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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 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) 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 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) 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 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) 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. 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) E ES S=1 2J ET S=0 J > 0 Ferromagnetic Interaction Figure 2 E ET S=1 2J 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
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