Structure order parameters (structure factors) are useful for monitoring the order-disorder transition
in the course of surface premelting. The structure order parameter is also related to low energy electron
diffraction (**LEED**) intensity [97], which can be measured experimentally. Atomic vibrations break
to some extent the periodicity of lattice and diffraction effects provide essentially direct
information about the vibration amplitude. The structure factor is defined as a Fourier
transformation of the atomic density of the system.

(6.4) |

In the case of a surface, the order parameter is often defined for each layer separately:

(6.5) |

, is a set of vectors which define a set of different directions (the order parameter is calculated along those directions), is the nearest-neighbor distance in direction (See table 5.1), is the instantaneous number of atoms in the layer , the sum extends over the particles in the layer , and the angular brackets denote averaging over time.

For an ordered crystalline surface the order parameter is a unity at zero temperature. The deviation of the from the unity originates from thermal vibrations and from formation of surface defect. The structure order parameters of the Va(001) sample at is shown in Fig. 5.14. Note the decrease of the order parameter in the surface region, which reflects enhanced atomic vibrations and adatom-vacancy pair creation. The existence of vacancies does not directly affects the order parameter, since a normalization procedure is employed during each measurement by using the instantaneous layer occupation of a layer. Nevertheless, vacancies have an indirect effect on the order parameter by introducing a lattice distortion around them.

As is evident from the Figs. 5.15 the structure order parameter of the Va(011) sample is lower along the -direction than along the -direction. The same effect is observed for the Va(111) sample, but is absent for the Va(001) sample.

This anisotropy actually arises from the anisotropic structure of the low-index faces Va(011) and Va(111), where the distances between the nearest-neighbors are different in the and directions . Assuming, in the first approximation, that each atom oscillates with the same amplitude in both the and directions,i.e. , and expanding the structure order parameter in term of (it is found in our MD simulations that mean square amplitude of vibration is order of while the lattice parameter squared is about ) we obtain:(6.6) |

The structure order parameter profiles at various temperatures are plotted in Figs. 5.16-5.18. Note a continuous decrease of the order parameter for the Va(111) sample, which begins to premelt first. In contrast, one can see a relatively abrupt decrease of the order parameter of the close packed Va(011) sample, which takes place only in vicinity of the melting transition.