6 Experiment: NbSe₂

In this chapter, recent $\mu$SR measurements of the $\hat{a}$-$\hat{b}$ plane magnetic penetration depth $\lambda_{ab}$and the vortex core radius r₀ in the conventional type-II superconductor NbSe₂ are presented.

Figures 6.1 and 6.2 show the Fourier transforms of the muon precession signal in NbSe₂ as functions of temperature and applied magnetic field, respectively. The horizontal axes are in terms of the internal magnetic field B relative to the average field of the background signal $B_{\rm bkgd}$, which by definition is centered at 0 G. As the temperature or magnetic field is lowered, the line shape broadens and the high-field tail becomes longer due mainly to a decrease in $\lambda_{ab}$. The high-field cutoff is clearly visible in all of the measured line shapes for NbSe₂. This implies that the vortex cores occupy a significant volume of the sample. This fractional volume depends on both the size of the vortex cores and the areal density of vortices.

 

 

In order to test the effects of the analysis procedure on the determined behaviour of $\lambda_{ab}$ and $\xi_{ab}$, three different models for the theoretical internal field distribution corresponding to the vortex lattice were considered:

1.

the modified London (ML) model given in Eq. (4.10) with a Gaussian cutoff factor $\exp (-K^2 \xi_{ab}^2/2(1-b))$.

2.

the ML model with a Lorentzian cutoff factor $\exp (-\sqrt{2} K \xi_{ab}/\sqrt{1-b})$.

3.

the analytical GL model given in Eq. (4.13).

In all three cases, a triangular vortex lattice was assumed. The cutoff factors assume circular symmetry, so that within these models a sixfold symmetry around a vortex line is generated solely from vortex-vortex interactions. The theoretical muon polarization function $\tilde{P}(t)$ is generated by assuming a field profile given by one of these three models, and then multiplying by a Gaussian relaxation function $G (t) \! = \! \exp (-\sigma_f^2 t^2/2)$ to account for random disorder in the vortex lattice and the contribution of the nuclear dipolar moments. In addition, a Gaussian broadened distribution of fields with width $\sigma_{\rm bkgd}$ and average field $B_{\rm bkgd}$ was used to fit the residual background signal independently.

 

Figure 6.3 shows the magnetic field dependence of $\lambda_{ab}$ at $T \! = \! 0.33~T_c$ (i.e. $T \! = \! 2.3$ K) obtained by fitting the $\mu$SR time spectra with a polarization function which assumes one of the three models for the field distribution due to the vortex lattice. From magnetization measurements, $H_{c2} (0.33~T_c) \! = \! 2.9$ T, so that the results extend over the field range $0.03 \! < \! h \! < \! 0.31$,where $h \! = \! H/H_{c2}$. A clear linear H-dependence for $\lambda_{ab} (H)$ is obtained for all three types of analysis, although there is some difference in the absolute value of $\lambda_{ab} (H)$ and the strength of the linear term. The solid lines in Fig. 6.3 are fits to the linear relation  

$$\lambda_{ab} (H) \! = \! \lambda_{ab} (0) [1 + \beta h ] \, .$$ (1)

The results of these fits are given in Table 6.2. At low magnetic fields there is good agreement between the ML model with a Lorentzian cutoff factor and the analytic GL model. This is reasonable since the Lorentzian cutoff is strictly valid only at low reduced fields $b \! = \! B/B_{c2}$. On the other hand, the ML model with a Gaussian cutoff gives a significantly higher value for $\lambda_{ab}$. The Gaussian cutoff is derived from the solution of the GL equations near B_c2 and is thus not valid at low reduced fields. Unfortunately, a Gaussian cutoff was used in some of our earlier work [2,3,50] and in various theoretical studies by others.

 

$$B ({\bf r}) \lambda_{ab}(0) \beta$$

ML: Gaussian Cutoff 1659(1) 1.85(4)

ML: Lorentzian Cutoff 1398(2) 0.81(3)

Analytical GL Model 1323(2) 1.62(3)

 

Figure 6.4 shows the magnetic field dependence of the quality of the fits at $T \! = \! 0.33~T_c$, obtained for the three different models. The ratio of $\chi^2$to the number of degrees of freedom (NDF) is significantly greater than 1.0 in most cases due to the high statistics of the measured magnetic field distribution. For a non-perfect fit, higher statistics magnify the value of $\chi^2$.Fits to the ML model with a Gaussian cutoff generally yield the worst $\chi^2$ value. On the other hand, fits assuming a Lorentzian cutoff are only slightly better than fits to the analytical GL model.

 

Figure 6.5 shows, in the frequency domain, how the quality of the fits obtained (in the time domain) from the ML model using a Gaussian cutoff factor and from the analytical GL model are virtually indistinguishable. One would expect the results from these two models to converge at higher magnetic fields. However, as shown in Fig. 6.3, $\lambda_{ab} (H)$ determined for the two different models appear to diverge slowly at high H. The reason is that the analytical GL model deviates significantly from the exact numerical GL solutions at high reduced fields [121], and also, according to Brandt [123], the ML model is really only applicable when $b \! < \! 0.25$.

 

Despite the quantitative differences between the three phenomenological models, which is related to their validity in different field ranges, the finding of a linear-H dependence for $\lambda_{ab} (H)$ is common to all. Since the analytical GL model properly accounts for the finite size of the vortex cores and our measurements are taken essentially at low reduced fields (especially in the case of YBa₂Cu₃O$_{7-\delta}$ which we consider later) the results obtained using this model should most faithfully reflect the behaviour of the fundamental length scales. Unless otherwise stated, results presented in the remainder of this thesis were obtained assuming this model.

Figure 6.6 shows a comparison between $\lambda_{ab} (H)$ in NbSe₂ at two different temperatures. A linear-H dependence is observed between $T \! = \! 0.33~T_c$ and 0.6 T_c. The field dependence at lower T was not investigated because the ⁴He gas flow cryostat limited us to temperatures above $T \! = \! 2.3$ K. In the Meissner state of a conventional s-wave superconductor, $\lambda_{ab}$ is expected to increase quadratically as a function of magnetic field, due to nonlinear effects. The nonlinear corrections to the supercurrent response are the same in both the Meissner and vortex states. However, the average supercurrent density $\langle J_s \rangle \! = \! (c/4 \pi) \langle \vert {\bf \nabla} \times {\bf B}({\bf r}) \vert \rangle$ scales quite differently in the Meissner and vortex states, as shown in Fig. 6.7. The curve in the top panel of Fig. 6.7 (i.e. the Meissner state) was generated assuming that the magnetic field decays exponentially [see Eq. (2.15)] and that $\lambda$ is field independent. Thus in the Meissner state, $\langle J_s \rangle \! \propto \! H$.It follows that if $\lambda_{ab} \! \propto \! H^2$, then $\langle J_s \rangle \! \propto \! \sqrt{\lambda_{ab}}$.

The solid curve in the bottom panel of Fig. 6.7 (i.e. the vortex state) was generated with the field profile $B ({\bf r})$ from the analytical GL model. The dashed curve in this figure shows that the average supercurrent density in the vortex state is approximately proportional to $H^{0.44} \! \approx \! \sqrt{H}$. Thus, if $\langle J_s \rangle \! \propto \! \sqrt{H}$ and $\lambda_{ab} \! \propto \! H$ (as measured here), then as in the Meissner state $\langle J_s \rangle \! \propto \! \sqrt{\lambda_{ab}}$.This suggests that the field dependence of $\lambda_{ab}$ in the vortex state of NbSe₂ is due to nonlinear effects. However, $\lambda_{ab}$ measured in our $\mu$SR experiment is by definition not the same as the penetration depth which appears in the nonlinear theory or which is measured in the Meissner state. Relating $\lambda$ from the nonlinear theory to the effective $\lambda_{ab}$ measured by $\mu$SR is nontrivial and requires a proper account of the vortex source term.

 

In the vortex state, the strength of the term which is linear in H is almost the same at both temperatures considered, when normalized with respect to the value of H_c2 (T) (see parameter $\beta$ in Table 6.3). As the temperature is increased, the energy gap in the quasiparticle excitation spectrum shrinks, leading to the thermal excitation of quasiparticles. The reduction in the size of the energy gap also means that quasiparticles can be excited by relatively smaller magnetic fields. For this reason, in the Meissner state, the strength of the term quadratic in H is found to increase with increasing T. Since the strength of the coefficient $\beta$ for the term linear in H in Eq. (6.1) does not appear to change over a large range of temperature in the vortex state, it seems unlikely that the mechanism responsible for the nonlinear Meissner effect can be solely responsible for the observed H-dependence of $\lambda_{ab}(H,T)$ in the vortex state. Furthermore, according to the calculations of Amin et al. [40], it seems unlikely that nonlinear corrections to the supercurrent response in the vortex state can result in a field dependence for the effective penetration depth measured by $\mu$SR which is as strong as that found here. However, as just mentioned, the calculation of the effective $\lambda$ is rather sensitive to the vortex source term, so that the size of the vortex cores should be included in such calculations.

 

$$\lambda_{ab}(0) \beta \lambda_{ab}(0) \beta$$

 [Å]   [Å] 

$$\lambda_{ab} (H) \! = \! \lambda_{ab} (0) [1 + \beta h ]$$

1323(2) 1.62(3) 1436(3) 1.56(2)

$$\kappa(0) \eta \kappa(0) \eta$$

$$\kappa (H) \! = \! \kappa (0) [1 + \eta h ]$$

8.4 (2) 7.4 (2) 5.7(2) 8.2(3)

$$\kappa^\prime(0) \gamma \kappa^\prime(0) \gamma$$

$$\kappa^\prime(H) \! = \! \kappa^\prime (0) [1 + \gamma h ]$$

6.9(2) 9.5(3) 5.1(2) 10.2(4)

 

 

Since $\lambda_{ab}$ is not the only parameter which contributes to the fitted $\mu$SR line width, it is necessary to monitor the behaviour of the additional broadening parameter $\sigma_{f}$. Besides disorder in the vortex lattice, the large ⁹³Nb nuclear moments ($3.34 \times 10^{-3} \mu_B$) in NbSe₂ also slightly broaden the $\mu$SR line shape. For instance, in the normal state the muon depolarization rate $\sigma_N$at $T \! = \! 10$ K is found to increase linearly from $\sigma_N \! = \! 0.19~\mu s^{-1}$at $H \! = \! 0.19$ T to $\sigma_N \! = \! 0.22~\mu s^{-1}$ at $H \! = \! 0.90$ T. To determine the degree of disorder in the vortex lattice, the contribution of the ⁹³Nb nuclear moments $\sigma_{\rm dip}$ to the muon depolarization rate can be subtracted in quadrature from the fitted value of $\sigma_{f}$

$$\sigma_{\rm dis}^2 = \sigma_f^2 - \sigma_{\rm dip}^2 \, .$$ (2)

Assuming $\sigma_{\rm dip}$ is temperature independent, the normal state value $\sigma_N$ is approximately the value of $\sigma_{\rm dip}$ in the vortex state. The parameter $\sigma_{\rm dis}$ due to disorder in the vortex lattice [plotted in Fig. 6.8(a)] is relatively small and is weakly dependent on magnetic field and temperature. This indicates that the determined behaviour of $\lambda_{ab}(H,T)$ does not arise from a systematic interplay between $\lambda_{ab}$ and $\sigma_{f}$ in the fitting procedure. Figure 6.8(b) shows the quality of the fits ($\chi^2$) normalized by the number of degrees of freedom (NDF), as a function of H for $T \! = \! 0.33~T_c$ and 0.6 T_c. The quality of the fits is essentially independent of H and T. An upper limit for the root mean square (RMS) displacement $\langle s^2 \rangle^{1/2}$ of the vortices from their ideal positions in the perfect vortex lattice, due to random pinning, can be obtained from $\sigma_{\rm dis}$ as follows [50]  

$$\langle s^2 \rangle^{\frac{1}{2}} \approx \frac{\sigma_{\rm ... ...^2 \lambda_{ab}^2/(1-b) \right]^2} \right)^{- \frac{1}{2}} \, .$$ (3)

The magnetic field dependence of $\langle s^2 \rangle^{1/2}$is shown in Fig. 6.9(a). The magnitude of $\langle s^2 \rangle^{1/2}$is essentially independent of H, except perhaps at low fields where the interaction between vortices is weakest. This suggests that there are no significant changes in the vortex lattice geometry as a function of magnetic field. The degree of disorder in the vortex lattice does appear somewhat larger for higher T.

 

For a perfect triangular vortex lattice the intervortex spacing is given by  

$$L = \left(\frac{2 \phi_0}{\sqrt{3} B_0} \right)^{\frac{1}{2}} \, .$$ (4)

Since the distance between vortices decreases with increasing field, the RMS deviation of the vortices from their ideal positions grows as a percentage of L [see Fig. 6.9(b)]. At low tempertures $\langle s^2 \rangle^{1/2}/L$ is less than $2~\%$,even at $H \! = \! 0.31 H_{c2} (T)$. This small disorder is consistent with the STM and SANS experiments on NbSe₂ discussed earlier.

 

The radius of a vortex core is not a uniquely defined quantity, since there exists no sharp discontinuity between a normal vortex core and the superconducting material. Nevertheless, a useful definition can be made taking into account the dramatic spatial changes observed in quantities such as the order parameter $\psi ({\bf r})$, the local density of states $N(E,{\bf r})$, the supercurrent density $J_s({\bf r})$ and the local magnetic field strength $B ({\bf r})$near the center of a vortex line. Since the supercurrent density $J_s({\bf r})$ can be easily obtained from the fitted field profile through the Maxwell relation ${\bf J} ({\bf r}) \! = \! (c/4 \pi) \mbox{\boldmath$\nabla$} \! \times \! {\bf B} ({\bf r})$, we define an effective core radius r₀ to be the distance from the vortex center for which J_s (r) reaches its maximum value. As shown in Fig. 6.10, J_s (r) rises steeply from zero at the vortex center to its maximum value $J_{s_{\rm max}}$ at r₀.

 

The magnetic field dependence of r₀ in NbSe₂ is shown in Fig. 6.11, where r₀ is obtained from J_s (r) profiles created from the fitted field profiles B (r). The deduced values of r₀ are less sensitive (than $\lambda$) to the choice of the theoretical model for B (r). This is because a good approximation of J_s (r) can be obtained by taking the curl of any function B (r) which fits the measured field distribution well. This includes an insensitivity to the assumed vortex-lattice geometry, provided a good fit is obtained. Since r₀ is fairly robust to the validity of the theoretical field distribution used to fit the data, then provided a good fit is obtained, the vortex-core radius can be determined from $\mu$SR with few theoretical assumptions. We note that discrepancies between the three models considered here do appear at low fields (see Fig. 6.11) because of the reduced statistics at the high-field tail of the measured internal field distribution. Since there are far fewer vortices in the sample at these low fields, there is a reduction in the number of events originating from muons which stop in the vicinity of the vortex cores. As a result, the high-field tail shows more ``statistical wiggles'', which in turn allows for a greater variation in the tail of the fitted B (r). Increasing the number of recorded muon decay events in the $\mu$SR spectra at low H would rectify this problem and should lead to better agreement between the three models in Fig. 6.11 at all magnetic fields.

 

Golubov and Hartmann [205] have shown that the shrinking of the vortex core radius with increasing magnetic field can be attributed to increased vortex-vortex interactions. They solved the microscopic equations in the dirty limit (i.e. the Usadel equations) self-consistently and showed that the order parameter $\psi (r)$ and the LDOS reach their maximum values closer to the vortex center when H is increased. From the LDOS, these authors calculated tunneling current I(r) profiles from the vortex center as a function of H in order to model STM measurements of r₀ (H) in NbSe₂ [206]. The magnetic field dependence of r₀ in NbSe₂ determined by STM at $T \! = \! 0.6~T_c$is shown in Fig. 6.12 along with that determined by $\mu$SR at $T \! = \! 0.6~T_c$ and 0.33 T_c. The definition of r₀ in the STM experiment was arbitarily chosen to be the radius at which the measured I(r) had diminished to $1/\sqrt{2}$ of its maximum value at the vortex-core center. It was shown in Ref. [205] that this gives a value of r₀ which is somewhat larger than the commonly used theoretical definition, i.e. the radius at which $\psi (r)$ rises from zero at the core center to $1/\sqrt{2}$ of its maximum value well away from the core. The different definitions of r₀ are the main reason for the difference in magnitude of r₀ between the STM and $\mu$SR experiments at $T \! = \! 0.6~T_c$. Also, we found from the microscopic theory (see Ref. [4]) that J_s (r) does not reach its maximum value at exactly the radius where $\psi (r)$ reaches $1/\sqrt{2}$ of its maximum value, at all temperatures and magnetic fields. For this reason, our definition of r₀ is robust to changes in T and H and should better reflect the actual H-dependence of the vortex-core radius.

 

The authors of Ref. [205] reported good agreement between the STM measurements of r₀ (H) and the dirty-limit microscopic theory. Although the magnitude of r₀ is reasonably predicted from their calculations, the error bars in the STM measurements are too large to conclude whether there is precise agreement with the theory. In a $\mu$SR experiment, typically $2 \times 10^8$ muons sample the local magnetic field of approximately 10¹¹ vortices in the bulk of a few mm² sample, as opposed to an STM experiment which averages r₀ from a few vortices at the surface. As a result of this statistical improvement, the $\mu$SR data shown in Fig. 6.12 have smaller error bars and less scatter. This improvement allowed us to show in Ref. [4] that r₀ decreases more strongly with magnetic field than predicted by the dirty-limit microscopic theory. This finding was not surprising since NbSe₂ is in the clean limit. To our knowledge, there have yet to be any calculations of the H-dependence of r₀ from the microscopic theory in the clean limit. However, in Ref. [4] we showed that the $\mu$SR results fit well to the simple phenomenological equation  

$$r_0(H) = \frac{\lambda_{ab} (H)}{\kappa^\prime (H)} = r_0(0)... ...c{\left[ 1 + \beta h \right]}{\left[ 1 + \gamma h \right]} \, ,$$ (5)

where $\kappa^\prime \! = \! \lambda_{ab}/r_0$ and r₀ (0), $\beta$ and $\gamma$ are temperature dependent constants. The excellent fits to this equation are shown as dashed curves in Fig. 6.12 where $r_0 (0) \! = \! 191$ Å  and 282 Å  at $T \! = \! 0.33~T_c$and 0.6 T_c, respectively. Equation (6.5) was derived from our observation that both $\lambda_{ab}$ and $\kappa^\prime$increased linearly with H. The latter is shown in Fig. 6.14(b) along with field dependence of $\kappa \! = \! \lambda_{ab}/\xi_{ab}$ in Fig. 6.14(a) [see Table 6.3 for the fitted parameters]. Since $h \! = \! H/H_{c2} \! = \! 3 \Phi_0 / \sqrt{2} L^2 B_{c2}$ for a triangular vortex lattice and H_c2 (T) can be obtained from measurement, for a given temperature r₀ in Eq. (6.5) is a function of only the distance between vortices, L. This observation supports the physical interpretation for the reduction in r₀, namely a shrinking of the vortex cores due to the increased interaction between vortices--which are closer together at high magnetic fields.

The effective coherence length $\xi_{ab}$ in Eq. (4.13) which best fits the data is plotted in Fig. 6.13 as a function of magnetic field. The variation of $\xi_{ab}(H,T)$ is similar to that of r₀(H,T), which was shown earlier to be model independent. The curves through the data points were generated from the fitted relations for $\lambda_{ab} (H)$ and $\kappa (H)$ given in Table 6.3. It should be kept in mind that $\xi_{ab}$ must be considered an ``effective'' coherence length. For instance, according to Eq. (2.73) of the GL theory, the coherence length near H<sub>c2</sub> at $T \! = \! 0.33~T_c$ should be 106.5 Å, where $H_{c2} (0.33~T_c) \! = \! 2.9$ T. However, at 2.9 T the fitted curve in Fig. 6.13 gives $\xi_{ab} \! = \! 49.1$ Å. Similarly, at $T \! = \! 0.6~T_c$, Eq. (2.73) yields $\xi_{ab} \! = \! 131.6$ Å, whereas the fitted curve in Fig. 6.13 gives 78.7 Å. A reduced value of $\xi_{ab}$ may be obtained if the fitted theoretical field distribution overestimates the length of the high-field tail in the $\mu$SR line shape. However, according to Fig. 6.5, it is highly unlikely that the fits are substantially overestimating the length of the high-field tail. The discrepancy between the measured $\xi_{ab}$ and that predicted in Eq. (2.73) is most likely due to the theoretical difference between $\xi_{ab}$ in Eq. (4.13) and the ``true'' GL coherence length. Given that GL theory is really only valid near the phase boundary, it is reasonable that deviations occur at low T and low H.

Assuming that the shrinking of the cores is associated with the strength of the vortex-vortex interactions, the increase in r₀ and $\xi_{ab}$ with decreasing magnetic field should saturate when the vortices are sufficiently far apart (i.e. when $\lambda_{ab} \! < \! L$). From Eq. (6.4) and the fitted expressions for $\lambda_{ab} (H)$ in Table 6.3, the field at which this saturation occurs can be estimated. In particular, at $T \! = \! 0.33~T_c$ there should be no change in the size of the vortex cores below $H \! \approx \! 0.12$ T, whereas this crossover field is $H \! \approx \! 0.10$ T at $T \! = \! 0.6~T_c$.

 

A few remarks are now necessary with regard to the behaviour of $\kappa (H)$ and $\kappa^\prime (H)$ in Fig. 6.14. The behaviour of $\kappa (H)$ implies that NbSe₂ becomes more type-II like with increasing magnetic field. In GL theory, $\kappa$ is independent of both H and T. However, this definition is strictly valid only near the superconducting-to-normal phase transition. Our results imply that the conventional GL equations with field-independent $\lambda$ and $\xi$ are not applicable deep in the superconducting state. Even if $\lambda_{ab}$ were field independent, an increase of $\kappa$ and $\kappa^\prime$ with H would still arise from the decrease of $\rho_0$ which has been independently observed in NbSe₂ by STM. Furthermore, attempts to fix $\kappa (H)$ and $\kappa^\prime (H)$ to constant values in the fitting procedure yield higher values of $\chi^2$ and unphysical results--such as a residual background signal which is $50~\%$ of the total signal amplitude.

 

Figure 6.15 shows a typical muon precession signal displayed for convenience in a reference frame rotating at about 1.5 MHz below the Larmor precession frequency of a free muon. The curves through the data points are examples of fits to the theoretical polarization function for fixed values of $\xi_{ab}$, where $\lambda_{ab}$ and all other parameters were free to vary. Only the first 3 $\mu$s of data are shown in Fig. 6.15 since the signal from the vortex lattice essentially decays over this time range--although the fits were actually performed over the first 6 $\mu$s. Figure 6.16 shows the difference between the data points and the fitted curve for the fits in Fig. 6.15. There is a clear oscillation for the fits corresponding to $\xi_{ab} \! = \! 80$ Å  and $\xi_{ab} \! = \! 160$ Å, indicating a missed frequency or frequencies. The ratio of $\chi^2$ to the number of degrees of freedom (NDF) is shown in Fig. 6.17(a) as a function of $\xi_{ab}$ for two different applied magnetic fields. Note that the value of $\xi_{ab}$for which $\chi^2$/NDF reaches its minimum value is quite different for the two fields. Figure 6.17(b) shows the behaviour of the free parameter $\kappa$ for these same fits. The best fits indicate that $\kappa$ is dependent on magnetic field.

 

 

 

The reduction in r₀ and $\xi_{ab}$ with increasing temperature which is shown in Fig. 6.12 and Fig. 6.13, respectively, is expected from theoretical predictions for a s-wave vortex [140,148,150]. However, as shown in Fig. 6.18(b), the vortex core radius does not decrease as steeply with temperature as predicted by theory. The dashed line in Fig. 6.18(b) is a fit to the theory of Kramer and Pesch [140] [see Eq. (4.11)] where $r_0 (T) \! = \! \alpha T/T_c$,with $\alpha \! = \! 299$ Å. Part of the problem is that these theoretical calculations pertain to a single isolated vortex. Given the apparent strong influence of vortex-vortex interactions, the vortex-lattice effect should not be ignored in theoretical calculations for r₀ (T). For a given magnetic field, $\lambda_{ab}$ will grow with increasing temperature [see Fig. 6.18(a)], whereas the intervortex spacing L remains constant. The strength of the vortex-vortex interactions will increase at higher field as the ratio $\lambda_{ab} / L$ increases, leading to additional changes in the electronic structure of the vortex cores. Since these interactions become stronger with increasing T, the difference between the measured value of r₀ and that predicted for an isolated vortex core will increase monotonically with temperature.

 

The solid line through the data in Fig. 6.18(a) is a fit to the empirical relation $\lambda_{ab}^{-2} (T) \! = \! \lambda_{ab}^{-2} (0) [1-t^p]$with $\lambda_{ab}^{-2} (0) \! = \! 50.7~\mu$m^-2, $p \! = \! 2.48$ and $t \! = \! T/T_c$.Although this is considered to be consistent with a weak-coupling BCS superconductor [1] (which shows a T-dependence of $\lambda^{-2}$ which is close to (1-t²)), there is no real low temperature data to obtain a proper fit to the BCS expression of Eq. (2.27). From the empirical fit and an observed weak linear-T dependence for $\kappa^\prime$,the temperature dependence of the vortex-core radius is given by  

$$r_0(T) = \frac{\lambda_{ab} (T)}{\kappa^\prime (T)} = r_0(0)... ...^p \right]^{-\frac{1}{2}} \left[ 1 - \gamma t \right]^{-1} \, ,$$ (6)

where $r_0(0) \! = \! 117$ Å  and $\gamma \! = \! 0.23$.This equation appears as the solid line fit in Fig. 6.18(b). Note that the change in r₀(T) is weakest at low temperatures where the strength of the interaction force between vortices is small. The measurements here are well above the quantum limit in NbSe₂ (see Ref. [148]) so a complete saturation of r₀ is not observed. As noted earlier, the weak increase in $\kappa$ (or $\kappa^\prime$) with decreasing temperature is predicted from the microscopic theory.