Ising model

statistical mechanics topic

1. Ising Model Presentation on “Advanced Statistical Mechanics ( PHY604)” to Dr. Khalid Mahmood Khattak By Muhammad Usama Daud (12987) Riphah International University Faisalabad

2. Introduction This model was given by Ising (student of Lenz) during his PHD. It is a dynamical model of phase transition. There is a model: “Array of lattice sites, with only nearest neighbor ( n.n ) interaction that depends upon the manner of occupation of neighboring sites.”

3. What can be explained by this model? This model can explain Ferromagnetism and anti ferromagnetism transition. Gas to liquid transition. Liquid to solid transition.

4. Study of Statistical mechanics of ising model To study the statistical mechanics: We disregard K.E of atoms sitting at various lattice sites. Phase transition is essentially result of interaction energy among atoms and we include only nearest neighbors interaction. To study properties such as magnetic susceptibility ( χ ) we subject the lattice to external magnetic field B. (Directed upward) The spin σ i , then possess additional potential energy Here μ =atomic magnetic moment= g= lande g Factor μ B = Bohr Magneton

5. Bohr Magneton

6. Statistical Mechanics of Ising model Consider a system in the configuration { σ 1 , σ 2 , σ 3 ,……, σ n } After applying external magnetic field So the Hamiltonian of above configuration

7. The 1 st term is due to interaction of spin with magnetic field. In the 2 nd term both of sigma represents nearest neighbors . The 2 nd term is due to exchange interaction term. To explain the concept of n.n in 2 nd term , consider the given triangle. All vertices have spin interaction with each other. If the spins are given by Then

8. The above equation is known as Ising model equation. Due to change in magnetic field: J is exchange energy/ Ising Interaction/ Ising Energy Exchange interaction occurs between identical particles. This effect is due to wave function of indistinguishable particles being subject to exchange symmetry. It is Quantum mechanical effect.

9. Different Notations Some authors write in different ways:

10. The Partition Function

11. When the partition function is found a lots of properties can be calculated. Helmholtz Free Energy: Internal Energy: Specific Heat: Net Magnetization:

12. Net Magnetization If T< Tc (Critical Temperature) and B=0 i.e in the absence of external field then net magnetization gives spontaneous magnetization of the system. If it is non zero i.e B>0 and system would be: Ferromagnetic T< Tc Paramagnetic T> Tc

13. The last property can be found by partition function is Magnetic Susceptibility “It is measure of the extent to which material can be magnetized in relation to applied field.