Curie Constant

The Curie constant is a key parameter in magnetism, especially concerning ferromagnetic and paramagnetic materials. Named after physicist Pierre Curie, it measures the magnetic susceptibility of these materials in relation to temperature. The value of the Curie constant varies among different materials, reflecting their unique magnetic properties. This variability is crucial for applications in physics, engineering, and materials science. A thorough understanding of the Curie constant offers valuable insights into how magnetic materials behave at varying thermal conditions, which is essential for advancing technologies in magnetic storage, electronics, and more.

1.0Paramagnetic Substance

• The materials which are weakly magnetized in the direction of applied magnetic field.
• These materials tend to lose their magnetic behavior with rise in temperature.
• Example- Aluminium, Chromium, Manganese

2.0Electron Theory of Paramagnetism

• Electrons in atoms exhibit both orbital and spin motion, contributing to the net magnetic dipole moment. In materials with permanent dipole moments, thermal energy at room temperature causes atomic dipoles to orient randomly, canceling each other out and resulting in a net magnetism of zero.
• When a magnetic field is utilized to paramagnetic materials, atomic dipoles attempt to align with the field. However, at ordinary temperatures and fields, this alignment is weak, resulting in a phenomenon known as paramagnetism. As temperature increases, the alignment is further disrupted, causing paramagnetic materials to lose their magnetism.

3.0Intensity of Magnetisation $(\vec{I}{)}^{\prime \prime }$

• When a magnetic material is placed in a magnetising field then the induced dipole moment per unit volume of that material is defined as the intensity of magnetisation.

$\vec{I}=\frac{\vec{M}}{V}$

• Unit of $(\vec{I}):\mathrm{A}/\mathrm{m}\left(\frac{\vec{M}}{V}=\frac{I\vec{A}}{V}=\frac{\text{ Ampere }\times {\text{ meter }}^2}{{\text{ meter }}^3}\right)$
• Induced magnetic field due to these induced dipoles in the material is given by ${\vec{B}}_{in}={\mu }_0\vec{I}$

4.0Magnetizing Field or Magnetic Intensity

• Field in which a material is placed for magnetization, called as magnetizing field. 

$\vec{H}=\frac{{\vec{B}}_0}{{\mu }_0}$

• Unit of $\vec{H}:A/m$
• ${\vec{B}}_0:\text{ Magnetic Field }$
• ${\mu }_0\text{ : Permeability of free space }$

5.0Magnetic Permeability $(\mu )$

• It represents the degree up to which a material can be penetrated by the magnetic field lines
• $\mu =\frac{B}{H}=\frac{\text{ Total magnetic field inside a material }}{\text{ Magnetising Field }}$
• Unit of $\mu :\mu =\frac{B_m}{H}=\frac{H}{m}$

6.0Magnetic Susceptibility $\left(x_m\right)$

• ${\chi }_m=\frac{I}{H}(\text{ It is a scalar quantity with no units and dimensions })$
• Physically it represents the ease with which a magnetic material can be magnetised.

Note:- A material with more ${\chi }_m$ can be changed into a magnet easily.

7.0Curie Law in Paramagnetism

• This law states that magnetisation of a paramagnetic material is inversely proportional to absolute temperature of the material.
• Magnetic Susceptibility $\left(x_m\right)$ of a paramagnetic material is inversely proportional to absolute temperature T of the material.

• It is found experimentally that Magnetisation of paramagnetic substance, is given by

$I=C\frac{B_0}{T}$        $\therefore B_0={\mu }_{0}H$

$I=C\frac{{\mu }_{0}H}{T}$

$\frac{I}{H}=C\frac{{\mu }_0}{T}\left(\because \frac{I}{H}={\chi }_m\right)$

${\chi }_m=C\frac{{\mu }_0}{T}(C=\text{ Curie Constant })$

${\chi }_m\propto \frac{1}{T}\Rightarrow {\chi }_m=\frac{C}{T}$

• In a paramagnetic substance, the alignment of magnetic dipoles is achieved only when the applied magnetic field is extremely strong and the temperature is very low.

For Example: Chromium potassium alum salt necessitates an extremely high magnetic field of 50,000 Gauss and very low temperatures of 1.3 K to reach 99.5% magnetism.

8.0Limitations of Curie Law in Paramagnetism

The limitation of Curie’s Law is that as the temperature decreases or the magnetic field increases, the magnetization of the material reaches a saturation point and cannot increase further. This results in a failure to accurately describe the material's behavior according to Curie’s Law.

Example: If two paramagnetic materials possess varying Curie constants, what can we deduce about their magnetic properties?

Solution: At a specific temperature, the material with the higher Curie constant will demonstrate increased magnetic susceptibility. Consequently, this material will have a stronger response to an external magnetic field. $\left({\chi }_m=\frac{C}{T}\right)$

Example: How does the Curie constant affect the magnetic susceptibility of a paramagnetic material as the temperature varies? What occurs to the susceptibility as the temperature approaches absolute zero?

Solution: The Curie constant is essential in determining the magnetic susceptibility of a paramagnetic material. As the temperature rises, susceptibility decreases due to heightened thermal agitation. In contrast, as the temperature approaches absolute zero, susceptibility increases because thermal agitation diminishes, facilitating better alignment of the magnetic moments.

9.0Sample Questions on Curie Constant

Q-1 How does the intensity of magnetization in a paramagnetic material change with temperature?

Solution: Paramagnetic material obey Curie law of magnetism, magnetism of such material decreases with increase in temperature ${\chi }_m\propto \frac{1}{T}$

Q-2 Write the dimensional formula for Curie Constant?

Solution: ${\chi }_m=C\frac{{\mu }_0}{T}$

$C=\frac{T}{{\mu }_0}=\frac{[K]}{\left[ML{T}^{-2}{A}^{-2}\right]}=\left[M^{-1}{L}^{-1}{T}^2{A}^2{K}^1\right]$

Q-3.Why does a paramagnetic sample exhibit increased magnetization at lower temperatures under the same magnetic field?

Solution: The alignment of atomic dipoles in a paramagnetic material occurs in a magnetic field when the temperature decreases. At low temperatures, the disruptive influence of thermal agitation is diminished, making the sample more susceptible to magnetization.

Q-4.Susceptibility of Mg at 300 K is 1.2 × 10^–5.  Find The temperature at which susceptibility will be 1.8 × 10^–5 .

Solution: ${\chi }_m\propto \frac{1}{T}$

${\chi }_1{T}_1={\chi }_1{T}_1$

$T_2=\frac{1.2\times 10^{-5}\times 300}{1.8\times 10^{-5}}=200\mathrm{K}$

Q-5. A paramagnetic gas consist of atoms each with a dipole moment of $1.5\times 10^{-23}\mathrm{J}/\mathrm{T}$ having magnetic field 3T,Temperature of the gas is $27^{\circ }\mathrm{C}$ and its number density is $2\times 10^{26}{\mathrm{m}}^{-3}$. Find the maximum magnetisation of the sample possible when placed in an external field?

Solution: Dipole moment (m)= $1.5\times 10^{-23}\mathrm{J}/\mathrm{T},B=3T,n=2\times 10^{26}{\mathrm{m}}^{-3},T=27^{\circ }\mathrm{C}=300\mathrm{K}$

Maximum Magnetisation 

$M_{\max }=nm=2\times 10^{26}{\mathrm{m}}^{-3}\times 1.5\times 10^{-23}$ $\mathrm{J}$$/T=3\times 10^3\mathrm{A}/\mathrm{m}$