Magnetization in fluid in Quantum Mechanical Case

Saturday, December 22, 2012

Magnetization in fluid in Quantum Mechanical Case

Magnetization in fluid in Quantum Mechanical Case

Previously, we found the equation for magnetization of a fluid in the classical case. Now, let`s find the equation for magnetization of a fluid for the quantum mechanical case.

Here, we consider the fluid to have N spins up (Nup) and N spins down (Ndown). In the presence of an external field B, the energy is given by where the positive z-axis points upwards.

Again, as in the classical case for magnetization of a fluid (Langevin`s equation), the relative probability of having any angle is, from statistical mechanics, exp(-energy/kT).

So,

Magnetization in fluid in Quantum Mechanical Case equation 2

and

Magnetization in fluid in Quantum Mechanical Case equation 3

. If N is the total number of dipoles, them M is the product of N with the average magnetic moment along the z-axis.

$Magnetization in fluid in Quantum Mechanical Case equation 4 - N = N_{up} + N_{down} \\ \\\mu = \frac{N_{up}(-\mu_{z}) + N_{down}(\mu_{z})}{N}\\ \\ M = N\mu = N\mu_{z}\frac{e^{\frac{+\mu_{z} B}{kT}} - e^{\frac{-\mu_{z} B}{kT}}}{e^{\frac{+\mu_{z} B}{kT}} + e^{\frac{-\mu_{z} B}{kT}}}\\ \\ M = N\mu_{z}\tanh(\frac{\mu_{z}B}{kT}) \\ \\ M = N\mu_{z}\tanh(u) \; \; ;\; \; u = frac{\mu_{z}B}{kT}$

In most normal cases, say, for typical moments, room temperatures, and the fields one can normally get (like 10 000gauss), the ratio represented by u is about 0.02. One must go to very low temperatures to see the saturation. For normal temperatures, tanh(u) can be replaced by u and

Magnetization in fluid in Quantum Mechanical Case equation 5

The graph of M against u for the quantum mechanical case of magnetization in fluids resembles that for the classical case.

Equation for Magnetization in Quantum Fluid graph

When B gets very large, the hyperbolic tangent approaches 1, and M approaches the limiting value (N x the average magnetic moment along the z-axis). So, at high fields, the magnetization saturates. We can see why that is,; at high enough fields, the moments are all lined up in the same direction . In other words, they are all in the spin-down state, and each atom contains the same moment.

Additionally, when we look at the expressions for magnetization in the fluid at room temperature for both the classical and quantum case, we notice that M is proportional to B/T. From this, we can derive Curie`s law which is M = kB/T, where k is a constant.