  The metal – metal distance  The electron – density

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Antiferromagnetic interaction is inversely proportional to the metal – metal distance. That is, when the metal – metal distance is short, the interaction will be greater. When the ligand supplies more electron density to the metal, the antiferromagnetic interaction will be greater. Interaction through -bond or superexchange In the case of copper(II) acetate monohydrate, exchange interaction increases with increase in Cu-Cu distance. This means that the interaction does not occur through direct metal – metal bond but occurs through super exchange via carboxylate bridges. The complex also does not exhibit any electronic spectral band characteristic of metal-metal interaction. The super exchange interaction depends on the angle between the high-spin metal ions.

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