4. DIAMAGNETISM OF METALS
It is shown that in quantum theory even free electrons, besides spin-paramagne-tism, have a non-vanishing diamagnetism originating from their orbits, which is due to the limitation of the electron orbits in the magnetic field. A few further possible inferences concerning this orbit limitation are indicated.
1. Up to now, it lias "been more or less quietly assumed that the magnetic properties of electrons, other than spin, are due exclusively to the binding of electrons in atoms. For free electrons, the classical zero-result is assumed for the orbital effect, on the basis that the Fermi integral of the con-esponding Hamiltonian, just as the Boltzmann function, is independent of the magnetic field. However, a quantum phenomenon is thereby allowed to be neglected. In the presence of a magnetic field, the motion of the electron is obviously finite in the plane perpendicular to the field. This leads, of necessity, to a partial discreteness {corresponding to the motion in this plane) of the eigenvalues of the system, which gives rise, as will be shown, to a non-vanishing orbitalmagne-tism.
The Hamiltonian of a free electron in a magnetic field can be written in the familiar form:
2.22 w
where
1 f       eB \ 1 (       eE \ 1
•m \        2c   j m \        2c   J rn
are the velocities of the systems (H is the absolute value of the magnetic field in the direction of the 2-axis). The motion in the direction of the field is independent of the field and of other components of motion, and can be split off by putting f2 simply equal to a constant, which corresponds to the Schrodinger function Vfc (3)
The energy value of the system will then be represented as the sum of two independent terms. Now, instead of having to solve the two corresponding Schrodinger equations for the a^-motion, we can use an artificial method for deriving the energy values by writing doMTi the commutation relationships of the component velocities v1 and v2. From equation (2) it follows directly
* eE
b\>     = *i v2 - v2vx =----, (4)
l cm"
L. Landau. Diamagnetismus der Metalle, Z. Phys. 64, 629 (1930),
31
32 collected papees of l. b. landau
since, as is well known {x, y\ = [px, p2] = 0, [p^ x] — [p2,y] = kji. The constant on the right-hand side of equation (4) is reminiscent of the usual p, q-commv.-tation relation. In order to come back to that case, we can now temporarily introduce the co-ordinates P and Q by means of
P el n
The commutation relation reduces into the usual form [P, Q] — hji. The equation referring to the energy can now be written in the form:
*«-™-^-- (6)
This, however, is none other than the Hamiltonian of a linear oscillator with mass m and frequency co — e Hjm c. The eigenvalues of such a system are, as is well-known, equal to
*K* + T)*ffl-(» + T)^*' (?)
where % can assume all positive integral values. Together with the 2-motion this gives
jSJn + ±)ULH + -P-, (8) V      2/ roc 2m
for the eigenvalue of the translational motion of the electron.
The eigenfunctions can also be determined in a simple manner. For this purpose we eliminate one of the co-ordinates, for example x, from the velocity operators (and thus also from the energy operator), by putting
leSxy
y = e~ 2«« x- (9)
This gives
v-.w—----v w = e  sac----y y
ir    i ox      2c * ' \i   8x -A
\ oy      2c i oy
The Schrodinger equation corresponding to this is written
ft   3 Y    /A   3      el V 1 .
+ -~---T—y   -2mlfU-0. (11)
i   dy J     \ i ox      %c   ) j
This equation does not contain x explicitly; thus, its solutions can be written in the exponential form i
X = e~*aX<p(y), (12)
BIAMAGNETISM OF METALS
33
where a is a constant and <p is no longer dependent on %. If we substitute equation (12) in equation (11), we obtain immediately for <p an oscillator equation
d3<p 2m d^
h2
r„ mfeH B---
2 \ fflc
eE
<p = 0.
(13)
which is just as we should expect from what has previously been said. The. "equilibrium point" of this oscillator is at the point r\ = c afeH. Thus, we obtain finally for the complete eigenfunction of the system
US
hc\V eHa
where    denotes the eigenfunction of the equation
d«2
4- (2n + 1 — u2) <pn — 0.
(14)
(15)
The quantity a does not enter into the eigenvalue. Since it can assume an arbitrary value, then our problem is still degenerate in a continuous way. In order to determine the density of the eigenvalues we replace, as usual, the infinite space by a finite container with the linear dimensions A, B and C in the x-, y- and z-directions. In the z-direetion the number of possible p3-values in the interval Ap is well known, and is equal to
jB
G
'4p
■Ap.
(16)
In a quite similar manner, we obtain for the ^-direction
Rax —
2n%
■Ac.
(17)
In the y-direction we require that the trajectory always lies in the container at a sufficient distance from the walls. Then we need not consider the influence of the "2/"-walls, because of the rapid damping of <pn with range. Since the number of trajectories colliding at the walls can be considered as small, with an adequately large container, then we can assume that this requirement gives us practically all the existing trajectories. On account of the large container dimensions, we can thus neglect also the radius of the trajectory and we can simply write
0 <-^—o < B
or
eH
0 <a<-H.
(18)
34 ■ COLLECTED JAPEBS OF L. D. LA3TDAT7
If, now, we -wish to obtain the total number of eigenvalues corresponding to the given non-degenerate quantum number n, then we have to substitute A a — (eBIc) S in equation (17). This gives
eH eH JRR = —T-AB = -—-S, 2ti ft c 2tz n c
where S is the area of the container sides. Altogether we have
eH
MAp_n^ EA7>Bn^j-^-7Ap, (19)
thus, as was to be expected, proportional to the volume. It can be easily cheeked that equation (19), as a result of the limiting transition H -» 0, converts into the usual eigenvalue distribution of free motion. Together with the spin, we have
E' = B ±-l!Lst (20) 2m c
that is to say
E =-n + -p-, (21)
so that to every n > 0, the double degeneracy
^-dnk™* <22a>
corresponds, and with n = 0, we have
^-'-4^77^- (22b)
2. In order to obtain the magnetic properties of the system, we require, as is well known, only to evaluate the summation
Q*= -kT%ln(l + e^T~) (23)
over all eigenvalues; co denotes the so-called chemical potential. The number of particles N is linked with a> through the expression
tf=-—, (24) oco
and the magnetic moment through
M--w- <25>
In our ease, we have a continuous and a discrete parameter, so that the summation in equation (23) can be represented by a sum of integrals. Thus, in
diamagotstism of metals
35
order to resolve the effect more clearly, we shall start from the orbital energies of equation (8) and consider to begin with the spin only in the multiplicity. If we put
e h
= P>
m c
then we hare
Q = -IT
i=0 J
1 + e
o>-(,ii+1/2) ŕiff
2n2 k2 o
Vdp£
If, now, for the sake of brevity we write
/        m \ mV
.u +.......
-hT
lni
2ji2Á3
■djj3 = /(co),
then Q assumes the form
co —
In order to determine this sum, we can use the f amiliar series expansion
i
X  í 1
/(*) d«-—|/'(*)|f.
Its admissibihty requires, in general, that
< I.
(26)
(27)
(28)
(29)
(30)
(31)
It can easily be seen in our case that this corresponds to
pH<hT. (32)
This condition is no longer fulfil led at very low temperatures and in strong fields. On account of this, the latter case should lead to a complicated, no longer linear dependence of the magnetic moment on H, which should have a very strong periodicity in the field. Because of this periodicity, it should be hardly possible to observe this phenomenon experimentally, since on account of the inhomogeneity of the existing field, an averaging will occur. If, however, we average the series in equation (29) over an interval A H, the condition for equation (31) will again be fulfilled, if in the "dangerous" part near co - [n + (1/2)] fi E = 0, the change of argument is considerably larger than the difference between the two successive arguments, i. e,
AS
or
AS fiB IT p ~
(33)
36
COLLECTED PAPEES OF L. D. LAKDAT7
Even with the strongest possible fields (H = 3 x 105 gauss), the right-hand side gives only 0-1 per cent with co — 3 eV. If we now use the summation formula (30) explicitly, we obtain
df((o — n n B)
3 co
- f/(a;)daf -f (<o) (34) J 24 c?&>
- »
{/ (oo) = 03. The first term of this summation is independent of the magnetic field. It represents the summation in the field-free state, so that in place of equation (34) we can write
0      24 do?
From this, we obtain
dQ     p.2 8*Q0 „
M--Hfm n~fiB- (35»
If we now put
6Q       ,T 3^ -— =-A,   co = ——, 0 co dN
where F = Q — 0){dQld<o) is the free energy of the system, then equation (35) becomes
i¥=--—= (36)
12—— 12—— dN N2
We have thus in reality a diamagnetism, which is exactly equal to one third of the Pauli spin paramagnetism1, for which we have, in the familiar form
Thus, free electrons are altogether still paramagnetic. , If the electrons are found in the periodic field of a lattice, then it is well known2 that their motion can still be considered as free, in a certain sense. The principal characteristic of the effect of the magnetic field therefore remains unchanged, although the above calculation is, of course, no longer quantitatively applicable. In particular, the ratio of para- and diamagnetism changes, and it is quite possible that in actual cases the latter can also exceed the former, so that we obtain a diamagnetic substance like bismuth. However, this is only possible with a relatively powerful lattice effect, so that a quantitative theory of this phenomenon should scarcely be possible. Another effect of the
DLA.MAGNETISM OF METALS
37
interaction consists in the fact that the diamagnetism loses its symmetry and now depends on direction, a property in which this type of diamagnetism differs from the normal atomic diamagnetism as well as from the necessarily symmetrical spin-paTamagnetism.
A similar phenomenon can also take place in non-conducting substances and indeed with paramagnetic substances, where we also have a continuous eigenTalue spectrum. Here, we also get discrete eigenvalues in the magnetic field and: as a result of this—diamagnetism. This diamagnetism is quite small compared with the paramagnetism which is present, but differs from it by its asymmetry, so that perhaps it forms the main basis of the observed asymmetry in paramagnetic crystals (another reason is the so-called magnetic or relativistic interaction between spins). On this account, it is of interest to estimate the order of magnitude of the effect. This is done in the simplest manner dimension-ally. The susceptibility is first of all proportional to (c/c)2, since the action of the magnetic field is always introduced by e Hjc. The mass of the electron m does not occur explicitly in the calculation, in this case. It plays its role in the exchange-integral, which characterises the exchange phenomena in the lattice. Moreover, only h and the density NjV can still make their appearance. Clearly this leads to the expression
The exchange integral J determines as is well-known, the Curie temperature &, and h Q is of the order of magnitude of so that in place of equation (38) we can write
The phenomena turn out to be qtiite different if the external effects are of a non-periodic nature. Such effects destroy the direction-degeneracy of the motion and consequently, if they cannot be assumed to be small, the possibility that the field produces an effect of the type investigated here. This requires that the "mean free path" corresponding to this effect is small compared with the diameter of the electron orbits in the magnetic field. Since this diameter in normal fields is of the order of magnitude of a tenth of a millimetre, then even very small impurities or even powdering of the substance can suffice. Such changes of susceptibility have been detected in bismuth, and for the first case in a whole range of substances. It would be of great interest to be able to observe in these cases a change of susceptibility with field, which ought to take place according to the present theory, when rs > A {rs is the radius of the orbit in the magnetic field, Á is the mean free path or the dimensions of the crystal) changes to rn <š x.
In conclusion, I should like to make the supposition that the phenomena which have been investigated might explain also the Kapitza effect of linear resistance changes in a magnetic field. For the admissibility of the presumed approximation of free electrons in a magnetic field, it is not necessary that rs
(38)
(39)
38
COLLECTED EAJPEBS OF L. X>. LANDAtT
be smaller than the mean free path corresponding to the lattice ('which, would be impossible at normal temperatures), because the interaction with the lattice oscillations involves, apart from momentum transfer, also energy transfer. However, according to the foregoing remarks, it is probably essential that r3 be considerably smaller than the mean free path of the lattice distortions, which leads, after short calculations, to the expression
N
H>ec~~R, (40) V
where R is the specific resistance (in electrostatic units) of the crystal. If inequality (40) is not fulfilled, then the method considered here is not applicable and it can he seen quite easily that all the effects of the field must be necessarily quadratic. The field in expression (40) is in good agreement with the critical field of the Kapitza experiments, which should lend support to the theory. I have not yet succeeded in presenting a quantitative development of the theory.
At this stage, I should like to thank sincerely Mr P. Kapitza- for discussions of the experimental results and for the communication of certain unpublished data.
Referekces
1. W. Paotjc, Z. Phya. 41, 81 (1927).
2. F. Bloch, Z. Phys. 52, 555 (1928).