Dynamics of a Square Lattice I. Frequency Spectrum

The distribution of frequencies of normal modes of vibration of a monatomic square lattice is obtained as a function of the force constants of the lattice. Since the forces between pairs of atoms in the lattice are short ranged, interactions between all atoms, other than nearest and next nearest neighbors, are neglected. Two constants, α and γ, are used to describe the lattice. α is a force constant derived from the interaction of nearest neighbors, and γ is that derived from the interaction of next nearest neighbors. According to the Debye continuum theory the frequency spectrum, or density of normal modes, should be a linear function of the frequency. In the Born‐Karman atomic model used in the present paper it is shown that this is only the case at very low frequencies and that there actually exist two sharp infinities in the frequency spectrum. We define g(ν) so that g(ν)dν is the number of normal modes of vibration with frequencies between ν and ν+dν. A closed expression involving complete elliptic integrals is obtained for g(ν) when τ=[1+(α/2γ)]⁻¹=⅓. For general values of τ<½, g(ν) is shown to have the following properties: (a) The largest frequency, ν_L, occurs at $${\nu }_L{\text{=[(4α+8γ)/4π}}^2{\mathrm{M]}}^{\frac{1}{2}},$$ where M=mass of each atom. (b) There is an infinity in g(ν) at ν=ν_Lτ^½. (c) The second infinity is at ${\mathrm{\nu =\nu }}_L{\text{(1−τ)}}^{\frac{1}{2}}\hspace{0.17em}\mathrm{if}\mathrm{\hspace{0.17em}\tau <}\frac{1}{5}$ , and ${\mathrm{\nu =\nu }}_L{\text{(1+3τ)/4τ}}^{\frac{1}{2}}\hspace{0.17em}\mathrm{if}\mathrm{\hspace{0.17em}\tau >}\frac{1}{5}$ . (d) As ν/ν_L→0, g(ν) is linear in ν.