**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
theYukawaParticle
Recentinvestigations
by
various
authors'
have
madeit
very
probable
that
the
hard
rays
ofthecosmic
radiation
(mesotrons),
nowidentifiedwiththe
particle
of
Yukawa'
of
massp~200m
(m=mass
of
the
electron),
areunstable
and
will
decayspontaneously
into
electronsandneutrinos.
Thelifetimefor
a
mesotron
at
resthas
beenestimated
from
experience
to
be
of
the
order
2-4&(10
'
sec.
Yukawahimself
calculatedthe
lifetimeonthe
basisof
hisideas
to
be
of
theorder0.
25&(10
'
sec.
,
a
resultnot
farfrom
the
observedvalue.However,
the
present
author'
obtainedonthe
same
assumptions
a
much
smallervalue.
The
importance
ofthis
question
may
justify
a
restatement
ofthe
theoreticalresult
andan
explanation
of
this
dif-
ference.
Thefinal
formulaefor
thelifetimeobtained
by
both
authors
is
the
same,
apart
from
differencesinnotation.
Itcan
be
writtenintheform
G2
m
4h
r
—
—
4~~
Ac
p
pcs
Gy~
Inthisformula
k,
m,
c
have
theusual
meaning,
and
p,
isthe
restmass
of
the
mesotron.G
is
the
constantof
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
ofthe
proton)
but
probably
somewhat
larger
thanthis
quotient.
Thelifetime
r
istherefore
essentially
proportional
to
p
4.
Gz
finally
is
theconstant
in
Fei:mi's
theory
of
P-decay,
normalized
to
be
a
pure
number.Theformofinteraction
assumedfor
the
coupling
between
proton,
neutronandthe
electron
neutrinofield
is
G~mc'(&/mc)'(4N*PPI
)
(v.*Ps.
)+c.
c.
(f~,
P~,
p„,
q,
being
thewavefunctionsof
neutron,
proton,
neutrino
and
electron,
respectively).
Thisleads
to
the
probability
for
emission
ofanelectronof
energy
e
Gy'
mc'
(ep
—
e)(e
mc
).&de
w(e)de=
(
M['
(2
)'
a
(mc2)
5
where
ep
isthemaximum
energy
oftheemittedelectrons
andM
a
matrix
elementfromthemotion
of
the
heavy
particles
insidethenucleus.
The
discrepancy
in
thecalculated
lifetimes
comes
from
the
differentvaluesusedfor
the
constant
G~.
Asdiscussed
by
Betheand
Bacher'
and
by
Nordheimand
Yost,
'
the
experimental
valueof
Gz
depends
quite
appreciably
on
the
group
ofelements
whicharetakenfor
comparison,
the
difference
being
due
inall
probability
to
thematrix
ele-
ment
M;
which
is
smallerthan
unity
for
heavy
elements
butcanbe
expected
to
be
unity
for
lightpositron
emitters.
The
value
for
Gz
used
by
Yukawa
(0.
87)&10
"
inour
units)corresponds
tothe
heavy
naturalradioactive
elements,
while
thevaluededuced
for
the
light
positron
emitters'
is
GJ
=5.
5&&10
".
It
seems
beyond
doubtthat
thislatervaluehas
tobetaken
forour
purpose.
Withthe
present
most
probable
values
G'/Ac
=
0.
3;
p=200m;
Gg=5.
5)&10
',
weobtain
from
(1)
v
=1.
6&(10
'
sec.
,
i.e.
,
a
valueabout10
'
timestoo
small.
A
decrease
in
the
assumed
valuefor
p,
to150m
wouldincrease
v
only
by
a
factor
of
order3.
Inview
ofthisdefinite
discrepancy
the
question
arises
whether
any
modifications
of
the
theory
could
give
a
better
result.
It'is
tobenoted
firstly
thattheintroduction
of
the
Konopinski-Uhlenbeckformofthe
P-decaytheory
would
only
make
matters
worse
asitwouldintroduce
roughly
anotherfactor
(m/p)'.
Areal
improvementcan
only
be
expected
by
a
complete
reformulationof
the
theory.
One
possible
suggestion
wouldbeto
assumethat
thedisintegration
of
a
freemesotron
is
infirst
order
approximation
aforbiddentransition,
while
innucleiitis
made
allowed
by
the
influenceoftheothernuclear
particles.
L.
W.
NoRDHEIM
Duke
University,
Durham,
North
Carolina,
February14,
1939.
~
H.Euler
andW.
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
andM.A.
Pomerantz,Phys.
Rev.
55,
105
(1939).
2
H.Yukawaandothers,
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).
Ithas
to
be
notedthattheformulafor7
o
on
p.
943
shouldbe
ro
~
=
(Gg2/(2x)3)
)&(mc~/$).
Thevalueof
Gg
isthendeterminedfromthe
empirical
value
vo
&—
10
4.
The
Scattering
of
Cosmic
Rays
by
the
Stars
of
a
Galaxy
The
problem
dealtwith
inthisnote
may
beformulated
in
thefollowing
way:
imagine
a
galaxy
ofN
stars,
each
carrying
a
magneticdipole
ofmoment
p,„(n
=
1,
2,
.
..
N)
and
assumethatthe
density,
definedasthe
number
of
stars
per
unit
volume,
varies
accordingto
any
givenlaw,
while
the
dipoles
are
oriented
at
randombecause
oftheir
very
weak
coupling.
Under
thisconditiontheresultant
field
of
the
whole
galaxy
almostvanishes.Lettherebean
isotropic
distributionof
charged
cosmicparticles
entering.
the
galaxy
fromoutside.Our
problem
is
to
findtheintensity
distributionin
alldirectionsaround
a
point
withinthe
galaxy.
Its
importance
arises
fromthefactthatifthe
dis-
tributionshould
prove
tobe
anisotropica
meanswould
beavailable
for
determining
whethercosmic
rays
come
from
beyond
the
galaxy,
independent
ofthe
galactic
rota-
tioneffect
already
considered
by
Compton
and
Getting.
'
Suppose
we
consideraparticle
sent
intoanelementof
volume
d
Vof
scattering
matter
in
a
direction
given
by
the
vector
R.Let
the
probability
of
emerging
in
thedirection
R'
be
given
by
a
scattering
function
f(R,
R')
per
unit
solid
angle.
Conversely
aparticle
entering
inthedirection
R'
will
havea
probability
f(R',
R)
of
emerging
inthe
direction
R.
Let
us
assume
thatthe
scatterer
(magnetic
fieldof
the
star)
hasthereciprocal
property
sothat
f(R,
R')
=f(R',
R).
In
our
case
this
property
issatisfie
provided
the
particle's
sign
isreversedat
thesame
time
as
itsdirection
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
foreach
element
of
volume
itwillalsohave
it
for
LETTERS
TOTHEEDITOR
any
extended
distributionof
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
thedirection
Ratthe
point
Pwill
emerge
in
thedirection
R'
at
the
point
P'.
Thisis
because
the
probability
of
following
any
route
is
equal
to
the
probability
of
followingthereverse
route,throughthe
same
elementsof
volume.Thusthe
probability
of
acertain
endresultfrom
a
numberof
possible
routeswill
equal
the
probability
of
thereversaloftheresult
occurring
through
thereverse
routes,
In
our
casethe
scatterer
(star)
is
to
a
large
extent
non-
absorbingand
noncapturing.
Theformer
is
true
except
for
particlescollidingwiththe
star,
whichcan
only
happen
whentheir
energy
is
sufficiently
great,
andthelatteris
true
except
for
particles
whichfollow
asymptotic
or
-periodicorbitsin
the
magnetic
fieldof
any
oneof
the
X
stars.
These
orbits,however,
almost
certainly
form
aset
ofzero
measurein
the
manifold
of
all
possibleorbits,
'
that
is,they
occur
only
exceptionally.
Thus,
while
a
dipole
magnetic
field
canimprison
chargedparticlesstarting
from
a
point
withinitand
can
also
keep
them
away
if
starting
from
infinity,
depending
ontheir
energy
and
angular
momentum,it
can
onlyexceptionallycapture
such
par-
ticles
starting
from
infinity.
In
our
case,
therefore,
all
particles
starting
in
adirectionRat
a
point
P,
sufficiently
farfrom
all
neighboring
stars,
have
only
a
small
chance
of
being
either
absorbedor
captured
in
aperiodic
orbit
(of
finiteorinfinite
period),
sothatthe
greatmajority
of
them
will
emerge
at
infinity.
Foralmostall
particles,
therefore,
the
probability
of
emerging
at
infinity
mustbe
unity,
or
J,
F(R,P;
R',
~
)dR'
=
1
almost
always.
Now
consider
a
beamof
particlesatinfinity
whose
in-
tensity
in
a
direction
R'
isI
(R').
The
intensity
at
P
observedinthedirection
R
will
be
I„(R)
=
f
F(R',
~;
R,
P)I„(R')dR'
(2).
Using
(1)
and
assuminganisotropic
distribution
at
infinity
such
that1„(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
sameandthedistributionis
isotropicat
Pifitis
isotropicatinfinity.
From
theremark
made
previously,
itis
clearthat
if
the
distribution
of
positive
and
negativeparticlesat
infinity
is
isotropic,
itwillalso
be
isotropic
at
any
point
P,
except
for
small
irregularitiesdue
toabsorption
by
collisionand
by
capture
into
periodic
orbits.
Weconcludethat
particle
scattering
by
magnetic
fieldsof
thestars
is
unable
to
contribute
anything
tothesolutionofthe
problem
whether
ornotcosmic
particles
comefrom
beyond
our
galaxy.
The
considerations
developed
in
thisnote
clearly
holdso
long
asthe
scatteringcenters
satisfy
theconditionsof
being
nonabsorbingand
noncapturing,
irrespective
of
the
lawof
forcewhichis
responsiblefor
the
scattering.It.
need
hardly
beemphasized
that
they
apply
only
to
thecase
in
which
thereisnoresultant
magnetic
field
for
the
whole
galaxy,
suchaswould
exist
ifthe
dipoles
were
oriented
along
preferentialdirections.
In
thiscase
particleswould
either
be
imprisoned
if
born
within
the
galaxy,
or
keptout,
if
coming
from
outside,
depending
ontheir
energy
and
angular
momentum.The
reciprocal
property
of
paths
wouldthenbreak
down
in
general,
butwouldstillholdfor
any
allowed
direction
at
any
point
within
the
galaxy.
M.S.
VALLARTA
R.P.
FEYNMAN
MassachusettsInstitute
of
Technology,
Cambridge,Massachusetts,
February
15,
1939.
'
A.H,
Comptonand
I.
A.
.
Getting,Phys.
Rev.
47,
817
(1935).
M.
S,
Vallarta,
C.
Graef
and
S.
Kusaka,
Phys.
Rev.
55,
1
(1939).
~
Seethe
discussion
by
E.
J.
Schremp,Phys.
Rev.
54,
153
(1938);
and
forthcoming
papers
by
O.Godart
and
by
A.
Banhs,
Jr.
Nuclear
Excitation
ofIndium
by
X-Rays
Ithasbeenshown
recently''
that
thestablenucleus
In"5,
when
excited
by
fastneutrons
or
protons,
may
be
left
in
a
metastable
excited
state,designated
by
In"'*,
fromwhich
it
decays,
emitting
negative
electrons,
with
a
half-life
timeof4.1hours.
Wehavenowobservedthatthe
same
metastable
state
can
beexcitedwhenindiumis
irradiated
by
x-rays.
The
x-rays
were
produced
by
bombarding
a
2-mm
thicklead
target
with
electronsfroman
electrostatic
generator.
Athickindium
foil,
1inchin
diameter,was
placeddirectly
behind
thelead
target.
After
30minutesirradiation
atan
electron
energy
of1.73Mevand
a
currentof
10
ya,
the
indiumfoil
showed
an
initial
activity,
recorded
ona
Geiger-Muller
counter,
of45counts
per
minute.
The
activity.
decayed
with
a
period
of
approximately4
hours.
Thewalls
ofthe
counterreduced
the
intensityofthe
rather
soft
P-raystoabout
one-half.
Until
more
is
known
abouttheeffective
x-rays,
nowell-defined
crosssection
can
bededuced
from
these
data.
By
varying
the
bombarding
voltage
it
was
established
that
theeAecthas
a
threshold
at
1.
35&0.
1Mev.This
result
might
beinterpreted
by
assumingthat
In"5
has
at
that
energy
an
excited
state
which
combines
both
with
the
ground
stateandthe
metastable
excited
state.
In
a
notewhichhas
just
becomeknown
to
usPontecorvo
and
Lazar'
also
report
theexcitation
ofindium
by
x-rays.
Intheir
experimentsthe
x-rayswere
produced
with
an
impulse
generator
workingata
peak
voltage
of1850
kv.
GEQRGE
B.
CoLLINs
BERNARD
WALDMAN
EDWARD
M.
STUBBLEFIELD
Departmentof
Physics,
University
ofNotre
Dame,
Notre
Dame,
Indiana.
1VI.
GQLDHABER
Department
ofPhysics,
University
of
Illinois
Urbana,
Illinois,
February
15,
1939.
~
Goldhaber,
Hilland
Szilard,
Phys.Rev.
55,
47
(1939).
~
Barnes
andAradine,
Phys.
Rev.
55,
50
(1939).
~
PontecorvoandLazar,
Compterendus
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.