Flux quantization
The London equation (21.21) was proposed to account for
the observed facts of superconductivity including the
Meissner effect.
In recent times, however, there have been some even more
dramatic predictions. One prediction made by London was so peculiar
that nobody paid much attention to it until recently. I will now
discuss it. This time instead of taking a single lump, suppose we take
a ring whose thickness is large compared to $1/\lambda$, and
try to see what would happen if we started with a magnetic field
through the ring, then cooled it to the superconducting state, and
afterward removed the original source of $\FLPB$. The sequence of
events is sketched in Fig. 21–4. In the normal state there will be a
field in the body of the ring as sketched in part (a) of the
figure. When the ring is made superconducting, the field is forced
outside of the material (as we have just seen). There will then
be some flux through the hole of the ring as sketched in part (b). If
the external field is now removed, the lines of field going through
the hole are “trapped” as shown in part (c). The flux $\Phi$ through
the center can’t decrease because $\ddpl{\Phi}{t}$ must be equal to
the line integral of $\FLPE$ around the ring, which is zero in a
superconductor. As the external field is removed a super current
starts flowing around the ring to keep the flux through the ring a
constant. (It’s the old eddy-current idea, only with zero resistance.)
These currents will, however, all flow near the surface (down to a
depth $1/\lambda$), as can be shown by the same kind of analysis that
I made for the solid block. These currents can keep the magnetic field
out of the body of the ring, and produce the permanently trapped
magnetic field as well. Now, however, there is an essential difference, and our equations
predict a surprising effect. The argument I made above that $\theta$
must be a constant in a solid block does not apply for a ring,
as you can see from the following arguments. Well inside the body of the ring the current density $\FLPJ$ is zero;
so Eq. (21.18) gives
\begin{equation}
\label{Eq:III:21:26}
\hbar\,\FLPgrad{\theta}=q\FLPA.
\end{equation}

Now consider what we get if we take the line integral of $\FLPA$
around a curve $\Gamma$, which goes around the ring near the center of
its cross-section so that it never gets near the surface, as drawn in
Fig. 21–5. From Eq. (21.26),
\begin{equation}
\label{Eq:III:21:27}
\hbar\oint\FLPgrad{\theta}\cdot d\FLPs=q\oint\FLPA\cdot d\FLPs.
\end{equation}

Now you know that the line integral of $\FLPA$ around any loop is
equal to the flux of $\FLPB$ through the loop
\begin{equation*}
\oint\FLPA\cdot d\FLPs=\Phi.
\end{equation*}

Equation (21.27) then becomes
\begin{equation}
\label{Eq:III:21:28}
\oint\FLPgrad{\theta}\cdot d\FLPs=\frac{q}{\hbar}\,\Phi.
\end{equation}

The line integral of a gradient from one point to another (say from
point $1$ to point $2$) is the difference of the values of the
function at the two points. Namely,
\begin{equation*}
\int_1^2\FLPgrad{\theta}\cdot d\FLPs=\theta_2-\theta_1.
\end{equation*}

If we let the two end points $1$ and $2$ come together to make a
closed loop you might at first think that $\theta_2$ would
equal $\theta_1$, so that the integral in Eq. (21.28) would
be zero. That would be true for a closed loop in a simply-connected
piece of superconductor, but it is not necessarily true for a
ring-shaped piece. The only physical requirement we can make is that
there can be only one value of the wave function for each
point. Whatever $\theta$ does as you go around the ring, when you get
back to the starting point the $\theta$ you get must give the same
value for the wave function
\begin{equation*}
\psi=\sqrt{\rho}e^{i\theta}.
\end{equation*}

This will happen if $\theta$ changes by $2\pi n$, where $n$ is any
integer. So if we make one complete turn around the ring the left-hand
side of Eq. (21.27) must be $\hbar\cdot2\pi n$. Using
Eq. (21.28), I get that
\begin{equation}
\label{Eq:III:21:29}
2\pi n\hbar=q\Phi.
\end{equation}

The trapped flux must always be an integer times $2\pi\hbar/q$!
If you would think of the ring as a classical object with an ideally
perfect (that is, infinite) conductivity, you would think that
whatever flux was initially found through it would just stay
there—any amount of flux at all could be trapped. But the
quantum-mechanical theory of superconductivity says that the flux can
be zero, or $2\pi\hbar/q$, or $4\pi\hbar/q$, or $6\pi\hbar/q$, and so
on, but no value in between. It must be a multiple of a basic quantum
mechanical unit. London14 predicted that the
flux trapped by a superconducting ring would be quantized and said
that the possible values of the flux would be given by
Eq. (21.29) with $q$ equal to the electronic charge.
According to London the basic unit of flux should be $2\pi\hbar/q_e$,
which is about $4\times10^{-7}$ $\text{gauss}\cdot\text{cm}^2$. To
visualize such a flux, think of a tiny cylinder a tenth of a millimeter
in diameter; the magnetic field inside it when it contains this amount
of flux is about one percent of the earth’s magnetic field. It should be
possible to observe such a flux by a sensitive magnetic measurement. In 1961 such a quantized flux was looked for and found by Deaver and
Fairbank15 at
Stanford University and at about the same time by Doll and
Näbauer16 in
Germany. In the experiment of Deaver and Fairbank, a tiny cylinder of
superconductor was made by electroplating a thin layer of tin on a
one-centimeter length of No. 56 ($1.3\times10^{-3}$ cm diameter)
copper wire. The tin becomes superconducting below $3.8^\circ$K while
the copper remains a normal metal. The wire was put in a small
controlled magnetic field, and the temperature reduced until the tin
became superconducting. Then the external source of field was
removed. You would expect this to generate a current by Lenz’s law so
that the flux inside would not change. The little cylinder should now
have magnetic moment proportional to the flux inside. The magnetic
moment was measured by jiggling the wire up and down (like the needle
on a sewing machine, but at the rate of $100$ cycles per second)
inside a pair of little coils at the ends of the tin cylinder. The
induced voltage in the coils was then a measure of the magnetic
moment. When the experiment was done by Deaver and Fairbank, they found that
the flux was quantized, but that the basic unit was only
one-half as large as London had predicted. Doll and Näbauer got the
same result. At first this was quite mysterious,17 but
we now understand why it should be so. According to the Bardeen,
Cooper, and Schrieffer theory of superconductivity, the $q$ which
appears in Eq. (21.29) is the charge of a pair
of electrons and so is equal to $2q_e$. The basic flux unit is
\begin{equation}
\label{Eq:III:21:30}
\Phi_0=\frac{\pi\hbar}{q_e}\approx2\times10^{-7}\text{ gauss}\cdot\text{cm}^2
\end{equation}

or one-half the amount predicted by London. Everything now fits
together, and the measurements show the existence of the predicted
purely quantum-mechanical effect on a large scale.