**This paper is Feynman's first published work.** The first offi...
This paragraph displays genius of Feynman. No equations or laws, ju...
!["Galaxy with N stars"](http://i.imgur.com/ZUAzqqe.png) **Figure ...
Feynman and Vallarta found that the **net effect of the scattering ...
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
LETTERS
TO THE EDITOR
any
extended
distribution of
matter,
that
is,
F(R,
P;
R',
P')
=
F(R',
P';
R,
P),
where
F(R,
P;
R', P')
is
the
probability
that
a
particle
going
in
the direction
R at the
point
P will
emerge
in
the direction
R'
at
the
point
P'.
This is
because
the
probability
of
following
any
route
is
equal
to
the
probability
of
following the reverse
route, through the
same
elements of
volume. Thus the
probability
of
a certain
end result from
a
number of
possible
routes will
equal
the
probability
of
the reversal of the result
occurring
through
the reverse
routes,
In
our
case the
scatterer
(star)
is
to
a
large
extent
non-
absorbing and
noncapturing.
The former
is
true
except
for
particles colliding with the
star,
which can
only
happen
when their
energy
is
sufficiently
great,
and the latter is
true
except
for
particles
which follow
asymptotic
or
-periodic orbits in
the
magnetic
field of
any
one of
the
X
stars.
These
orbits, however,
almost
certainly
form
a set
of zero
measure in
the
manifold
of
all
possible orbits,
'
that
is, they
occur
only
exceptionally.
Thus,
while
a
dipole
magnetic
field
can imprison
charged particles starting
from
a
point
within it and
can
also
keep
them
away
if
starting
from
infinity,
depending
on their
energy
and
angular
momentum, it
can
only exceptionally capture
such
par-
ticles
starting
from
infinity.
In
our
case,
therefore,
all
particles
starting
in
a direction R at
a
point
P,
sufficiently
far from
all
neighboring
stars,
have
only
a
small
chance
of
being
either
absorbed or
captured
in
a periodic
orbit
(of
finite or infinite
period),
so that the
great majority
of
them
will
emerge
at
infinity.
For almost all
particles,
therefore,
the
probability
of
emerging
at
infinity
must be
unity,
or
J,
F(R, P;
R',
~
)dR'
=
1
almost
always.
Now
consider
a
beam of
particles at infinity
whose
in-
tensity
in
a
direction
R'
is I
(R').
The
intensity
at
P
observed in the direction
R
will
be
I„(R)
=
f
F(R',
~;
R,
P)I„(R')dR'
(2).
Using
(1)
and
assuming an isotropic
distribution
at
infinity
such
that 1„(R')
is
a
constant
(independent
of
R'),
we
find
that
Eq.
(2)
becomes
Ip(R)
=I
J
F(R,
P;
R',
~)dR'=I„(3)
by (1).
Therefore
the
intensity
in
any
direction
at
P
is
the
same and the distribution is
isotropic at
P if it is
isotropic at infinity.
From
the remark
made
previously,
it is
clear that
if
the
distribution
of
positive
and
negative particles at
infinity
is
isotropic,
it will also
be
isotropic
at
any
point
P,
except
for
small
irregularities due
to absorption
by
collision and
by
capture
into
periodic
orbits.
We conclude that
particle
scattering
by
magnetic
fields of
the stars
is
unable
to
contribute
anything
to the solution of the
problem
whether
or not cosmic
particles
come from
beyond
our
galaxy.
The
considerations
developed
in
this note
clearly
hold so
long
as the
scattering centers
satisfy
the conditions of
being
nonabsorbing and
noncapturing,
irrespective
of
the
law of
force which is
responsible for
the
scattering. It.
need
hardly
be emphasized
that
they
apply
only
to
the case
in
which
there is no resultant
magnetic
field
for
the
whole
galaxy,
such as would
exist
if the
dipoles
were
oriented
along
preferential directions.
In
this case
particles would
either
be
imprisoned
if
born
within
the
galaxy,
or
kept out,
if
coming
from
outside,
depending
on their
energy
and
angular
momentum. The
reciprocal
property
of
paths
would then break
down
in
general,
but would still hold for
any
allowed
direction
at
any
point
within
the
galaxy.
M. S.
VALLARTA
R. P.
FEYNMAN
Massachusetts Institute
of
Technology,
Cambridge, Massachusetts,
February
15,
1939.
'
A. H,
Compton and
I.
A.
.
Getting, Phys.
Rev.
47,
817
(1935).
M.
S,
Vallarta,
C.
Graef
and
S.
Kusaka,
Phys.
Rev.
55,
1
(1939).
~
See the
discussion
by
E.
J.
Schremp, Phys.
Rev.
54,
153
(1938);
and
forthcoming
papers
by
O. Godart
and
by
A.
Banhs,
Jr.
Nuclear
Excitation
of Indium
by
X-Rays
It has been shown
recently''
that
the stable nucleus
In"5,
when
excited
by
fast neutrons
or
protons,
may
be
left
in
a
metastable
excited
state, designated
by
In"'*,
from which
it
decays,
emitting
negative
electrons,
with
a
half-life
time of 4.1 hours.
We have now observed that the
same
metastable
state
can
be excited when indium is
irradiated
by
x-rays.
The
x-rays
were
produced
by
bombarding
a
2-mm
thick lead
target
with
electrons from an
electrostatic
generator.
A thick indium
foil,
1 inch in
diameter, was
placed directly
behind
the lead
target.
After
30 minutes irradiation
at an
electron
energy
of 1.73 Mev and
a
current of
10
ya,
the
indium foil
showed
an
initial
activity,
recorded
on a
Geiger-Muller
counter,
of 45 counts
per
minute.
The
activity.
decayed
with
a
period
of
approximately 4
hours.
The walls
of the
counter reduced
the
intensity of the
rather
soft
P-rays to about
one-half.
Until
more
is
known
about the effective
x-rays,
no well-defined
cross section
can
be deduced
from
these
data.
By
varying
the
bombarding
voltage
it
was
established
that
the eAect has
a
threshold
at
1.
35&0.
1 Mev. This
result
might
be interpreted
by
assuming that
In"5
has
at
that
energy
an
excited
state
which
combines
both
with
the
ground
state and the
metastable
excited
state.
In
a
note which has
just
become known
to
us Pontecorvo
and
Lazar'
also
report
the excitation
of indium
by
x-rays.
In their
experiments the
x-rays were
produced
with
an
impulse
generator
working at a
peak
voltage
of 1850
kv.
GEQRGE
B.
CoLLINs
BERNARD
WALDMAN
EDWARD
M.
STUBBLEFIELD
Department of
Physics,
University
of Notre
Dame,
Notre
Dame,
Indiana.
1VI.
GQLDHABER
Department
of Physics,
University
of
Illinois
Urbana,
Illinois,
February
15,
1939.
~
Goldhaber,
Hill and
Szilard,
Phys. Rev.
55,
47
(1939).
~
Barnes
and Aradine,
Phys.
Rev.
55,
50
(1939).
~
Pontecorvo and Lazar,
Compte rendus
208, 99
(1939).

Discussion

This paragraph displays genius of Feynman. No equations or laws, just defining the problem in a better way, with making reasonable assumptions or ruling out possibilities, and he reached an equation. Pure delight! !["Galaxy with N stars"](http://i.imgur.com/ZUAzqqe.png) **Figure 1:** *The dashed line represents an element of Volume $dV$, the black dots represents the stars. The solid line depicts the trajectory of a cosmic ray inside that element of volume when it is scattered by a star.* Consider a set of stars (dipoles) distributed inside an element of volume $dV$. We assume that all the stars are: - non-absorbing: we consider the CRs have sufficient energy to collide with the star to be negligeable - non-capturing: the set of these type of orbits is negliegeable compared to all the other orbits Thanks to these assumption we know that all particles entering the distribution of matter will emerge at infinity, thus we have summing over all the possible directions $R'$ we can write: \begin{eqnarray*} \int F(R,P;R',\infty) \, dR' = 1 \end{eqnarray*} We want to find the intensity of particles in the direction $R$ at point $P$: $I_p(R)$. Considering a beam of particles at infinity with constant intensity $$I_{\infty}(R') = I_{\infty}$$ (independent of R') and taking into account that the probability of following any route is equal to the probability of following the reverse route we can write that: \begin{eqnarray*} I_p(R) &=& \int F(R',\infty;R,P)I_{\infty}(R') \, dR' \\ &=& I_{\infty} \int F(R,P;R',\infty) \, dR' \\ &=& I_{\infty} \end{eqnarray*} We can conclude that the intensity in any direction at any point $P$ is the same and the distribution is isotropic at $P$ if it is isotropic at infinity. **This paper is Feynman's first published work.** The first official scientific publications by Richard Feynman first appeared in 1939 in the Physical Review. He published two papers as an undergraduate student: - "The Scattering of Cosmic Rays by the Stars of a Galaxy", co-written with Manuel Vallarta - ["The Forces in Molecules"](http://faculty.washington.edu/miller/51816/Feynman.pdf), Feynman's senior thesis This paper was published in 1939 in his final year as an undergraduate at the Massachusetts Institute of Technology (MIT). Feynman and his Professor Manuel Vallarta were investigating whether cosmic rays came from inside or outside the galaxy by studying their scattering by the magnetic fields of the galaxy's stars. Their result is simple and straightforward but their reasoning is very clever: they stated that the probability of a particle’s emerging from a clump of scattering matter in a certain direction must be equivalent to the probability of an antiparticle's taking the reverse path. (like an anti-particle going backwards in time) Feynman went on to receive his Bachelor's degree from MIT and attained a perfect score on the graduate school entrance exams to Princeton University in physics. !["Feynman 1939"](https://qph.ec.quoracdn.net/main-qimg-5f3903392fe56957ff50821bff2186ed-c?convert_to_webp=true) *1939 - Picture of Richard Feynman at the Princeton Library* Feynman and Vallarta found that the **net effect of the scattering of cosmic rays by stars was zero.** They have shown that that the intensity of cosmic rays is isotropic and thus the magnetic fields of the stars are unable to contribute anything to the solution of the problem whether or not cosmic rays come from beyond our galaxy. If cosmic rays seem to come from all directions, it is not because the stars are interfering and disguising their original orientation.