Question

Paramagnetism can occur when some atoms in a crystalline solid possess a magnetic dipole moment owing to an unpaired electron with its associated orbital angular momentum. For simplicity, assume that (1) each paramagnetic atom has a magnetic dipole moment \(\mu\) and (2) magnetic interactions between unpaired electrons can be neglected. When a magnetic field is applied, the magnetic dipoles will align themselves either parallel or antiparallel to the direction of the magnetic field. If the magnetic moment is parallel to a magnetic field of induction \(\vec{B}\), the potential energy is \(-\mu B\), when the magnetic moment is antiparallel to \(\vec{B}\), the potential energy is \(+\mu B\). a. Prove that the probability for an atomic magnetic dipole moment to point parallel to the magnetic field is given at temperature \(T\) by $$ P_{\mathrm{a}}=\left(1+e^{-2 t}\right)^{-1} $$ where \(x=\mu B / k T\). Give a physical explanation for the value of \(P\) as \(T \rightarrow 0\) and as \(T \rightarrow \infty\) Hint: Determine the partition function for the system. b. Show that, for \(N\) independent magnetic dipoles, the mean effective magnetic moment parallel to the magnetic field is $$ m=N \mu \tanh (x) . $$ c. Demonstrate that the mean magnetic moment at high temperatures and/or weak magnetic fields ( \(x \& 1\) ) is proportional to \(1 / T\). This is Curie's law. d. Show that the contribution from paramagnetism to the internal energy of a crystalline solid is \(U=-m B\). Determine this paramagnetic contribution at \(T=\) \(\infty\). Why should this result have been expected? e. Develop an expression for the entropy of this paramagnetic system.

Short answer

a. The probability is derived from the partition function. As T approaches 0, Pa approaches 1, as T approaches infinity, Pa approaches 1/2. b. The mean effective magnetic moment is obtained using tanh(x). c. Curie's law is demonstrated at high temperatures where tanh(x) approximates to x. d. The internal energy from paramagnetism at infinite temperature is zero, expected due to random orientation of dipoles. e. Entropy expression requires differentiation of the partition function with respect to temperature.

Step-by-step solution

Calculate the partition function

The partition function, denoted as Z, for a two-state system is given by summing over all possible states: Z = e^(-E_1/(kT)) + e^(-E_2/(kT)), where E_1 = -B and E_2 = B are the energies of the states parallel and antiparallel to the magnetic field, respectively, and k is the Boltzmann constant.

Express the probability for the magnetic dipole moment to point parallel

The probability Pa for the dipole to align parallel to the magnetic field is the ratio of the exponential of the negative energy of that state to the partition function: Pa = e^(-E_1/(kT)) / Z = 1 / (1 + e^(-2c/kT)). Setting x = B / kT simplifies the probability to Pa = (1 + e^(-2x))^(-1).

Physical explanation for P as T approaches 0 and infinity

As T approaches 0, x approaches infinity since x is inversely proportional to T. Thus, e^(-2x) approaches 0 and Pa approaches 1, meaning all dipoles align parallel to the field. As T approaches infinity, x approaches 0 making e^(-2x) approach 1, and Pa approaches 1/2, resulting in an equal probability of dipoles being parallel or antiparallel.

Mean effective magnetic moment for N dipoles

The mean effective magnetic moment m is the number of dipoles N multiplied by the magnetic moment and the difference in probabilities of alignments, which is tanh(x): m = N(tanh(x)).

Show that mean magnetic moment m proportionality at high temperatures

At high T or weak B, x << 1, the Taylor expansion of tanh(x) is approximated as tanh(x) ≈ x, therefore m ≈ Nx = N(B / kT), which shows that m is proportional to 1/T. This is Curie's law.

Paramagnetism contribution to internal energy

By the definition of potential energy in a magnetic system, the internal energy U is: U = -mB. At T = infinity, the mean magnetic moment m equals zero due to random orientations of dipoiles, hence U = 0. This result is expected because at infinite temperature there is no preferential alignment of magnetic dipoles due to the high thermal agitation.

Expression for the entropy of the paramagnetic system

The entropy S can be found from the partition function Z. The general expression for S is given by S = k(ln Z + T (d ln Z/dT)). In our two-state system, Z = 2 cosh(B/kT), and thus we have to differentiate ln Z with respect to T and then use the result to find the entropy.

Key concepts

Paramagnetism

Paramagnetism is a form of magnetism which occurs only in the presence of an external magnetic field. In paramagnetic materials, atoms or ions contain permanent magnetic moments, often due to unpaired electrons spinning in their atomic orbitals. When an external magnetic field is applied, these magnetic moments tend to align themselves with it, resulting in a net magnetization.

Each magnetic dipole has a certain energy depending on its orientation in the magnetic field. The possibility of each orientation, and hence the overall magnetic behavior, can be described statistically using temperature and the field-strength as parameters. As the temperature increases, the thermal agitation overcomes the aligning effect of the magnetic field, leading to random orientation of the dipoles and decreased net magnetization.

Partition Function

The partition function is a central concept in statistical thermodynamics, denoted by the letter Z. It encompasses the sum of the exponentials of the negative energy states for a system, each divided by the product of the Boltzmann constant (k) and temperature (T). Essentially, Z is a sum over all possible states that a system can occupy, and it serves as a kind of bookkeeping tool to account for all the different ways a system can be arranged.

The partition function is directly related to other thermodynamic quantities such as the free energy, the average energy, and, as in the case of a paramagnetic material, the probability of a magnetic dipole moment being aligned with an external magnetic field.

Curie's Law

Curie's law relates the magnetization of a paramagnetic material to the applied magnetic field and the absolute temperature. According to this law, the magnetization (M) is directly proportional to the magnetic field strength (B) and inversely proportional to the temperature (T). In a formula, it is expressed as M = C/T, where C is the Curie constant.

This law reveals that as the temperature approaches infinity, or in simpler terms gets very hot, the effects of the magnetic field weaken and the magnets' ability to align with the field decreases. Conversely, at lower temperatures, the alignment increases, leading to greater magnetization. The concept reflects the balance between the tendency of magnetic dipoles to align with the magnetic field and the disruptive influence of thermal energy.

Magnetic Dipole Moment

The magnetic dipole moment is a vector quantity that represents the strength and direction of a magnet's power. It is a crucial concept when studying magnetism in materials, especially in the context of paramagnetism. The magnetic dipole moment arises mainly from the orbital and spin motions of electrons within an atom. When subjected to an external magnetic field, these dipoles attempt to align themselves with the field, each having potential energies associated with their orientation parallel or antiparallel to the field.

Curie's law describes how these dipoles lead to a net macroscopic magnetization in the material. The higher the magnetic dipole moments per atom and the stronger the magnetic field, the greater the magnetization effect. However, as temperature rises, thermal motion counteracts this alignment, reducing the magnetic moment's effect.

Entropy

Entropy is a measure of the disorder or randomness in a system and is a fundamental concept in thermodynamics and statistical mechanics. In the context of paramagnetism, entropy can be related to the degree of alignment of the magnetic dipoles. At low temperatures, the dipoles are well-aligned with the external magnetic field, meaning the system has low entropy. As the temperature increases, the dipoles become more randomly oriented, increasing the system's entropy.

The partition function plays a pivotal role in finding the entropy of a system. By analyzing the partition function and its temperature derivative, the entropy of the paramagnetic system can be expressed mathematically. This relationship underscores the microscopic underpinnings of magnetic behavior in terms of the collective states accessible to the system's dipoles, given the thermal energy in their environment.