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Application of Canonical Ensemble: Spin Models in Paramagnetism

Explore the quantum system of spins in paramagnetic materials, considering identical magnetic ions with spin J. Study the magnetic moment, Brillouin functions, and Curie Law in the canonical ensemble. Understand magnetization and dipole moments at different temperatures.

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Application of Canonical Ensemble: Spin Models in Paramagnetism

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  1. Applications of the Canonical Ensemble: Simple Models of Paramagnetism A Quantum System of Spins J

  2. Paramagnetic Materials: Spin J • Consider a solid in which all of the magnetic ions • are identical, having the same value of spin J. • Every value of Jz is equally likely, so the • average value of the ionic dipole moment is • zero. • When a magnetic field is applied in the • positive z direction, states of differing values • of Jz will have differing energies and differing • probabilities of occupation.

  3. The equation for the magnetic moment of an atom is: Where g is the Lande’ splitting factor given as, Also

  4. Let N be the number of atoms or ions/ m3 of a paramagnetic material. The magnetic moment of each atom is, In presence of magnetic field, according J is quantized Where MJ = –J, -(J-1),…,0,…(J-1), J i.e. MJ will have (2J+1) values.

  5. If the dipole is kept in a magnetic field B then potential energy of the dipole is: In the Canonical Ensemble, the mean magnetic moment at temperature T is formally:

  6. Therefore, magnetization is: Let,

  7. Mj = -J, -(J-1),….,0,….,(J-1), J, therefore, Simplifying this

  8. Let a = xJ, above equation may be written as, Here, BJ(a) = Brillouin function.

  9. Brillouin Function Brillouin Function As a result of these probabilities, the average dipole moment is given by

  10. The maximum value of magnetization is Thus, For J = 1/2 For J = 

  11. Special case: But Thus above equation becomes,

  12. Thus where, where, This is curie law. Further, Thus Peff is effective number of Bohr Magnetons. C is Curie Constant. Obtained equation is similar to the relation obtained by classical treatment.

  13. High T ( x << 1 ): Curie-Brillouin law: Brillouin function:

  14. High T ( x << 1 ): Curie law = effective number of Bohr magnetons Gd (C2H3SO4)  9H2O

  15. Brillouin Function

  16. Curie Law The Curie constant can be rewritten as where p is the effective number of Bohr magnetons per ion.

  17. The J=1/2 case Two spins, J=1/2, just two states (parallel or AP), to average statistically Several similarities Estimate the paramagnetic susceptibility

  18. Generic J and the Brillouin function

  19. Lande’ g-value and effective moment J=1/2 J=3/2 J=5 Curie law: c=CC/T

  20. (2.828)2χT=g2S(S+1)

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