8.1 Diamagnetism and paramagnetism

Magnetic susceptibility is a key concept in solid state physics, measuring how materials respond to magnetic fields. It helps classify materials as diamagnetic or paramagnetic, with diamagnetic materials being repelled and paramagnetic materials attracted to magnetic fields.

Diamagnetism and paramagnetism are two fundamental types of magnetism in materials. Diamagnetism opposes applied fields and occurs in all materials, while paramagnetism aligns with fields and requires unpaired electrons. Understanding these properties is crucial for many technological applications.

Magnetic susceptibility

• Magnetic susceptibility is a measure of how strongly a material responds to an applied magnetic field
• It is a fundamental property in the study of magnetic materials and their behavior in solid state physics
• The sign and magnitude of magnetic susceptibility can be used to classify materials as diamagnetic, paramagnetic, or ferromagnetic

Diamagnetic vs paramagnetic materials

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• Diamagnetic materials have a negative magnetic susceptibility, meaning they are slightly repelled by an applied magnetic field (bismuth, copper, water)
• Paramagnetic materials have a positive magnetic susceptibility, meaning they are slightly attracted to an applied magnetic field (aluminum, platinum, oxygen)
• The magnitude of the susceptibility is typically much larger for paramagnetic materials compared to diamagnetic materials

Magnetic susceptibility tensor

• In anisotropic materials, the magnetic susceptibility is a tensor quantity, meaning it depends on the direction of the applied magnetic field
• The susceptibility tensor is a 3x3 matrix that relates the induced magnetization to the applied magnetic field
• The principal axes of the susceptibility tensor correspond to the directions of maximum and minimum susceptibility

Measuring magnetic susceptibility

• Magnetic susceptibility can be measured using various techniques, such as the Faraday balance, vibrating sample magnetometer (VSM), and superconducting quantum interference device (SQUID)
• These techniques involve applying a known magnetic field to the sample and measuring the resulting magnetization or force on the sample
• Temperature-dependent susceptibility measurements can provide insights into the magnetic properties and phase transitions of materials

Diamagnetism

• Diamagnetism is a weak form of magnetism that opposes an applied magnetic field
• It arises from the induced magnetic moments of the electrons in an atom or molecule
• Diamagnetic materials have no unpaired electrons and do not possess a permanent magnetic moment

Origin of diamagnetism

• Diamagnetism originates from the change in the orbital motion of electrons in an atom or molecule when an external magnetic field is applied
• According to Lenz's law, the induced current in the electron orbits creates a magnetic field that opposes the applied field
• This results in a negative magnetic susceptibility and a repulsive force on the material

Diamagnetic susceptibility

• The diamagnetic susceptibility is negative and typically on the order of -10^-5 to -10^-6 (SI units)
• It is temperature-independent and does not vary with the strength of the applied magnetic field
• The diamagnetic susceptibility can be calculated using Langevin's theory of diamagnetism, which considers the induced magnetic moments of the electrons

Diamagnetism in metals

• In metals, the conduction electrons contribute to the diamagnetic susceptibility
• The Landau diamagnetism arises from the quantization of the electron orbits in a magnetic field (Landau levels)
• The Landau susceptibility is proportional to the density of states at the Fermi level and depends on the Fermi surface topology

Diamagnetism in insulators

• In insulators, the diamagnetic susceptibility is determined by the bound electrons in the atoms or molecules
• The susceptibility can be calculated using the Langevin-Larmor formula, which considers the induced magnetic moments of the electrons
• The diamagnetic susceptibility of insulators is typically smaller than that of metals due to the absence of conduction electrons

Diamagnetic levitation

• Diamagnetic materials can be levitated in strong magnetic fields due to their negative susceptibility
• Stable levitation occurs when the diamagnetic force balances the gravitational force on the object
• Diamagnetic levitation has applications in materials processing, such as containerless processing of high-purity materials (graphite, bismuth, water droplets)

Paramagnetism

• Paramagnetism is a form of magnetism where materials are attracted to an applied magnetic field
• Paramagnetic materials have unpaired electrons in their atomic or molecular orbitals, which give rise to a net magnetic moment
• The magnetic moments in paramagnetic materials are randomly oriented in the absence of an external field and align partially with the field when one is applied

Origin of paramagnetism

• Paramagnetism arises from the presence of unpaired electrons in the atomic or molecular orbitals of a material
• Each unpaired electron has a magnetic moment associated with its spin angular momentum
• In the absence of an external magnetic field, these magnetic moments are randomly oriented due to thermal agitation

Paramagnetic susceptibility

• The paramagnetic susceptibility is positive and typically on the order of 10^-3 to 10^-5 (SI units)
• It is temperature-dependent, with the susceptibility decreasing as temperature increases
• The paramagnetic susceptibility can be described by the Curie law or the Curie-Weiss law, depending on the presence of magnetic interactions between the moments

Curie's law

• Curie's law states that the paramagnetic susceptibility is inversely proportional to the absolute temperature: $\chi = C/T$
• The Curie constant, $C$, depends on the effective magnetic moment of the paramagnetic ions and the density of the material
• Curie's law assumes that the magnetic moments are non-interacting and that the thermal energy is much larger than the magnetic interaction energy

Curie-Weiss law

• The Curie-Weiss law is a modification of Curie's law that accounts for the presence of magnetic interactions between the moments
• It states that the paramagnetic susceptibility follows the relation: $\chi = C/(T - \theta)$, where $\theta$ is the Weiss constant
• A positive Weiss constant indicates ferromagnetic interactions, while a negative Weiss constant indicates antiferromagnetic interactions

Paramagnetic materials

• Examples of paramagnetic materials include transition metal ions (Fe^3+, Mn^2+), rare earth ions (Gd^3+, Er^3+), and organic radicals (DPPH, TEMPO)
• Paramagnetic salts, such as Mohr's salt (ammonium iron(II) sulfate), are commonly used in magnetic susceptibility measurements
• Liquid oxygen is paramagnetic due to the presence of unpaired electrons in the oxygen molecules

Langevin theory of paramagnetism

• The Langevin theory of paramagnetism is a classical approach to describing the behavior of paramagnetic materials in an applied magnetic field
• It assumes that the magnetic moments are non-interacting and that their orientation is determined by the competition between the magnetic energy and the thermal energy
• The theory provides a good approximation for the magnetization and susceptibility of paramagnetic materials at high temperatures

Classical approach

• In the classical approach, the magnetic moments are treated as classical vectors that can orient in any direction
• The orientation of the moments is determined by the balance between the magnetic energy, which tends to align the moments with the field, and the thermal energy, which tends to randomize their orientation
• The distribution of the magnetic moments is described by the Boltzmann distribution, which gives the probability of a moment having a particular orientation

Langevin function

• The Langevin function, $L(\alpha) = \coth(\alpha) - 1/\alpha$, describes the average magnetization of a paramagnetic material as a function of the applied magnetic field
• The parameter $\alpha$ is the ratio of the magnetic energy to the thermal energy: $\alpha = \mu B / k_B T$, where $\mu$ is the magnetic moment, $B$ is the applied field, $k_B$ is the Boltzmann constant, and $T$ is the absolute temperature
• The Langevin function saturates to unity for large values of $\alpha$, indicating a complete alignment of the magnetic moments with the field

Limitations of Langevin theory

• The Langevin theory assumes that the magnetic moments are non-interacting, which is not always the case in real materials
• It does not account for the quantum mechanical nature of the magnetic moments, which becomes important at low temperatures or for materials with strong spin-orbit coupling
• The theory fails to describe the behavior of materials with localized magnetic moments, such as rare earth ions, where the crystal field effects play a significant role

Quantum theory of paramagnetism

• The quantum theory of paramagnetism takes into account the quantum mechanical nature of the magnetic moments and their interactions with the crystal environment
• It provides a more accurate description of the magnetic properties of paramagnetic materials, especially at low temperatures or for materials with strong spin-orbit coupling
• The theory considers the contributions of both the spin and orbital angular momenta to the magnetic moment

Spin and orbital angular momentum

• In atoms and ions, the magnetic moment arises from the spin and orbital angular momenta of the electrons
• The spin angular momentum, $S$, is an intrinsic property of the electron and has a value of 1/2
• The orbital angular momentum, $L$, is associated with the motion of the electron around the nucleus and can take integer values
• The total angular momentum, $J$, is the vector sum of the spin and orbital angular momenta: $J = L + S$

Hund's rules

• Hund's rules are a set of empirical rules that determine the ground state electronic configuration of an atom or ion
• The first rule states that the electrons occupy orbitals to maximize the total spin, $S$
• The second rule states that the electrons occupy orbitals to maximize the total orbital angular momentum, $L$, consistent with the first rule
• The third rule states that the total angular momentum, $J$, is equal to $|L - S|$ for a less than half-filled shell and $L + S$ for a more than half-filled shell

Effective magnetic moment

• The effective magnetic moment, $\mu_\text{eff}$, is a measure of the strength of the paramagnetic response of an atom or ion
• It is related to the total angular momentum, $J$, by the equation: $\mu_\text{eff} = g_J \sqrt{J(J+1)} \mu_B$, where $g_J$ is the Landé g-factor and $\mu_B$ is the Bohr magneton
• The effective magnetic moment can be determined experimentally from the Curie constant, $C$, obtained from susceptibility measurements

Van Vleck paramagnetism

• Van Vleck paramagnetism is a form of temperature-independent paramagnetism that arises from the mixing of the ground state with excited states by the applied magnetic field
• It occurs in materials with a non-magnetic ground state and low-lying excited states that can be mixed by the field
• The Van Vleck susceptibility is given by the Van Vleck formula, which considers the matrix elements of the magnetic moment operator between the ground state and the excited states

Applications

• The magnetic properties of diamagnetic and paramagnetic materials have numerous applications in various fields, ranging from medicine to materials science
• These applications exploit the unique response of these materials to applied magnetic fields and their temperature-dependent behavior
• Understanding the fundamental principles of diamagnetism and paramagnetism is crucial for developing new technologies and optimizing existing ones

Magnetic resonance imaging (MRI)

• MRI is a non-invasive medical imaging technique that relies on the magnetic properties of hydrogen atoms in the body
• It uses strong magnetic fields and radio waves to manipulate the spin of the hydrogen nuclei and generate detailed images of the body's internal structures
• Paramagnetic contrast agents, such as gadolinium complexes, are often used to enhance the contrast between different tissues in MRI scans

Magnetic levitation

• Magnetic levitation exploits the repulsive force experienced by diamagnetic materials in strong magnetic fields
• It has applications in high-speed transportation, such as maglev trains, where the train is levitated above the track by powerful superconducting magnets
• Diamagnetic levitation is also used in materials processing, such as containerless processing of high-purity materials, to avoid contamination from the container walls

Magnetic refrigeration

• Magnetic refrigeration is an energy-efficient cooling technology that utilizes the magnetocaloric effect in paramagnetic materials
• It involves the cyclic magnetization and demagnetization of a paramagnetic material, which causes a temperature change due to the coupling between the magnetic and thermal properties
• Magnetic refrigeration has the potential to replace conventional vapor-compression refrigeration systems, offering higher efficiency and reduced environmental impact

Magnetic separation

• Magnetic separation is a process that uses magnetic fields to separate magnetic materials from non-magnetic materials
• It has applications in mineral processing, where paramagnetic minerals (ilmenite, wolframite) are separated from diamagnetic gangue minerals (quartz, calcite)
• Magnetic separation is also used in biotechnology for the isolation and purification of biomolecules, such as proteins and nucleic acids, using paramagnetic beads coated with specific ligands