1. of the Canonical Ensemble: A Quantum System of Spins J
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. 2

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

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

10. Brillouin Function
As a result of these probabilities, the average dipole moment is given by
Brillouin
Function 10

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

12. Special case:
But
Thus above equation becomes,

13. 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.

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

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

16. Brillouin Function 16

17. For the second case, in weak fields the magnetization is linear with the applied field and the sample is linearly paramagnetic. Of course, there is also a diamagnetic response which could be included, but thee are generally small compared to the paramagnetic response.
The equation for the magnetization is valid for most paramagnetic salts at room temperature under typical magnetic fields in the lab. 17

18. Curie Law The Curie constant can be rewritten as
where p is the effective number of Bohr magnetons per ion. 18

19. 19

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

21. Generic J and the Brillouin function

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

23. (2.828)2χT=g2S(S+1)