Vibrations of a solid sphere

The lowest vibrational modes of the cluster are those which have wavelengths comparable with the dimensions of the cluster. For these modes a good approximation is to treat the cluster as an elastic continuum, and then the eigenfrequencies may be calculated by standard techniques from classical elastodynamics. For simplicity, we approximate the shape of the cluster by a sphere. Then the displacements are expressed in terms of compression and shear potentials, which are expanded in spherical Bessel functions j_n(r) and spherical harmonics (see, for example, Pao and Mow (1973) for the relevant expressions). The eigenfrequencies are given by the zeros of the determinant

 

$$\left \vert \matrix{ S_{\rm rp} (n, \alpha a, \beta a ) & S_... ... \cr S_{\rm tp} (n, \alpha a ) & S_{\rm ts} (n, \beta a ) } \right\vert$$ (4)

where $\alpha$ and $\beta$ are the compression and shear wavenumbers $\omega$/c_p and $\omega$/c_s respectively, a is the radius of the sphere, and n is the index of the mode, which may take any non-negative integer value. The functions involved in the determinant are

$$\left ( {n^2 - n - {y^2 \over 2}} \right)^$$ S_rp(n,x,y) = (5)

$$\left ({ (n-1) j_n (x) - x j_{n+1} (x) } \right)$$ S_rs(n,x) = (6)

S_tp(n,x) = (n - 1)j_n(x) - xj_{n + 1}(x) (7)

$$\left ( {n^2 - n - {x^2 \over 2}} \right)^$$ S_ts(n,x) = (8)

and the notation is that subscripts r and t denote radial and tangential variations, p and s denote compression and shear waves.

For the potentials used in the simulations, the bulk wave speeds in LaF₃ are 6000 m s ^{- 1} and 2400 m s ^{- 1}. The density of the material is 5981 kg m ^{- 3}. Then, solving equation 4, we find that the two lowest modes of a sphere of LaF₃ of radius 10 Å have periods of 0.7 ps and 0.5 ps . The periods of the vibrations in region A of Figure 1, where the cluster is solid, are clustered about 0.6 ps.

Now the displacement in the lowest mode is purely radial, and has the form

$$\alpha \omega$$ (9)

when the amplitude of the mode is A . The time average of the kinetic energy in this mode is

$$\left \langle KE \right\rangle= \pi \rho \omega^{2}_{} \int_{0}^{a} \alpha$$ (10)

which will be equal to ${\frac{1}{2}}$kT if the amplitude is approximately 10^{- 12} m at a temperature of 100 K , 10^{- 11} m at 1500 K . The number of purely radial modes will be similar to the number of atoms along a radius of the cluster, that is, between 3 and 7 for the two smaller clusters considered. Thus we can expect the overall amplitude of vibration to be about 0.3% of the radius of the cluster at a temperature of 100 K , and approaching 1% of the radius at 1500 K . The figure of 0.3% is quantitatively similar to the amplitudes seen in region A of Figure 1.