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Spin wave
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Spin wave
In condensed matter physics, 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, μB 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 the 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.
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Spin wave
In condensed matter physics, 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, μB 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 the 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.