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