Surface Diffusion at Finite Coverage: The Characteristic Function Method

Abstract

In this work, the so-called characteristic function method is proposed as a new approach to describe and interpret the diffusion process with interacting adsorbates in terms of surface coverage. In this context, the intermediate scattering function is identified as a characteristic function that is very well defined in probability theory. From this function, the generating functions of the moments and cumulants of the jump probability distribution are straightforwardly obtained at any order. This analysis is carried out in two stages. First, the dilute limit, corresponding to non-interacting adsorbates or very low surface coverage, is briefly reviewed. Second, the method is extended to low and intermediate coverages, where adsorbate-adsorbate interactions become relevant. A further consequence of the present analysis is that the static structure factor is also a characteristic function of the adsorbate separation distance distribution. This method thus provides a compact and physically transparent route for connecting scattering observables, diffusion coefficients, and coverage-dependent structural correlations.

1. Introduction

Surface diffusion is a fundamental, elementary, and primary step underlying a wide range of processes occurring on surfaces, such as, for example, chemical reactions, epitaxial growth, and thin-film formation. Above all, diffusion is a stochastic process that has to be studied and analyzed within the theory of open classical and quantum systems and, more broadly, within non-equilibrium statistical mechanics.

Two well-known surface experimental techniques are mainly used, namely, quasi-elastic neutron scattering (QENS) [1] and quasi-elastic helium atom scattering (QHAS) with their respective spin echo (SE) variants [2,3]. Probe particles (light, neutrons, or helium atoms) are interacting weakly with the system of interest (adsorbates), which is coupled to a surface considered as a reservoir (or thermal bath), and their linear response is measured. The original theory dates back to van Hove and Vineyard when dealing with neutron scattering by crystals and liquids. The nature of the particles probing the system formed by moving and interacting adsorbates on a surface is largely irrelevant when the Born approximation is assumed [4,5,6,7]. Unlike neutron scattering, which can be coherent or incoherent, atom-surface scattering is fully coherent.

As is well known, two linear response functions are central in this context: the dynamic structure factor (DSF), $S(\mathbf{K},\omega )$, and the intermediate scattering function (ISF), $I(\mathbf{K},t)$, which are expressed in terms of the energy exchange, $\hslash \omega$, and the momentum transfer parallel to the surface, $\mathbf{K}$, undergone by the probing particles in the scattering process. In the QHAS and HeSE surface techniques, the DSF and ISF are the observables (or response functions) measured, respectively. Both of them are related by a Fourier transform as follows:

$$\begin{array}{ccc}\hfill I(\mathbf{K},t)& =& \int d\omega e^{i\omega t}S(\mathbf{K},\omega )\hfill \\ & =& {\displaystyle \frac{1}{N}}⟨{\rho }_{\mathbf{K}}{\rho }_{-\mathbf{K}}\left(t\right)⟩,\hfill \end{array}$$

(1)

where the particle density operator in the reciprocal space is given by

$${\rho }_{\mathbf{K}}\left(t\right)=\sum _{j=1}^{N}exp(-i\mathbf{K}.{\mathbf{R}}_j\left(t\right)),$$

(2)

${\mathbf{R}}_j\left(t\right)$ being the position operators of the $j=1,\dots ,N$ adsorbates on the surface as a function of time. Within the Born approximation, the DSF and ISF are expressed in terms of the generalized pair-distribution function $G(\mathbf{R},t)$ or the van Hove space-time correlation function. Given an adsorbate at the origin and at time $t=0$, this correlation function gives the probability that any adsorbate, including the same one, can be found at the surface position $\mathbf{R}$ and at time t. The ISF is then written as (the first line of the equation) [4]:

$$\begin{array}{ccc}\hfill I(\mathbf{K},t)& =& \int d\mathbf{R}{e}^{i\mathbf{K}.\mathbf{R}}G(\mathbf{R},t),\hfill \\ & =& \int d\mathbf{R}{e}^{i\mathbf{K}·\mathbf{R}}\rho (\mathbf{R},t),\hfill \\ & =& ⟨e^{i\mathbf{K}·\mathbf{R}\left(t\right)}⟩.\hfill \end{array}$$

(3)

Quantum probabilities, $\rho (\mathbf{R},t)$ (as written in the second line of this equation), are given by the diagonal elements of the reduced density matrix, which are obtained after solving the well-known Liouville–von Neumann equation [8,9]. And, finally, the third line of this equation can be seen as an average of the exponential function, $expi\mathbf{K}·\mathbf{R}\left(t\right),$ [10]. In probability theory, this average is known as the characteristic function of a given probability distribution [11,12]. Thus, the ISF, which is a time-dependent, real observable, is the characteristic function (CF) of the probability distribution of the adsorbate positions on the surface as a function of time, with the real parameter given by the modulus of the $\mathbf{K}$-vector (the value of K and its direction of observation are chosen by the experimental technique used). This Fourier transform can be reduced to a Fourier sum when a tight-binding description is used, and a jump model is assumed.

There is a vast amount of literature on characteristic functions because it is often easier to evaluate a characteristic function for a random variable than to evaluate the moments and cumulants (at any order) of the corresponding probability distribution function. In this regard, several properties and theorems can be used [11,12] to better understand this approach to surface diffusion:

• A CF completely characterizes a random variable and uniquely determines the corresponding probability distribution function.
• If the CF is known, the diagonal elements of the reduced density matrix, or probabilities, are obtained from its inverse Fourier transform, which determines them unequivocally.
• A CF is real if and only if it is even.
• A distribution function is symmetric if and only if its CF is real and even.
• $I(\mathbf{0},t)=1$.
• $I(\mathbf{K},t)$ is a uniformly continuous function of its argument for all values of $\mathbf{K}$.
• The moments of the probability distribution are obtained by Taylor expansion of the exponential function to any order, and therefore one can speak of the moment generating function.
• The natural log of the CF is the corresponding cumulant generating function. A similar Taylor expansion at any order can also be applied.
• When time-translational invariance is assumed, the CF is an exponential function of time, $exp(-t\alpha )$, where $\alpha$ is the decay rate, which is also known as the characteristic exponent [13].

On the other hand, when the diffusive regime is reached at very long times (times much greater than the inverse of the friction coefficient), the experimental ISF displays an exponential function of time according to [2]

$$I(\mathbf{K},t)=B{e}^{-\alpha \left(\mathbf{K}\right)t}+C,$$

(4)

where B and C are constants and $\alpha \left(\mathbf{K}\right)$ is the so-called decaying/dephasing rate. After Equation (1), the DSF or the quasi-elastic peak around $\omega =0$ is a Lorentzian function:

$$S(\mathbf{K},\omega )\propto {\displaystyle \frac{\alpha \left(\mathbf{K}\right)}{{\omega }^2+\alpha {\left(\mathbf{K}\right)}^2}}.$$

(5)

This rate also depends implicitly on the friction coefficient, surface structure, and surface temperature.

Thus, the ISF is a fundamental dynamical measurement which clearly provides a compact and physically transparent route for relating diffusion dynamics, structural correlations, and scattering observables. There are very few theoretical works considering surface diffusion at finite surface coverages, $\theta$. At very low coverages, or in the dilute limit, the adsorbate-adsorbate interaction can be safely neglected, and one can then speak of the tracer diffusion coefficient. At higher values, this interaction is critical to better interpret the ISF and the dephasing rate. Because the coverage is fixed at the very beginning of the experiment, the coverage should be considered as a parameter of the theory, so that the main functions can be written as follows: $\alpha (\mathbf{K};\theta )$, $I(\mathbf{K},t;\theta )$, and $S(\mathbf{K},\omega ;\theta )$. The main purpose of this work is to formulate surface diffusion at finite coverage within the language of CFs. In addition, it is shown that the so-called static structure factor (SSF) is again a CF of the adsorbate separation distance distribution.

This work is then organized as follows. Section 2 introduces the CF method in a general tight-binding framework, starting from a master/rate or Pauli equation in the diffusive regime. This master equation is solved for a simple Bravais lattice by assuming that adsorbates remain longer at the surface sites than while traveling to another site. The solution provides the ISF, which is a generating function for the moments and cumulants of the jump distribution at any order. In particular, we focus on the second moment or cumulant because it is related to the diffusion coefficient, whether tracer or collective, depending on the value of the surface coverage. As additional information, the frequency moments, or sum rules, are also calculated from the ISF. Section 3 applies the formalism to the dilute limit and then extends it to low and intermediate surface coverages, including simple interaction models. Finally, Section 4 summarizes the main conclusions and discusses the advantages of the present approach.

2. Diffusion Dynamics in the Presence of Surface Coverages: The Characteristic Function Method

Surface coverage is the fraction of a surface that is occupied by adsorbates, whether atoms or molecules. In a tight-binding description, it is defined as the ratio of occupied adsorption sites to the total number of available sites and is denoted by $\theta$. This quantity is fixed by the experiment from the beginning, and the diffusion dynamics are studied for a given value of this external parameter. In the dilute limit, adsorbates can be considered effectively non-interacting.

Once the diffusive regime is established at long times, the ISF is well represented by Equation (4), but now this function and the dephasing rate depend parametrically on the surface coverage and are written as $I(\mathbf{K},t;\theta )$ and $\alpha (\mathbf{K};\theta )$. If the Chudley–Elliott (CE) jump diffusion model [14] is assumed, and for simplicity, the surface is taken to be a simple Bravais lattice with instantaneous jumps between adsorption sites, the Pauli master equation for the diagonal elements of the reduced density matrix is:

$${\displaystyle \frac{𝜕\rho (\mathbf{R},t;\theta )}{𝜕t}}=\sum _{\mathbf{L}}{\mathrm{\Gamma }}_{\mathbf{L}}\left[\rho (\mathbf{R}+\mathbf{L},t;\theta )-\rho (\mathbf{R},t;\theta )\right],$$

(6)

where the summation runs over all two-dimensional vectors $\mathbf{L}$. The partial jumping rate, ${\mathrm{\Gamma }}_{\mathbf{L}}\left(\theta \right)$, is usually considered as a fitting parameter, and its inverse, ${\mathrm{\Gamma }}_{\mathbf{L}}^{-1}\left(\theta \right)$, is interpreted as the average time between successive jumps. In this model, the time for a single jump is very short compared to the time between successive jumps. The total jump rate is given by $\mathrm{\Gamma }\left(\theta \right)={\sum }_{\mathbf{L}}{\mathrm{\Gamma }}_{\mathbf{L}}\left(\theta \right)$, with ${\mathrm{\Gamma }}_{\mathbf{L}}\left(\theta \right)={\mathrm{\Gamma }}_{-\mathbf{L}}\left(\theta \right)$. Due to the linearity of the Fourier transform, the ISF is governed by the following:

$${\displaystyle \frac{𝜕I(\mathbf{K},t;\theta )}{𝜕t}}=-4I(\mathbf{K},t;\theta )\sum _{\mathbf{L}>\mathbf{0}}{\mathrm{\Gamma }}_{\mathbf{L}}\left(\theta \right){sin}^2\left({\displaystyle \frac{\mathbf{K}·\mathbf{L}}{2}}\right).$$

(7)

The solution of this differential equation is given by the following:

$$I(\mathbf{K},t;\theta )=I(\mathbf{K},0;\theta )e^{-2\left|t\right|{\sum }_{\mathbf{L}>\mathbf{0}}{\mathrm{\Gamma }}_{\mathbf{L}}\left(\theta \right)\left(1-cos\left(\mathbf{K}·\mathbf{L}\right)\right)},$$

(8)

where the dephasing rate is expressed as

$$\alpha (\mathbf{K};\theta )=2\mathrm{\Gamma }\left(\theta \right)\sum _{\mathbf{L}>\mathbf{0}}{P}_{\mathbf{L}}\left(\theta \right)\left(1-cos\left(\mathbf{K}·\mathbf{L}\right)\right).$$

(9)

The jump probabilities are then written as $P_{\mathbf{L}}\left(\theta \right)={\mathrm{\Gamma }}_{\mathbf{L}}\left(\theta \right)/\mathrm{\Gamma }\left(\theta \right)$. Experimental dephasing rates have to be fitted to Equation (9), where, as mentioned above, the fitting parameters in this model are the partial jumping rates, ${\mathrm{\Gamma }}_{\mathbf{L}}\left(\theta \right)$. With this fitting procedure, one can express the ISF in an analytical form and easily derive the moments and cumulants of the jump distribution.

The observation direction is also selected experimentally, $\mathbf{K}$. In general, the scalar product in Equation (3) can be written as $\mathbf{K}·\mathbf{R}\left(t\right)=K{L}_{\Vert }(t;\theta )$, where K is the modulus of $\mathbf{K}$ and $L_{\Vert }(t;\theta )$ gives us the total sum of nonzero projections of the $\mathbf{L}$-vectors along $\mathbf{K}$ (parallel to the direction of observation) as a function of time. The jumps along the different projections are obviously conditioned by the surface coverage. In general, several jump/transition vectors $\mathbf{L}$ are contributing to the total diffusion process, and one can write the total sum as $L_{\Vert }(t;\theta )={\sum }_m{L}_{\Vert ,m}(t;\theta )$. Equation (3) can then be expressed as follows [10]:

$$I(K,t;\theta )=⟨e^{iK{L}_{\Vert }(t;\theta )}⟩=\sum _{n=0}^{\infty }{\displaystyle \frac{{\left(iK\right)}^n}{n!}}⟨L_{\Vert }^n(t;\theta )⟩.$$

(10)

The $n^{th}$-derivative of the CF with respect to K, at $K=0$, provides the $n^{th}$ moment of the jump/transition distribution [15,16]

$$⟨L_{\Vert }^n(t;\theta )⟩={\left(-i{\displaystyle \frac{𝜕}{𝜕K}}\right)}^{n}I(K,t;\theta ){|}_{K=0},$$

(11)

as well as the $n^{th}$-cumulant of the same distribution given by

$${⟨L_{\Vert }^n(t;\theta )⟩}_c={\left(-i{\displaystyle \frac{𝜕}{𝜕K}}\right)}^{n}lnI(K,t;\theta ){|}_{K=0}.$$

(12)

The second moment is of particular importance because it yields the diffusion coefficient, namely, the tracer diffusion coefficient in the dilute regime and the collective diffusion coefficient at finite coverage.

Using the Fourier transform between the ISF and DSF, the ISF may also be expanded in frequency moments of $S(K,\omega ;\theta )$ (frequency sum rules) by Taylor expansion with respect to time around $t=0$ [17]

$$I(\mathbf{K},t;\theta )=\sum _{n=0}^{\infty }{\displaystyle \frac{{\left(it\right)}^n}{n!}}⟨{\omega }^n(\mathbf{K};\theta )⟩,$$

(13)

with

$$⟨{\omega }^n(\mathbf{K};\theta )⟩={\displaystyle \frac{1}{2\pi }}{\int }_{-\infty }^{+\infty }d\omega {\omega }^{n}S(\mathbf{K},\omega ;\theta ),$$

(14)

and, therefore,

$$⟨{\omega }^n(\mathbf{K};\theta )⟩={\left(i{\displaystyle \frac{𝜕}{𝜕t}}\right)}^{n}I(\mathbf{K},t;\theta ){|}_{t=0},$$

(15)

which only depend on the projectile along the direction of observation through Equation (8). The so-called quasi-elastic peak, given by Equation (5), but now expressed in terms of surface coverage, is an even function of frequency; therefore, only even frequency moments are different from zero.

Interestingly, the ISF at $t=0$ is precisely the SSF, which, in terms of coverage, is written as

$$I(\mathbf{K},0;\theta )=F(\mathbf{K};\theta ),$$

(16)

This factor is again a characteristic function associated with the distribution of adsorbate separation distances.

3. Results

3.1. Very Low Surface Coverages

In a tight-binding description, the relevant quantity is the probability $P_n\left(t\right)$ that an adsorbate occupies site n at time t. This description is valid when the thermal energy is much smaller than the activation energy, so that adsorbates remain localized at adsorption sites. Furthermore, the time between two consecutive jumps is greater than the duration of a given jump. In the following, we consider different jump models to illustrate how the characteristic function method is implemented in the dilute regime.

3.1.1. Nearest Neighbors

In order to simplify the discussion, we reduce the diffusion dynamics to only one high-symmetry direction on a periodic substrate, with the projection along $\mathbf{K}$ written as $L_{\Vert }$. Within the formalism of the master equation [15], if a simple Bravais lattice is assumed, the Pauli master equation reads as follows (note that this starting point was already used by CE in neutron scattering [14]) [9,18,19]:

$${\dot{P}}_n\left(t\right)={\mathrm{\Gamma }}_{n-1}^{+}{P}_{n-1}\left(t\right)+{\mathrm{\Gamma }}_{n+1}^{-}{P}_{n+1}\left(t\right)-({\mathrm{\Gamma }}_n^{+}+{\mathrm{\Gamma }}_n^{-})P_n\left(t\right),$$

(17)

where the dot on ${\dot{P}}_n\left(t\right)$ denotes the time derivative. Here, ${\mathrm{\Gamma }}_{n\pm 1}^{\pm }$ denotes the hopping transition rates from the $(n\mp 1)$-th well to the n-th well, and ± indicates whether diffusion goes to the right or to the left, respectively. Usually, these transition rates are fitting parameters. If the initial condition is such that $P_n\left(0\right)={\delta }_{n0}$ and $\mathrm{\Gamma }={\mathrm{\Gamma }}^{+}+{\mathrm{\Gamma }}^{-}$ describes the total rate (with ${\mathrm{\Gamma }}_n^{+}={\mathrm{\Gamma }}_n^{-}$, ${\mathrm{\Gamma }}_n^{+}={\mathrm{\Gamma }}^{+}$, and ${\mathrm{\Gamma }}_n^{-}={\mathrm{\Gamma }}^{-}$), one obtains

$$P_n\left(t\right)=I_n\left(\mathrm{\Gamma }t\right)e^{-\mathrm{\Gamma }t},$$

(18)

where $I_n\left(x\right)$ is the modified Bessel function of integer order n.

It is also very instructive to introduce the so-called probability generating function [15], which is very useful for solving a master equation when a jump model is assumed. In this context, this function is defined as an infinite series:

$$G(z,t)=\sum _{n=0}^{\infty }{P}_n\left(t\right)z^n,$$

(19)

which converges absolutely for all complex numbers z with $\left|z\right|\le 1$. Equation (18) could also be derived by using the probability generating function $G(z,t)$.

Now, the ISF can be rewritten as [9]:

$$\begin{array}{ccc}\hfill I(K,t)& =& \sum _{n=-\infty }^{n=+\infty }{P}_n\left(t\right)e^{inKacos\beta }=e^{-\mathrm{\Gamma }t}\sum _{n=-\infty }^{+\infty }{I}_n\left(\mathrm{\Gamma }t\right)e^{inKacos\beta }\hfill \\ & =& e^{-\mathrm{\Gamma }t[1-cos(Kacos\beta \left)\right]},\hfill \end{array}$$

(20)

where a is the unit cell length along the symmetry direction chosen, and $\beta$ is the angle formed with the observation direction. Notice that the ISF is an even function of K. Now, after Equation (20), one has that the total jumping/transition rate is expressed as

$$\mathrm{\Gamma }\left(K\right)={\displaystyle \frac{\alpha \left(K\right)}{1-cos(Kacos\beta )}}.$$

(21)

This rate is a function of the projection along the direction of $\mathbf{K}$ and depends implicitly on the surface temperature and friction coefficient.

Once the ISF for a given surface diffusion process is known analytically, it is straightforward to calculate the moments and cumulants; in particular, the second moment, which is related to the tracer diffusion coefficient D (for this case, $⟨L_{\Vert }\left(t\right)⟩={⟨L_{\Vert }\left(t\right)⟩}_c=0$):

$$⟨L_{\Vert }^2\left(t\right)⟩={⟨L_{\Vert }^2\left(t\right)⟩}_c=Dt,$$

(22)

where

$$D=a^2\mathrm{\Gamma }co{s}^2\beta .$$

(23)

The $cos$-factor reflects the projection onto the observation direction, which is very often disregarded in the literature.

3.1.2. Beyond Nearest Neighbors

When considering jumps beyond nearest neighbors, the probability distribution is given by [20]:

$$P_n\left(t\right)=e^{-2({\gamma }_1+{\gamma }_2+{\gamma }_3)t}\sum _{k=-\infty }^{+\infty }{I}_k\left(2{\gamma }_{3}t\right)\sum _{j=-\infty }^{+\infty }{I}_j\left(2{\gamma }_{2}t\right)I_{n-2j-3k}\left(2{\gamma }_{1}t\right),$$

(24)

where ${\gamma }_1,{\gamma }_2$, and ${\gamma }_3$ denote the single, double, and triple jump rates, respectively. More generally, arbitrary jump lengths may also be considered, leading to the master equation:

$${\dot{P}}_n\left(t\right)=\sum _{l=1}^{\infty }\left[{\mathrm{\Gamma }}_l^{+}{P}_{n-l}\left(t\right)+{\mathrm{\Gamma }}_l^{-}{P}_{n+l}\left(t\right)-({\mathrm{\Gamma }}_l^{+}+{\mathrm{\Gamma }}_l^{-})P_n\left(t\right)\right].$$

(25)

The ISF can then be written as a product of backward and forward diffusion:

$$I(K,t)=\prod _{n=1}^{\infty }{I}_n^{+}(K,t)I_n^{-}(K,t),$$

(26)

with

$$I_n^{\pm }(K,t)=e^{t(e^{\pm inKacos\beta }-1){\mathrm{\Gamma }}_n^{\pm }},$$

(27)

where a and $\beta$ are, respectively, the unit-cell length along the symmetry direction under consideration and the angle formed by this direction and $\mathbf{K}$. An alternative expression for the ISF can also be written as follows within the CE model:

$$I(K,t)=e^{-2t\mathrm{\Gamma }{\sum }_{n>0}{\overline{P}}_n[1-cos(nKacos\beta )]},$$

(28)

where ${\sum }_n{\mathrm{\Gamma }}_n=\mathrm{\Gamma }$, ${\overline{P}}_n={\mathrm{\Gamma }}_n/\mathrm{\Gamma }$, and the total jump rate is now given by the following:

$$\mathrm{\Gamma }\left(K\right)={\displaystyle \frac{\alpha \left(K\right)}{2{\sum }_{n>0}{\overline{P}}_n[1-cos(nKacos\beta )]}}.$$

(29)

Equation (29) clearly shows the relationship between the surface structure and the dynamics through the total jump rate. The moments and cumulants of the probability distribution function can again be straightforwardly calculated. At second order, one has the following:

$${⟨L_{\Vert }^2\left(t\right)⟩}_c=⟨L_{\Vert }^2\left(t\right)⟩=Dt,$$

(30)

where the tracer diffusion coefficient D is expressed as [10]

$$\begin{array}{ccc}\hfill D& =& 2\mathrm{\Gamma }{a}^{2}co{s}^2\beta \sum _{n=1}{\overline{P}}_n{n}^2,\hfill \\ & =& 2\mathrm{\Gamma }{\displaystyle \frac{b^2+2b+2}{b^3}}{e}^{-b}{a}^{2}co{s}^2\beta .\hfill \end{array}$$

(31)

The first line of this equation indicates that only the second moment of ${\overline{P}}_n$ is involved, whereas in the second line, one assumes that the jump probabilities decrease exponentially with jump length n, ${\overline{P}}_n=exp(-bn)$. The summation over n can be replaced by an integral from 1 to infinity. Similar expressions can also be found in the literature [21].

3.2. Low and Intermediate Surface Coverages

In this subsection, we now incorporate the effect of finite coverage. For simplicity, only one projected jump vector is again assumed here, although the extension to multiple contributions is immediate. In this regime, diffusion must be interpreted as a collective phenomenon involving interacting adsorbates. Let $P_n^c\left(t\right)$ denote the probability of finding one or more adsorbates at site n at time t. Conservation of the total number of adsorbates requires the following:

$$\sum _{n=1}^N{P}_n^c\left(t\right)=N_a,$$

(32)

where $N_a$ denotes the number of adsorbates, and N denotes the total number of adsorption sites. The surface coverage is therefore defined as follows:

$$\theta ={\displaystyle \frac{N_a}{N}}.$$

(33)

At this point, one has to distinguish between the tracer diffusion coefficient, D, and the collective diffusion coefficient, $D_c$.

Following the same tight-binding description of the diffusion process, one can start from Equation (25), now considering the explicit $\theta$-dependence of the jump/transition rates:

$${\dot{P}}_n^c\left(t\right)=\sum _{l=1}^{\infty }\{{\mathrm{\Gamma }}_l^{+}\left(\theta \right)P_{n-l}^c\left(t\right)+{\mathrm{\Gamma }}_l^{-}\left(\theta \right)P_{n+l}^c\left(t\right)-({\mathrm{\Gamma }}_l^{+}\left(\theta \right)+{\mathrm{\Gamma }}_l^{-}\left(\theta \right))P_n^c\left(t\right)\}.$$

(34)

Equation (8) is expressed now as

$$I(K,t;\theta )=e^{-2t\mathrm{\Gamma }(K;\theta ){\sum }_{n>0}{\overline{P}}_n\left(\theta \right)[1-cos(nKacos\beta )]},$$

(35)

where ${\sum }_n{\mathrm{\Gamma }}_n\left(\theta \right)=\mathrm{\Gamma }\left(\theta \right)$ and ${\overline{P}}_n\left(\theta \right)={\mathrm{\Gamma }}_n\left(\theta \right)/\mathrm{\Gamma }\left(\theta \right)$. After Equation (35), the total jumping/transition rate is given by

$$\mathrm{\Gamma }(K;\theta )={\displaystyle \frac{\alpha (K;\theta )}{2{\sum }_{n>0}{\overline{P}}_n\left(\theta \right)[1-cos(nKacos\beta )]}}.$$

(36)

Equation (36) again shows the relationship between the surface structure and the diffusion process through the total jump rate in terms of the surface coverage. Now, following the same procedure as above, at second order, one has the following:

$${⟨L_{\Vert }^2(t;\theta )⟩}_c=⟨L_{\Vert }^2(t;\theta )⟩=D_{c}t,$$

(37)

where the collective diffusion coefficient $D_c$ is expressed as

$$D_c(K;\theta )=2\mathrm{\Gamma }(K;\theta )a^{2}co{s}^2\beta \sum _{n=1}{\overline{P}}_n\left(\theta \right)n^2,$$

(38)

and the second moment of ${\overline{P}}_n\left(\theta \right)$ also depends now on $\theta$. As shown previously, jump probabilities are assumed to decrease exponentially with n according to

$${\overline{P}}_n\left(\theta \right)=exp[-b\left(\theta \right)n],$$

(39)

and by integration from 1 to infinity, the collective diffusion coefficient $D_c$ is expressed as

$$D_c=2\mathrm{\Gamma }\left(\theta \right){\displaystyle \frac{b{\left(\theta \right)}^2+2b\left(\theta \right)+2}{b{\left(\theta \right)}^3}}{e}^{-b\left(\theta \right)}{a}^{2}co{s}^2\beta ,$$

(40)

where $b\left(\theta \right)$ could be extracted from a fitting procedure.

Several modeling routes are possible at this stage. First, because the dephasing rate has been assumed to be a parametric function of the surface coverage, this dependence can be extracted from the experiment. Alternatively, one could also assume an analytical function for the dephasing rate in terms of surface coverage and K and derive the associated total jump rate. This approach can be regarded as the most general case. A third alternative is to incorporate explicit interaction models together with a suitable ansatz for ${\overline{P}}_n\left(\theta \right)$. In all cases, the CF method remains directly applicable.

3.2.1. Hard Core Interaction

The simplest interaction model is hard-core exclusion, where double occupancy is forbidden [22]. The corresponding master/rate equation can be written as

$$\begin{array}{ccc}\hfill {\dot{P}}_n^c\left(t\right)& =& \sum _{l=1}^{\infty }\{{\mathrm{\Gamma }}_l^{+}{P}_{n-l}^c\left(t\right)[1-P_n^c\left(t\right)]+{\mathrm{\Gamma }}_l^{-}{P}_{n+l}^c\left(t\right)[1-P_n^c\left(t\right)]\hfill \\ & -& ({\mathrm{\Gamma }}_l^{+}[1-P_{n-1}^c\left(t\right)]+{\mathrm{\Gamma }}_l^{-}[1-P_{n+l}^c\left(t\right)])P_n^c\left(t\right)\},\hfill \end{array}$$

(41)

where $[1-P_n^c\left(t\right)]$ gives the probability that site n is empty as a function of time. Equation (41) is then reduced to Equation (25), but now for $P_n^c\left(t\right)$:

$${\dot{P}}_n^c\left(t\right)=\sum _{l=1}^{\infty }\left[{\mathrm{\Gamma }}_l^{+}{P}_{n-l}^c\left(t\right)+{\mathrm{\Gamma }}_l^{-}{P}_{n+l}^c\left(t\right)-({\mathrm{\Gamma }}_l^{+}+{\mathrm{\Gamma }}_l^{-})P_n^c\left(t\right)\right].$$

(42)

The same expression obtained previously for D (first line of Equation (31)) should now be valid for $D_c$.

If one wants to relate D to $D_c$, the new master equation in terms of $P_n\left(t\right)$ is

$$\begin{array}{ccc}\hfill {\dot{P}}_n\left(t\right)& =& \sum _{l=1}^{\infty }\{{\mathrm{\Gamma }}_l^{+}{P}_{n-l}\left(t\right)[1-P_n^c\left(t\right)]+{\mathrm{\Gamma }}_l^{-}{P}_{n+l}\left(t\right)[1-P_n^c\left(t\right)]\hfill \\ & -& ({\mathrm{\Gamma }}_l^{+}[1-P_{n-1}^c\left(t\right)]+{\mathrm{\Gamma }}_l^{-}[1-P_{n+l}^c\left(t\right)])P_n\left(t\right)\},\hfill \end{array}$$

(43)

and by assuming a mean-field approximation, we replace any $P_n^c\left(t\right)$ by the surface coverage $\theta$, so that Equation (43) is now written as follows:

$${\dot{P}}_n\left(t\right)=(1-\theta )\sum _{l=1}^{\infty }\{{\mathrm{\Gamma }}_l^{+}{P}_{n-l}\left(t\right)+{\mathrm{\Gamma }}_l^{-}{P}_{n+l}\left(t\right)-({\mathrm{\Gamma }}_l^{+}+{\mathrm{\Gamma }}_l^{-})P_n\left(t\right)\},$$

(44)

and from Equations (31) and (38), we have the well-known Darken equation:

$$D=(1-\theta )D_c,$$

(45)

where the factor $(1-\theta )$ is known as the blocking factor. Both diffusion coefficients become equal as the coverage approaches zero.

3.2.2. Binomial Probability Distribution

For single-particle diffusion, ${\overline{P}}_n$ gives the jump probabilities. At finite surface coverage, let ${\overline{P}}_n^c$ denote the corresponding collective jump probabilities. If a binomial jump distribution is assumed:

$$P_n^c=\sum _{k=1}^{\infty }{\overline{P}}_k\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right){\theta }^{n-k}{(1-\theta )}^k$$

(46)

the corresponding first two moments are then

$$\begin{array}{ccc}\hfill {⟨n⟩}_c& =& \sum _{n=1}^{\infty }n{P}_n^c={\displaystyle \frac{⟨n⟩}{1-\theta }}\hfill \\ \hfill {⟨n^2⟩}_c& =& \sum _{n=1}^{\infty }{n}^2{P}_n^c={\displaystyle \frac{⟨n^2⟩+\theta ⟨n⟩}{{(1-\theta )}^2}},\hfill \end{array}$$

(47)

where $⟨n⟩={\sum }_{n=1}^{\infty }n{\overline{P}}_n$ and $⟨n^2⟩={\sum }_{n=1}^{\infty }{n}^2{\overline{P}}_n$. Thus, the diffusion coefficient is now expressed as

$$\begin{array}{ccc}\hfill D_c& =& 2\mathrm{\Gamma }{a}^{2}co{s}^2\beta \sum _{n=1}{P}_n^c{n}^2,\hfill \\ & =& 2\mathrm{\Gamma }{a}^{2}co{s}^2\beta {\displaystyle \frac{⟨n^2⟩+\theta ⟨n⟩}{{(1-\theta )}^2}},\hfill \end{array}$$

(48)

which correctly reduces to the dilute-limit expression, Equation (31).

3.3. Sum Frequency Rules

Our next step is to calculate the frequency moments (or frequency rules) when the ISF is given by Equation (20). Because this function is an exponential function of time, the DSF is a Lorentzian function, which is an even function of frequency, and, therefore, only even frequency moments are allowed. Thus, after Equation (4), the ISF is expressed as follows:

$$I(K,t;\theta )=1-{\displaystyle \frac{t^2}{2}}⟨{\omega }^2(K;\theta )⟩+{\displaystyle \frac{t^4}{4!}}⟨{\omega }^4(K;\theta )⟩+\dots ,$$

(49)

with

$$\begin{array}{ccc}\hfill ⟨{\omega }^0⟩& =& 1,\hfill \\ \hfill ⟨{\omega }^{2n}⟩& =& \mathrm{\Gamma }{\left(\theta \right)}^{2n}{[1-cos(Kacos\beta )]}^{2n}.\hfill \end{array}$$

(50)

These moments depend explicitly on the total jump rate and therefore retain dynamical content. Similar expressions can be easily derived when longer jumps are included; see Equation (35). In this sense, surface diffusion analyzed by He atom scattering differs from other scattering experimental techniques, such as neutron and light scattering, where the low-order frequency moments admit different interpretations. For example, in neutron scattering, the zeroth moment provides the static structure factor and the second moment provides the square of the product of the momentum transfer and thermal velocity [17].

3.4. The Static Structure Factor

According to Frenken and Hinch [23], a simple expression for the SSF of a low-density adatom overlayer with a lattice-gas structure can be written as

$$F\left(\mathbf{K}\right)=⟨e^{i\mathbf{K}.\mathbf{L}}⟩=\sum _{\mathbf{L}}{P}_{\mathbf{L}}{e}^{i\mathbf{K}.\mathbf{L}},$$

(51)

where again the 2D lattice vector $\mathbf{L}$ runs over the 2D adatom lattice, and $P_{\mathbf{L}}$ is the probability of finding an occupied site at $\mathbf{L}$ given that the site at the origin is occupied. At a very low surface coverage, one could assume that $P_{\mathbf{L}\ne 0}=\theta$, and therefore

$$F\left(\mathbf{K}\right)=1-\theta +\theta \sum _{\mathbf{L}}{e}^{i\mathbf{K}.\mathbf{L}},$$

(52)

where the sum over position vectors can be transformed into a sum over reciprocal lattice vectors, leading to a series of diffraction peaks given by $\delta (\mathbf{K}-\mathbf{G})$.

As discussed above, in Equation (51), the scalar product in the exponential can again be written as $\mathbf{K}.\mathbf{L}=K{L}_{\Vert }$, $L_{\Vert }$ being a sum over all the $\mathbf{L}$-vectors with a nonzero projection along $\mathbf{K}$. Following the same procedure used for the ISF, one has the following:

$$F(K;\theta )=⟨e^{iK{L}_{\Vert }\left(\theta \right)}⟩=\sum _{n=0}^{\infty }{\displaystyle \frac{{\left(iK\right)}^n}{n!}}⟨L_{\Vert }^n\left(\theta \right)⟩.$$

(53)

The $n^{th}$ derivative of the SSF at $K=0$ provides the $n^{th}$ moment of the static adsorbate distribution:

$$⟨L_{\Vert }^n\left(\theta \right)⟩={\left(-i{\displaystyle \frac{𝜕}{𝜕K}}\right)}^{n}F(K;\theta ){|}_{K=0},$$

(54)

as well as the $n^{th}$-cumulant:

$${⟨L_{\Vert }^n\left(\theta \right)⟩}_c={\left(-i{\displaystyle \frac{𝜕}{𝜕K}}\right)}^{n}lnF(K;\theta ){|}_{K=0}.$$

(55)

These moments and cumulants are critical for determining the adsorbate distribution as well as the separation distances between adsorbates on the surface. From the experiment, the SSF is easily extracted as a function of K. Furthermore, as before, a jump distribution as a function of coverage could be assumed for $P_{\mathbf{L}}$.

4. Conclusions

The ISF is the central dynamical observable in surface diffusion, and its interpretation as a CF provides a natural and efficient route to the moments and cumulants of the jump distribution. At finite surface coverage, the CF method allows adsorbate interactions to be incorporated in a compact manner through coverage-dependent jump probabilities and dephasing rates.

The ISF decays exponentially with time according to Equation (4). The dephasing rate, $\alpha (K;\theta )$, is a periodic function of the momentum transfer K for each surface coverage and mainly characterizes the diffusion rate in each measurement. Thus, both functions play a fundamental role in any diffusion process. In the dilute regime, this rate displays maxima and minima, leading to faster or lower exponential decays, respectively. The corresponding width of the quasi-elastic peak thus increases or decreases according to Equation (5). In particular, when the rate reaches a minimum, the corresponding K value coincides with the first Bragg diffraction peak [2]. This is the condition of the so-called de Gennes narrowing [6] in the quasi-elastic peak. Furthermore, after Equations (21) and (29), the total jump/transition rate follows the same trend as the dephasing rate, as well as the tracer diffusion coefficient after Equations (23) and (31) (in order to simplify the discussion, only one high-symmetry direction with nonzero projection along $\mathbf{K}$ is assumed).

The increase in surface coverage implies higher dephasing rates, faster decays of the ISF, and broadening of the quasi-elastic peak, as observed, for example, in the systems Na/Cu(001) and CO/Pt(111) [2,3]. Moreover, at least for CO/Pt(111), this is true for the K-values used in the experiment. Following the theoretical aspects developed here, the moments and cumulants at any order as a function of coverage are easily calculated analytically with the CF method. Concerning the collective diffusion coefficient, this method is simpler than the standard way to calculate it theoretically. When using, for example, Langevin calculations with increasing coverage, one needs to know the positions and velocities of all adsorbates as functions of time. From this information, one calculates the corresponding mean square displacements or the velocity/total flux autocorrelation functions. Sometimes, the center-of-mass displacement is also calculated for this purpose once the so-called thermodynamic factor is known (or even from a Darken equation). Obviously, in order to carry out such classical calculations, several points should be taken into account when increasing the coverage: first, the activation energy and possibly the friction coefficient should be changed; second, the interaction potential as well as the interaction between adsorbates should also change, and third, longer jumps should be more important.

A second major result of this analysis is that the SSF is also a characteristic function, but now associated with the distribution of adsorbate separation distances. This extends the usefulness of the CF method beyond dynamical observables and provides a unified description of both diffusion dynamics and structural correlations.

When very high surface coverages are involved, some two-dimensional phase transitions could take place. This is not the regime where the CF method could readily be applied. Furthermore, the method proposed here could be easily extended to neutron scattering, given that this scattering is not fully coherent. Finally, it would be interesting to explore this method for grain boundary diffusion in metals [24,25] and diffusion in polycrystalline alloys [26].