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Application A Simple Model of Paramagnetism Section 6.3 ( Spin = ½ ) Section 7.8 ( Spin = S )

Application A Simple Model of Paramagnetism Section 6.3 ( Spin = ½ ) Section 7.8 ( Spin = S ). Paramagnetic Materials: Spin J. Consider a solid in which all of the magnetic ions are identical, having the same value of J (appropriate for the ground state).

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Application A Simple Model of Paramagnetism Section 6.3 ( Spin = ½ ) Section 7.8 ( Spin = S )

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  1. Application A Simple Model of Paramagnetism Section 6.3 (Spin = ½) Section 7.8 (Spin = S)

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

  3. Recall the equation for the magnetic moment of an atom, i. e. Where g is the Lande’ splitting factor given as, Consider only spin,

  4. Consider only orbital motion, 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 to space quantization. Where MJ = –J, -(J-1),…,0,…(J-1), J i.e. MJ will have (2J+1) values.

  5. The magnetic moment of an atom along the magnetic field corresponding to a given value of MJ is thus, If dipole is kept in a magnetic field B then potential energy of the dipole would be Therefore, Boltzmann factor would be, Represents fraction of dipoles with energy MjgBB. The magnetic moment of such atoms would be Thus, average magnetic moment of atoms of the paramagnetic material would be

  6. Average magnetic moment Therefore, magnetization would be Case 1: Let,

  7. Since Mj = -J, -(J-1),….,0,….,(J-1), J, therefore, Simplifying this equation, we get

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

  9. The maximum value of magnetization would be Thus, For J = 1/2 For J = 

  10. Case 2: But Thus above equation becomes,

  11. 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.

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

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

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

  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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