Derivation of the Van Vleck Formula for Susceptibility

 

Derivation of the Van Vleck Formula for Susceptibility The Van Vleck formula provides a quantum mechanical expression for the magnetic susceptibility of a material. It is derived from the interaction of magnetic dipoles with an external magnetic field, incorporating the principles of quantum mechanics and statistical physics. Magnetic susceptibility and quantum mechanical framework Magnetic susceptibility (χ) measures the degree to which a material is magnetized in response to an external magnetic field. In quantum mechanics, the energy of a system in an external magnetic field B is described by the Hamiltonian: H = H₀ − μ ⋅ B where H₀ is the field-free Hamiltonian, and μ is the magnetic moment operator. The magnetic moment is related to the total angular momentum operator J by: μ = −g μB J where g is the Landé g-factor, and μB is the Bohr magneton. The susceptibility arises from the induced magnetic moment per unit volume. For a quantum system, the induced moment is proportional to the expectation value of μ. Perturbative approach In the absence of a magnetic field, the eigenstates and eigenvalues of the Hamiltonian H₀ are given by: H₀ |n⟩ = Eₙ |n⟩ where Eₙ are the energy eigenvalues, and |n⟩ are the eigenstates. The magnetic field perturbs the system, and the perturbation is treated using first-order perturbation theory: H′ = −μ ⋅ B The perturbed energy levels and wavefunctions are calculated, leading to a modified partition function. Derivation of the Van Vleck formula The magnetic susceptibility can be expressed as: χ = ∂M/∂B where M is the magnetization per unit volume. Using quantum mechanics, the magnetization is: M = (1/Z) ∑ₙ ⟨n| μz |n⟩ e⁻ᴱⁿ/ᵏᵀ Here, Z is the partition function, μz is the z-component of the magnetic moment operator, k is the Boltzmann constant, and T is the temperature. For systems with no permanent magnetic moment (such as in Van Vleck paramagnetism), the magnetization results from virtual transitions between states. The magnetic susceptibility is derived by summing over contributions from all such states: χ = (1/V) (g² μB² / kT) ∑ₙ ≠ₘ |⟨n| Jz |m⟩|² (Eₘ − Eₙ) / (Eₘ − Eₙ)² e⁻ᴱⁿ/ᵏᵀ The above expression accounts for the thermal population of states and the energy gaps between them. Van Vleck paramagnetism In systems where Eₙ and Eₘ are closely spaced, the summation simplifies. The resulting susceptibility is called Van Vleck susceptibility, named after J.H. Van Vleck, who first derived this expression. χ = μ₀ g² μB² / h ∑ₙ,ₘ |⟨n| Jz |m⟩|² / (Eₘ − Eₙ) e⁻ᴱⁿ/ᵏᵀ This formula shows that the susceptibility is proportional to the energy differences and the matrix elements of the angular momentum operator, weighted by the Boltzmann distribution of states. Applications and implications The Van Vleck formula is significant for understanding paramagnetic and diamagnetic behavior in materials without a permanent magnetic moment. It is particularly relevant for explaining susceptibilities in rare-earth ions and other systems where higher-order magnetic effects are important. References Van Vleck, J. H. (1932). The theory of electric and magnetic susceptibilities. Oxford University Press. Kittel, C. (2004). Introduction to solid state physics (8th ed.). Wiley. Blundell, S. (2001). Magnetism in condensed matter. Oxford University Press.

Derivation of the Van Vleck Formula for Susceptibility  


The Van Vleck formula provides a quantum mechanical expression for the magnetic susceptibility of a material. It is derived from the interaction of magnetic dipoles with an external magnetic field, incorporating the principles of quantum mechanics and statistical physics.  


Magnetic susceptibility and quantum mechanical framework  


Magnetic susceptibility (χ) measures the degree to which a material is magnetized in response to an external magnetic field. In quantum mechanics, the energy of a system in an external magnetic field B is described by the Hamiltonian:  


H = H₀ − μ ⋅ B  


where H₀ is the field-free Hamiltonian, and μ is the magnetic moment operator. The magnetic moment is related to the total angular momentum operator J by:  


μ = −g μB J  


where g is the Landé g-factor, and μB is the Bohr magneton.  


The susceptibility arises from the induced magnetic moment per unit volume. For a quantum system, the induced moment is proportional to the expectation value of μ.  


Perturbative approach  


In the absence of a magnetic field, the eigenstates and eigenvalues of the Hamiltonian H₀ are given by:  


H₀ |n⟩ = Eₙ |n⟩  


where Eₙ are the energy eigenvalues, and |n⟩ are the eigenstates. The magnetic field perturbs the system, and the perturbation is treated using first-order perturbation theory:  


H′ = −μ ⋅ B  


The perturbed energy levels and wavefunctions are calculated, leading to a modified partition function.  


Derivation of the Van Vleck formula  


The magnetic susceptibility can be expressed as:  


χ = ∂M/∂B  


where M is the magnetization per unit volume. Using quantum mechanics, the magnetization is:  


M = (1/Z) ∑ₙ ⟨n| μz |n⟩ e⁻ᴱⁿ/ᵏᵀ  


Here, Z is the partition function, μz is the z-component of the magnetic moment operator, k is the Boltzmann constant, and T is the temperature.  


For systems with no permanent magnetic moment (such as in Van Vleck paramagnetism), the magnetization results from virtual transitions between states. The magnetic susceptibility is derived by summing over contributions from all such states:  


χ = (1/V) (g² μB² / kT) ∑ₙ ≠ₘ |⟨n| Jz |m⟩|² (Eₘ − Eₙ) / (Eₘ − Eₙ)² e⁻ᴱⁿ/ᵏᵀ  


The above expression accounts for the thermal population of states and the energy gaps between them.  


Van Vleck paramagnetism  


In systems where Eₙ and Eₘ are closely spaced, the summation simplifies. The resulting susceptibility is called Van Vleck susceptibility, named after J.H. Van Vleck, who first derived this expression.  


χ = μ₀ g² μB² / h ∑ₙ,ₘ |⟨n| Jz |m⟩|² / (Eₘ − Eₙ) e⁻ᴱⁿ/ᵏᵀ  


This formula shows that the susceptibility is proportional to the energy differences and the matrix elements of the angular momentum operator, weighted by the Boltzmann distribution of states.  


Applications and implications  


The Van Vleck formula is significant for understanding paramagnetic and diamagnetic behavior in materials without a permanent magnetic moment. It is particularly relevant for explaining susceptibilities in rare-earth ions and other systems where higher-order magnetic effects are important.  


References  


Van Vleck, J. H. (1932). The theory of electric and magnetic susceptibilities. Oxford University Press.  


Kittel, C. (2004). Introduction to solid state physics (8th ed.). Wiley.  


Blundell, S. (2001). Magnetism in condensed matter. Oxford University Press.  

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