Uncovering Flux Line Correlations in Superconductors by Reverse Monte Carlo Refinement of Neutron Scattering Data

We describe the use of reverse Monte Carlo refinement to extract structural information from angle-resolved data of a Bragg peak. Starting with small-angle neutron scattering data, the positional order of an ensemble of flux lines in superconducting Nb is revealed. We discuss the uncovered correlation functions in the light of topical theories, in particular, the “Bragg glass” paradigm.

Figure 1

Angular dependence of the diffracted neutron intensity from the flux line ensemble. The circles depict the experimental rocking curve $I_{\exp }(\phi )$, while the solid line indicates $I_{\mathrm{calc}}(\phi )$ calculated from a RMC ensemble of $N=40015$ flux lines, after smoothing over the instrument resolution $Y(\phi )$ illustrated by the thin dashed line. The inset replots these data on a double logarithmic scale; the thick dashed line is a guide to the eye showing an algebraic decay.

Figure 2

2D (in-plane) maps from a RMC ensemble of $N=40015$ flux lines after $980N$ moves. In (a) the structure factor $S(\mathbf{k})$ defined by (1) is plotted on a logarithmic color scale. The solid white arcs depict the reciprocal space encompassed by our experimental rocking curve (Fig. 1). The hexagonal FL lattice symmetry is preserved by duplicating our data at three angles 60° apart. Between the first-order Bragg peaks zero scattered intensity was observed within experimental error. Thus, in these regions (marked by dotted orange arcs), the intensity to be fitted was set to zero with ${\chi }^2$ determined by the error. In calculating the neutron intensity $I_{\mathrm{calc}}$, $S(\mathbf{k})$ is first integrated radially as shown by the white radial line, to account for the poor instrument resolution in this direction, before convolution with the much better tangential resolution $Y(\phi )$. In (b) we show the displaced FL positions in a typical portion of the RMC ensemble. The color of each FL denotes the magnitude (lightness) and direction (hue) of its in-plane displacement $\mathbf{u}(\mathbf{R})$ from the corresponding perfect lattice, as described by the color wheel (inset) where white circles mark displacements of $0.2a_0$ and $0.4a_0$, and white radial lines depict the lattice basis vectors.

Figure 3

Correlation functions derived from RMC ensembles of $N=40015$ FL’s with the scale factor $F$ in (1) chosen to minimize ${\chi }^2$ after $610N$ moves (red diamonds) and $1740N$ moves (blue squares), and with $F$ bounded as described in the text after $980N$ moves (green circles). The displacement correlator $B(R)$ is shown in (a); the inset shows the goodness-of-fit statistic ${\chi }^2$ describing how well the calculated intensity $I_{\mathrm{calc}}(\phi )$ reproduces the experimental data $I_{\exp }(\phi )$. (b) shows the translational order $c_G(R)$. Here only one data set is plotted as the results overlap on the linear scale; the three data sets are replotted on a double logarithmic scale in the inset of (b), where the dashed line is a guide to the eye highlighting an algebraic decay. (c) shows the orientational order $g_6(R)$. Note that for clarity, the points $c_G(0)=1$ and $g_6(0)=1$ lie outside the scales used here. The colored solid lines in these plots are fits as described in the text.