Data Analysis Techniques

Abstract

In the following sections the concepts and techniques used in the data analysis are illustrated. Section 4.1 shortly explains the calculation of charge and screening parameter by wave spectra analysis. The structural properties of the two-dimensional system, and the methods to obtain them are described in Sect. 4.2 starting with the defect analysis in Sect. 4.2.1. The long range translational and orientational order of the system are described by means of the pair- and bond-correlation functions in Sect. 4.2.2 and 4.2.3. A measure for local order is introduced in Sect. 4.2.4 with the bond order parameter which can be defined at each respective particle position within the lattice. The last Sect. 4.3 concludes this section with the statistical description of the dynamics of a system of particles with regard to distribution functions of displacements and velocities.

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In the following sections the concepts and techniques used in the data analysis are illustrated. Section 4.1 shortly explains the calculation of charge and screening parameter by wave spectra analysis. The structural properties of the two-dimensional system, and the methods to obtain them are described in Sect. 4.2 starting with the defect analysis in Sect. 4.2.1. The long range translational and orientational order of the system are described by means of the pair- and bond-correlation functions in Sect. 4.2.2 and 4.2.3. A measure for local order is introduced in Sect. 4.2.4 with the bond order parameter which can be defined at each respective particle position within the lattice. The last Sect. 4.3 concludes this section with the statistical description of the dynamics of a system of particles with regard to distribution functions of displacements and velocities.

4.1 Estimation of Particle Charge and Screening Parameter

The particle charge $Q$ and the screening parameter $\kappa=\Updelta/\lambda_{D}$ can be derived from the sound velocities of phonons in a plasma crystal. Here $\Updelta$ is the interparticle spacing and $\lambda_{D}$ the Debye length of the particles. Phonons are natural waves which exist in any plasma crystal due to random particle motions without the need of excitation. Their wave numbers $k$ and the wave frequencies $\omega$ for longitudinal (index “l”) and transverse (index “tr”) waves are connected through the dispersion relations $\omega=c_{l,tr}k$, with the respective sound velocities $c_{l}$ and $c_{tr}$. The following relations show the dependence of the sound velocities on the charge and $\kappa$ in the limit of $k\rightarrow 0\;\hbox{and}\; \kappa \rightarrow 0$[1, 2]:

$$ {\frac{c_{tr}}{c_{0}}} = 0.51317-0.0226 (\kappa/l_{{\rm corr}})^{2} \approx 0.51317 $$

(4.1)

$$ {\frac{c_{l}}{c_{0}}} = {{2.585}\over \sqrt{\kappa/l_{{\rm corr}}}} $$

(4.2)

$$ c_{0} =\Upomega_{0}\Updelta = {{Q}\over \sqrt{4\pi\epsilon_{0}m\Updelta^{3}}}\Updelta $$

(4.3)

$\Upomega_{0}$ is the plasma frequency and $l_{\rm corr}=\sqrt{2/\sqrt{3}}\approx1.075$ is a correction factor accounting for the geometrical arrangement in a 2D hexagonal lattice.

With $c_{0}$ the particle charge can be calculated as

$$ Q=c_{0}\sqrt{4\pi\epsilon_{0}m\Updelta} $$

(4.4)

The screening parameter $\kappa$ is given by the direct dependency of $c_{l}$ on $(\sqrt{\kappa})^{-1}$ in (4.2). The sound velocities can be obtained from the wave spectra $V(k,\omega )$ of the phonons by the following procedure [3–6]:

The particle velocities $v_{x,y}(t)$ are averaged in spatial bins of the width $\delta x$ and in temporal bins of the width $\delta t$ for the directions $x\;\hbox{and}\;y$ in the image separately. The bin widths have to be chosen accordingly to get a good resolution. This gives matrices $V(x,t),V(y,t)$ with equally spaced entries—each component $(x,t),(y,t)$ corresponds to an equally spaced range in space and time, with the quantity of the entry being the average of the velocity components of all particles falling into that range.

$V(x,t)\;\hbox{and}\; V(y,t)$ are then Fourier transformed to the frequency space to obtain the spectra $V(k_{x},\omega),V(k_{y},\omega)$. The two matrices for $x\;\hbox{and}\; y$ are then squared and added to get the final spectrum $V(k,\omega)$, dependent on the frequency $\omega$ and the wave number $k$. The left panel of Fig. 4.1 shows an example of such a matrix in the frequency space. Each pixel in the plot corresponds to an entry of $V(k,\omega)$ at the respective position $(k,\omega)$. The brightness is a measure for the quantity of the matrix entry, where brighter pixels mean higher values. The two wave branches $\omega_{l},\omega_{tr}$ can be seen as the brighter accumulations of pixels.

Fig. 4.1

a Example of a wave spectrum for one of the performed experiments. Brighter pixels mean a higher density of velocities at the respective wave number $k$ and frequency $\omega$. Yellowdiamonds mark the points used for the direct linear fit (white lines) to the two branches. The blue and red dashed lines correspond to the fits of the theoretical model. b Measured sound velocity vs. $k$. The error bars are calculated from the dispersion of intensity values at one $k$ from the spectrum in (a). The red and blue lines are fits of theoretical curves giving $\kappa=0.56\pm0.23$ from the longitudinal, and $Q=10500\pm300$e from the transverse sound velocities, respectively

Both branches of the spectrum have to be fitted by lines for small $k$ as it is implied by (4.1) and (4.2). From the theory follows [1], that the transverse branch $\omega_{tr}$ (lower branch in Fig. 4.1), giving the particle charge, is linear over a wider range, while the longitudinal branch is linear only for very small $k$. Due to the often very poor resolution and contrast at low $k$, which arises from the limited system size and length of the time series in real space, the fitting of a line to $\omega_{l}(k)$ is usually subject to large errors. Note that the large uncertainty in the particle velocities discussed in Sect. 3.3 further diminishes the quality of the spectra.

Instead of directly applying fits to the spectra, first the positions $(k,\omega_{l,tr})$ defining the wave branches are identified. For one value of $k$ and a restricted range of $\omega$ at approximately the position of one of the wave branches, all pixels brighter than a threshold are chosen. This is done for both branches separately. $\omega_{l,tr}$ is then the intensity weighted center of these pixels. The dispersion of intensity values gives error bars for $\omega_{l,tr}$. The positions $(k,\omega_{l,tr})$ define the sound velocities $v_{l,tr}=\omega_{l,tr}/k$. The obtained sound velocities are plotted vs. $k$ in Fig. 4.1b, with error bars are transformed from the error bars of $\omega_{l,tr}$.

It is now possible to fit the theoretical model $c_{tr}=0.51317c_{0}$ to the transverse branch $v_{tr}$ (lower curve in Fig. 4.1b) with $c_{0}$ as fit parameter, and with that to obtain the particle charge $Q$ from (4.4) (the interparticle spacing $\Updelta$ is usually known from the pair correlation function, see Sect. 4.2.2). The longitudinal branch in that range of $k$ is clearly not linear. To find a function which could be fitted over the whole range of $k$, theoretical curves $c_{l,tr}/c_{0}$ for different $\kappa/l_{\rm corr}$ between 0.5 and 3 were calculated for a similar range of $k$ with the $c_{o}$ obtained from the transverse wave. The transverse velocities shown in the Fig. 4.2b are only weakly dependent on $\kappa$,as expected. From the longitudinal velocities $c_{l}/c_{0}$ (shown in Fig. 4.2a) a polynomial fit connecting the curves for different $\kappa$ was used to create coefficients depending on $\kappa$. Then a fifth grade polynomial with those coefficients was fitted to $v_{l}/c_{0}$ (red line in Fig. 4.1b), yielding a best estimate for $\kappa$.

Fig. 4.2

Theoretical curves $c_{l}/c_{0}$ (a) and $c_{tr}/c_{0}$ (b) vs. wave number $k$ for values of $\kappa=0.5,0.65,0.8,1,1.2,1.5,2,3$. The wave polarization direction is $0^{\circ}$

4.2 Structural Analysis

4.2.1 Defects

Defects are disruptions of the crystal structure. Possible defects in a two-dimensional system are point defects—or disclinations—consisting of a vacancy or an interstitial. The lattice around such an isolated defect is distorted so that the crystal structure is maintained. In a hexagonal lattice with typically six nearest neighbors to each lattice site, the most common point defects are 5-folded (vacancy) or 7-folded (interstitial) lattice sites, one of which is illustrated in Fig. 4.3a.

Point defect positions are found by performing a Delauney triangulation on the particle coordinates $(x,y)$ in an image. In the later analysis, the Triangle-algorithm described in [7] was used. The triangulation covers the two-dimensional xy-surface with a mesh of triangles between neighboring particles under the condition that lines never cross. Each lattice site is then connected by $n$ lines to the adjacent $n$ lattice sites. Those bonds define the number and position of all nearest neighbor particles. In the hexagonal 2D lattice, the lattice site is a point defect if $n\neq6$.

Dislocations Another type of defect is the dislocation which can be understood as an additional row of particles inserted into an ideal lattice [8]. At the end of this row there will be a pair of disclinations to adjust the lattice, usually a pair of a 5- and 7-fold defect as seen in Fig. 4.3b. In the following the expression “dislocation” will mean a pair of a 5-fold and 7-fold disclination, since this is the most prominent in the hexagonal lattice. Dislocations can form pairs as shown in Fig. 4.3c. This configuration, also called “dislocation pair”, is used in the theories for dislocation-mediated two-dimensional melting, which will be introduced later.

Dislocations can be described by means of their Burgers vector [8, 9]. If one draws a closed path around a dislocation, jumping from one lattice site to the next, the same path (the equal number of jumps in the same directions as before) will not close in an ideal lattice. The Burgers vector is the additional vector needed to close that path. It is perpendicular to the dislocation line, i.e. the vector connecting the two disclinations. In the case of a dislocations pair as in Fig. 4.3c, the net Burgers vector will be zero. For this, the two dislocations do not necessarily have to be adjacent in the lattice, only their orientation (the direction of the Burgers vector) is important.

Fig. 4.3

Defects in a two-dimensional hexagonal lattice. a Free disclination (5-fold). b Free dislocation (pair of a 5- and a 7-fold disclination). c Pair of dislocations. The lines indicate the nearest neighbor bonds with dashed lines symbolizing 5-fold and dotted lines 7-fold lattice sites

Defect Analysis The arrangement of defects in a lattice, for example as dislocations, can provide valuable information on the system in addition to the absolute number of disclinations. To investigate dislocations, one has to assign adjacent point defects to each other. In experimental data the following defect configurations (for 5- and 7-folds) can usually be seen:

• 1. isolated, “free” disclinations (either 5- or 7-fold)

• 2. isolated, “free” dislocations

• 3. open chains with alternating 5-folds and 7-folds

• 4. closed chains with alternating 5-folds and 7-folds (“loops”, loops of four adjacent defects are an unique dislocation pair)

• 5. clusters of defects (a larger amount of disclinations in one region, without apparent structure)

The points 2–5 always contain at least two defect lattice sites directly adjacent to each other. Point 1 does not need further examination, and for the second case the assignation of the defect pair is straightforward. There are two approaches for analysis of the other cases:

• 1. Always pair those 5- and 7-fold defects with the smallest distance between them. This is a very simple procedure, but it can only identify free dislocations, not complex structures such as chains or loops.

• 2. After finding a defect pair, search all neighboring lattice sites of each defect for other defects successively until no further adjacent defects are found. With this procedure, chains and closed loops can be identified. Dislocations can be associated within this structures. Defect clusters can not be specified correctly, since they are misinterpreted as chains or loops.

The second procedure is preferable, because it finds both free dislocations and chains or loops. The only problem are the identification of clusters with no apparent structure, or the misinterpretation of randomly distributed disclinations, which often appear in liquid-like states of high disorder, as structures.

Point defects with less than five, or more than seven neighbors can be observed in complex plasmas, but their numbers are very small, especially in the crystalline state they are practically absent.

4.2.2 Pair Correlation Function

The pair—or translational correlation function $g(r)$, also called radial density distribution, shows the probability to find a particle in a distance $r$ from another particle [10]. It is computed for an image by choosing consecutively each particle as a center particle $i$ and counting the number of particles $j$ found in a ring with radius $r$ and width ${\it dr}$ around that particle. $r$ goes up to a maximum radius $r_{\max}$. The results for the different center particles $N_{cp}$ are averaged and then normalized by the particle density $N_{\it cp}/(\pi r^{2})$ times the ring area $2\pi {\it rdr}$ for each $r$. The normalization factor was further improved by taking into account that rings might cross the edge of the analyzed region, or the image edges. Therefore a correction factor $\rho_{{\rm corr}}$ was calculated specifically for each center particle and $r$ from a simple geometric construction, and the bin counts were normalized by it. This ensured that for large $r,g(r)$ goes to 1.

$$ g(r)={\frac{1}{2\pi \it r dr}}{\frac{\pi r_{\max}^{2}}{N_{cp}}}{\frac{1}{N_{cp}}}\sum_{i=1}^{N_{cp}} {\frac{1}{\rho_{{\rm corr},\,i}(r)}}\sum_{r-dr < r_{j}-r_{i}\,\leq r+dr} 1 $$

(4.5)

The shape of $g(r)$ in an ideal lattice is composed of singular peaks at the distinct distances given by the lattice constant, and of an decaying envelope of the peakamplitudes due to the normalization. For $r\rightarrow\infty$, $g(r)$ goes to 1. In the real crystal, the thermal motion of the particles caused by the finite particle temperature $T$ broadens and lowers the peaks. The measured curve then looks like a series of Gaussian functions, each centered around the respective distance given by the lattice constant (or interparticle distance). The peak amplitudes decrease with increasing $r$, as shown in the example in Fig. 4.4a. In a liquid-like state, Fig. 4.4b, the peaks become wider and begin to overlap while their amplitude decays very fast exponential with $r$.

Fig. 4.4

Examples of $g(r)$ found in two-dimensional complex plasmas. $r$ is normalized by the mean particle distance $\Updelta$. The solid lines correspond to fits using (4.6) and (4.7). a Solid state with the fit function proposed by Beresinskii. b Liquid state with exponential fit

The following fit function for $g(r)$ was proposed in [11] and used in 2D complex plasma analysis [12]; the free fit parameters ($k,\sigma_{0},\Updelta$) are highlighted:

$$ g_{{\rm fit},1}(r)=\left[ {\frac{k}{\sqrt{2\pi}}} {\frac{1}{\sigma_{0}}}\sum_{i}{\frac{g_{id}(x_{i})}{\Delta x_{i}}} \exp{\left(-{\frac{(r-\Delta x_{i})^{2}}{{2\sigma_{0}}^{2}}}\right)}-1 \right]\times \exp{\big(\!{-}r/\xi\big)}+1 $$

(4.6)

with a prefactor $k$ depending on the normalization of $g(r)$ by the particle density, the peak width $\sigma_{0}$, the mean interparticle distance $\Updelta$ and an exponential decaying envelope $\propto e^{-r/\xi}$. $\xi$ is also called translational correlation length. $g_{id}(x_{i})$ is the total number of particles found on a ring with radius $x_{i}$ around a center particle in the ideal hexagonal lattice. The positions $x_{i}$ are distinct and defined by the translation vectors (1, 0) and $(1/2,\sqrt(3)/2)$ of the lattice. From this, $g_{id}(x_{i})$ can be calculated. The summation in (4.6) goes over all calculated $x_{i}$, here $n=120$ positions were used. This function is valid in liquid states, where an exponential decaying $g(r)$ is expected [13]. Another function was introduced by Beresinskii in [14, 15] for solid systems in one and two dimensions. The main difference to (4.6) is a peak width ${\tilde \sigma}$ depending on the distance $r$. With the correct normalization factors the following fit function was constructed (with the fit parameters $A_{\rm hex}$, $\sigma_{0},\Updelta,\xi$):

$$ g_{\rm fit,2}(r)=\left[{\frac{A_{\rm hex}} {(2\pi)^{3/2}}}{\frac{1}{\tilde\sigma}}\sum_{i=1}^{n}{\frac{g_{id}(x_{i})} {\Delta x_{i}}}\exp\left({\frac{-(r-\Delta x_{i})^{2}}{2\tilde\sigma^{2}}}\right)-1\right]\times\exp{\big(\!{-}r/\xi\big)}+1 $$

(4.7)

with $\tilde\sigma =\sigma_{0}\sqrt{\ln{\frac{r}{r_{0}}}}, r_{0}=0.3\Delta$

The prefactor is composed of the inverse particle density $A_{\rm hex}$, which should be equal to the area of one Voronoi cell around a particle in a hexagonal lattice, the factor $1/\sqrt{2\pi}\tilde\sigma$ from the Gaussian shape of the peaks, and $1/(2\pi)$ from the normalization of $g(r)$. Additional, a parameter $r_{0}$ is introduced, which was theoretically estimated to be between 0.2 and 0.4 for the investigated system. It was chosen to be 0.3 in the fit. The peaks become wider with increasing $r$ in this model, which agrees well with the observations in the experimental data. In fact, this function fitted the data better than (4.6) even in liquid-like states of high disorder. The following paragraph gives some details on the fit procedure itself to explain the interpretation of the outcome of the fit.

Fit Procedure The fit procedure used here is implemented in IDL (Interactive Data Language) and described in [16]. It performs a Levenberg-Marquardt least-squares minimization on a given set of points $y(x)$ using a user-supplied function $f(x,{\mathbf a})$. Two functions were written using (4.6) and (4.7), with the free parameters ${\mathbf a}=(A_{\rm hex},\Updelta,\sigma_{0},\xi)$. One set of guessed, initial parameters have to be supplied to the fit procedure. The best set of parameters is then the set which minimizes $\sum_{x}[y(x)-f(x,{\mathbf a})]^{2}$. In each iteration, the parameters are varied in direction of their negative gradient until the minimum is reached. In general this is a very effective method, but in case of the rather complex function $g_{\rm fit}(r)$ with four free parameters, the choice of the initial starting parameters can become crucial. There is always the danger that local minima of the parameter set appear for certain starting values, which then lead to ambiguous results. Therefore the fit should be repeated with different configurations to ensure the validity of the result.

The fit procedure also provides the $1\text{-}\sigma$ uncertainty for each fit parameter, which is calculated during the fit as described in [17], and the $\chi^{2}$ as a measure of the goodness of the fit. $\chi^{2}$ is the weighted sum of squared distances between data and fit, divided by the squared uncertainties of the data. Often the reduced $\chi^{2}$ is stated as $\chi^{2}_{\nu}=\chi^{2}/N_{\rm dof}\;{\hbox{with}}\; N_{\rm dof}$ being the number of degrees of freedom (number of fitted points minus number of fit parameters). The fit procedure assumes that the supplied uncertainties reflect the deviations of the fit model to the real data, therefore only by supplying valid uncertainties a meaningful $\chi^{2}$ and $1\text{-}\sigma$ can be obtained. Since the uncertainty in the single points of $g(r)$ is of purely statistical nature (only numbers of particles were counted), and the number of particles used in the analysis is very high (>1000), the statistical error bars are very small, which makes $\chi^{2}$ very large. With this kind of uncertainties it is not possible to take into account a deviation of the experiment from the model. Therefore, $\chi^{2}$ should not be interpreted as the real probability of the goodness of the fit, but the ratio $\chi^{2}_{\nu,B}/\chi^{2}_{\nu,E}$ (index $B$: Beresinskii fit, index $E$: exponential fit) can be used as an estimate to choose the best fitting function. To obtain more realistic values for the $1\text{-}\sigma$ uncertainty of each fit parameter, the reduced $\chi^{2}_{\nu}$ can be manually set to 1, implying that the fit is the best possible. With that, the $1\text{-}\sigma$ uncertainties are calculated again, and should now represent the actual deviation of data to fit function [12].

Interpretation The pair correlation function provides a good estimate for the mean interparticle spacing $\Updelta$ as the position of the first peak. Further is gives information on the range of translational order in a system, expressed by the correlation length $\xi$, which is of importance for some established theories of phase transitions, as will be discussed in later chapters.

Aside from the structural information immanent in $g(r)$, the fit parameter $\sigma_{0}$ is correlated to the particle temperature and with that to the dynamics of the particle motion. $\sigma_{0}$ is the dispersion of the particle separation, or lattice constant. This quantity is related to the radius $\sigma_{r}$ of the area a single particle occupies in average while it oscillates with the frequency $\Upomega_{E}$ around its lattice site ($\sigma_{r}$ is the width of the particle displacement distribution):

$$ \sigma_{r}=\sqrt{{\frac{{{\it k}_{\it B}{\it T}}}{m\Upomega_{E}^{2}}}}={\frac{\sigma_{0}}{\sqrt{2}}} $$

(4.8)

The factor $\sqrt{2}$ comes from the fact that $\sigma_{0}$ is the width of the Gaussian distribution $f({\varvec{\Delta}}_{\it ik})$ of the vectors ${\varvec{\Delta}}_{\it ik}$ between two particles $i,k$. The mean $\bar{\varvec{\Delta}}$ of the values ${\varvec{\Delta}}_{\it ik}$ is the distance between the mean lattice sites of the two particles, and ${\bf r}_{\it i,k}$ are the respective displacement vectors of the particles $\it i,k$ from their mean lattice site. Since ${\varvec{\Delta}}_{ik}-\bar{\varvec{\Delta}}= {\bf r}_{\it k}-{\bf r}_{\it i}$, it holds:

$$ \begin{aligned} f({\varvec{\Delta}}_{ik}) \propto \exp{ \left\{ -{\frac{ ({\varvec{\Delta}}_{ik}-\bar{\varvec{\Delta}})^{2}}{2\sigma_{0}^{2}}}\right\}} ={}& \exp{ \left\{ -{\frac{2{\mathbf r}^{2}}{2\sigma_{0}^{2}}}\right\}} = \exp{ \left\{ -{\frac{{\mathbf r}^{2}}{2\sigma_{r}^{2}}}\right\}} \propto f({\mathbf r}_{i}) f({\mathbf r}_{k}) \\ \Rightarrow \sigma_{0}^{2}/2={}&\sigma_{r}^{2} \end{aligned} $$

In the equality it was used that all ${\mathbf r}_{\it i,k}$ have the same distribution, which is only shifted in space by a constant factor, therefore ${\mathbf r}_{i}^{2}+{\mathbf r}_{k}^{2}=2{\mathbf r}^{2}$, and that the motion of the two particles are uncorrelated. Then ${\mathbf r}_{i}{\mathbf r}_{k}\,{\approx}\,0$. In other words, $f({\varvec{\Delta}}_{\it ik})$ is the convolution of two independent Gaussian distributions. Due to the convolution invariance of Gaussian distributions, the relation between the widths also follows directly.

4.2.3 Bond Correlation Function

An ideal hexagonal crystal has angles of multiples of $60^{\circ}$ between any two nearest-neighbor bonds, no matter how far the bonds are separated in space. This defines an orientational long range order which can be described by the bond correlation function $g_{6}(r)$ [13, 18]. To calculate $g_{6}(r)$, the nearest neighbor bonds have to be identified by a Delauney triangulation. The bond correlation function $g_{6}(r)$ is then defined as

$$ g_{6}(r) = \left|{\frac{1}{N_{B}}}\sum_{l=1}^{N_{B}}{\frac{1}{n(l)}}\sum_{k=1}^{n(l)} \exp\{i 6(\theta({\mathbf r}_{k})-\theta({\mathbf r}_{l}))\} \right| $$

(4.9)

Here $N_{B}$ is the total number of bonds in the crystal, $n(l)$ is the number of bonds at the distance $r$ from bond $l$, $\theta({\mathbf r}_{k,l})$ are the respective angles of bonds at ${\mathbf r}_{\it k,l}$ to an arbitrary axis. Note that $g_{6}(r)$ is always 1 for the perfect hexagon by definition. In the solid state, $g_{6}(r)$ should be constant and close to 1. Further, power-law and exponential decays for large $r$ are predicted in hexatic and liquid states, respectively [12, 13]. The hexatic state is an intermediate two-dimensional state assumed to appear between the solid and liquid phase according to some theories. This state will be addressed later in Chap. 6.

The following three models were fitted to $g_{6}(r)$:

• 1. Exponential decay $g_{6}(r)=A_{1} e^{-r/\xi_{6}}$ with the fit parameters $A_{1}$ and $\xi_{6}$

• 2. Power-law decay $g_{6}(r)=A_{2} r^{-\eta_{6}}$ with the fit parameters $A_{2}$ and $\eta_{6}$

• 3. Linear decay $g_{6}(r)=c_{6} r + A_{3}$ with the fit parameters $A_{3}$ and $c_{6}$.

The third model, the linear decay, is not mentioned in the theories, but it was added due to the findings in the experimental data.

The fits are applicable for large $r$ only, since the bond correlation function describes the long range behavior of the system. Values of $r\,{<}\,2\text{--}3\Updelta$ have therefore to be omitted in the fit. The degree of order goes roughly from unordered (exponential) to ordered (linear). $\xi_{6}$ serves as a correlation length comparable to $\xi$ of the pair correlation function. The decay of the power-law is slower than the exponential decay and could be applicable for a better ordered system.

The linear decay can be seen as a practically constant $g_{6}(r)$, but under the influence of long range effects not considered in the common theories of two-dimensional melting. In all experimental data a substantial slope $c\,{\neq}\,0$ was found in states appearing rather crystalline with regard to other properties, while power-law decays where hard to find at all. To examine the effect leading to a linear decay, an artificial particle lattice was generated and modified to simulate different deviations from the ideal lattice (compression, domain forming, rotation of domains). The bond correlation function was calculated and a linear decay of $g_{6}(r)$ appeared in such cases when larger adjacent domains where formed which were rotated relative to one another by large angles, while the unit cells around particles within each domain were kept ideal hexagonal. The domain boundaries consisted necessarily of defect strings to compensate for the deformation. The complete procedure and graphical results are given in detail in Chap. 10.

This result coincides with the observation in the experimental data, where the same kind of rotated domain structures could be identified. It should be noted that the strong influence of the domains on $g_{6}(r)$ could mask any power-law or exponential decay as well as prevent $g_{6}(r)$ from being constant for large $r$.

Examples for $g_{6}(r)$ are given in Fig. 4.5 for the two states crystalline (Fig. 4.5a) and liquid-like (Fig. 4.5b). The different models of linear (black line), power-law (blue line) and exponential (red line) decay were fitted and are shown for comparison.

Fig. 4.5

Examples of $g_{6}(r)$ as found in experimental data of two-dimensional complex plasmas. $r$ is normalized by the mean particle separation $\Updelta$. a The linear decay (black line) fits best, the exponential (red line) fits well but has to be omitted because $\xi_{6}\approx 26\Updelta$ which is larger than the actual system size. The power-law (blue line) fits only poorly. b Examples of different states of order, distinguished by different plot symbols ($\times,\diamond,+$). Colored lines correspond to fits with the decay power-law (blue) and exponential (red). The fits were performed for $r>2\Updelta$ only as indicated by the change from dotted lines to solid lines

4.2.4 Bond Order Parameter

A useful quantity to examine the lattice in terms of the local orientational order is ${\varvec{\Uppsi}}_{6}$ which is defined as (following the definition given in [19]):

$$ \begin{aligned} {\varvec{\Uppsi}}_{6,k} &= {\frac{1}{n}}\sum_{j=1}^{n} e^{6i\theta_{kj}}=\left|{\varvec{\Uppsi}}_{6,k}\right| e^{\left\{i\phi\right\}} \end{aligned} $$

(4.10)

$$ \begin{aligned} \phi &= \arctan{\left\{\Im(\varvec{\Uppsi}_{6,k})/\Re(\varvec{\Uppsi}_{6,k})\right\}} \end{aligned} $$

(4.11)

over the $n$ nearest neighbors of each particle $k\;\hbox{with}\;\theta_{kj}$ being the angle between the nearest-neighbor-bond of the particles $k\;\hbox{and}\;j$ and the $x$-axis (Fig. 4.6a) The axis can in fact be chosen arbitrary as long as it is fixed, but typically the image $x$-axis is taken.

The modulus $\left|{\varvec{\Uppsi}}_{6,k}\right|$ of this complex quantity is the bond order parameter [11] which is 1 by definition for an ideal hexagonal structure. It is often averaged over all particles in the lattice and then used as a measure for the mean local order of the crystal. In Fig. 4.6b it is represented as the length of the dashed line in the complex number plane.

The argument $\phi=\arg({\varvec{\Uppsi}}_{6,k})$ is the angle indicated in Fig. 4.6b. It is a measure for the unit cell orientation with respect to the $x$-axis (the unit cell is the Wigner-Seitz or Voronoi cell, not the larger cell spanned by the neighboring particles). For rotations of a unit cell around a center particle with respect to the fixed axis, i.e. a common rotation of all nearest neighbors, the angle between the axis and bonds modulo $\pi/3$ is mapped like:

$$ \begin{aligned} (\theta_{kj} \mod \pi/3) \in [0,\pi/6] \rightarrow{}& [0,\pi] \\ (\theta_{kj} \mod \pi/3) \in [\pi/6,\pi/3] \rightarrow{}& [-\pi,0] \end{aligned} $$

The dependence of $\phi$ on the degree of rotation is shown in Fig. 4.7. The black dots correspond to the ideal hexagonal unit cell. In the non-ideal crystal the angles might deviate from $60^{\circ}$, and this dependence is slightly shifted, but still gives an idea of the orientation of the cell. Only for defect lattice sites and strongly deformed hexagons the information in $\phi$ is not reliable when comparing it to other unit cell orientations.

Fig. 4.6

a Hexagonal cell around a particle $k$ defined by its nearest neighbors. The bond between particles $k\;\hbox{and}\; j$ has an angle $\theta_{kj}$ to the x-axis. b Imaginary plane with the dashed line being the modulus of ${\varvec{\Uppsi}}_{6,k}$ of one center particle $k$, and the angle $\phi$ being the argument of ${\varvec{\Uppsi}}_{6,k}$. In the ideal hexagon, the modulus is always 1. $\phi$ lies in the interval $[0,\pi]$ for rotations of the unit cell from $0\text{--}30^{\circ}$ from the x-axis, and in $[-\pi,0]$ for angles between $30\text{--}60^{\circ}$

Fig. 4.7

The argument $\phi\;\hbox{of}\;{\varvec{\Uppsi}}_{6}$ vs. rotation of a unit cell with respect to the x-axis. The black dots refer to an ideal hexagon, the open circles are a cell with angles between the bonds varying slightly from $60^{\circ}$

4.3 Statistical Evaluation of Particle Dynamics

If the crystalline particle system is in local equilibrium, the particle interaction with the plasma is balanced by the neutral gas friction. The particle motion around its mean lattice site can then be described by a Langevin equation [20]

$$ m\ddot{\mathbf {r}} = -m\Upomega_{E}^{2}{\mathbf r} -m\nu_{Ep}{\mathbf v}+\zeta (t) $$

(4.12)

with the particle mass $m$, displacement ${\mathbf r}$ from the mean lattice site and the velocities ${\mathbf v}$. The Einstein frequency $\Upomega_{E}$ is the frequency of the particle oscillation around its equilibrium position [9], and $\nu_{ep}$ the Epstein drag coefficient. $\zeta(t)$ is a stochastical force which is the driving thermal force originating from the finite temperature $T$ of the particles. It causes the particles to perform a random Brownian motion and counteracts the damping and the restoring forces.

The first term on the right hand side of (4.12) describes the restoring force, which drives the particle towards its mean lattice site, as a repulsive electric force between the equally charged particles. A mean lattice site with respect to the surrounding particles is defined as the position of the minimum of the electric potential of all neighboring particles. A particle oscillates around this center with a frequency $\Upomega_{E}$, depending on the shape of the potential and the particle charge. The basic interaction potential for the complex plasma was introduced as the Yukawa-type potential $\Upphi(r)= Q_{D}e^{-r/\lambda_{D}}/(4\pi\epsilon_{0}r)$ with the screening length $\lambda_{D}$, in Sect. 2.1.

The particle motion is damped mainly by collisions with neutral gas atoms. This process is described by the second rhs term in (4.12) and was explained in Sect. 2.2 as well approximated by Epstein damping. The rate of collisions and therefore the rate of damping, $\nu_{Ep}$, depends on the neutral gas properties pressure $p$, gas temperature $T_{g}$, mass of the gas atoms $m_{g}$ and on dust particle properties and can be calculated as [3]:

$$ \nu_{\it Ep} = \delta \sqrt{{\frac{8{\it m}_{\it g}}{\pi {\it k}_{\it B}{\it T}_{\it g}}}}{\frac{p}{\rho {\it r}_{\it p}}} $$

(4.13)

with the radius and mass density ${\it r}_{\it p}$ and $\rho$ of the particles and the Boltzmann constant $k_{B}$. The coefficient $\delta$ depends on the mechanism of the reflection of gas atoms from the particle surface. For thermal nonconductive, spherical particles, Epstein calculated $\delta=1.442$ in the case of diffuse reflection [21]. The coefficient $\delta$ was also measured in experiments from horizontal oscillations of particles in a potential well to be $1.48\pm0.05$ [22], and $1.26\pm0.13$ [23]. A vertical resonance method yielded $1.44\pm0.19$ [23].

The Langevin equation of motion can be solved by the Fokker-Planck equation which yields a particle ensemble that obeys a Maxwell-Boltzmann distribution at a particle temperature $T$. The Hamiltonian for one cell in the lattice is

$$ H=E_{\rm kin}({\mathbf v})+E_{\rm pot}({\mathbf r}) = {\frac{1}{2}}m({\mathbf v}-\langle{\mathbf v}\rangle)^{2} + W({\mathbf r}) $$

(4.14)

The mean value $\langle{\mathbf v}\rangle$ is subtracted to eliminate contributions to the kinetic energy due to motions of the center of the lattice site itself. The probability distribution becomes

$$ P({\mathbf r},{\mathbf v}) = C \exp{\left\{-{\frac{H}{{{\it k}_{\it B}{\it T}}}}\right\}} = C\exp{\left\{-{\frac{m({\mathbf v}{}-\langle{\mathbf v}\rangle)^{2}}{2{{\it k}_{\it B}{\it T}}}}\right\}} \exp{\left\{-{\frac{m\Upomega_{E}^{2}{\mathbf r}^{2}}{2{{\it k}_{\it B}{\it T}}}}\right\}} $$

(4.15)

with $C$ a constant factor depending on the normalization.

The probability distribution can be separated into functions for the several components of the displacement and velocity vectors as long as they are independent:

$$ P({\bf r},{\bf v}) = p({\bf r})p({\bf v}) = p(r_{x})p(r_{y}) p(v_{x})p(v_{y})\propto e^{-{\frac{r_{x}^{2}}{2\sigma_{r}}}}\, e^{-{\frac{r_{y}^{2}}{2\sigma_{r}}}} \, e^{-{\frac{(v_{x}-\bar v_{x})^{2}}{2\sigma_{v}}}}\, e^{-{\frac{(v_{y}-\bar v_{y})^{2}}{2\sigma_{v}}}} $$

(4.16)

The above distribution functions are provided directly by the measurement of particle coordinates in the images, provided there is a high enough spatial and temporal resolution to resolve the particle oscillation amplitude as well as the velocity.Of importance here is the quantity of the measurement error which puts a limit on the smallest measurable distance.

The calculation of the velocity was given in (3.2). The displacements ${\mathbf r}(t)=\left( {r_{x}(t) \atop r_{y}(t)}\right)$ can be estimated by defining a mean lattice site for each particle $i$ in one frame at time $t$ as the mean of all $n$ nearest neighbor coordinates ${\mathbf x}_{k}(t)$:

$$ {\mathbf r}(t) = {\mathbf x}_{i}(t)-{\frac{1}{n}}\sum_{i=1}^{n}{\mathbf x}_{k}(t) $$

(4.17)

where the coordinates ${\mathbf x}_{i}(t),{\mathbf x}_{k}(t)$ are with respect to the image axis. The mean lattice site is then time-dependent. Motions not related to the particle oscillation like rotations or shifts of the whole system are therefore eliminated in ${\mathbf r}_{i}(t)$. This method is only applicable as long as the crystal structure is well defined. If the nearest neighbors of a particle change from one frame to the next, e.g. due to diffusion, it is not possible to define the mean lattice site clearly, and large jumps in ${\mathbf r}$ can be the consequence.

The displacement and velocity distributions are given by the histograms of the respective quantity. Gaussian fits $A \propto e^{-(\xi-\bar \xi)^2/(2\sigma_{\xi}^{2})}$ with $\xi=r_{x},r_{y},v_{x},v_{y}$ to each component of those quantities provide the widths $\sigma_{r}, \sigma_{v}$ separately for the x- and y-direction. In an isotropic system, the fits produce the same outcome for both directions. Such fits were performed with the same Levenberg-Marquardt-Algorithm as it was introduced in Sect. 4.2.2 for the pair correlation function, giving the goodness of the fit $\chi^{2}$ and the $1\text{-}\sigma$ uncertainty for the fit parameters for a qualitative evaluation.

In the case of problems with the mean lattice site identification, another method to at least obtain the dispersion of the displacement was also presented Sect. 4.2.2 through the relation $\sigma_{r}=\sigma_{0}/\sqrt{2}$ of $\sigma_{r}$ to the dispersion of the interparticle distances (4.8). $\sigma_{0}$ can either be obtained from the fit to the pair correlation function, or directly from a Gaussian fit to the histogram of interparticle distances in one image.

The obtained widths $\sigma_{r}, \sigma_{v}$ are connected to the particle temperature and Einstein frequency:

$$ \sigma_{r}=\sqrt{{\frac{{{\it k}_{\it B}{\it T}}}{m\Upomega_{E}^{2}}}}, \quad\sigma_{v}=\sqrt{{\frac{{{\it k}_{\it B}{\it T}}}{m}}} $$

(4.18)

It is therefore possible to obtain averaged quantities like the particle temperature by a simple measurement of all particle coordinates in one image. On the other hand, the same distributions can be calculated for a single particle from a long enough time series. This then yields a locally defined temperature as an average over time.

In an ergodic system, the averages over the particle ensemble are equal to those over time. Assuming that the ergodic hypothesis holds for plasma crystals, one could simply record images of a few particles out of a larger ensemble, at a spatial resolution high enough to calculate velocities. From long time series of single particle trajectories then a temperature representing that of the particle ensemble could be estimated. This way one could avoid the bad spatial resolution accompanying the recording of a huge ensemble, where velocities are subject to large uncertainties.An analysis of the ergodicity of (small) plasma crystals showed that the assumption of ergodicity might be wrong [24], it might be valid though in large enough systems, where external parameters change only slowly compared to the spatial scale of the system.

The above interpretation of the distribution functions as thermodynamic characterization of the particle system by means of a temperature is not strictly valid, because the complex plasma is an open system, not in thermal equilibrium with its surroundings. However, the continuous interaction of the particles with plasma constituents establish a certain equilibrium with regard to the particle energy (damping by collisions with neutral atoms, heating by inelastic collisions with ions). Then the system can be described by a kinetic temperature behaving similar to an equilibrium temperature as was argued in [20, 25]. In fact, experimentally obtained velocity distributions in two-dimensional complex plasmas are often found to be Gaussian distributed, defining a particle temperature from the Maxwellian model (e.g. in [4, 26, 27]). A theoretical approach to the velocity distribution yielded an Maxwellian distribution with an effective particle temperature two times larger than the temperature of the ions, following from inelastic collisions with ions in the calculations [28].