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A **spin wave** is a propagating disturbance in the ordering of a magnetic material. These low-lying collective excitations occur in magnetic lattices with continuous symmetry. From the equivalent quasiparticle point of view, spin waves are known as magnons, which are bosonic modes of the spin lattice that correspond roughly to the phonon excitations of the nuclear lattice. As temperature is increased, the thermal excitation of spin waves reduces a ferromagnet's spontaneous magnetization. The energies of spin waves are typically only μeV in keeping with typical Curie points at room temperature and below.

The simplest way of understanding spin waves is to consider the Hamiltonian for the Heisenberg ferromagnet:

where J is the exchange energy, the operators S represent the spins at Bravais lattice points, g is the Landé g-factor, *μ*_{B} is the Bohr magneton and **H** is the internal field which includes the external field plus any "molecular" field. Note that in the classical continuum case and in 1 + 1 dimensions Heisenberg ferromagnet equation has the form

In 1 + 1, 2 + 1 and 3 + 1 dimensions this equation admits several integrable and non-integrable extensions like the Landau-Lifshitz equation, the Ishimori equation and so on. For a ferromagnet *J* > 0 and the ground state of the Hamiltonian is that in which all spins are aligned parallel with the field **H**. That is an eigenstate of can be verified by rewriting it in terms of the spin-raising and spin-lowering operators given by:

resulting in

where z has been taken as the direction of the magnetic field. The spin-lowering operator *S*^{−} annihilates the state with minimum projection of spin along the z-axis, while the spin-raising operator *S*^{+} annihilates the ground state with maximum spin projection along the z-axis. Since

for the maximally aligned state, we find

where N is the total number of Bravais lattice sites. The proposition that the ground state is an eigenstate of the Hamiltonian is confirmed.

One might guess that the first excited state of the Hamiltonian has one randomly selected spin at position i rotated so that

but in fact this arrangement of spins is not an eigenstate. The reason is that such a state is transformed by the spin raising and lowering operators. The operator will increase the z-projection of the spin at position i back to its low-energy orientation, but the operator will lower the z-projection of the spin at position j. The combined effect of the two operators is therefore to propagate the rotated spin to a new position, which is a hint that the correct eigenstate is a **spin wave**, namely a superposition of states with one reduced spin. The exchange energy penalty associated with changing the orientation of one spin is reduced by spreading the disturbance over a long wavelength. The degree of misorientation of any two near-neighbor spins is thereby minimized. From this explanation one can see why the Ising model magnet with discrete symmetry has no spin waves: the notion of spreading a disturbance in the spin lattice over a long wavelength makes no sense when spins have only two possible orientations. The existence of low-energy excitations is related to the fact that in the absence of an external field, the spin system has an infinite number of degenerate ground states with infinitesimally different spin orientations. The existence of these ground states can be seen from the fact that the state does not have the full rotational symmetry of the Hamiltonian , a phenomenon which is called spontaneous symmetry breaking.

In this model the magnetization

where V is the volume. The propagation of spin waves is described by the Landau-Lifshitz equation of motion:

where γ is the gyromagnetic ratio and λ is the damping constant. The cross-products in this forbidding-looking equation show that the propagation of spin waves is governed by the torques generated by internal and external fields. (An equivalent form is the Landau-Lifshitz-Gilbert equation, which replaces the final term by a more "simply looking" equivalent one.)

The first term on the right hand side of the equation describes the precession of the magnetization under the influence of the applied field, while the above-mentioned final term describes how the magnetization vector "spirals in" towards the field direction as time progresses. In metals the damping forces described by the constant λ are in many cases dominated by the eddy currents.

One important difference between phonons and magnons lies in their dispersion relations. The dispersion relation for phonons is to first order linear in wavevector k, namely *ώ* = *ck*, where ω is frequency, and c is the velocity of sound. Magnons have a parabolic dispersion relation: *ώ* = *Ak*^{2} where the parameter A represents a "spin stiffness." The *k*^{2} form is the third term of a Taylor expansion of a cosine term in the energy expression originating from the **S**_{i} ⋅ **S**_{j} dot-product. The underlying reason for the difference in dispersion relation is that the order parameter (magnetization) for the ground-state in ferromagnets violates time-reversal symmetry. Two adjacent spins in a solid with lattice constant a that participate in a mode with wavevector k have an angle between them equal to ka.

Spin waves are observed through four experimental methods: inelastic neutron scattering, inelastic light scattering (Brillouin scattering, Raman scattering and inelastic X-ray scattering), inelastic electron scattering (spin-resolved electron energy loss spectroscopy), and spin-wave resonance (ferromagnetic resonance). In the first method the energy loss of a beam of neutrons that excite a magnon is measured, typically as a function of scattering vector (or equivalently momentum transfer), temperature and external magnetic field. Inelastic neutron scattering measurements can determine the dispersion curve for magnons just as they can for phonons. Important inelastic neutron scattering facilities are present at the ISIS neutron source in Oxfordshire, UK, the Institut Laue-Langevin in Grenoble, France, the High Flux Isotope Reactor at Oak Ridge National Laboratory in Tennessee, USA, and at the National Institute of Standards and Technology in Maryland, USA. Brillouin scattering similarly measures the energy loss of photons (usually at a convenient visible wavelength) reflected from or transmitted through a magnetic material. Brillouin spectroscopy is similar to the more widely known Raman scattering, but probes a lower energy and has a superior energy resolution in order to be able to detect the meV energy of magnons. Ferromagnetic (or antiferromagnetic) resonance instead measures the absorption of microwaves, incident on a magnetic material, by spin waves, typically as a function of angle, temperature and applied field. Ferromagnetic resonance is a convenient laboratory method for determining the effect of magnetocrystalline anisotropy on the dispersion of spin waves. One group at the Max Planck Institute of Microstructure Physics in Halle, Germany proved that by using spin polarized electron energy loss spectroscopy (SPEELS), very high energy surface magnons can be excited. This technique allows one to probe the dispersion of magnons in the ultrathin ferromagnetic films. The first experiment was performed for a 5 ML Fe film.^{ [1] } With momentum resolution, the magnon dispersion was explored for an 8 ML fcc Co film on Cu(001) and an 8 ML hcp Co on W(110), respectively.^{ [2] } The maximum magnon energy at the border of the surface Brillouin zone was 240 meV.

When magnetoelectronic devices are operated at high frequencies, the generation of spin waves can be an important energy loss mechanism. Spin wave generation limits the linewidths and therefore the quality factors *Q* of ferrite components used in microwave devices. The reciprocal of the lowest frequency of the characteristic spin waves of a magnetic material gives a time scale for the switching of a device based on that material.

In quantum mechanics, the **Hamiltonian** of a system is an operator corresponding to the total energy of that system, including both kinetic energy and potential energy. Its spectrum, the system's *energy spectrum* or its set of *energy eigenvalues*, is the set of possible outcomes obtainable from a measurement of the system's total energy. Due to its close relation to the energy spectrum and time-evolution of a system, it is of fundamental importance in most formulations of quantum theory.

The **Schrödinger equation** is a linear partial differential equation that governs the wave function of a quantum-mechanical system. It is a key result in quantum mechanics, and its discovery was a significant landmark in the development of the subject. The equation is named after Erwin Schrödinger, who postulated the equation in 1925, and published it in 1926, forming the basis for the work that resulted in his Nobel Prize in Physics in 1933.

In physics, a **phonon** is a collective excitation in a periodic, elastic arrangement of atoms or molecules in condensed matter, specifically in solids and some liquids. Often referred to as a quasiparticle, it is an excited state in the quantum mechanical quantization of the modes of vibrations for elastic structures of interacting particles. Phonons can be thought of as quantized sound waves, similar to photons as quantized light waves.

In physics and probability theory, **mean-field theory** studies the behavior of high-dimensional random (stochastic) models by studying a simpler model that approximates the original by averaging over degrees of freedom. Degrees of freedom (statistics) Such models consider many individual components that interact with each other. In MFT, the effect of all the other individuals on any given individual is approximated by a single averaged effect, thus reducing a many-body problem to a one-body problem.

The **classical XY model** is a lattice model of statistical mechanics. In general, the XY model can be seen as a specialization of Stanley's *n*-vector model for *n* = 2.

In condensed matter physics, the **Fermi surface** is the surface in reciprocal space which separates occupied from unoccupied electron states at zero temperature. The shape of the Fermi surface is derived from the periodicity and symmetry of the crystalline lattice and from the occupation of electronic energy bands. The existence of a Fermi surface is a direct consequence of the Pauli exclusion principle, which allows a maximum of one electron per quantum state.

A **magnon** is a quasiparticle, a collective excitation of the electrons' spin structure in a crystal lattice. In the equivalent wave picture of quantum mechanics, a magnon can be viewed as a quantized spin wave. Magnons carry a fixed amount of energy and lattice momentum, and are spin-1, indicating they obey boson behavior.

In quantum mechanics, a **parity transformation** is the flip in the sign of *one* spatial coordinate. In three dimensions, it can also refer to the simultaneous flip in the sign of all three spatial coordinates :

In quantum mechanics, a **two-state system** is a quantum system that can exist in any quantum superposition of two independent quantum states. The Hilbert space describing such a system is two-dimensional. Therefore, a complete basis spanning the space will consist of two independent states. Any two-state system can also be seen as a qubit.

In quantum physics, the **spin–orbit interaction** is a relativistic interaction of a particle's spin with its motion inside a potential. A key example of this phenomenon is the spin–orbit interaction leading to shifts in an electron's atomic energy levels, due to electromagnetic interaction between the electron's magnetic dipole, its orbital motion, and the electrostatic field of the positively charged nucleus. This phenomenon is detectable as a splitting of spectral lines, which can be thought of as a Zeeman effect product of two relativistic effects: the apparent magnetic field seen from the electron perspective and the magnetic moment of the electron associated with its intrinsic spin. A similar effect, due to the relationship between angular momentum and the strong nuclear force, occurs for protons and neutrons moving inside the nucleus, leading to a shift in their energy levels in the nucleus shell model. In the field of spintronics, spin–orbit effects for electrons in semiconductors and other materials are explored for technological applications. The spin–orbit interaction is one cause of magnetocrystalline anisotropy and the spin Hall effect.

In chemistry and physics, the **exchange interaction** is a quantum mechanical effect that only occurs between identical particles. Despite sometimes being called an **exchange force** in an analogy to classical force, it is not a true force as it lacks a force carrier.

In quantum mechanics, **Landau quantization** refers to the quantization of the cyclotron orbits of charged particles in a uniform magnetic field. As a result, the charged particles can only occupy orbits with discrete, equidistant energy values, called Landau levels. These levels are degenerate, with the number of electrons per level directly proportional to the strength of the applied magnetic field. It is named after the Soviet physicist Lev Landau.

In solid-state physics, the **k·p perturbation theory** is an approximated semi-empirical approach for calculating the band structure and optical properties of crystalline solids. It is pronounced "k dot p", and is also called the "**k·p** method". This theory has been applied specifically in the framework of the Luttinger–Kohn model, and of the Kane model.

The **Bohr–Van Leeuwen theorem** states that when statistical mechanics and classical mechanics are applied consistently, the thermal average of the magnetization is always zero. This makes magnetism in solids solely a quantum mechanical effect and means that classical physics cannot account for paramagnetism, diamagnetism and ferromagnetism. Inability of classical physics to explain triboelectricity also stems from the Bohr–Van Leeuwen theorem.

The **empty lattice approximation** is a theoretical electronic band structure model in which the potential is *periodic* and *weak*. One may also consider an empty irregular lattice, in which the potential is not even periodic. The empty lattice approximation describes a number of properties of energy dispersion relations of non-interacting free electrons that move through a crystal lattice. The energy of the electrons in the "empty lattice" is the same as the energy of free electrons. The model is useful because it clearly illustrates a number of the sometimes very complex features of energy dispersion relations in solids which are fundamental to all electronic band structures.

**Magnonics** is an emerging field of modern magnetism, which can be considered a sub-field of modern solid state physics. Magnonics combines the study of waves and magnetism. Its main aim is to investigate the behaviour of spin waves in nano-structure elements. In essence, spin waves are a propagating re-ordering of the magnetisation in a material and arise from the precession of magnetic moments. Magnetic moments arise from the orbital and spin moments of the electron, most often it is this spin moment that contributes to the net magnetic moment.

This is a glossary for the terminology often encountered in undergraduate quantum mechanics courses.

In quantum mechanics, **orbital magnetization**, **M**_{orb}, refers to the magnetization induced by orbital motion of charged particles, usually electrons in solids. The term "orbital" distinguishes it from the contribution of spin degrees of freedom, **M**_{spin}, to the total magnetization. A nonzero orbital magnetization requires broken time-reversal symmetry, which can occur spontaneously in ferromagnetic and ferrimagnetic materials, or can be induced in a non-magnetic material by an applied magnetic field.

**Magnetic resonance** is a quantum mechanical resonant effect that can appear when a magnetic dipole is exposed to a static magnetic field and perturbed with another, oscillating electromagnetic field. Due to the static field, the dipole can assume a number of discrete energy eigenstates, depending on the value of its angular momentum quantum number. The oscillating field can then make the dipole transit between its energy states with a certain probability and at a certain rate. The overall transition probability will depend on the field's frequency and the rate will depend on its amplitude. When the frequency of that field leads to the maximum possible transition probability between two states, a magnetic resonance has been achieved. In that case, the energy of the photons composing the oscillating field matches the energy difference between said states. If the dipole is tickled with a field oscillating far from resonance, it is unlikely to transition. That is analogous to other resonant effects, such as with the forced harmonic oscillator. The periodic transition between the different states is called Rabi cycle and the rate at which that happens is called Rabi frequency. The Rabi frequency should not be confused with the field's own frequency. Since many atomic nuclei species can behave as a magnetic dipole, this resonance technique is the basis of nuclear magnetic resonance, including nuclear magnetic resonance imaging and nuclear magnetic resonance spectroscopy.

In condensed matter and atomic physics, **Van Vleck paramagnetism** refers to a positive and temperature-independent contribution to the magnetic susceptibility of a material, derived from second order corrections to the Zeeman interaction. The quantum mechanical theory was developed by John Hasbrouck Van Vleck between the 1920s and the 1930s to explain the magnetic response of gaseous nitric oxide and of rare-earth salts. Alongside other magnetic effects like Paul Langevin's formulas for paramagnetism and diamagnetism, Van Vleck discovered an additional paramagnetic contribution of the same order as Langevin's diamagnetism. Van Vleck contribution is usually important for systems with one electron short of being half filled and this contribution vanishes for elements with closed shells.

- ↑ Plihal, M.; Mills, D. L.; Kirschner, J. (1999). "Spin wave signature in the spin polarized electron energy loss spectrum in ultrathin Fe film: theory and experiment".
*Phys. Rev. Lett*.**82**(12): 2579–2582. Bibcode:1999PhRvL..82.2579P. doi:10.1103/PhysRevLett.82.2579. - ↑ Vollmer, R.; Etzkorn, M.; Kumar, P. S. Anil; Ibach, H.; Kirschner, J. (29 September 2003). "Spin-Polarized Electron Energy Loss Spectroscopy of High Energy, Large Wave Vector Spin Waves in Ultrathin fcc Co Films on Cu(001)" (PDF).
*Physical Review Letters*.**91**(14): 147201. Bibcode:2003PhRvL..91n7201V. doi:10.1103/PhysRevLett.91.147201. PMID 14611549.

- Anderson, Philip W. (1997).
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*Basic notions of condensed matter physics*. Cambridge, Mass.: Perseus Publishing. ISBN 0-201-32830-5. - Ashcroft, Neil W.; Mermin, N. David (1977).
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- Spin Waves Biennial International Symposium for discussion of the latest advances in fundamental studies of dynamic properties of various magnetically ordered materials.
- List of labs performing Brillouin scattering measurements.

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