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LETTERS
TO
THE
EDITOR
Lifetime
of
the Yukawa Particle
Recent investigations
by
various
authors'
have
made it
very
probable
that
the
hard
rays
of the cosmic
radiation
(mesotrons),
now identified with the
particle
of
Yukawa'
of
mass p~200m
(m=mass
of
the
electron),
are unstable
and
will
decay spontaneously
into
electrons and neutrinos.
The lifetime for
a
mesotron
at
rest has
been estimated
from
experience
to
be
of
the
order
2-4&(10
'
sec.
Yukawa himself
calculated the
lifetime on the
basis of
his ideas
to
be
of
the order 0.
25&(10
'
sec.
,
a
result not
far from
the
observed value. However,
the
present
author'
obtained on the
same
assumptions
a
much
smaller value.
The
importance
of this
question
may
justify
a
restatement
of the
theoretical result
and an
explanation
of
this
dif-
ference.
The final
formulae for
the lifetime obtained
by
both
authors
is
the
same,
apart
from
differences in notation.
It can
be
written in the form
G2
m
4h
r
—
—
4~~
Ac
p
pcs
Gy~
In this formula
k,
m,
c
have
the usual
meaning,
and
p,
is the
rest mass
of
the
mesotron. G
is
the
constant of
dimension
of
a
charge
in
the
potential
between
nuclear
particles
V(r)
=
(G'/r)e
rw&l&
following
from
Yukawa's
theory.
G'/kc
is
of
the
order'
4
p,
/3II (&=mass
of the
proton)
but
probably
somewhat
larger
than this
quotient.
The lifetime
r
is therefore
essentially
proportional
to
p
4.
Gz
finally
is
the constant
in
Fei:mi's
theory
of
P-decay,
normalized
to
be
a
pure
number. The form of interaction
assumed for
the
coupling
between
proton,
neutron and the
electron
neutrino field
is
G~mc'(&/mc)'(4N*PPI
)
(v.*Ps.
)+c.
c.
(f~,
P~,
p„,
q,
being
the wave functions of
neutron,
proton,
neutrino
and
electron,
respectively).
This leads
to
the
probability
for
emission
of an electron of
energy
e
Gy'
mc'
(ep
—
e) (e
mc
).&de
w(e)de=
(
M['
(2
)'
a
(mc2)
5
where
ep
is the maximum
energy
of the emitted electrons
and M
a
matrix
element from the motion
of
the
heavy
particles
inside the nucleus.
The
discrepancy
in
the calculated
lifetimes
comes
from
the
different values used for
the
constant
G~.
As discussed
by
Bethe and
Bacher'
and
by
Nordheim and
Yost,
'
the
experimental
value of
Gz
depends
quite
appreciably
on
the
group
of elements
which are taken for
comparison,
the
difference
being
due
in all
probability
to
the matrix
ele-
ment
M;
which
is
smaller than
unity
for
heavy
elements
but can be
expected
to
be
unity
for
light positron
emitters.
The
value
for
Gz
used
by
Yukawa
(0.
87)&10
"
in our
units) corresponds
to the
heavy
natural radioactive
elements,
while
the value deduced
for
the
light
positron
emitters'
is
GJ
=5.
5&&10
".
It
seems
beyond
doubt that
this later value has
to be taken
for our
purpose.
With the
present
most
probable
values
G'/Ac
=
0.
3;
p=200m;
Gg=5.
5)&10
',
we obtain
from
(1)
v
=1.
6&(10
'
sec.
,
i.e.
,
a
value about 10
'
times too
small.
A
decrease
in
the
assumed
value for
p,
to 150m
would increase
v
only
by
a
factor
of
order 3.
In view
of this definite
discrepancy
the
question
arises
whether
any
modifications
of
the
theory
could
give
a
better
result.
It'is
to be noted
firstly
that the introduction
of
the
Konopinski-Uhlenbeck form of the
P-decay theory
would
only
make
matters
worse
as it would introduce
roughly
another factor
(m/p)'.
A real
improvement can
only
be
expected
by
a
complete
reformulation of
the
theory.
One
possible
suggestion
would be to
assume that
the disintegration
of
a
free mesotron
is
in first
order
approximation
a forbidden transition,
while
in nuclei it is
made
allowed
by
the
influence of the other nuclear
particles.
L.
W.
NoRDHEIM
Duke
University,
Durham,
North
Carolina,
February 14,
1939.
~
H. Euler
and W.
Heisenberg,
Ergebn. d.
Exakt.
Naturwiss.
(1938);
P.
Blackett,
Phys.
Rev.
54,
973
(1938);
P. Ehrenfest
and
A.
Freon,
J.
d.
Phys.
9,
529
(1938);
T.
H.
Johnson
and M. A.
Pomerantz, Phys.
Rev.
55,
105
(1939).
2
H. Yukawa and others,
I
—
IV,
Proc.
Phys.
Math. Soc.
Japan
17,
58
(1935);
19,
1084
(1937);
20, 319,
720
(1938).
o
L. W. Nordheim
and
G.
Nordheim,
Phys.
Rev.
54,
254
(1938).
4
R.
Sachs
and
M.
Goeppert-Mayer, Phys.
Rev.
53,
991
(1938).
o
H. Bethe
and
R.
Bacher,
Rev. Mod.
Phys.
8,
82
(1936).
6
L. W. Nordheim
and
F.
Yost, Phys.
Rev.
51,
942
(1937).
It has
to
be
noted that the formula for 7
o
on
p.
943
should be
ro
~
=
(Gg2/(2x)3)
)&(mc~/$).
The value of
Gg
is then determined from the
empirical
value
vo
& —
10
4.
The
Scattering
of
Cosmic
Rays
by
the
Stars
of
a
Galaxy
The
problem
dealt with
in this note
may
be formulated
in
the following
way:
imagine
a
galaxy
of N
stars,
each
carrying
a
magnetic dipole
of moment
p, „(n
=
1,
2,
.
. .
N)
and
assume that the
density,
defined as the
number
of
stars
per
unit
volume,
varies
according to
any
given law,
while
the
dipoles
are
oriented
at
random because
of their
very
weak
coupling.
Under
this condition the resultant
field
of
the
whole
galaxy
almost vanishes. Let there be an
isotropic
distribution of
charged
cosmic particles
entering.
the
galaxy
from outside. Our
problem
is
to
find the intensity
distribution in
all directions around
a
point
within the
galaxy.
Its
importance
arises
from the fact that if the
dis-
tribution should
prove
to be
anisotropic a
means would
be available
for
determining
whether cosmic
rays
come
from
beyond
the
galaxy,
independent
of the
galactic
rota-
tion effect
already
considered
by
Compton
and
Getting.
'
Suppose
we
consider a particle
sent
into an element of
volume
d
V of
scattering
matter
in
a
direction
given
by
the
vector
R. Let
the
probability
of
emerging
in
the direction
R'
be
given
by
a
scattering
function
f(R,
R')
per
unit
solid
angle.
Conversely
a particle
entering
in the direction
R'
will
have a
probability
f(R',
R)
of
emerging
in the
direction
R.
Let
us
assume
that the
scatterer
(magnetic
field of
the
star)
has the reciprocal
property
so that
f(R,
R')
=f(R',
R).
In
our
case
this
property
is satisfie
provided
the
particle's
sign
is reversed at
the same
time
as
its direction
of
motion.
That
is,
the
probability
of
elec-
tron's
going
by
any
route
is
equal
to
the
probability
of
positrons
going
by
the
reverse
route,
If
it
has
the
reciprocal
property
for each
element
of
volume
it will also have
it
for