The magnetic properties of some superconducting mercury colloids By C. S. W hitehead The Royal Society Mond Laboratory, University of Cambridge (Communicated by D. Shoenberg, F.R.S.—Received 15 June 1956) Magnetization curves have been measured for fourteen colloidal mercury specimens whose average radius varied from 3 x 10-6 to 15 x 10-6 cm. For small applied fields the magnetic moment is proportional to the field and the observed temperature dependence of the slope is as predicted by the London theory. However, the dependence of these initial slopes on the average particle size cannot be explained by the London theory; the values found experi mentally, extrapolated to 0° K, are numerically in better agreement with Pippard’s modification of the London equation, but this does not correctly describe the temperature dependence. Measurements of the area of the magnetization curves give a mean value of — 5 + 12 A for the surface-energy parameter ft, which means that the influence of this surface energy is negligible for most specimens. It was found that the magnetization curves of drops small compared with the penetration depth can be scaled to have the same shape at all temperatures. This property has been explained quantitatively by a suitable modification of the two-fluid model of the superconducting state, but the theory does not seem adequate to explain some of the hysteresis features in the magnetization curves of larger drops.
176 C. S. Whitehead phenomenological theories. Essentially the experiment consists in measuring the magnetization curves of sufficiently uniform colloid specimens of various average particle sizes at different temperatures. The dependence of the initial slopes of the magnetization curves on particle size and temperature can be compared with the predictions of the various theories.
Superconducting colloids '' agents and hydrazine as the reducing agent. On being acidified with acetic acid, most of the protecting agents coagulated and precipitated out, bringing down with them all the mercury in the form of drops of colloidal size dispersed in the organic material. The precipitate was then filtered off and dried under vacuum.
178 C. S. Whitehead The important characteristics of the various specimens are listed in table 1 together with the results of some of the magnetic measurements to be described in the next section. It will be convenient to postpone discussion of the accuracy of the estimates of the average radii given in table 1 until some of the magnetic data have been discussed. The values of the transition temperatures shown in table 1 are about 1 % greater than the average value of 4*153°K found by Laurmann & Shoenberg (1949) for pure strain-free mercury in bulk; this increase may be due to strains set up in the mercury on solidification.
Superconducting colloids 179 the volume of the sphere. If a/A < n, this equation can be expanded as a power series in a2/A2 to give XlX o= “ A a4M4 + • • • • In practice each specimen contained about 1013 drops, so that to interpret the results, average sizes must be substituted in (3). If a" is the average value of the nth power of the radius, it is easily shown that if (3) is valid for a single drop, the appropriate expression for a specimen of many spheres is xlx° = 15A2 (p) “ 3l5A* (p) + l 575A« (p) ' <4) 600 H (gauss) Figure 2. Magnetization curves for Hg 36. •, field increasing at 1-225° K; o, field decreasing at 1-225° K; +, field increasing at 2-947° K; x , field decreasing at 2-947°K; -----, residual paramagnetic susceptibility.
180 C. S. Whitehead To investigate the validity of (2), xlX aot the absolute zero is plotted logarithmic ally against~a6]as (figure 5);* as can be seen from figures 3 and 4 only slight extra polation of the experimental data is required to obtain o at °° The theoretical curve (a) in figure 5 is based on (2), assuming the value A0 = 4*3 x 10 6 cm found by Laurmann & Shoenberg (1949) from experiments on macroscopic mercury speci- Figure 3. Temperature variation of the Figure 4. Temperature variation of the initial slopes. + , Hg 38; •, Hg56; initial slopes. +, Hg36; •, Hg36B; o, Hg37. The non-zero intercept on the o, Hg44A.
Superconducting colloids 181 mens. (The curve (6) is based on Pippard’s theory and will be discussed later (p. 184).) In spite of the scatter of the points it is clear that the experimental values of xlXo are much smaller than the theory predicts, which suggests that either the London equation (1) is not valid for a sphere of small size or that there are large systematic errors in the experiment.
C. S. Whitehead 182 be in error if the spread of the distribution is too great. Suppose that the number of drops with radii between a and a + Saisf(a) then (6) Thus iff {a) is proportional to a~p for large a, the upper integral of (&jrwill converge only if p >6.* Consequently, in some cases the measured value of (a5/u3)2 could be much less than the correct average size a to be substituted in formulae such as (2). In fact the histograms of the size distributions, of which figure 1 is an example, show a decrease off (a) for increase of a when a is large, that cannot be fitted very well by functions of the type a~p. Instead, for the tails of the histograms f(a) seems to decrease roughly exponentially with increasing radius as shown in figure 6. Thus the convergence of the sums used to estimate a5 and a3 seems reasonably certain, since j' /? an e~Pad a is finite if and n are positive. However, it is important to notice that slow convergence leads to too small an estimate of the average sphere size, so that any error is in the wrong direction to account for the discrepancy with the London theory. Some indication of the possible uncertainty of the estimate of a5/a3 is given by the observation that it usually fell by about 3 % if the largest drop observed was omitted from a distribution of a thousand, while the random error in a5/a3 as estimated from the scatter of the points in figure 5 is about 12 %.
Superconducting colloids 183 measurements on the penetration depth in impure tin. His basic equation for the current density at a point P is _3__ Jf r(r.A) j = e-^dr, (7) 477-£0 Ac r4 where r is the position vector joining P to the volume element dr of the super conductor in which the vector potential is A. It is assumed that £0 and A vary with temperature and direction of current flow relative to the crystallographic axes, while £ depends also on impurity content. Pippard solved this equation for the case of an applied field parallel to a plane surface and found that as £->oo, a (arbitrary units) Figure 6. The variation of /(a) with the radius a for the larger drops in the size distributions: +, Hg36; •, Hg45; o, Hg50.
C. S. Whitehead 184 Pippard (1953, p. 564) from a consideration of Shoenberg’s (1940) data and has not yet been resolved. The observed values of extrapolated to 0°K are, how ever, in better numerical agreement with (8) than with the London theory formula (3), as may be seen from curve (b) of figure 5. The correct average size forjhe application of (8) is (a«/^)*, which was found to be about 4 % greater than (a5/a3)I, and curve (6) in figure 5 has been corrected to allow for this; A^ has been assumed to be 4-3 x 10-6cm. Actually Pippard’s theory predicts that Aro should be slightly less than the observed value A0 for macroscopic specimens, and allowance for this would bring the experimental points and the theoretical curve rather closer together. Although the discrepancy between the two could be attributed to the experimental errors, it is perhaps significant that the experimental points for small a fit the empirical law ^ _ 0-019(a/Ao)«(1-f) (9) rather better than a cube law. A rather speculative interpretation of this result would be to suppose that there is significance in expressing it as xlx ^)2(alv)K o= A W where 8 is the temperature-dependent penetration depth (mc2/47me2)^ predicted by the London theory, and tjsi a characteristic length, independent of temperature, which enters into the true theory in such a way that the factor {a/7})? is needed to correct the London formula. Numerically comes out as of order 5 x 10-5 cm, and it may be noted that this is not very different from the parameter <r, which enters into Frohlich’s (1950, 1951) theory, and is the de Broglie wavelength of an electron moving with twice the velocity of sound (cr ~ 5 x 10~6 cm for mercury). It is also of the same order of magnitude as Pippard’s coherence length £0 which at T = 0° K is 2 x 10~5em for tin and 12 x 10-5 cm for aluminium (no estimate has been made for mercury) (Faber & Pippard 1955); unfortunately, however, £0 appears to vary with temperature, while 7) does not.