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Applications of the Canonical Ensemble : Simple Models of Paramagnetism

Applications of the Canonical Ensemble : Simple Models of Paramagnetism A Quantum System of Spins J. 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 J z is equally likely, so the

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Applications of the Canonical Ensemble : Simple Models of 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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