Structure order parameters

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.

$$\eta_{\vec q}=\left< \frac{1}{N^2} \left \vert \sum_{j} exp{(i\vec q \vec r_j)} \right \vert^2 \right >$$ (6.4)

where $N$ is the number of atoms, and the vector $\vec r_j$ describes the position of the atom $j$, while the vector $\vec q$ is related to elastic moment transfer (diffractive scattering).

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

$$\eta_{l,\alpha}=\left< \frac{1}{n_l^2} \left \vert \sum_{j \in l } exp{(i\vec g_{\alpha} \vec r_j)} \right \vert^2 \right >$$ (6.5)

where $\alpha = 1,2,3\equiv x,y,z$ are indices of the Cartesian axis $x,y,z$ and
$\vec g_1,\vec g_2,\vec g_3=\frac {2\pi}{a_x}\hat x,\frac {2\pi}{a_y}\hat y,\frac {2\pi}{a_z}\hat z$, is a set of vectors which define a set of different directions (the order parameter is calculated along those directions), $a_{\alpha}$ is the nearest-neighbor distance in $\alpha$ direction (See table 5.1), $n_l$ is the instantaneous number of atoms in the layer $l$, the sum extends over the particles in the layer $l$, and the angular brackets denote averaging over time.

Figure: Order parameters of $Va(011)$ at $2000$ K. The difference between $\eta_{l,x}$ and $\eta_{l,y}$ reflects the anisotropy of the Va(011) surface.

For an ordered crystalline surface the order parameter is a unity at zero temperature. The deviation of the $\eta_{l,\alpha}$ from the unity originates from thermal vibrations and from formation of surface defect. The structure order parameters of the Va(001) sample at $T=2000~ K$ 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 $n_l$ 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 $y$-direction than along the $x$-direction. The same effect is observed for the Va(111) sample, but is absent for the Va(001) sample.

Figure: Comparison of order parameter of the layers of $Va(011)$ as a function of temperature.

This anisotropy $\eta_{l,y}< \eta_{l,x}$ 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 $x$ and $y$ directions $a_x>a_y$. Assuming, in the first approximation, that each atom oscillates with the same amplitude in both the $x$ and $y$ directions,i.e. $<u_x^2> \simeq <u_y^2>$, and expanding the structure order parameter in term of $<u^2>/a_\alpha^2$ (it is found in our MD simulations that mean square amplitude of vibration is order of $<u^2>\simeq 10^{-2} \dot A^2$ while the lattice parameter squared is about $a_0^2 \simeq 3.05^2 \dot A^2$) we obtain:

$$\eta_{l,\alpha} \simeq 1- \sum_{j \in l } \frac{4\pi^2 \left< u^2\right > }{n_l^2a_\alpha^2}+...$$ (6.6)

Hence it follows that if $a_x>a_y$ then $\eta_x > \eta_y$, because the smaller $<u^2>/a_\alpha^2$, the less the decrease of the order parameter. The same consideration can be applied to explain the difference between in-plane components of the order parameter ($x$ and $y$ directions) and the out-of-plane component ($z$-direction), but one has to take into account that mean square vibrational amplitude in the plane direction is larger than in the out-of-plane ones.

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.

Figure: $x$ component of the order parameter of $Va(001)$ at various temperatures.

Figure: $x$ component of the order parameter of $Va(111)$ at various temperatures.

Figure: $x$ component of the order parameter of $Va(011)$ at various temperatures.