628
Letters to the Editor
On the Accuracy of the Moliere
Funetion, II
J.
Nishimura.
Department of Phyoics, Kobe University
K. Kamata
Scientific Research Institute
July 9, 1951
In our recent letter to the editor, I) it was
shown that the Moliere's2) angular distribution
function does not represent the .structure of
cascade showers accurately. In this paper
we evaluate the integral lateral distribution
function* IIl21trdr for a few shower ages
8 from the formula (5) of the reference (3),
and compare them with Moliere's results**.
As shown in Fig. 1, our distribution functions3) become less. steep with increasing
shower .age, as it must be. In spite of his,
crude approximation,. his function agrees.
<
fairly well with ours of s=l for ~
2.
r
rl
5 seems
The discrepancy appearing for -
r1
>
to be surely due to the difference of the
approximations used by each author, because
the average number of particles lying in this
5)
region (~>
is much underestimated in
rl
the Arley's approximation used by him.
Nevertheless, Roberg-Nordheim4 ) compared
their calculated mean square deviation
<r~> A *** with Moliere's one, and found
that his function is larger than exact one
v
for large~.
rl
If we adopted this argument,
629
Letters to the Editor
less significant. However, remembering the
fact that the average number of the particles
at shower maximum having energies less than
0.05 E is about 20% of the total number of
shower particles and that the lateral deviations
from shower axis of such low energy particles
are much larger than that of high energy
particles, the difference of the mean square
deviations between two cases mentioned above,
would become considerably important. In
fact, the results calculated by the method of
Roberg-Nordheim show that <r2) A I" E=O is
about two times as large as <r2) A I" E=o.ose •
ours would not also be free from such adverse criticism, since ours is larger than
Moliere's .results for .!:..rl
> 5.
However, this
conclusion must be revised because of the
The <r2) A I" used by
following reason.
Roberg-Nordheim for the comparison with
Moliere's result corresponds to that of the
particles having energies larger than 0.05
e****, while Moliere's function and ours
contain the contribution from all particles
having energies larger than zero.
At first sight, the difference between
<r2) A I" E=O.05e and <r2) A I" E=o seems to be
I"
i
L
- - - - :\lohcn.:
~=1
- - - - Ours
S=I
-
S=l.t
- - Our!>
10-'
10-'4
~.L-
__
~
________
~
______
~
6
o
______
~
____
---7
Fig. 1. Lateral distribution of shower electrons.
These functions are normalized as
i "" H12n~d(~) = 1.
j0
"1"1
It should be noted that the contribution of single scattering are not taken into
account in our case, while his function contains this contribution.
*
**
In this paper we limit ourselves to the shower
initiated by an electron of very high energy.
Integral lateral distribution function nl (E, r)
2n,. dr represents the total number of electrons
***
with the energy larger than E and having
the lateral distance from the axis between r
and r+dr.
(rIl)AI" is given by the formula
630
Letters to the Editor
~'" III (E', r,
(r'l)AV
****
1)
2)
3)
4)
t)r2 21l'r dr
~-------.
III (E', r, t) 21l'r dr
L
E'=8'=-
represents the critical energy of the transversing material.
E
J. Nishimura and K. Kamata, Prog. Theor.
Phys. 6 (1951), 262.
G. Moliere, Cosmic Radiation edited by W.
Heisenberg, (Dover Publication, N. Y. 1946)
Chap. 3.
J. Nishimura and K. Kamata, Prog. Theor.
Phys. 5 (1950), 899.
J. Roberg and L. W. Nordheim, Phys. Rev.
75 (1949), 444.