The
Debye frequency (Symbol: \omega_{\rm Debye} or \omega_{\rm D} ) is a parameter in the Debye model that refers to a cut-off
angular frequency for
waves of a harmonic chain of masses, used to describe the movement of
ions in a
crystal lattice and more specifically, to correctly predict that the
heat capacity in such crystals is constant at high temperatures (Dulong–Petit law). The concept was first introduced by Peter Debye in 1912. Throughout this section,
periodic boundary conditions are assumed.
Definition Assuming the
dispersion relation is : \omega = v_{\rm s} |\mathbf k| , with v_{\rm s} the
speed of sound in the crystal and
k the wave vector, the value of the Debye frequency is as follows: For a one-dimensional monatomic chain, the Debye frequency is equal to : \omega_{\rm D} = v_{\rm s} \pi / a = v_{\rm s} \pi N / L = v_{\rm s} \pi \lambda , with a as the distance between two neighbouring atoms in the chain when the system is in its
ground state of energy, here being that none of the atoms are moving with respect to one another; N the total number of atoms in the chain; L the size of the system, which is the length of the chain; and \lambda the
linear number density. For L, N, and a, the relation L = N a holds. For a two-dimensional monatomic square lattice, the Debye frequency is equal to : \omega_{\rm D}^2 = \frac {4 \pi}{a^2} v_{\rm s}^2 = \frac {4 \pi N}{A} v_{\rm s}^2 \equiv 4 \pi \sigma v_{\rm s}^2 , with A \equiv L^{2} = N a^{2} is the size (area) of the surface, and \sigma the
surface number density. For a three-dimensional monatomic
primitive cubic crystal, the Debye frequency is equal to : \omega_{\rm D}^3 = \frac {6 \pi^2}{a^3} v_{\rm s}^3 = \frac {6 \pi^2 N}{V} v_{\rm s}^3 \equiv 6 \pi^2 \rho v_{\rm s}^3 , with V \equiv L^3 = N a^3 the size of the system, and \rho the
volume number density. The general formula for the Debye frequency as a function of n, the number of dimensions for a (hyper)cubic lattice is : \omega_{\rm D}^n = 2^n \pi^{n/2} \Gamma\left(1+\tfrac{n}{2}\right) \frac {N}{L^n} v_{\rm s}^n , with \Gamma being the
gamma function. The speed of sound in the crystal depends on the mass of the atoms, the strength of their interaction, the
pressure on the system, and the
polarisation of the spin wave (longitudinal or transverse), among others. For the following, the speed of sound is assumed to be the same for any polarisation, although this limits the applicability of the result. The assumed
dispersion relation is easily proven inaccurate for a one-dimensional chain of masses, but in Debye's model, this does not prove to be problematic.
Relation to Debye's temperature The Debye temperature \theta_{\rm D} , another parameter in Debye model, is related to the Debye frequency by the relation \theta_{\rm D}=\frac{\hbar}{k_{\rm B}}\omega_{\rm D}, where \hbar is the reduced Planck constant and k_{\rm B} is the
Boltzmann constant.
Debye's derivation Three-dimensional crystal In Debye's derivation of the
heat capacity, he sums over all possible modes of the system, accounting for different directions and polarisations. He assumed the total number of modes per polarization to be N, the amount of masses in the system, and the total to be :\sum_{\rm modes}3=\frac {3 V}{(2 \pi)^3} \iiint d \mathbf k, where \mathbf k \equiv (k_x, k_y, k_z) ; V \equiv L^3 is the size of the system; and the integral is (as the summation) over all possible modes, which is assumed to be a finite region (bounded by the cut-off frequency). The triple integral could be rewritten as a single integral over all possible values of the absolute value of \mathbf k (see
Jacobian for spherical coordinates). The result is :\frac {3 V}{(2 \pi)^3} \iiint d \mathbf k = \frac {3 V}{2 \pi^2} \int_0^{k_{\rm D}} |\mathbf k|^2 d \mathbf k , with k_{\rm D} the absolute value of the wave vector corresponding with the Debye frequency, so k_{\rm D} = \omega_{\rm D}/v_{\rm s}. Since the dispersion relation is \omega =v_{\rm s}|\mathbf k|, it can be written as an integral over all possible \omega : : \frac {3 V}{2 \pi^2} \int_0^{k_{\rm D}} |\mathbf k|^2 d \mathbf k = \frac {3 V}{2 \pi^2 v_{\rm s}^3} \int_0^{\omega_{\rm D}} \omega^2 d \omega , After solving the integral it is again equated to 3 N to find : \frac {V}{2 \pi^2 v_{\rm s}^3} \omega_{\rm D}^3 = 3 N . It can be rearranged into : \omega_{\rm D}^3 =\frac {6 \pi^2 N}{V} v_{\rm s}^3 .
One-dimensional chain in 3D space The same derivation could be done for a one-dimensional chain of atoms. The number of modes remains unchanged, because there are still three polarizations, so :\sum_{\rm modes}3=3 N. The rest of the derivation is analogous to the previous, so the left hand side is rewritten with respect to the Debye frequency: :\sum_{\rm modes}3=\frac {3 L}{2 \pi} \int_{-k_{\rm D}}^{k_{\rm D}}d k = \frac {3 L}{\pi v_{\rm s}} \int_{0}^{\omega_{\rm D}}d \omega. The last step is multiplied by two is because the integrand in the first integral is even and the bounds of integration are symmetric about the origin, so the integral can be rewritten as from 0 to k_D after scaling by a factor of 2. This is also equivalent to the statement that the volume of a one-dimensional ball is twice its radius. Applying a change a substitution of k=\frac{\omega}{v_s} , our bounds are now 0 to \omega_D = k_Dv_s, which gives us our rightmost integral. We continue; : \frac {3 L}{\pi v_{\rm s}} \int_{0}^{\omega_{\rm D}}d \omega = \frac {3 L}{\pi v_{\rm s}} \omega_{\rm D} = 3 N . Conclusion: : \omega_{\rm D} = \frac {\pi v_{\rm s} N}{L} .
Two-dimensional crystal The same derivation could be done for a two-dimensional crystal. The number of modes remains unchanged, because there are still three polarizations. The derivation is analogous to the previous two. We start with the same equation, :\sum_{\rm modes}3=3 N. And then the left hand side is rewritten and equated to 3N : \sum_{\rm modes}3=\frac {3 A}{(2 \pi)^2} \iint d \mathbf k = \frac {3 A}{2 \pi v_{\rm s}^2} \int_{0}^{\omega_{\rm D}} \omega d \omega = \frac {3 A \omega_{\rm D}^2}{4 \pi v_{\rm s}^2} = 3 N , where A \equiv L^2 is the size of the system. It can be rewritten as : \omega_{\rm D}^2 = \frac {4 \pi N}{A} v_{\rm s}^2 .
Polarization dependence In reality, longitudinal waves often have a different wave velocity from that of transverse waves. Making the assumption that the velocities are equal simplified the final result, but reintroducing the distinction improves the accuracy of the final result. The dispersion relation becomes \omega_i = v_{s,i}|\mathbf k|, with i = 1, 2, 3 , each corresponding to one of the three polarizations. The cut-off frequency \omega_{\rm D} , however, does not depend on i. We can write the total number of modes as \sum_{i}\sum_{\rm modes} 1 , which is again equal to 3 N. Here the summation over the modes is now dependent on i.
One-dimensional chain in 3D space The summation over the modes is rewritten : \sum_{i}\sum_{\rm modes} 1 = \sum_i \frac {L}{\pi v_{s,i}} \int_0^{\omega_{\rm D}} d \omega_i = 3 N . The result is : \frac {L \omega_{\rm D}}{\pi} (\frac {1}{v_{s,1}} + \frac {1}{v_{s,2}} + \frac {1}{v_{s,3}}) = 3 N . Thus the Debye frequency is found : \omega_{\rm D} = \frac{ \pi N}{L} \frac{3}{\frac {1}{v_{s,1}} + \frac {1}{v_{s,2}} + \frac {1}{v_{s,3}}} = \frac {3 \pi N}{L} \frac {v_{s,1} v_{s,2} v_{s,3}}{v_{s,2} v_{s,3} + v_{s,1} v_{s,3} + v_{s,1} v_{s,2}} = \frac{\pi N}{L} v_{\mathrm{eff}}\,. The calculated effective velocity v_{\mathrm{eff}} is the harmonic mean of the velocities for each polarization. By assuming the two transverse polarizations to have the same phase speed and frequency, : \omega_{\rm D} = \frac {3 \pi N}{L} \frac {v_{s,t}v_{s,l}}{2v_{s,l} + v_{s,t}} . Setting v_{s,t} = v_{s,l} recovers the expression previously derived under the assumption that velocity is the same for all polarization modes.
Two-dimensional crystal The same derivation can be done for a two-dimensional crystal to find : \omega_{\rm D}^2 = \frac {4 \pi N}{A} \frac{3}{\frac {1}{v_{s,1}^2} + \frac {1}{v_{s,2}^2} + \frac {1}{v_{s,3}^2}} = \frac {12 \pi N}{A} \frac {(v_{s,1} v_{s,2} v_{s,3})^2}{(v_{s,2} v_{s,3})^2 + (v_{s,1} v_{s,3})^2 + (v_{s,1} v_{s,2})^2} = \frac{4 \pi N}{A} v_{\mathrm{eff}}^2\,. The calculated effective velocity v_{\mathrm{eff}} is the square root of the harmonic mean of the squares of velocities. By assuming the two transverse polarizations to be the same, : \omega_{\rm D}^2 = \frac {12 \pi N}{A} \frac {(v_{s,t} v_{s,l})^2}{2 v_{s,l}^2 + v_{s,t}^2} . Setting v_{s,t} = v_{s,l} recovers the expression previously derived under the assumption that velocity is the same for all polarization modes.
Three-dimensional crystal The same derivation can be done for a three-dimensional crystal to find (the derivation is analogous to previous derivations) : \omega_{\rm D}^2 = \frac{6 \pi^2 N}{V} \frac{3}{\frac {1}{v_{s,1}^3} + \frac {1}{v_{s,2}^3} + \frac {1}{v_{s,3}^3}} = \frac {18 \pi^2 N}{V} \frac {(v_{s,1} v_{s,2} v_{s,3})^3}{(v_{s,2} v_{s,3})^3 + (v_{s,1} v_{s,3})^3 + (v_{s,1} v_{s,2})^3 } = \frac{6 \pi^2 N}{V} v_{\mathrm{eff}}^3\,. The calculated effective velocity v_{\mathrm{eff}} is the cube root of the harmonic mean of the cubes of velocities. By assuming the two transverse polarizations to be the same, : \omega_{\rm D}^3 = \frac {18 \pi^2 N}{V} \frac {(v_{s,t} v_{s,l})^3}{2 v_{s,l}^3 + v_{s,t}^3} . Setting v_{s,t} = v_{s,l} recovers the expression previously derived under the assumption that velocity is the same for all polarization modes.
Derivation with the actual dispersion relation points matter, two different waves could render the same physical manifestation (see
Phonon). This problem could be made more applicable by relaxing the assumption of linearity of the dispersion relation. Instead of using the dispersion relation \omega = v_{\rm s} k , a more accurate dispersion relation can be used. In classical mechanics, it is known that for an equidistant chain of masses which interact harmonically with each other, the dispersion relation is This is possible because the system consists of
discretized points, as is demonstrated in the animated picture. Dividing the dispersion relation by k and inserting \pi / a for k, we find the speed of a wave with k = \pi / a to be v_{\rm s}(k = \pi / a) = \frac {2 a}{\pi} \sqrt {\frac {\kappa}{m}} . By simply inserting k = \pi/a in the original dispersion relation we find \omega(k = \pi / a) = 2 \sqrt {\frac {\kappa}{m}} = \omega_{\rm D} . Combining these results the same result is once again found \omega_{\rm D} = \frac {\pi v_{\rm s}}{a} . However, for any chain with greater complexity, including diatomic chains, the associated cut-off frequency and wavelength are not very accurate, since the cut-off wavelength is twice as big and the dispersion relation consists of additional branches, two total for a diatomic chain. It is also not certain from this result whether for higher-dimensional systems the cut-off frequency was accurately predicted by Debye when taking into account the more accurate dispersion relation.
Alternative derivation For a one-dimensional chain, the formula for the Debye frequency can also be reproduced using a theorem for describing
aliasing. The
Nyquist–Shannon sampling theorem is used for this derivation, the main difference being that in the case of a one-dimensional chain, the discretization is not in time, but in space. The cut-off frequency can be determined from the cut-off wavelength. From the sampling theorem, we know that for wavelengths smaller than 2a , or twice the sampling distance, every mode is a repeat of a mode with wavelength larger than 2a , so the cut-off wavelength should be at \lambda_{\rm D} = 2 a . This results again in k_{\rm D} = \frac{2 \pi}{\lambda_D} = \pi / a , rendering \omega_{\rm D} = \frac {\pi v_{\rm s}}{a} . It does not matter which dispersion relation is used, as the same cut-off frequency would be calculated. == See also ==