The surface impedance of superconductors and normal metals at high frequencies. IV. Impedance at 9400 Mc. /sec. of single crystals of normal and superconducting tin

1950 ◽  
Vol 203 (1072) ◽  
pp. 98-118 ◽  
Keyword(s):  
Single Crystals ◽  
Surface Impedance ◽  
Surface Resistance ◽  
Skin Effect ◽  
Scaling Factors ◽  
High Frequencies ◽  
Normal Metals ◽  
Anomalous Skin Effect ◽  
The Difference ◽  
Fundamental Equations

The measurements described in the earlier papers of this series have been extended to 9400 Mc. /sec., a resonance technique being employed to determine the surface resistance of normal and superconducting tin, and the difference between the surface reactances of the material in the two states. Measurements on single crystals of different orientations have brought to light a marked anisotropy of all these quantities, of a kind which shows clearly the non-tensorial nature of the fundamental equations relating the field vectors. The prediction of the theory of the anomalous skin effect in normal metals, that the surface resistance should vary with frequency as ω ⅔ , is confirmed. The temperature variation of the resistance and reactance of superconducting tin has been studied in detail for a number of specimens of different orientations, and it has been found that over certain ranges of temperature the shapes of corresponding curves for different specimens are similar, apart from scaling factors depending on the orientation; the values of these scaling factors are used to characterize the surface impedance of each orientation.

scholarly journals The theory of the anomalous skin effect in metals

1948 ◽  
Vol 195 (1042) ◽  
pp. 336-364 ◽  
Keyword(s):  
Surface Impedance ◽  
Low Temperatures ◽  
Skin Effect ◽  
Frequency Variation ◽  
High Frequencies ◽  
Microwave Region ◽  
Conduction Electrons ◽  
Relaxation Effects ◽  
Infra Red ◽  
Anomalous Skin Effect

The problem of metallic conduction at high frequencies and low temperatures, recently discussed by Pippard, is reformulated using the general methods of the theory of metals, and exact solutions are obtained which are valid for all frequencies and temperatures. It is shown that, for large values of the free path of the conduction electrons, the electric field is propagated through the metal as a ‘surface wave’ which differs considerably from the classical exponential solution. The temperature variation of the surface impedance in the microwave region is considered in detail. Pippard’s simplified theory is shown to be qualitatively correct, and a quantitative discussion of his experimental results is given. The frequency variation of the surface impedance at low temperatures is also discussed, and it is shown that relaxation effects are negligible in the microwave region but become important in the infra-red and eventually restore the validity of the classical theory. The theory predicts that, as the frequency is increased, the reflexion coefficient of metals passes through a minimum in the far infra-red.

scholarly journals The surface impedance of superconductors and normal metals at high frequencies III. The relation between impedance and superconducting penetration depth

1947 ◽  
Vol 191 (1026) ◽  
pp. 399-415 ◽  
Keyword(s):  
Magnetic Field ◽  
Penetration Depth ◽  
Surface Impedance ◽  
Simple Extension ◽  
High Frequencies ◽  
Resistivity Measurements ◽  
Direct Measurements ◽  
Normal Metals ◽  
Velocity Of Propagation ◽  
Theory Of Superconductivity

The theory developed in II is extended to cover the case of a superconductor, and a formula is derived relating the r. f. resistivity to the superconducting penetration depth and other parameters of the metal. It is shown how the penetration depth may be deduced directly from measurements of the skin reactance, and a method of measuring reactance is described, based essentially on the variation of the velocity of propagation along a transmission line due to the reactance of the conductors. For technical reasons it is not convenient to measure the reactance absolutely, but a simple extension of the technique described in I enables the change in reactance to be accurately measured when superconductivity is destroyed by a magnetic field. The method has been applied to mercury and tin. In the former case the results are in agreement with Shoenberg’s direct measurements, and confirm that the penetration depth at 0° K is of the order of 7 x 10 –6 cm. The theory developed at the beginning of the paper is used to deduce the variation of penetration depth with temperature from the resistivity measurements of I, and it is shown that agreement with other determinations and with the reactance measurements is fairly good, but not perfect. Some of the assumptions used in developing the theory are critically discussed, and a qualitative account is given to show how Heisenberg’s theory of superconductivity offers an explanation of some of the salient features of superconductivity and inparticular indicates the relation between superconducting and normal electrons.

The anomalous skin effect

1952 ◽  
Vol 215 (1123) ◽  
pp. 481-497 ◽  
Keyword(s):  
Free Path ◽  
Low Temperatures ◽  
Skin Effect ◽  
Mean Free Path ◽  
Skin Depth ◽  
Surface Conductance ◽  
High Frequencies ◽  
Surface Imperfections ◽  
Anomalous Skin Effect ◽  
Cadmium Lead

The anomalous skin effect arises in good conductors at low temperatures and high frequencies when the electronic mean free path becomes comparable with or greater than the classically calculated skin depth. Measurements have been made on a number of metals at frequencies of 1200 and 3600 Mc/s, and the form of variation of r. f. surface conductance with d. c. conductivity agrees well with that predicted theoretically by Reuter & Sondheimer, assuming that the electrons are scattered diffusely when they hit the surface of the metal. From the results, estimates are made of the effective value of σ/ l , the ratio of d. c. conductivity to mean free path, and hence of the free surface area of the occupied region of k -space. The estimate for copper agrees well with that expected theoretically; those for silver and gold are rather lower than the theoretical values. For the other metals investigated, tin, cadmium, lead and aluminium, no theoretical estimates are available. The results are very sensitive to the presence of surface imperfections; the effect of these is discussed.

The theory of the anomalous skin effect in anisotropic metals

1954 ◽  
Vol 224 (1157) ◽  
pp. 260-272 ◽  
Keyword(s):  
Surface Impedance ◽  
Skin Effect ◽  
Energy Bands ◽  
Crystal Axis ◽  
Metal Crystal ◽  
Anomalous Skin Effect ◽  
Degree Of Anisotropy ◽  
Explicit Formulae ◽  
Energy Surfaces ◽  
High Degree

The theory of the anomalous skin effect in metals is extended to a uniaxial metal crystal containing two energy bands in each of which the energy surfaces are ellipsoids of revolution about the crystal axis. Explicit formulae are obtained, for the extreme anomalous limit, giving the dependence of the surface impedance on the orientation of the crystal axis, both for a plane metal surface and for a circular wire. The form of the anisotropy of the surface impedance is found to depend upon the axial ratios of the spheroidal energy surfaces and upon the ratio of the electron free paths in the two bands. Wide variations in behaviour are possible, and the surface impedance may show a high degree of anisotropy even when the d.c. conductivity is almost isotropic (as with tin at low temperatures). The results are evaluated numerically for tin, and the surface conductivity of a circular wire is found to show the minimum observed by Pippard (1950); the parameters can be chosen to give reasonable agreement with Pippard’s results.

scholarly journals The surface impedance of superconductors and normal metals at high frequencies II. The anomalous skin effect in normal metals

1947 ◽  
Vol 191 (1026) ◽  
pp. 385-399 ◽  
Keyword(s):  
Free Path ◽  
Skin Effect ◽  
Classical Model ◽  
Mean Free Path ◽  
Skin Depth ◽  
Normal Metals ◽  
Anomalous Skin Effect ◽  
Skin Conductivity ◽  
The Mean ◽  
Dimensional Argument

Measurements on the skin conductivity of the normal metals silver, gold, and tin show that at low temperatures the skin conductivity tends to become independent of the d. c. conductivity, which is at variance with the predictions of classical skin effect theory. Following a suggestion of H. London that this anomalous behaviour is due to the mean free path of the electrons becoming much greater than the skin depth, an attempt is made to calculate the effect for a semi-classical model of a metal. Although a rigorous solution has not been found, it is shown that the model predicts constancy of skin conductivity when the mean free path becomes very long. Moreover, there is reason to suppose that under these conditions only a small proportion of the conduction electrons contribute effectively to the high-frequency current, and an exact solution is given for a model based on this concept, which also predicts that the skin conductivity should be independent of the d. c. conductivity. A simple dimensional argument may be applied to enable values of the mean free path in copper, gold, aluminium and tin, relative to the value in silver, to be deduced from the experimental results. These values are not in good agreement with theoretical estimates by Mott and Jones. The behaviour of mercury is different from that of the other metals in­vestigated, in that the skin conductivity does not tend to a constant value. It is suggested that the theory based on a crude classical model is inapplicable to a metal such as mercury, in which the anomalous skin effect appears at such temperatures that the ideal resistance is still many times greater than the residual resistance.

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