Field-dependent specific heat of the canonical underdoped cuprate superconductor \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hbox {YBa}_2\hbox {Cu}_4\hbox {O}_8$$\end{document}YBa2Cu4O8

The cuprate superconductor \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hbox {YBa}_2\hbox {Cu}_4\hbox {O}_8$$\end{document}YBa2Cu4O8, in comparison with most other cuprates, has a stable stoichiometry, is largely free of defects and may be regarded as the canonical underdoped cuprate, displaying marked pseudogap behaviour and an associated distinct weakening of superconducting properties. This cuprate ‘pseudogap’ manifests as a partial gap in the electronic density of states at the Fermi level and is observed in most spectroscopic properties. After several decades of intensive study it is widely believed that the pseudogap closes, mean-field like, near a characteristic temperature, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T^*$$\end{document}T∗, which rises with decreasing hole concentration, p. Here, we report extensive field-dependent electronic specific heat studies on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hbox {YBa}_2\hbox {Cu}_4\hbox {O}_8$$\end{document}YBa2Cu4O8 up to an unprecedented 400 K and show unequivocally that the pseudogap never closes, remaining open to at least 400 K where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T^*$$\end{document}T∗ is typically presumed to be about 150 K. We show from the NMR Knight shift and the electronic entropy that the Wilson ratio is numerically consistent with a weakly-interacting Fermion system for the near-nodal states. And, from the field-dependent specific heat, we characterise the impact of fluctuations and impurity scattering on the thermodynamic properties.

www.nature.com/scientificreports/ but very weak in the pure; and another at 120 K, present only in the pure sample. The sample variability indicates unidentified impurities and these anomalies are ignored in the following). Our first task is to identify the normal-state coefficient, γ n , that would occur in the absence of superconductivity. This is very much constrained by the displayed data for γ (T) because γ n must follow each of the three data sets above their respective T c values. This is the black dashed curve. It is further tightly constrained by the requirement for entropy balance. Because the area under a γ (T) curve is entropy then integrating γ (T) from T = 0 to some T 0 > T c must give the same result as integrating γ n (T) from T = 0 to T 0 . The fit function which satisfies these two requirements is where E * = 13.44 meV (or T * = E * /k B = 156 K) and the exponent α = 1.7 . The general form of this equation for γ (T) arises analytically from inserting into Eq. (3), below, a triangular gap in the density of states (DOS) with a finite DOS at the Fermi level 3 . The amplitude 0.913 (being less than unity) reflects the finite DOS at E F . This residual DOS is manifested in the finite value of γ n (0) = 0.183 mJ/g.at.K 2 and is a signature of the ungapped Fermi arcs, or hole pockets, of a reconstructed Fermi surface 18,19 .
T * = 156 K is typical of values reported for Y124 from transport 20 and NMR relaxation 21 measurements but we emphasize this reflects an energy scale not a temperature 4 . An important implication of Fig. 1 is that there is no coupling between superconductivity and the pseudogap in the sense that the onset of superconductivity does not weaken the pseudogap. This is evident from the fact that a single γ n (T) curve fits all three samples The measured electronic specific heat coefficient γ (T) for YBa 2 Cu 4 O 8 for 0%, 2% and 4% planar Zn concentration at 0, 1, 2, 3, 5, 7, 9, 11 and 13 tesla. The same colour coding is used for each sample. Inset: the same data plotted versus ln(T) highlighting the impurity scattering term (dotted lines). (b) the same data with the impurity term subtracted. The black dashed curve is the fitted normal-state specific heat coefficient, γ n which satisfies entropy balance.
Scientific Reports | (2020) 10:22288 | https://doi.org/10.1038/s41598-020-79017-3 www.nature.com/scientificreports/ i.e. γ n (T) is the same for 4% and 0% Zn even in the temperature range below T c for 0% Zn so that the onset of superconductivity in the latter case does not alter the underlying pseudogap energy scale, E * . Close scrutiny of the k-dependent gap in Bi2212, as measured by angle-resolved photoelectron spectroscopy (ARPES) 22 , (which allows separation of the antinodal pseudogap from the nodal superconducting gap on the Fermi arcs) confirms that the pseudogap amplitude does not alter on cooling below T c .

NMR Knight shift and entropy
Next, we note that the spin susceptibility and electronic entropy are closely related. To see this consider the entropy for a weakly-interacting Fermi liquid 23 : where f(E) is the Fermi function and N(E) is the electronic DOS for one spin direction. This is just a weighted integral of the DOS with the 'Fermi window' On the other hand, the spin susceptibility for a weakly-interacting Fermion system is: Therefore, like the entropy, the susceptibility is an integral of the DOS where the Fermi window is now the function ∂f /∂E . It turns out that if χ s in the former is stretched in temperature by a factor 1.187 24 . It is therefore not surprising that S/T and χ s are related. This relationship is expressed by the Wilson ratio, a W , such that S(T)/T = a W χ s (T) , where We will now test this relationship in the present case of Y124. By integrating γ n (T) from T = 0 to T we obtain the normal-state entropy and this is plotted as S n (T)/T by the black dashed curve in Fig. 2. For fully-oxygenated Y123 S n (T)/T is essentially independent of temperature 25 , reflecting the fact that the pseudogap has closed at maximal doping ( p ≈ 0.19 holes/Cu). But for Y124 there is a large pseudogap present which suppresses S/T at low T. This is also seen in the T-dependent NMR Knight shift which is linearly related to the spin susceptibility. To illustrate, we show in Fig. 2 26 . These authors also report the bulk susceptibility as a function of the Knight shift, thus enabling calibration of the spin susceptibility from the Knight shift. As a final step we multiply the spin susceptibility by the Wilson ratio, a W , in order to express the T-dependent part of the Knight shift in entropy/T units. www.nature.com/scientificreports/ It can be seen in Fig. 2 that, not only the shape, but the absolute magnitude concurs remarkably well with the derived S n (T)/T suggesting, as already noted for the bulk susceptibility 27 , that the near-nodal states are consistent with a weakly interacting Fermionic system. Of especial interest is the fact that the O2 and O3 Knight shifts begin to diverge below 200 K. The difference in shift, 17 K 2,c − 17 K 3,c , expressed in entropy units, is also shown in the figure (10× ). This shows an abrupt onset in nematicity, consistent with that reported by the Matsuda group 10 using torque magnetometry. Its location at (p = 0.13, T = 200) is precisely consistent with the three data points in the (p, T) plane reported by the Matsuda group for Y123. (For the doping state of 0.13 see "Materials"). These are plotted in the inset to Fig. 2 by the red data points. Their susceptibility data was presented as evidence for a thermodynamic "phase transition at the onset of the pseudogap" however it is clear from Fig. 2 that the pseudogap is already open far above T nematic , having already depleted half of the spin susceptibility. We observe no anomaly in γ (T) at or near 200 K to suggest a phase transition. It must be very weak. We will see below that the pseudogap in fact extends at least to well above 400 K. Consequently this nematic phase transition occurs within a preexisting pseudogap state that extends far above and is not a transition into the pseudogap state, contrary to what has been claimed 10 .

Volovik effect
We now consider the field-dependent low-T behaviour of γ (T) for the pure sample. Figure 3 shows γ (T, H) plotted as a function of √ µ 0 H for T = 6, 8, 10, 12 and 14 K. Above 1 tesla the behaviour is linear, consistent with the expected phenomenology of the Volovik effect. Such behaviour has been seen in the specific heats of single-crystal Y123 28 and La 2−x Sr x CuO 4 29 , and in interlayer tunneling in Bi2212 30 . For d-wave symmetry, in the superconducting state the finite ground-state specific heat coefficient is 29 in units of mJ/g.at.K 2 , where V M is the molar volume, d the unit cell length, φ 0 the flux quantum, 0 is the antinodal amplitude of the d-wave gap, and k F the Fermi wave vector at the node along the ( π,π ) direction. For 0 = 30 meV the dash/dot line in Fig. 3 shows the expected slope. The data is very consistent with this expectation. The averaged slope of 0.0443 mJ/g.at.K 2 T 1/2 implies a gap amplitude of 0 = 34 meV. We have used the value k F = 0.436 × 10 10 m −131 . Figure 4a shows the normal-state and superconducting state entropy below 100 K obtained by integrating γ n (T) and γ s (T) , respectively (as displayed in Fig. 1b) from 0 to T. The curves shown are for pure Y124 in zero field and they are denoted S n (T) and S s (T) , respectively. The difference is the condensation entropy �S ns = S n (T) − S s (T) . This is plotted underneath for fields of µ 0 H = 0, 1, 3, 5, 7, 9, 11 and 13 T, colour-coded as in panel (b). Clearly the condensation entropy is rapidly suppressed in field. Another feature of note is the presence of fluctuations around T c which broaden the transition somewhat. This will be discussed later.

Free energy and superconducting gap
By integrating �S ns (T) from a temperature T 0 , sufficiently above the fluctuation regime that �S ns (T 0 ) = 0 , down to a temperature, T, one obtains the condensation free energy �F ns (T) . However, a more efficient way of calculating the condensation free energy using just a single integration is given by: www.nature.com/scientificreports/ where the first term is the condensation internal energy, −�U ns (T) , and the second term is the condensation entropy term, T�S ns (T) . The condensation free energy calculated in this way is plotted in Fig. 4b for 0%, 2% and 4% Zn and for the various annotated fields. �F ns (T) and its components �U ns (T) and −T�S ns (T) are plotted in Fig. 5. Also plotted in Fig. 4b is the condensation energy for fully oxygenated Y123 which rises to a ground-state value of 3400 mJ/g.at-a full six-fold greater than for pure Y124. This shows the full impact of the pseudogap for Y124 in weakening superconductivity. Also evident is the dramatic effect of impurity scattering in further reducing the condensation energy (which is particularly marked in underdoped cuprates where the pseudogap is present 16 ). �F ns (0) for the 4% Zn-doped sample in zero field is just 28 mJ/g at-125 times smaller than for pure Y123. The curves for 2% Zn are dashed below 30 K and this is because �S ns (T) is a little noisy at low T and in some cases does not fall exactly to zero, as it must. We find the first 20 K of the data scales precisely with �S ns (T) for the 0% Zn sample, and so we assumed that this scaling continues down to T = 0 thus enforcing �S ns (T) to fall to zero as T → 0 . Any errors introduced are very small-of the order of the thickness of the curves and of no consequence in the following analysis. As mentioned, Fig. 5a shows �F ns (T) and its components �U ns (T) and −T�S ns (T) . It is striking that fluctuations persist high above T c in both �U ns (T) and −T�S ns (T) while they are almost completely cancelled in The T-dependence of the condensation free energy F ns = F n − F s for pure Y124 obtained by integrating �S ns (T) from above T c for each field as annotated. Below these are shown the condensation energy for the 2% Zn and 4% Zn samples showing a rapid suppression with scattering and applied field. The condensation energy for fully-oxygenated Y123 is also shown and is much higher. www.nature.com/scientificreports/ �F ns (T) . The superconducting gap function �(T) may be calculated from these components of the free energy using 32 where ζ = 1 for s-wave and 1/2 for d-wave. The calculated �(T) values are plotted in Fig. 5b. The gap is the magnitude of the order parameter, ′ , rather than the spectral gap, , which is higher 13 . Roughly speaking,  Fig. 5b. The values of � ′ (T) are seen to descend towards zero at T c then persist with a more slow decline above T c . Here the pairing is incoherent 33,34 and is a feature of the strong superconducting fluctuations above T c .

Comparison of Y124 with Y123
It is highly instructive to compare the measured data for Y124 with that for Y123. This is shown for γ (T) in Fig. 6a and for S(T) in Fig. 6b. The Y123 data is taken from ref. 24 which, in contrast to earlier reports 13,35 , includes a small correction for the background DOS in the undoped state. We also show the entropy-conserving normal-state functions as well. Two features are prominent. (1) the size of the specific heat jump for Y124 is much smaller than for Y123 due to the presence of the pseudogap in the former. (2) while the γ (T) curves converge above T c the S(T) curves remain separated and parallel to the highest temperature. This is clear evidence that the pseudogap remains present in Y124 to the highest temperatures measured-here 400 K.  Fig. 6b, and again in Fig. 7b. In contrast, because γ ≡ ∂S/∂T , the Fermi window for γ (T) is the T-derivative of that for S(T). This is a double-peaked function as shown in the inset to Fig 6a. It can be seen that at high enough temperature this Fermi window falls outside of the gap. Therefore γ (T) recovers its full ungapped magnitude at high T, despite the presence of the gap. The data in Fig. 6a,b are entirely consistent with this picture. The two systems have essentially the same background DOS. In the normal-state Y123 is ungapped while Y124 is gapped to the highest temperature as evidenced by the parallel suppression of S(T). If the gap were to close with increasing T then S(T) for Y124 would recover to that for Y123. It does not. By extrapolating S(T) back to the ordinate axis one can read off the normal-state gap magnitude. For a triangular gap, as in the insets to Fig. 6, but with a finite DOS of N 1 at E F and a constant DOS of N 0 above E * this negative intercept is 2 ln 2 k B (N 0 − N 1 )V M E * , where V M is the molar volume. The finite value of γ n at T = 0 is given by γ 0 n = 4 ln 2 × 2.374k 2 B N 1 V M , while at high T we have γ n = (2/3)π 2 k 2 B N 0 V M . From these we obtain E * = 19.1 meV and N 0 = 6.14 × 10 −3 states/meV/cell.

High-temperature susceptibility and entropy
We now extend this comparison to 400 K, a range never previously achieved in differential measurements. We combine this data with 89 Y Knight shift data, bulk susceptibility measurements and 1/ 63 T 1 data in such a way that demonstrates, individually and collectively, the persistence of the pseudogap to 400 K and beyond. 89 K s (T) www.nature.com/scientificreports/ probes the spin susceptibility of the CuO 2 planes that sandwich the Y atom in Y123 and Y124 36 . The 89 Y nucleus has the additional benefit of having no quadrupole moment so there is no quadrupole splitting of the resonance arising from electric field gradients. Further, the use of magic-angle spinning (MAS) enables extremely narrow line widths as will be used below 37 . The 1/ 63 T 1 relaxation rate is a weighted sum over q of the imaginary part of the spin susceptibility, χ ′′ (q, ω) , where, for the 63 Cu nucleus, the weighting form factor is strongly enhanced near the antiferromagnetic wave vector, q = (π, π) 38 and hence 1/ 63 T 1 is dominated by the antinodal pseudogap. Figure 7a shows the 89 Y Knight shift 89 K s (T) for YBa 2 Cu 3 O 6+x as reported by Alloul et al. 36 , with values of x annotated. Also plotted is MAS 89 K s for samples of Y124 from our laboratory 39 (red spheres), along with the 1/ 63 T 1 data from Raffa et al. 21 (up triangles) which we showed 39 scales precisely with 89 K s , as can be seen. (The conversion scale is 89 K s = 48.8 × 1/ 63 T 1 − 164.5 .) Notably, there is an excellent match over the entire temperature range with Alloul's data for x = 0.75 (green up-triangles) where the doping state (0.13) and T c for Y123 are much the same as those of Y124. As in Fig. 2, we convert 89 K s to spin susceptibility using the calibration of Alloul et al. 36 (see "Methods" for more detail) and multiply by a W to express in S/T units. We find an excellent agreement between Alloul's 89 K s and the measured entropy for Y123 across a wide range of doping and temperature (see Fig. S4 in Supplementary Information, SI).
In Fig. 7b, we assemble four distinct data sets (S, 89 K s , 1/ 63 T 1 , and χ s from the bulk susceptibility) for Y123 at near full oxygenation ( x = 0.97 ), and for Y124. All are expressed in entropy units, in this case using the factor a W T to convert susceptibilities (including the 1/ 63 T 1 data expressed as a susceptibility). The χ s data is shown for Y123 with x = 0.97 and x = 0.73 (purple and green dashed curves, respectively, the latter as a proxy for Y124) and is taken from Loram et al. 24 . Evidently S (blue solid curve), a W 89 χ s T (purple squares) and a W χ s T (purple www.nature.com/scientificreports/ dashed curve) for Y123 at full oxygenation all track linearly to the origin indicating the absence of the pseudogap. For Y124 on the other hand, S (red solid curve-this work), a W 89 χ s T (red spheres) and T/ 63 T 1 (open black triangles), together with Alloul's a W 89 χ s T (solid green triangles) for Y123 with x = 0.75 and a W χ s T (dashed green curve) for Y123 with x = 0.73 , both as proxies for Y124, all reveal a linear high-temperature behaviour that extrapolates to a negative intercept on the y-axis indicating the presence of a gap-the pseudogap. For Y124 the entropy curve almost completely overlays the bulk susceptibility green-dashed curve which is barely visible, so the figure is reproduced in the SI with the green-dashed curve overlaying the entropy. Evidently, the agreement over the full temperature range is excellent. Importantly, our entropy data extends to 400 K as does the χ s data, and the 89 K s data extends to 370 K. Figure 7b represents the central result of this work. There is no indication, at any temperature, of the entropy recovering to the gap-less curve observed for fully-oxygenated Y123 that would signify the closing of the pseudogap at, or around, some T * value. The small upturn in S(T) near 400 K simply represents the limitations of the present differential technique at such a high temperature and is not seen in the χ s data. We conclude that the pseudogap does not close at some postulated T * in the range 150-200 K but remains open to the highest temperature investigated-400 K. A similar conclusion has recently been drawn from 1/ 17 T 1 planar oxygen NMR relaxation data for a number of cuprates 41 . We showed the same long ago 3 for the in-plane resistivity and similarly for the c-axis resistivity 4,42 .

Scattering resonance
Finally, we wish to discuss the upturn in the raw γ (T) data at low T seen in Fig. 1a. This is in fact a peak rather than an upturn and may be identified with an impurity resonance 43 . Scanning tunneling spectroscopy (STS) measurements in lightly Zn-doped Bi 2 Sr 2 CaCu 2 O 8+δ reveal resonance spots in spatial maps at low energy and low temperature 43 . Away from these spots, tunneling spectra reveal a well-formed d-wave superconducting gap with sharp coherence peaks. Tunneling spectra collected on the spots (the location of individual Zn atoms) show a nearly full suppression of both the gap and the coherence peaks with, instead, a sharp resonance appearing at ε r = −1.5 meV. To calculate the entropy contribution arising from this resonance we replace the DOS in Eq. (2) by a delta function, N(E) = N r δ(ε − ε r ) . The equation integrates to give: where the amplitude A res = 4k 2 B /ε r N r . In view of the relationship between entropy and spin susceptibility discussed above, it is highly instructive to contrast this resonance component of γ (T) with that of the susceptibility. Again, replacing the DOS in Eq. (3) by N(E) = N r δ(ε − ε r ) we find: where the amplitude B res = µ 2 B /2ε r N r . The interesting point in relation to Eqs. (8) and (9) is that �γ res falls off rapidly as T −3 while χ s,res falls off more slowly as T −1 . This is borne out by our experimental data. Experimental evidence from the magnetic susceptibility for a Zn-induced resonance within the pseudogap was discussed previously in relation to Y124 15 and La 2−x Sr x CuO 4 44 . Figure 8 shows the as-measured field-induced change in specific heat coefficient, �γ (13, T) = γ (13, T) − γ (0, T) . Recall that this difference contains no correction for the residual phonon contribution so is free of any imposed model. As well as the suppression of fluctuations around the specific heat anomaly near T c the low-temperature resonance is evident. We fit this using Eq. (8) with the parameters A res = 0.61 mJ/g.at.K 2 and ε r = −1.55 meV, the latter value being nicely consistent with the STS result for Bi 2 Sr 2 CaCu 2 O 8+δ 43 . This fit is the red curve in Fig. 8 and the difference is shown by the blue curve. The calculated resonance response is an excellent fit and shows the rapid decay at higher temperatures associated with the T −3 tail. The difference is close to entropy conserving and requires a straightforward extrapolation below 2.8 K of its trend above 2.8 K to achieve exact entropy conservation. This rapid decay of the resonance in γ (T) contrasts the predicted much slower T −1 decay in the resonance part of the spin susceptibility. We have previously investigated the 89 Y NMR Knight shift in Zn-doped Y124 15 . The Zn resonance contribution to the spin susceptibility is seen in a satellite peak which has a slowly-decaying Curie temperature dependence observable all the way up to 300 K, thus nicely confirming the behaviour predicted by Eq. (9).

Conclusions
In conclusion, we have measured the electronic specific heat of YBa 2 Cu 4−x Zn x O 8 using a precision differential technique that allows separation of the electronic term from the lattice term up to an unprecedented 400 K. The pure sample reveals the expected Volovik effect which is fully suppressed in the Zn-doped samples. We show that the pseudogap, characteristic of underdoped cuprates, always remains open to above 400 K, far above the nominal pseudogap onset temperature usually proposed, T * ≈ 150-200 K. Weak thermodynamic transitions reported at T * , for example the onset of susceptibility nematicity, occur within the already fully established pseudogap and are not transitions into the pseudogap state as widely claimed. The spin susceptibility, derived from the 17 O and 89 Y Knight shift and expressed in entropy units, is numerically the same as the electronic entropy divided by temperature, indicating that the near-nodal states are those of weakly-interacting Fermions. We derive the fielddependent condensation energy and superconducting energy gap. These expose the presence of strong pairing fluctuations extending well above T c as well as canonical impurity scattering behaviour, including the expected low-temperature resonance response which in the entropy channel decays rapidly as T −3 , but in the spin channel  8 . Thermoelectric power measurements at 290 K give 6.95 µV/K (0% Zn), 6.89 µV/K (2% Zn) and 6.93 µV/K (4% Zn). From the correlation of thermoelectric power with doping 45 this amounts to essentially identical doping states of 0.130 holes/Cu for each. The suppression of T c with Zn substitution is very much in line with that for underdoped Y123 at the same doping state. Supplementary Fig. S2 shows T c as a function of doping for 0, 2, 4 and 6% planar Zn substitution for Y 0.8 Ca 0.2 Ba 2 Cu 3 O 7−δ while the red stars show the data for Zn-substituted Y124. They are very consistent. The rapid suppression of T c for underdoped samples (open symbols) compared with overdoped (filled symbols) is a signature of the pseudogap which lowers the DOS at the Fermi level and hence raises the scattering rate 16 . Supplementary Fig. S3 shows T c as a function of planar Zn concentration for Y 0.8 Ca 0.2 Ba 2 Cu 3 O 7−δ at various doping states while red stars show the same data for Y124. Again they are very consistent. Typically, for higher Zn concentrations, the T c value sits higher than that expected from Abrikosov-Gorkov pairbreaking due to the statistical overlap of nearest neighbours to the Zn substituent 16 .
Spin susceptibility and Knight shift. The spin susceptibility, χ s , is related to the bulk magnetic susceptibility, χ m , by χ m = χ s + χ 0 where the constant χ 0 comprises diamagnetic and van Vleck terms and is evaluated for Y123 by Alloul et al. 36 . The measured Knight shift, 89 K s , is given by 89 K s = σ 0 (x) + a(x) 89 χ s , where σ 0 (x) is the T-independent chemical shift and a(x) is the relevant hyperfine coupling constant and, as indicated, both change only with oxygen content, x. For each x therefore, χ s , χ m and 89 K s are linearly related to within an additive constant. We used values of a(x) reported by Alloul 36 , while for each x the additive constant was determined by matching the 89 χ s data to our bulk susceptibility data, χ s 24 . This fixed the value of the constant σ 0 (x) which differed somewhat from those of Alloul but other literature values also reflect those differences 39 . The overall T-dependence (independent of σ 0 (x) ) was an excellent match.