1. Weiss developed a molecular field theory to explain paramagnetism, proposing that atomic magnetic moments align due to an internal molecular field proportional to the magnetization. 2. The molecular field arises from exchange interactions between neighboring atomic moments due to the Pauli exclusion principle and Coulomb interaction. 3. The Curie-Weiss law describes the paramagnetic susceptibility above the Curie temperature, relating it inversely to temperature minus the Curie-Weiss temperature.

1. Wiess field model of Paramagnetism

2. Wiess field model of paramagnetism
• In the ferromagnetic materials the magnetic moments (spins) are magnetized
spontaneously. (Mean field theory)
• In 1907, Weiss developed a theory of effective fields
• Magnetic moments (spins) in ferromagnetic material aligned in an internal
(Weiss) field. (Molecular field theory)
• Weiss assumed that this field is proportional to the magnetization BE = 𝜆M
• 𝜆 is the Weiss constant,which is temperature independent.
• Weiss called this field the molecular field and thought that this field results
from all the molecules in the sample.
• In reality, the origin of this field is the exchange interaction

3. Exchange mechanism
• The exchange interaction is the consequence of the Pauli exclusion principle and
the Coulomb interaction between electrons.
• For two electron system there are two possible arrangements for the spins of the
electrons
• either parallel or antiparallel
• If parallel, the exclusion principle requires the electrons to remain far apart.
• If they are antiparallel, the electrons may come closer together and their wave
functions overlap considerably.
• electrostatic energy of an electron system depends on the relative orientation of
the spins.

4. • The difference in energy defines the exchange energy
• The exchange interaction is short ranged. Therefore, only nearest
neighbour atoms are responsible for producing the molecular field.
• The magnitude of the molecular (exchange) field is very large of the order
of 10⁷G or 10³T.
• It is not possible to produce such
field in laboratories.
Coulomb repulsion
energy high
Coulomb repulsion
energy lowered

5. Curie-Weiss law
• Consider the paramagnetic phase, an applied magnetic field B0 causes a finite magnetization
( 𝜆M)
• This in turn causes a finite exchange field BE. If χ ₚ is the paramagnetic susceptibility, the induced
magnetization is given by
• M = χₚ (B0 +BE) = χₚ( B0+𝜆M)
•
• M=
𝜒ₚ+B0
1−𝜒ₚ𝜆
C=
𝑁
3𝐾в
𝜇ₑ , 𝜇ₑ=gJ 𝜇в 𝐽(𝐽 + 1)
•
• χ ₚ = C/T Curie law
• TC =
𝑁 𝜆
3𝐾в
𝜇ₑ
• 𝜒=
𝑀
B0
=
𝐶
𝑇−𝐶𝜆
=
𝐶
𝑇−𝑇ᴄ
Curie-Weiss law

6. The paramagnetic region
• Consider magnetization in the region well above the curie
temperature
• For T>Tc the spontaneous magnetization become zero
• An external field is required to produce magnetization
• This field should weak enough to avoid the saturation state

7. • The Curie-Weis law describes fairly well the observed susceptibility
variation in the paramagnetic region above the Curie point .
• Only in the vicinity of the Curie temperature a notable deviations are
observed.
• This due to the fact that strong fluctuations of the magnetic moments
close to the phase transition temperature can not be described by the
mean field theory which was used for deriving the Curie-Weiss law.
Accurate calculations predict that at temperatures very close to TC.

8. Curie temperatures of some ferromagnetic materials
• iron (Fe) 1,043 K
• cobalt (Co) 1,394 K
• nickel (Ni) 631 K
• gadolinium (Gd) 293 K
• manganese arsenide (MnAs) 318 K

10. 1. Langevin Theory of Paramagnetism- Curie Law
• Postulating non interacting localized atomic moments
2. Curie -Weiss law 𝝌=
𝑴
B 𝟎
=
𝑪
𝑻−𝑪𝝀
=
𝑪
𝑻−Tᴄ
• Postulating a Molecular Field which is internal interaction between
localized moments

11. • Weiss theory is a good phenomenological theory of magnetism , But does not explain source of
large Weiss field.
• Heisenberg and Dirac showed that ferromagnetism is a quantum mechanical effect that
fundamentally arises from Coulomb interaction.(Exchange model)
• The Weiss theory gives information about the magnitude of magnetization, but nothing can be
said about the direction
• Ferromagnetic Paramagnetic Curie Temperature
• Antiferromagnetic Paramagnetic Neel Temperature
• At these temperatures, the available thermal energy simply overcomes the interaction energy
between the spins.
• Paramagnetic effects are quite small: the magnetic Susceptibility is of the order of 10−3 to
10−5 for most paramagnets
• Some paramagnetic material aluminium ,oxygen ,titanium and iron oxide..

12. • In the case of Gd (Gadolinium)
• TC = 292 K, J = S = 7/2
• g = 2 , N = 3.0 1028 m¯³
• C = 4.9 K
• 𝜆 = 59
• In case of iron 𝜆 = 100
• E = -2(J/ћ²)s1.s2
• The operator s1.s2 is 1/2[(s1 + s2) ² - s² 1 - s²2]
• Value of S = s1+ s2 will be either 0 or 1

13. • Obviously spontaneous ordering of magnetic moments minimizes the
entropy and consequently it cannot happen just at any temperature.
At certain temperature the thermal energy kBT becomes greater than
the exchange energy Eex and the material becomes disordered
(entropy wins) and behaves as paramagnetic.
name symbol
energy
between
parallel
neighbours (J)
iron Fe -1.2110
cobalt Co -5.1510
nickel Ni -4.4610

14. By
Muhammad Dawood Khan
Department of Physics
University of Peshawar KPK Pakistan
date :22/05/2019