silsbee_rule

# Silsbee's rule

 Stan Zurek, Silsbee's rule, Encyclopedia-Magnetica.com, {accessed 2020-12-03}

## Definition

Silsbee's rule (also Silsbee rule or Silsbee criterion)1) - a criterion describing the critical (maximum) value of magnetic field, above which a superconductor looses its superconducting state, and becomes a normal resistive conductor.

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Critical magnetic field Hc for a type I superconductor

by S. Zurek, Encyclopedia Magnetica, CC-BY-3.0

There are two critical magnetic field values for a type II superconductor

by S. Zurek, Encyclopedia Magnetica, CC-BY-3.0

The phenomenon is thought to occur due to the magnetic field destroying the symmetry between up and down spins in Cooper pairs (pairons).2) The density of current capable of such action is called depairing current density.3)

The original rule was proposed in 1916 by F.B. Silsbee, and regarded only the value of transport current, without any other external magnetic field applied. However, the behaviour can be extended to the condition where the collapse of superconductivity is caused by the superposition of magnetic field generated by the transport current and the externally applied field and it is then sometimes referred to as the generalised Silsbee's rule (or generalised Silsbee's criterion).4)

The value of critical field depends on the type of material, but it is also decreases with increasing temperature.

## Type I and type II superconductors

Silsbee rule is directly applicable to type I superconductors, where the collapse of the superconducting state is sharp. For the type II superconductors the transition is gradual and the rule denotes only a point after which the vortex state (mixed) develops. For type II there are two critical values: one after which a superconductive state begins reducing (Hc1), and another one above which there is only a fully resistive state (Hc2). The region between them is the mixed state.

Therefore, some authors talk about the “breakdown of Silsbee rule”.5)

In type I superconductors the critical fields are lower, not exceeding around 2 T.

In type II they can reach much greater values, for instance 44 T for Nb3Al. Such materials are therefore much more useful for practical applications like superconducting electromagnets.6)

## Equation

For type I superconductors the critical value of the transport current Ic (i.e. the main current in the conductor) can be calculated as:7)

 $$I_c = 2 · \pi · r · ( H_c - 2 · H_t)$$ (A)

where: r - radius of the superconductor (m), Hc - value of critical magnetic field strength (A/m), Ht - value of applied transverse magnetic field strength (A/m)