491
lattice, calculated from DEBIJE' s law with II = 413. The additional heat
capacity can, for the range 1.1 to 9.0 0 K. be represented by C = 0.001744 T .
It follows another law and considerably surpasses the contribution that
will be due to the interaction energy of the electrons that are responsible
for ferromagnetism. The hypothesis is made that it is connected with the
energy of the conduction electrons that give rise to a heat capacity
proportional to T. The fact that it is many times larger than follows from
SOMMERFELD'S formula for free electrons then shows that in the corresponding energy band, at least at thelevel of the limiting energy, the
density of the possible energy states is particularly large.
Physics. - Mean va lues of the electC'Îc force in a random distribution
of charges. By Prof. L. S. ÜRNSTEIN. (Communication from the
Physical Laboratory of the University of Utrecht).
(Communicated at the meeting of April 27. 1935).
HOLTSMARK has given a deduction of the frequency law of the electric
BeId caused by a random distribution of electric charges ions. dipoles.
quadrupoles surrounding an atom of given radius a. His deduction is
rather complicated and it is perhaps of some importance that the mean
values of the electric BeId excerted on the atom can be deduced in a
quite elementary way. We assume a random distribution of ions surrounding the atom and suppose that there is no correlation of the
probability of the numbers in the elements of volume, then if the mean
density is e and the mean number of ions in an element dv of volume
e dv = n. the probability of a deviation of the mean value ~ will be
given by the GAUSSian law. From this law we can easily deduce the
mean values of the deviation in every element of volume. the odd means
are zero and the even mean values are given by the formula
t5 2p = 1.3 ... (2p-l) n P
= 1 .3
2p-l (edv)p.
The force x in any direction can be expressed if we know the numher
of ions n" in every element of volume dv" situated at a di stance r" from
the centre of the atom of radius a in a direction forming an angle
{}y. with the direction x by the formula
wh ere e is the charge of the ions.
N ow n" can he expressed by n 15" wh ere n is the numher of ions
in a homogeneous distribution and one sees easily from the symmetry
+
492
that the force x caused by the homogeneous distribution is zero that
therefore
~!5x cos {}x
Xe
2'
t'x
The mean value X therefore is zero, and if we neglect the correlation
from the fact that !5~p + 1 etc. are zero we easily see that
X2p+l
=0.
For X2 we get:
as the mean values !5 x de are zero.
We obtain th us applying the value of
c5~
For Xi we obtain
_
X - e1
~_
2 2
615 x de
)
2
2
~ (~
?"x cos{}x1 + ~7cOS
{}x cos {}e + ...
x
x
e
or applying the values for tbe means of the d:
=3e1e2~J dVC~:2{}x
f=1.3(4;:ey=1.3X22
a
By analogeous transformation we obtain
-X2p =
1 . 3 . (2 p - 1)
~4e2e~p
(3a ~ =
-p
1. 3 ... 2 p - 1 x2 .
The mean values of x correspond with those of a GAuSsian distribution, however the frequency law ought not necessary to be GAuSsian,
as the reg ion of X is not infinite.
Now it is easy to give an analogeous deduction for the mean force
in the case of dipoles.
493
If we e.g. suppose that the potentialof the dipoles is given by
7ax
for a dipole in the point X r the force in the x direction is
Por the force excerted by a homogeneous distribution we get again
zero. the force by a distribution with deviations c5 x is again:
X is again zero and the same is true for
)(2p+l
Por
)(2
=0.
we Gnd
•
Por the mean values of x 2p we get
X2p
= 1 .3 2p-l x2.
Por other types of dipoles and for quadrupoles the analogeous result
holds.
It is possible to Gnd the mean values also for the case that a corre~
lation of probabilities is assumed. Let r represent the distance of two
elements of volume du x and dUe than we will assume that c5 x c5e is given
by a correlation function (Ct ORNSTEIN and ZERNIKE these Proceedings
1914). such that
dx c5 e
= g (r) e du. du,..
Por )(2 we have:
In order to Gnd its value we assume that g (r) shows only Gnite values
494
for small values of r than we can introducing polar coordinates rp
with XY. yY. ZY. as a centre develop. We have
+r
+ r cos rp
Ye = yY. + r sin rp cos x
Ze = ZY. + r sin rp sin x.
Xe = XY.
For )(2 we get up to the second order in rafter integration for the
element du. and putting
f
2
n
J'f
e2 e
"
g (r) r 2 du =
2n cos 2 rp
~
7f
(8---r
1-
n r
2
)
•
sm
rp drp
=
0
or
2
4n ( 1 -8nr
2 e)(2=e
- -)
3a
3a 2
if we assume a positive correlation and
if we assume a negative one.
In analogeous way it is possible to determine the mean values of
the electric field itself. It is of importance that R2 the mean square of
the field is for the case of ions when the correlation is neglected given
by 3)(2 thus
the square of the force is -therefore proportional to the same density.
the same is true for the field of dipoles and quadrupoles. If it is possible
to measure R2 we can find eta. wh en a is given e could be determined.