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ass1: Switch B -> H for external magnetic field

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Kees van Kempen
2022-02-18 08:18:03 +01:00
parent 42f4f2a8ca
commit fb43a72cde
3 changed files with 120 additions and 58 deletions
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Lecture-2-slides-for-printing-type-i-vs-ii.pdf
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Lecture-2-slides-for-printing-type-i-vs-ii.svg
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14
superconductivity_assignment1_kvkempen.tex
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@ -129,27 +129,27 @@ such that the net field is zero.
This is called the Meissner effect. This is called the Meissner effect.
The perfect diamagnetism is quantized as having susceptibility $\chi = -1$. The perfect diamagnetism is quantized as having susceptibility $\chi = -1$.
There is, however, a limit to how large a field can be completely expelled. There is, however, a limit to how large a field can be completely expelled.
This is called the critical field $B_c$. This is called the critical field $H_c$.
If $B_c$ is exceeded, the superconductivity breaks down, thus the material will cease to expell the field, If $H_c$ is exceeded, the superconductivity breaks down, thus the material will cease to expell the field,
the diagmagnetism drops to $\chi = 0$. the diagmagnetism drops to $\chi = 0$.
This effect is seen in the bottom-left plot in figure \ref{fig:typevs}. This effect is seen in the bottom-left plot in figure \ref{fig:typevs}.
The relation between these two critical values is given as The relation between these two critical values is given as
\[ \[
B_c(T) = B_c(0) \left[ 1 - \frac{T}{T_c}^2 \right]. H_c(T) = H_c(0) \left[ 1 - \frac{T}{T_c}^2 \right].
\] \]
An example of this curve is plotted in the top-left of figure \ref{fig:typevs}. An example of this curve is plotted in the top-left of figure \ref{fig:typevs}.
Two phases can be distinguished: Two phases can be distinguished:
the Meissner (or superconducting) state under the graph, the Meissner (or superconducting) state under the graph,
and the normal state outside it. and the normal state outside it.
The transition between these states is a first-order phase transition due to the discontinuity in the magnetization $M = \frac{dF}{dB}$, the first derivative of the free energy to the applied field. The transition between these states is a first-order phase transition due to the discontinuity in the magnetization $M = \frac{dF}{dH}$, the first derivative of the free energy to the applied field.
Then there are type-II sc, for example, niobium (Nb). Then there are type-II sc, for example, niobium (Nb).
Instead of only having a Meissner or sc phase, they have another phase: Instead of only having a Meissner or sc phase, they have another phase:
the vortex state. the vortex state.
Below $T_c$ and lower critical field $B_{c1}(T)$, the type-II sc is in the Meissner state, Below $T_c$ and lower critical field $H_{c1}(T)$, the type-II sc is in the Meissner state,
and the material thus completely cancels the externally applied field. and the material thus completely cancels the externally applied field.
Above the upper critical field $B_{c2}(T)$, the material is in the normal state. Above the upper critical field $H_{c2}(T)$, the material is in the normal state.
When the field is between these two critical fields, $B_{c1}(T) < B < B_{c2}(T)$, When the field is between these two critical fields, $H_{c1}(T) < H < H_{c2}(T)$,
the material is in the vortex state. the material is in the vortex state.
In the vortex state, the externally applied magnetic field is not completely expelled. In the vortex state, the externally applied magnetic field is not completely expelled.
Instead, the material consists of normal and sc regions, the former called vortices. Instead, the material consists of normal and sc regions, the former called vortices.