7.5 Magnetic susceptibility

Magnetic susceptibility is a key concept in understanding how materials respond to magnetic fields. It measures a material's ability to become magnetized, revealing crucial insights into its atomic structure and electron behavior.

Different types of magnetic materials exhibit varying susceptibilities. From diamagnetic to ferromagnetic, these classifications help us predict and utilize material properties in applications ranging from MRI machines to data storage devices.

Types of magnetic materials

• Magnetic materials exhibit different behaviors in response to applied magnetic fields, depending on their atomic structure and electron configuration
• The three main categories of magnetic materials are diamagnetic, paramagnetic, and ferromagnetic, each with distinct magnetic properties
• Understanding the differences between these types of magnetic materials is crucial for selecting appropriate materials for various applications in electromagnetism and magnetic devices

Diamagnetic vs paramagnetic vs ferromagnetic

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• Diamagnetic materials have a weak, negative magnetic susceptibility, meaning they are slightly repelled by magnetic fields (bismuth, copper, water)
• Paramagnetic materials have a small, positive magnetic susceptibility, causing them to be weakly attracted to magnetic fields (aluminum, platinum, oxygen)
• Ferromagnetic materials exhibit a strong, positive magnetic susceptibility and can retain magnetization even after the external field is removed (iron, nickel, cobalt)
• The magnetic behavior of these materials arises from the alignment of magnetic moments of atoms or ions within the material
• Ferromagnetic materials have a high degree of magnetic moment alignment, leading to strong magnetization and potential for permanent magnets

Antiferromagnetic and ferrimagnetic materials

• Antiferromagnetic materials have magnetic moments that align antiparallel to each other, resulting in a net zero magnetization (manganese oxide, chromium)
• Ferrimagnetic materials have opposing magnetic moments with different magnitudes, leading to a net magnetization in one direction (magnetite, ferrites)
• These materials exhibit unique magnetic properties and are used in applications such as data storage, magnetic sensors, and high-frequency devices
• The magnetic ordering in these materials is temperature-dependent, with the Néel temperature being the critical point above which antiferromagnetic order is lost

Magnetic susceptibility

• Magnetic susceptibility is a measure of how strongly a material responds to an applied magnetic field
• It is a dimensionless quantity that relates the magnetization of a material to the applied magnetic field strength
• Understanding magnetic susceptibility is essential for characterizing magnetic materials and predicting their behavior in various applications

Definition of magnetic susceptibility

• Magnetic susceptibility ($\chi$) is defined as the ratio of the magnetization ($M$) induced in a material to the applied magnetic field strength ($H$): $\chi = M/H$
• It is a measure of how easily a material can be magnetized in response to an external magnetic field
• Positive values of $\chi$ indicate paramagnetic or ferromagnetic behavior, while negative values indicate diamagnetic behavior
• The magnitude of $\chi$ depends on the material's composition, crystal structure, and temperature

Magnetic susceptibility tensor

• In anisotropic materials, the magnetic susceptibility is a tensor quantity, represented by a 3x3 matrix
• The tensor describes the directional dependence of the magnetic response, with different values along the principal axes
• The tensor components can be determined experimentally using techniques such as torque magnetometry or single-crystal susceptibility measurements
• Knowledge of the susceptibility tensor is crucial for understanding the magnetic anisotropy of materials and designing devices that exploit this anisotropy

Relation to permeability

• Magnetic susceptibility is related to the magnetic permeability ($\mu$) of a material, which describes how the material affects the magnetic field inside it
• The relative permeability ($\mu_r$) is given by: $\mu_r = 1 + \chi$
• For vacuum, $\chi = 0$ and $\mu_r = 1$, meaning the magnetic field is unaffected
• Materials with high magnetic susceptibility have a higher relative permeability, indicating a stronger response to magnetic fields
• The relationship between susceptibility and permeability is important for understanding the propagation of electromagnetic waves in materials and designing magnetic devices

Curie's law

• Curie's law describes the temperature dependence of magnetic susceptibility for paramagnetic materials
• It states that the susceptibility is inversely proportional to the absolute temperature, assuming no interaction between magnetic moments
• Curie's law is essential for understanding the magnetic behavior of materials at different temperatures and for determining the Curie constant and Curie temperature

Temperature dependence of susceptibility

• According to Curie's law, the magnetic susceptibility ($\chi$) of a paramagnetic material is given by: $\chi = C/T$, where $C$ is the Curie constant and $T$ is the absolute temperature
• As temperature increases, the thermal energy overcomes the alignment of magnetic moments, leading to a decrease in susceptibility
• Deviations from Curie's law can occur due to interactions between magnetic moments or the presence of other magnetic phases
• The temperature dependence of susceptibility is used to study phase transitions, such as the Curie temperature in ferromagnetic materials

Curie constant and Curie temperature

• The Curie constant ($C$) is a material-specific parameter that depends on the effective magnetic moment of the atoms or ions in the material
• It is related to the number of unpaired electrons and the total angular momentum of the magnetic species
• The Curie temperature ($T_C$) is the critical temperature above which a ferromagnetic material becomes paramagnetic
• At $T_C$, the thermal energy overcomes the exchange interaction between magnetic moments, destroying the spontaneous magnetization
• Determining the Curie constant and Curie temperature is crucial for characterizing magnetic materials and understanding their temperature-dependent behavior

Measuring magnetic susceptibility

• Various experimental techniques are used to measure the magnetic susceptibility of materials, each with its own advantages and limitations
• These techniques provide valuable information about the magnetic properties of materials and help in the development of new magnetic devices and applications
• Understanding the principles behind these measurement methods is essential for interpreting the results and selecting the appropriate technique for a given material or application

Faraday balance method

• The Faraday balance method measures the force experienced by a sample in a non-uniform magnetic field
• The sample is suspended from a balance and placed between the poles of an electromagnet with a field gradient
• The force on the sample is proportional to its magnetic susceptibility and the field gradient
• By measuring the force at different field strengths, the susceptibility can be determined
• This method is suitable for measuring the susceptibility of small samples and is sensitive to both paramagnetic and diamagnetic materials

Vibrating sample magnetometer

• A vibrating sample magnetometer (VSM) measures the magnetization of a sample by detecting the voltage induced in a pickup coil
• The sample is vibrated at a fixed frequency in a uniform magnetic field, generating an alternating magnetic flux in the pickup coil
• The induced voltage is proportional to the sample's magnetization and the vibration amplitude
• VSM is a versatile technique that can measure magnetization as a function of applied field, temperature, and time
• It is widely used for characterizing the magnetic properties of materials, including hysteresis loops, saturation magnetization, and magnetic anisotropy

SQUID magnetometer

• A superconducting quantum interference device (SQUID) magnetometer is the most sensitive instrument for measuring magnetic fields and susceptibility
• It consists of a superconducting loop with one or two Josephson junctions, which are sensitive to changes in magnetic flux
• The sample is moved through a pickup coil, inducing a current in the SQUID loop that is proportional to the sample's magnetization
• SQUID magnetometers can detect extremely weak magnetic signals, making them ideal for studying materials with low susceptibility or small sample sizes
• They are used in a wide range of applications, including fundamental research, materials characterization, and biomedical imaging

Applications of magnetic susceptibility

• Magnetic susceptibility measurements have numerous applications across various fields, including materials science, chemistry, physics, and engineering
• These applications rely on the ability to characterize the magnetic properties of materials and exploit their behavior in specific environments
• Understanding the practical applications of magnetic susceptibility is crucial for developing new technologies and improving existing ones

Identifying unknown materials

• Magnetic susceptibility measurements can be used to identify unknown materials based on their characteristic magnetic response
• By comparing the measured susceptibility with known values for different materials, the composition and purity of a sample can be determined
• This technique is particularly useful in fields such as archaeology, where the identification of ancient artifacts and their provenance is of great interest
• Magnetic susceptibility can also be used for quality control in manufacturing processes, ensuring that materials meet the required specifications

Studying phase transitions

• Magnetic susceptibility measurements provide valuable insights into phase transitions in materials, such as the Curie temperature in ferromagnets or the Néel temperature in antiferromagnets
• By monitoring the change in susceptibility as a function of temperature, the critical points and the nature of the phase transition can be determined
• This information is essential for understanding the fundamental properties of materials and their behavior under different conditions
• Magnetic susceptibility studies of phase transitions are crucial in the development of new magnetic materials for applications such as data storage, sensors, and actuators

Magnetic resonance imaging (MRI)

• Magnetic susceptibility differences between tissues are exploited in magnetic resonance imaging (MRI) to generate contrast in medical images
• The local magnetic field experienced by protons in tissues is influenced by the susceptibility of the surrounding materials, affecting their resonance frequency and relaxation times
• By applying magnetic field gradients and radiofrequency pulses, MRI can map the spatial distribution of susceptibility-induced field variations, creating detailed images of the body
• Susceptibility-weighted imaging (SWI) is a specific MRI technique that enhances the contrast between tissues with different susceptibilities, such as blood vessels and iron-rich regions in the brain

Magnetic separation techniques

• Magnetic susceptibility differences can be used to separate materials based on their magnetic properties
• In magnetic separation, a mixture of materials is exposed to a magnetic field gradient, causing the more magnetic components to be attracted towards the field source
• This technique is widely used in the mining industry for separating magnetic ores from non-magnetic gangue minerals
• Magnetic separation is also employed in environmental remediation for removing magnetic contaminants from soil or water
• In biotechnology, magnetic nanoparticles with high susceptibility are used to label and separate specific biomolecules or cells from complex mixtures

Quantum mechanical description

• The magnetic properties of materials, including susceptibility, have their origins in the quantum mechanical behavior of electrons in atoms and molecules
• A quantum mechanical description is necessary to fully understand the magnetic behavior of materials at the atomic and molecular level
• This description takes into account the contributions of electron spin and orbital angular momentum, as well as the interactions between magnetic moments in a material

Atomic and molecular magnetism

• The magnetic properties of atoms and molecules arise from the magnetic moments associated with the spin and orbital angular momentum of electrons
• In an atom, the electronic configuration determines the number of unpaired electrons and their respective spin and orbital angular momenta
• The total magnetic moment of an atom is the vector sum of the spin and orbital magnetic moments of all its electrons
• In molecules, the magnetic properties are influenced by the coupling between the magnetic moments of the constituent atoms and the formation of molecular orbitals

Hund's rules and electron configuration

• Hund's rules are a set of principles that determine the ground state electronic configuration of an atom or ion, taking into account the spin and orbital angular momenta of electrons
• The first rule (maximum multiplicity) states that electrons occupy orbitals to maximize the total spin, minimizing the electron-electron repulsion
• The second rule (maximum orbital angular momentum) states that electrons occupy orbitals to maximize the total orbital angular momentum, consistent with the first rule
• The third rule (spin-orbit coupling) states that the total angular momentum is determined by the coupling between the spin and orbital angular momenta, depending on the shell being less than or more than half-filled
• The electronic configuration dictated by Hund's rules determines the magnetic properties of an atom, including its effective magnetic moment and susceptibility

Spin and orbital angular momentum contributions

• The magnetic susceptibility of a material is determined by the contributions of both spin and orbital angular momenta of its electrons
• The spin angular momentum arises from the intrinsic spin of electrons, which can be either up (ms = +1/2) or down (ms = -1/2)
• The orbital angular momentum originates from the motion of electrons around the nucleus, characterized by the quantum number l
• The spin and orbital magnetic moments are proportional to their respective angular momenta, with the gyromagnetic ratio determining the proportionality constant
• The relative contributions of spin and orbital angular momenta to the total magnetic moment and susceptibility depend on the electronic configuration and the strength of the spin-orbit coupling in the material

Magnetic anisotropy

• Magnetic anisotropy refers to the directional dependence of a material's magnetic properties, including susceptibility, magnetization, and coercivity
• It arises from the crystal structure, shape, or interfacial effects in the material, leading to preferred directions for the alignment of magnetic moments
• Understanding magnetic anisotropy is crucial for designing materials with desired magnetic properties and for optimizing their performance in various applications

Magnetocrystalline anisotropy

• Magnetocrystalline anisotropy originates from the coupling between the spin and orbital angular momenta of electrons and the crystal lattice
• The crystal structure creates a non-spherical electric field that interacts with the orbital angular momentum, leading to preferred orientations for the magnetic moments
• The magnetocrystalline anisotropy energy is the energy required to rotate the magnetization away from the easy axis (the preferred direction) towards the hard axis
• The strength and direction of the magnetocrystalline anisotropy depend on the crystal symmetry and the electronic configuration of the material
• Materials with high magnetocrystalline anisotropy, such as rare-earth permanent magnets, are used in applications requiring strong and stable magnetic fields

Shape anisotropy

• Shape anisotropy arises from the geometry of a magnetic material, particularly in non-spherical particles or thin films
• The demagnetizing field, which opposes the magnetization, is stronger along the shorter dimensions of the material, creating a preferred direction for the magnetization along the longer dimensions
• The shape anisotropy energy depends on the difference between the demagnetizing factors along the principal axes of the material
• In elongated particles (such as nanorods) or thin films, the shape anisotropy can dominate over the magnetocrystalline anisotropy, leading to a preferred magnetization direction in the plane of the film or along the long axis of the particle
• Shape anisotropy is exploited in applications such as magnetic recording media, where the magnetic bits are stored in elongated particles or patterned thin films

Anisotropy in nanostructures and thin films

• Magnetic anisotropy becomes particularly important in nanostructures and thin films, where the surface and interface effects can significantly influence the magnetic properties
• In magnetic nanoparticles, the surface anisotropy arises from the broken symmetry and reduced coordination of atoms at the surface, leading to a different magnetic environment compared to the bulk
• The surface anisotropy can compete with or even dominate over the magnetocrystalline and shape anisotropies, depending on the size and shape of the nanoparticles
• In magnetic thin films, the interfacial anisotropy originates from the interaction between the magnetic layer and the substrate or adjacent layers, which can induce a preferred magnetization direction perpendicular or parallel to the film plane
• The control of magnetic anisotropy in nanostructures and thin films is crucial for developing novel magnetic devices, such as spin valves, magnetic tunnel junctions, and magnetoresistive random-access memory (MRAM)