Inorganic Chemistry-II

Subject Chemistry

Paper No and Title 7; Inorganic Chemistry-II (Metal-Ligand Bonding, Electronic Spectra and Magnetic Properties of Transition Metal Complexes) Module No and Title 36; Magnetic Exchange Interaction Part – II

Module Tag CHE_P7_M36

CHEMISTRY PAPER No. : 7; Inorganic Chemistry-II (Metal-Ligand Bonding, Electronic Spectra and Magnetic Properties of Transition Metal Complexes) MODULE No. : 36 (Magnetic Exchange Interaction Part–II)

TABLE OF CONTENTS

1. Learning outcomes

2. Models for magnetic exchange interaction

2.1 Heisenberg model

2.2 Ising model

2.3 Sign of exchange integral J

3. Antiferromagnetism

3.1 Intramolecular antiferromagnetism

3.2 Intermolecular antiferromagnetism

4. Mean field theory of antiferromagnetism

4.1 Susceptibility below the Néel temperature

4.2 Antiferromagnetic exchange pathways

5. Summary

Prerequisite Before going into the details of this module we should be aware of the fact that anomalous magnetic moment may also arise in magnetically concentrated complex. The magnetic interaction in such a complex occurs when the neighboring centers are close enough for direct or indirect orbital overlap. This interaction affects the magnetic property of a complex and swamps the ligand field effect. In fact, almost all paramagnetic complexes are involved in exchange interaction to a certain extent which is dominant only at low temperatures. When the exchange interaction energy is greater than kT, the result is a cooperative phenomenon of ferromagnetism or antiferromagnetism. There are two types of magnetic exchange coupling; 1. Ferromagnetism 2. Antiferromagnetism For ferromagnetic substances, we should also be aware of mean field theory of ferromagnetism and ferrimagnetism. In this module we will discuss about the Heisenberg Model and Ising Model of spin-spin interaction alongwith antiferromagnetism.

1. Learning Outcomes

After studying this module, you shall be able to

• Know about models for magnetic exchange interaction. • Learn Heisenberg model, Ising model and also the sign of exchange integral J • Learn about the antiferromagnetism. • Identify various examples of intramolecular antiferromagnetism and intermolecular antiferromagnetism • Evaluate mean field theory of antiferromagnetism and antiferromagnetic exchange pathways.

2. Models for Magnetic Exchange Interaction

Two models have been proposed for the magnetic exchange interactions. These are

(1) Heisenberg Model

(2) Ising Model

Let us discuss each of the model in details.

2.1 Heisenberg Model

The spin-spin interaction may occur between the neighboring metals ions with the effect being propagated throughout the chain of interacting nuclei. A linear chain is a polymetallic system with an infinitely long array of metal ions held up linearly by the ligand system. If the chains are isolated, interaction occurs

within each chain neglecting interchain interaction. The exchange field gives an approximate representation of the quantum mechanical exchange interaction. According to Heisenberg model, the energy of interaction of atoms i,j bearing electron spins Si, Sj (Figure 1) is given by

� = −2� �!. �!

i j Si Sj

A B

Figure 1

Where J is the exchange integral and is related to the overlap of charge distributions of the atoms i, j. The charge distribution of a system of two spins depends on whether the spins are parallel or antiparallel for the Pauli principle excludes two electrons of the same spin from being at the same place at the same time. It does not exclude two electrons of opposite spin. Thus the electrostatic energy of a system will depend on the relative orientation of spins: the difference in energy defines the exchange energy. The exchange energy of two electrons may be written in the form 2� �!. �!, just as if there were a direct coupling between the directions of the two spins. For many purposes in ferromagnetism it is a good approximation to treat the spins as classical angular momentum vectors. When a system contains more than two electrons, the exchange energy takes the form

� = −2 �!" �!. �! !!! Where the sum is taken over all pair-wise interactions of spins i and j. Note that the exchange coupling constant J here is different from the total angular momentum quantum number J. It is necessary to appreciate that the exchange forces come into play only between the paramagnetic ions. These forces vanish if one of the interacting ions has a resulting non vanishing spin, say, Si and if the other ion has a closed shell of n similar electrons (i.e., j =1,2……..n). Then, the exchange energy can be expressed as

� = −2�!"�! �!

Since �! is zero for a closed shell, the exchange energy has to be zero. Thus, a closed shell has no influence on the spin of other electron.

We can establish an approximate connection between the exchange integral J and the Curie temperature Tc. Suppose that the atom under consideration has z nearest neighbors, each connected with the central atom by the interaction J. For more distant neighbors we take J as zero. The mean field theory result is

3� � � = ! ! 2��(� + 1)

2.2 Ising Model

The pair-wise spin-spin coupling is considered between the nearest neighbours only, and summation is taken over the entire lattice. In a lattice ferromagnet, each individual spin couples with the magnetic field produced (the magnetisation) by the net alignment of other spins in the lattice, as well as with any applied magnetic field. The same arguments apply to lattice antiferromagnet. The detailed theory for such exchange interactions in various types of lattice is quite complicated, but is usually not relevant to the magnetic behavrour of transition metal complexes, since in most complexes organic ligands separating the metal atoms prevent the formation of a lattice of closely linked paramagnetic atoms. One type of lattice exchange interaction that does apply to some transition metal complexes, is the Ising model which describes interactions between an infinite linear chain of neighbouring paramagnetic atoms. lf s = 1/2 for each member of the chain, the Ising model yields expression for the magnetic susceptibilities

parallel (χ||) and perpendicular (χ⊥) to the magnetic field direction, which reduces to

2��!�! = �! !" … … … … … (1) 4�� And ��!�! � = tan ℎ� + � �� ℎ!� �! !" … … … … … (2), (� = ) 8� ��

The theoretical value of χM is obtained by summing equations 1 and 2 over all orientations; 1 2 ! = + 3 3 or ��!�! �!! + 2 + �!! �!! − �!!�!!! + 5 = × … … … … . . (3) ! 12�� !! + �!!! + 2

The Ising model should apply fairly well to any system which can be visualised as a linear chain of interacting paramagnetic atoms, and comparison of observed magnetic properties with Equation (3) can help to elucidate the structure of an unknown compound.

2.3 Sign of Exchange Integral J

The sign of J may be positive or negative. When the electron spins couple in a manner such that they become parallel, J is positive (Figure 2) and when the electron spins are anti parallel, J is negative (Figure 3). The temperature dependence of J is not expected to be important and J is usually considered to be temperature independent. Let us consider general situation of n interacting paramagnetic ions, each having the spin quantum number S. These n ions can then generate the new quantum number Sʹ, that can have following values Sʹ = nS, nS−1, nS−2,…….0 or ½.

ES S=1

ET S=0 J > 0 Ferromagnetic Interaction

Figure 2

ET S=1

ES S=0 J < 0 Antiferromagnetic Interaction

Figure 3 When J is positive, larger the value of Sʹ, lower the exchange energy −2 �!"�!�!, i.e., the state having highest value of Sʹ will be the ground state. This leads to a magnetic moment higher than for the system with no interaction. This is the ferromagnetic situation. However, when J is negative, the smaller the value of Sʹ, the smaller the value of exchange energy −2 �!"�!�!, since it becomes positive i.e., the state having the smallest value of Sʹ becomes the ground state. This is the antiferromagnetic situation which leads to a lower magnetic moment for the system with no interaction. The exchange forces arise due to an overlap of the orbitals and the origin of J is electrostatic not magnetic. J is usually expressed in cm-1 (unit of energy).

3. Antiferromagnetism

In an antiferromagnetic substance, the neighboring magnetic centers are opposed to each other, but in ferromagnetic substance, they are aligned parallel. For antiferromagnetic substances, the neighboring spins align anti-parallel with one another below a certain critical temperature (TN, Néel temperature) and below TN the material tends to remain spin aligned and has a lower moment than expected. The coupling forces are responsible for the alignment of the magnetic spins. Spontaneous alignment of the magnetic dipoles in ferro- / antiferromagnetic states needs some positive energy of interaction between the neighboring spins. The origin of this coupling is quantum mechanical. Antiferromagnetic interactions can be divided into two categories. 1. Intramolecular Antiferromagnetism 2. Intermolecular Antiferromagnetism

3.1 Intramolecular Antiferromagnetism CHEMISTRY PAPER No. : 7; Inorganic Chemistry-II (Metal-Ligand Bonding, Electronic Spectra and Magnetic Properties of Transition Metal Complexes) MODULE No. : 36 (Magnetic Exchange Interaction Part–II)

Intramolecular antiferromagnetism arises when the interacting paramagnetic centers are situated within the same molecule. In other words, intramolecular antiferromagnetic exchange occurs between two or more centers within the same molecule (necessarily dimeric or polymeric). It is often associated with direct metal-metal bonding between the magnetic centers. The common example of this class is copper(II)acetate monohydrate dimer, [Cu2(OAc)4].(H2O)2 .The copper centers in this complex are antiferromagnetically coupled resulting in diminishing of the magnetic moment near 90 K. [Cu2(OAc)4].(H2O)2 is essentially diamagnetic due to the cancellation of two opposing spins. For this complex, the room temperature magnetic moment per Cu2+ ion is 1.4 B.M., however, if there were no antiferromagnetic interaction, the magnetic moment would have been at least 1.73 B.M. 4 Another example is K4[Ru2OCl10] or K4[Cl5Ru−RuCl5]. The monometallic low spin d Ru(IV) complex is expected to have two unpaired spin. But the complex K4[Ru2OCl10] is diamagnetic due to the coupling of the neighboring spins leading to the antiferromagnetic exchange. One more interesting example is d1 molybdenum(V) complex of ethyl xanthate. This complex is also diamagnetic due to the coupling of the neighboring spins leading to antiferromagnetic exchange. If there were no antiferromagnetic interaction, the magnetic moment of the complex would have been 1.73 B.M. per Mo5+.

3.2 Intermolecular Antiferromagnetism Intermolecular antiferromagnetism arises due to the exchange between many centers in a crystal lattice. In other words, intermolecular antiferromagnetic exchange occurs between two or more centers within different molecules in a crystal lattice. The transition metal oxides and halides belong to this class, e. g., 2+ 2+ 2+ 2+ perovskite fluorides, KMF3, where M (M = Mn , Fe , Co , Ni etc) stands for almost any bivalent transition metal ion. A weak intermolecular antiferromagnetic coupling occurs in many transition metal complexes. The extent of such an effect may be checked by measuring the magnetic susceptibility in solution where it should be poorer than in the solid state.

4. Mean Field Theory of Antiferromagnetism

In an antiferromagnet, the spins are ordered in an antiparallel arrangement with zero net moment at temperatures below ordering or Néel temperature. The susceptibility of an antiferromagnet is not infinite at T = TN. An antiferromagnet is a special case of a ferrimagnet for which both sublattices A and B have equal saturation magnetizations. Thus CA= CB in magnetic susceptibility equation of ferrimagnet

� + � � + � � − 2�� ! ! ! ! ! ! = = ! ! … … … … … . (1) �! � − �!

And the Néel temperature in the mean field approximation is given by

�! = ��

Where C refers to a single sublattice. The susceptibility in the paramagnetic region T > TN is obtained from equation (1)

2�� − 2��! 2� = = �! − �� ! � + ��

2� = … … … . (2) � + �!

The experimental results at T > TN are of the form: 2� = … … … . . (3) � + �

Table 1

Substance Paramagnetic Transition Curie- θ/ TN Ion Lattice Temprature TN Weiss (K) constant θ (K) MnO fcc 116 610 5.3 FeO fcc 198 570 2.9 CoO fcc 291 330 1.14

NiO fcc 525 2000 4

MnS fcc 160 528 3.3

MnTe hex. layer 307 690 2.25 MnF2 bc tetragonal 67 82 1.24 FeF2 bc tetragonal 79 117 1.48 FeCl2 hex. layer 24 48 2 CoCl2 hex. layer 25 38.1 1.53 NiCl2 hex. layer 50 68.2 1.37

Experimental values of θ/TN listed in Table 1 often differ substantially from the values expected from equation (2) and (3). Values of θ/TN of the observed magnitude may be obtained when next-nearest- neighbor interactions are provided for, and when possible sublattice arrangements are considered. If a mean field constant −ε is introduced to describe interactions within a sublattice, then � � + � = �! � − �

4.1 Susceptibility below the Néel Temperature

Below the Néel temperature of an antiferromagnet, the spins have antiparallel orientations; the

susceptibility attains its maximum value at TN where there is a well-defined k in the curve of χ versus T (Figure 4). There are two situations: 1) With the applied magnetic field perpendicular to the axis of the spins and 2) With the field parallel to the axis of the spins. At and above the Néel temperature the susceptibility is nearly independent of the direction of the field relative to the spin axis. For Ba perpendicular to the axis of the spins we can calculate the susceptibility by elementary considerations. The energy density in the presence of the field is, with M = |MA| = |MB|, 1 � = �� . � − � . � + � ≅ −��! 1 − 2� ! − 2� �� … … … … … . (4) ! ! ! ! ! 2 ! Where 2ϕ is the angle that spins make with each other (Figure 4). The energy is minimum when

�� = 0 = 4��!� − 2� � �� ! This gives

� � = ! 2�� So that, 2�� 1 = = �! �

In the parallel orientation (Figure 4), the magnetic energy is not changed if the spin systems A and B make equal angles with the field. Thus the susceptibility at T= 0K is zero. The parallel susceptibility increases smoothly with temperature up to TN.

0 = 0

Figure 4

4.2Antiferromagnetic Exchange Pathways Antiferromagnetic interaction may take place through one of the following two processes: 1. Direct metal-metal interaction via the 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. The Super-exchange phenomenon usually involves three atoms.

8. Summary

In this module, we discussed that,

• There are two models for magnetic exchange interactions. These are Heisenberg model and Ising model. • Ising model describes interactions between an infinite linear chain of neighbouring paramagnetic atoms • The sign of exchange integral J may be positive or negative. When the electron spins couple in a manner such that they become parallel, J is positive and when the electron spins are anti-parallel, J is negative. • The mean field theory of antiferromagnetism and antiferromagnetic exchange pathways. • Antiferromagnetic interaction may take place either through direct metal-metal interaction or through super-exchange