Finite-temperature crossover from a crystalline to a cluster phase for a confined finite chain of ions

Employing Monte Carlo simulation techniques we investigate the statistical properties of equally charged particles confined in a one-dimensional box trap and detect a crossover from a crystalline to a cluster phase with increasing temperature. The corresponding transition temperature depends separately on the number of particles $N$ and the box size $L$, implying nonextensivity due to the long-range character of the interactions. The probability density of the spacing between the particles exhibits at low temperatures an accumulation of discrete peaks with an overall asymmetric shape. In the vicinity of the transition temperature it is of a Gaussian form, whereas in the high-temperature regime an exponential decay is observed. The high-temperature behavior shows a cluster phase with a mean cluster size that first increases with the temperature and then saturates. The crossover is clearly identifiable also in the nonlinear behavior of the heat capacity with varying temperature. The influence of the trapping potential on the observed results as well as possible experimental realizations are briefly addressed.

Figure 1

The difference of the expected equidistant position of each particle from its minimum energy position as a function of the index of the particle for $N=100$ particles confined in a box with unit length $L=1$.

Figure 2

The minimum energy configurations of particles as obtained by the minimization of the potential $V_C(x_1,x_2,...,x_N)$ for $N=6,10$ and 20. The particles are represented with the (blue) full dots, and the equidistant positions are marked with the (red) vertical line segments.

Figure 3

Particle configurations for $N=100$: (a) A single configuration of particles at $\tau ={10}^{-3}$, (b) the same for $\tau ={10}^3$, (c) the mean configuration of particles for $\tau ={10}^{-3}$, (d) the same for $\tau ={10}^3$.

Figure 4

The mean difference ${\Delta }_{\text{eq}}{\xi }_i$ of the position of each particle from its expected equidistant position as a function of the particle's index $i$ for temperatures $\tau ={10}^{-5}$ (blue circles), ${10}^{-3}$ (green line), ${10}^{-1}$ (magenta squares), 1 (cyan triangles), ${10}^2$ (black line with circles), and ${10}^4$ (red stars). Note that the $\tau ={10}^{-1},1$ curves for ${\Delta }_{\text{eq}}{\xi }_i$ are on top of those for $\tau ={10}^{-5},{10}^{-3}$, and therefore hardly visible.

Figure 5

The density of charge $\langle \rho \left(\xi \right)\rangle$ as a function of the particles’ positions $\xi$ for temperatures $\tau ={10}^{-5}$ (blue line), 1 (green circles), ${10}^3$ (black line with triangles), ${10}^5$ (estimated mean red line and fluctuations red stars). The particle number is $N=100$.

Figure 6

The distributions of $\Delta {\xi }_i$, $\rho \left(\Delta {\xi }_i\right)$, for different temperatures: (a) $\tau ={10}^{-5}$, (b) $\tau ={10}^{-3}$, (c) $\tau ={10}^{-1}$, (d) $\tau =10$ (the red line is a Gaussian fit), (e) $\tau ={10}^3$, (f) $\tau ={10}^5$ (the red line is an exponential fit). In each figure the mean value of the interspace distance $\langle \Delta \xi \rangle$ and its most probable value ($\max \Delta {\xi }_i$) are provided.

Figure 7

The skewness of the distributions $\rho \left(\Delta {\xi }_i\right)$ as a function of the temperature $\tau$.

Figure 8

The probability distribution of cluster sizes for different temperatures (semilogarithmic scale): $\tau =5$ (blue line with squares), $\tau =10$ (green line), $\tau =50$ (red line with circles), $\tau =5\times {10}^2$ (cyan line with triangles), $\tau ={10}^3$ (purple line with diamonds), $\tau =5\times {10}^3$ (yellow line with circles), $\tau =5\times {10}^4$ (brown line), $\tau ={10}^5$ (black line with stars). The blue dashed line indicates the probability distribution of cluster sizes in the case of noninteracting particles confined in the box (purely random configurations).

Figure 9

The mean cluster size as a function of the reduced temperature $\tau$ for systems with different numbers of particles: (a) $N=50$, (b) $N=100$, (c) $N=200$. The horizontal dashed lines indicate the values of the mean cluster size for the respective numbers $N$ of noninteracting particles confined in the box.

Figure 10

The normalized correlation functions $A\left(m\right)/A\left(0\right)$ for $\tau ={10}^{-5}$ (blue line), $\tau ={10}^{-1}$ (cyan circles), $\tau =10$ (gray triangles with black line), $\tau ={10}^5$ (red line with stars), and for two different quantities: (a) the absolute positions of the particles ${\xi }_i$, (b) the interspace distance $\Delta {\xi }_i$. Note that for (a) all the curves for different $\tau$ are on top of each other and therefore not distinguishable.

Figure 11

(a) The dependence of the dimensionless specific mean energy $\langle \varepsilon \rangle$ on the reduced temperature $\tau$ on a semilog scale for $N=100$ (Metropolis: cyan circles; Wang-Landau: blue line). (b–d) The dependence of the dimensionless specific mean energy $\langle \varepsilon \rangle$ on $N$ on a log-log scale for (b) $\tau ={10}^{-2}$ (circles for numerical values, blue line fit), (c) $\tau =10$ (triangles, red line linear fit), (d) $\tau ={10}^5$ (squares, black line linear fit).

Figure 12

The temperature dependence of the specific heat capacity $\frac{c_L}{k_B}$ for $N=100$ on a semilogarithmic scale: (blue line) Wang-Landau results, (orange circles) Metropolis results. The dashed purple line indicates the temperature ${\tau }_1$ at which the skewness becomes zero.

Figure 13

Statistical properties of a system of $N=100$ ions confined in a harmonic trap with $m{\omega }^2=32$: (a) The skewness of the distributions $\rho \left(\Delta x_i\right)$ as a function of the temperature $k_{B}T$. (b) The temperature dependence of the chain's mean length $\langle L\rangle$. (c) The mean cluster size as a function of the temperature $k_{B}T$. The horizontal dashed line indicates the value of the mean cluster size for $N$ noninteracting particles confined in the same harmonic trap.