Figure 1

(Color online) Schematic of the cryostat insert for the heat-capacity measurements showing how the sample holder connects to the 1 K Pot via the temperature control plate. All of these were mounted inside a vacuum can, with no exchange gas. Heaters and thermometers were present on the sample, on the temperature control plate and on the 1 K Pot.Reuse & Permissions

Figure 2

(Color online) The ratio of the superconducting and normal-state heat capacities, $C_s/C_n$, measured as a function of temperature in the niobium sample Nb1. The magnetic field was applied parallel to the [111] crystal axis. Normal-state measurements, $C_n$, were taken 20 mT higher in field. The temperature has been plotted as $T\text{−}{T}_{c2,MFT}$, where $T_{c2,MFT}\left(H\right)$ is the value the transition temperature would have in the absence of fluctuations and is taken as the temperature at which the fluctuation heat capacity is a fraction, 0.2, of the mean-field jump in $C$ (Ref. 18). When a small hysteresis is seen, measurements taken on warming show the higher transition temperature.Reuse & Permissions

Figure 3

(Color online) (a) Schematic to illustrate the geometry of a SANS experiment, where ${\mathbf{k}}_i$ and ${\mathbf{k}}_f$ represent the incident and scattered neutron beams, respectively. (b) The FWHM of Gaussian fits to the size of diffraction spots at the detector in the temperature regime of critical fluctuations at 200 mT (applied parallel to the [111] axis of Nb1). Measurements were taken on cooling and are shown for all temperatures at which there was sufficient diffracted intensity to fit the spot shape. Backgrounds taken at 5.610 K were subtracted before fitting. For measurements of ${\omega }_{rad}$ and ${\omega }_{tang}$, $\varphi$ was fixed at the angle that brought the top two first-order spots from the hexagonal lattice, left (●) and right $(\star )$, simultaneously onto the Bragg diffraction condition and maximized their intensity. Horizontal red lines mark the FWHM expected for a perfect lattice, where the width is due solely to instrument resolution.Reuse & Permissions

Figure 4

(Color online) The square root of the diffracted intensity minus background, $\sqrt{I}\left(\propto \left|F_{hk}\right|\right)$, as a function of temperature in a field of 200 mT, measured during the same scan that ${\omega }_{rad}$ and ${\omega }_{tang}$ [Fig. 2b] were recorded. Negative values of $\sqrt{I}$ [calculated as $\text{sgn}\left(I\right)\sqrt{\left|I\right|}$] result from the background subtraction. The mean-field theory linear behavior is obtained from a fit to the temperature dependence outside the critical region. Insets show the diffraction pattern observed on the detector (with direct beam masked and background subtracted) for two temperature points.Reuse & Permissions

Figure 5

(Color online) Scaling of the heat capacity and SANS intensity. (a) The heat-capacity measurements of Fig. 1, normalized to the mean-field behavior as $\left(C_s-C_n\right)/\left(C_{s,MFT}-C_n\right)$, plotted on the Thouless reduced temperature scale, $a_T$. $C_{s,MFT}$ was extrapolated through the transition from a linear fit of $C_s/C_n$ versus temperature below the fluctuation region (Ref. 14). The measurement at 44 mT is shifted vertically by 0.2 units, to avoid overlap with the Thouless theoretical prediction (Ref. 18), which is plotted as the black line. All successive plots (at the same sequence of fields as in Fig. 1) are shifted by 0.1 units. (b) SANS intensity, normalized as $I/I_{MFT}$, where $I_{MFT}$ was found from a linear fit of $\sqrt{I}$ versus temperature below the critical region (as shown, for example, in Fig. 2) plotted versus the Thouless reduced temperature $a_T$. Measurements are shown for three samples, Nb1: barrel shaped, Nb2: spherical, and Nb3: ellipsoidal. All samples were prepared under similar condition with similar $RRR$ values. As no significant difference was observed between the left- and right-hand diffraction spots, or on warming and cooling, these measurements have been plotted with the same marker style for clarity. The error bars become large at high temperatures because the data points represent the ratio of two quantities, both of which are tending to zero.Reuse & Permissions

Figure 6

(Color online) The magnetization extracted from the heat capacity and the SANS intensity. (a) $C_s-C_n$ from Thouless theory integrated with respect to temperature plotted versus the Thouless reduced temperature, $a_T$. (b) $C_s-C_n$ from experiment, integrated with respect to temperature and plotted versus $a_T$. Measurements taken from Nb1 in 198 mT. (c) The SANS diffraction intensity, presented as $\sqrt{I}\left(\propto M\right)$ versus $a_T$. Measurements taken from Nb1 in 200 mT. Red lines show the extrapolation of linear mean-field-theory behavior.Reuse & Permissions

Figure 7

(Color online) Investigation of a small oscillatory field applied in addition to the main field. (a) Raw LIA output $\left(\propto 1/C\right)$ for Nb1 in a field of 308 mT. In one measurement, an oscillating field of amplitude 0.7 G, frequency 100 Hz was continually applied to the sample during the measurement. (b) Diffracted intensity from Nb3 taken with and without a small decaying oscillatory magnetic field applied to the sample before each measurement. The main field applied was 300 mT.Reuse & Permissions