In the case of a low concentration of impurities, their spin relaxation rate is defined mainly by interactions with conduction electrons and phonons. The exchange coupling of impurities with conduction electrons in the case of the axial symmetry can be presented by the following Hamiltonian:

where

${J}_{\perp},{J}_{\parallel}$ are the exchange integrals,

${S}_{i}^{x,y,z}$ are the spin components of the impurity, and

${\sigma}^{x,y,z}({r}_{i})$ are the spin density components of the conduction electrons at the

${r}_{i}$ position. The spin–lattice relaxation rate of conduction electrons

${\mathsf{\Gamma}}_{\sigma L}$ is defined usually by their scattering on different defects of the lattice and on phonons due to their spin–orbital and spin–phonon interactions. If this relaxation is effective enough, the spin temperature of conduction electrons remains in the equilibrium state with other degrees of freedom. The EPR line width is defined in this case mainly by the relaxation rate

${\mathsf{\Gamma}}_{SS}$ (the so-called Korringa relaxation rate) due to the exchange interaction (Equation (1)). The spin–lattice relaxation rate of impurities due to other interactions

${\mathsf{\Gamma}}_{SL}$ in metals can be usually neglected. In the simplest case where

${J}_{\perp}={J}_{\parallel}=J$, the Korringa relaxation rate is given by the following equation:

where

${\rho}_{F}$ is the density of electronic orbital states at the Fermi energy level,

${k}_{B}$ is the Boltzmann constant, and

$T$ is the temperature. The vice versa spin relaxation rate of conduction electrons to the equilibrium state of impurities (the Overhauser relaxation rate)

${\mathsf{\Gamma}}_{\sigma \sigma}$ and

${\mathsf{\Gamma}}_{SS}$ satisfy the following detailed balance equation:

where

${\chi}_{S}^{0},{\chi}_{\sigma}^{0}$ are the spin susceptibilities of non-interacting impurities and the conduction electrons,

${g}_{S},{g}_{\sigma}$ are the corresponding

g-factors, and

N and

S are the concentration and spin of impurities, respectively. However, a very different situation appears if the spin systems of impurities and conduction electrons are strongly coupled and their Zeeman frequencies

$\hslash {\omega}_{S}={g}_{S}{\mu}_{B}{H}_{0},\hspace{0.17em}\hslash {\omega}_{\sigma}={g}_{\sigma}{\mu}_{B}{H}_{0}$ are very close (

${H}_{0}$ is the external magnetic field). This case can be represented by the following relations:

In this situation, both spin systems cannot be considered in the equilibrium states. The motion equations of magnetic impurities and conduction electrons are coupled by the additional coefficients

${\mathsf{\Gamma}}_{S\sigma}$ and

${\mathsf{\Gamma}}_{\sigma S}$, which coincide in the isotropic case with

${\mathsf{\Gamma}}_{\sigma \sigma}$ and

${\mathsf{\Gamma}}_{SS}$ correspondingly. The spin relaxation of the whole system is realized then by the two steps as follows: first, the spin systems of magnetic impurities and conduction electrons achieve the common spin temperature and, second, they both relax to the equilibrium state of the lattice (the electron bottleneck regime). The spin dynamics of both spin systems in the bottleneck regime is also sufficiently changed since the collective spin excitations of impurities and conduction electrons appear. We are interested in the EPR signal of the mode, corresponding to the spin oscillations of impurities and conduction electrons in the same phase. In the simplest case

${\omega}_{S}={\omega}_{\sigma}$ (

${g}_{S}={g}_{\sigma}$) under the relaxation-dominated bottleneck regime, the spin relaxation rate of this mode

${\mathsf{\Gamma}}_{\mathrm{coll}}$ can be presented as follows (neglecting

${\mathsf{\Gamma}}_{SL}$):

where

B is the bottleneck factor—the less this factor, the stronger the bottleneck regime; in the case

$B=1$, the bottleneck is absent. If the impurities’ concentration is large enough (

${\chi}_{S}^{0}\gg {\chi}_{\sigma}^{0}$), then

${\mathsf{\Gamma}}_{\mathrm{coll}}$ is strongly reduced and proportional to temperature:

Since the spin–lattice relaxation of conduction electrons

${\mathsf{\Gamma}}_{\sigma L}$ does not depend on temperature, the behavior according to (Equation (6)) imitates the Korringa relaxation rate (Equation (2)), despite the bottleneck situation. Nevertheless, it should be mentioned that in the strongly anisotropic case (

${J}_{\perp}\ne {J}_{\parallel}$) it was found that the EPR line width narrowing is rather weak despite the bottleneck regime [

4]. In the case of a parallel orientation of the external magnetic field to the symmetry axis for the spin relaxation rate of the collective mode (instead of Equation (6)), roughly the following result was obtained:

If distances between impurities become relatively short, one should also take into account their magnetic dipole–dipole interactions and the Ruderman–Kittel–Kasuya–Yosida (RKKY) spin–spin interactions via conduction electrons. The latter indirect interaction between two impurities with spins

**S**_{1} and

**S**_{2} at the distance

R between them is as follows:

where

${k}_{F}$ is the electron wave vector at the Fermi surface,

$Z$ is the number of conduction electrons per lattice atom, and

${E}_{F}$ is the Fermi energy. Although this interaction is rather long range, it should be limited by the free path distance

${l}_{p}$ of the conduction electrons:

$f(x)\to f(x)\mathrm{exp}(-R/{l}_{p})$. The dipole–dipole interactions give an additional broadening of the EPR signal from impurities, and the RKKY interaction leads to its narrowing of the Anderson–Weiss type. This contribution to the EPR line width is independent of temperature being responsible for the residual line width (at

$T=0$).