ANSALS
OF
PHYSICS:
The
14: 333-345 (1961)
Exponential
Decay
Law
of Unstable
Systems*
ROGEK G. KEWTOX
Ph!/sics
Department,
Indiana
L’niversitg,
Bloomington,
Indiana
It is shown
that
associated
with any sutficientl~
sharp
resonance
there
is a
delayed
emergence
of particles
that under
suitable
conditions
follows
an exponential
law of decay to an excellent
approximation
over many lifetimes.
The
dependence
of the duration
of the exponential
curve on the resonance
widt,h,
on
the excitation
widt,h,
and on the observation
distance,
is esamined
in detail.
The time develcFment of unst,able quantum mechanical systems has lately
received renewed atStention (1-g). Although some of the recent papers have considerably elucidated the problem, however, the issue is unfortunately
almost
always t,reated by means which unnecessarily encumber it. Whether we are talking about the de-excit’ation of broadened energy levels of atomic or nuclear systems, or about, the decay of unstable particles, the prohlrm is always of the same
nat,ure. It may be summarized briefly as follows.
In the unsophisticated approach to t,hc decay problem one assumes that the
initial state of t,he system is stationary, except, for the fart’ that it, may decay.
The immediat’e result is a constant decay probability and an exponential time
dependence of the remaining number of part,icles. The original assumption is, of
course, self-contradict’ory. The initial state is &her a steady state, i.e., an cigenst,at,e of t,hr full Hamiltonian, or it, is not; if it is, it does not decay; if it is not.
it changes in time and so will the decay probability. It is t#hen argued that, the
naive approach should be a good approximation for long-lived systems,
that, is
for small transition rates. With that justification, however, one really has no
right, to cxpcct t’he result to hold for longer than times of the order of a halflife.
The csperiment#al verification of the exponential decay law over many life times
t,hus remains somewhat myst#erious.
Remnank of the troubles that afflict the naive procedure can be found even
in many sophisticated discussions of decay. Whenever it is treated as an initial
value problem, the question “What is the initial st,at,e?” necessarily arises. It is
in the very nature of t,he problem that this question has no uniclue answer. In
the context of a specific model with an “unperturbed
Hamiltonian”
Ho which
has a bound stat’e, and a “perturbation”
Hr which produces the dcray, one is
* Supported
in part
by the
?;ntional
Science
333
Foundation
334
NEWTON
inclined to use an eigenstate of Ho as the inkial state. Rut that, choice depends
ent’irely 011 t’he model. Kature knows only t.he total Hamilt,onian H; the divisiori
int’o Ho and HI is no less arhihrary t,han is that of the init,ial stat,e to begin with,’
In order t’o dissociat’e the discussion from point,s that are not dire&ly relevant
it is of advant’age to distinguish two problems clearly and treat, them separat,ely.
One is to explain a resonance; the ot,her, t’o obtain the time development. The
first can he done in a time independent framework. Its aim is always to derive
in various context from relevant models and fundamental t,heories, a resonance
formula of t,he Rreit-Wigner t,ype (or a modification of it). There are many different, techniques of various degrees of generality that lead t,o the derivation of
a resonance appearing in a scat’tering or react,ion cross seckion.2 Some of t,hesc
make use of analytic coruinuat,ion into t#he complex plane, others do not,; some
make use of actual poles of the T-mat,rix in t)he second sheet of its Riemann surface as a function of t’he energy, ot,hers never need to leave the first sheet. Rut.
apart from shape modifications in rather exceptional circumst8ances, all of these
resonance t#heories end up with essentially t’he same result for real physical energies, namely, t)hat a resonance is described by an energy dependence of the
S-matrix t,hat, looks (from the real ask) as though a number of it’s elements had
a pole closely below t’he real line. The t,heory is then usually able, in one way 01
anot’her, t,o associat)e wit)h such a resonance a bound stat’e that would exist at
a nearby energy if t,he interparticle forces were slight)ly altered.
The second part of t’he problem, t’o find the development of the syst)em in
t,ime, is usually trcat,ed in t’he context of the first. As a result it is not only burdened with much irrelevant detail t,hat happens to be necessary for the resonance
theory, but, t’he answer also appears to he less general than it, really is. The fact.
is that the decay problem is quite independent of t’he resonance problem, if by
the lat,t,er is meant the explanation of a given resonance curvr in terms of undcrlying structure. Its solution depends on the cxistSenccand shape of the resonance
only, and not at all on the model or theory which products it. In order t)o est,ablish this independence, specifically of such quest’ions as the analyt’ic continuat,ion of the S-matrix into the complex plane or the various sheets of its Riemann
surface, we shall not make use of any rcsonancc theory at all but st)art purely
from the behavior of the S-mat,rix for physically possible energies, assuming
merely the existence of a typical resonance energy dependence. The result, is
then cert,ain to hold no matter what the underlying structure is.”
1 Such an approach
is, of course,
quite justified
for a perturbation
calculation
of the tran
aition
probability
itself.
2 See, for example,
Fonda
and Newton
(10); other references
t,o the literature
on resonance
reactions
can be found
there.
3 We do make use of complex
variable
t,echniques,
to be sure, in order to manipulate
the
ensuing
expressions.
This is, however,
merely
a mathematical
device
for the evaluation
of
integrals
after appropriate
approximations
have been made on the real line. It must not be
confused
with statements
about the actual
behavior
of the S-matrix
in the complex
plane.
USSTABLE
SYSTEMS
3%
It is well known that the decay curve of au unstable system is not, exactly an
exponential, that,, particularly, it,s asympt,ot,ic t,ail is usually of the t-” type.
Onr of t,he reasons for t,his can be understood very easily on purely classical
grounds.
Suppose that a classical system were to start decaying e~poncntially at the
time t = 0, emit,t,ing particles of the origin along the .r-nsis wit,h a velocity dist8ribution P(P). Then the number of particles observed at the point, .r at the time
t is
I’(R., t) = /““t dvp(u) cxp [- (1 - X/l!)F1l.
It is easy t,o see that. hwausc of the strongly singular esp (X/IV) at. u = 0, the
asympt,otic behavior of P(.c, t) as t -+ p is
I’(s, t) ,- p(O)(.r1.‘T)(T:f)4
and not at, all exponential. I;or a ~((3) strongly peaked near I! = [lo , the decay
curve observed at .r is very nearly exponential startSing at t = fF , tp = .r,‘/lo
bring the time uf flight of the distribution center, so long as-l
R = p(O)(.r ’ T)(T ‘t )” esp (t - f F) ;‘7
remains very small. Sotiw that) the length of the rxponcntial part depends on
the distance .r from the decay center. If the observation takes plwe t,oo far away,
an axponential law is never ser1i.
The physical reason for this nonesponeltt.ial tail is, of course, the lat,c arrival
of very slow p&icles. Classically that trouble is easily gotten rid of by out8ting
t,he velocit)y distribution p(r)) off at some I’,,,;,, > 0. Quantum mechanically one
cannot, expect to he able to avoid it so readily heceause of the uncert,ainty priw
ciple.
There is in addition another reason for t.he nonexponential tail. Bwause
cluant,um mechanics does not have the full range of all frcqucncics at its disposal,
it, is impossible to cut off the exciting signal cleanly. As a result, there is an
ambient excitation of small experimental (wnwluellces that camlot, be completely
avoided.
We lwm turn to the quantitat’ivc discussion. The following process is to 1,~
described.
We want to trace the time development, of an unstable system through the
ent,irt course of it,s hist’ory. That means t,hat we start at the time t = - 00 wit,h
stable particles far apart. They approach one another, interact and form the unstable st,ate. The system then disintegrates into product,s which may or may not
he the same as the initial particles. At some distance from t,hr cuenter of mass
the decay is observed as a funct,ion of the time. Evidently the appropriate frame4 p is assumed
normalized
to unity.
336
NEU’TON
work for t,he description is time dependent scat’tering theory (elastic or inelastic).
The advantage of this approach over t#he more common initial value method is
t’hat t,he question of the exact state at t = 0 never arises. We deal wit,h entities
of expcriment)al significance only, far away from t,he intrract,ion region itself. It
is more realistic as well as more manageable.
In order to apply scatt(rring theory we assume t,hat t,he interparticle distance
T at observation satisfies
1;r >> 1
LA)
for all relevant relative wave numbers. In practice t,he inequality (a) is of course
usually satisfied extremely well. Under these conditions we may writme for the
cent’er-of-mass wave function5
where k, = II&, and k = x-k/r are the initial and final relative wave vectors,
respect#ively, and allowance is made for the inelastic case with 1~ # I;i. As a
funct#ion of the t,ime t#herefore a wave packet looks like this:
\L(r,
t),-s
(dk;) j(k;
wt)
+r--l
s
- k,) exp i(k;.r
-
(,dki) .f(k< -
kg)ei(k’--wf)O(k, k,),
provided t,hat (A) holds for all 1; appreciably contained in it. The fur&on f is
assumed to be sharply peaked at k, = k. .
As a first orientation we go t,hrough the well-known stationary phase argument. For t < 0 and large r, the second term contributes little and only the
plane wave part, is of significant size, namely, at the t’ime
t = nL.r/(dw,/dk)ki=ko
= nj.r/rl,,
if zjOis t,he magnitude of the sum of the incident part#icles’ velocities at t,he wave
packet center. For t > 0, on the other hand, t#he second term also contributes,
there now being a t,ime for which the phase of the integrand is stat#ionary. If we
write
0 = IoIPzq,
then the outgoing spherical wave is observed more or less at the time
t = (T + dp/dli)/(dw/dk)//;=I;,
= (r/ad
6 We shall
for
simplicity
restrict,
+
ourselves
(dp,‘dw)u,
to two-body
decays
I-NSTABLE
337
SYSTEMS
The scattered wave is t,herefore delayed by t,he amount tD = &/CL evaluated
at k = ko.
It must, now be remembered t,hat a resonance is charact)erized by t’wo features.
One is t)hat t,hc T-matrix elements in yuest’ion (or 0) exhibit a sharp peak in
magnit~dc; t,hr ot,her, that they vary rapidly upward in phase. The first aspect,
is responsible for the large cross section and is usually the only one e.xpli&ly
measured in an experiment,. The second causes t,he time delay which is an intrgral part) of the mechanism to which the resonance peak in t,he cross sectiou is
attributed. In principle these two effect)s are quitme independent and only when
they occur together may we speak legitimately of a resonance. It) is equally
possible to produce a peak in the magnitude of 0 accompanied by an increase
or a decrease in it.s phase; however, only the case of the sharply r-ising phase is
a real resonance.6 As a consequence, t’he t,rue shape of 8 in the vicinity of a
Brr+Wigner
resonance in t,hc cross s&ion, gi\ren by
is of nec,essity
with r > 0.7, 8
We now assume that, t.he T-matrix
6 The falling
phase leads to
may choose a “short-cut”
path
t,he expectation
of a limitation
definite
radius.
Such a limitation
(If).
See also Newton
(11).
7 In order t.o exhibit
the fact.
pole can occur only on the“other
to write
0 a 2
cont.ains a well isolat.ed sharp resonance.
a time adzwzce
which
is causally
possible
hecause
the wave
along the surface
of the interaction.
This naturally
leads to
on the negative
slope of a phase shift if the interaction
has a
has indeed
been derived
by Wigner
(11) and by Liiders
that under restrictive
assumpt,ions
sheet,”
of it,s Riemann
surface
p k (k +
i (3 -
a)-’
(k +
on the interaction
it would
he bet,trr
a true
inste:td
i p + a)-‘.
However,
this small change
in phase has no discernible
consequences
in the tinw develoljment.
d In the resonance
theory
of Ref. 10 it, appears
as though
one could have r < 0; see bottom
of p. 507 of that, paper.
Ivow one can show hy means of (2.10) of Ref. 10 that. if I,,(,!?,,) = 0,
then R,,’ of (4.2) is negatioe.
The reason
is that
if f(r’)
>= 0 everywhere
andf(.ro)
= 0. Consequently
for small
and hence rib > 0. When
the curve
R,,(E)
crosses
t,he straight.
may have a sharp
peak in the cross section
without
the dela,v
I,, we certainly
have IL,,’ < 1
linef(E)
= E upwards,
\n;e
features
of a resonance.
338
NEWTON
Since \ve are int’crest,ed in t,he det,ailed time development, of the delayed signal
it) is clear what, the relewnt crit,erion of sharpness ought to be: The delay t,ime
at hhe resonance peak must, he large relative to those at neighboring energies.
In ot’her words, the slope of ‘p at 6, must he much larger t,hall at, ot,her energies
nearby. Since for the Breit-Wigner shape
tD/h
= (dpkp:,dE)E=E”= 2,/r
t,hat means that (2/r) must he large relative to the slope of cpelsewhere in t,his
region. The relevant criterion of isolat,ion of the resonance must be su& t,hat we
can form a wave packet inside whose energy range only one resonance line appears. It, is to be expected that, that means that the next one is at, least, sever;11
widt’hs away. A more precise conclusion is to he drawn from the analysis below.9
Finally, a word about the shape of the exciting wave packet. It is physically
clear that, it) should do two mut,ually contjradictory things as best, it (*an: In order
not to interfere more t#han necessary wit,h the time curve of the decaying system
it should be of short duration and he cut, off sharply; in order to isolate t,he
resonance products from other delayed signals (such as from other resonances)
it should be of narrow energy rangt and be cut off sharply there. In the extreme
case of a monochromat,ic beam there is oh\-iously no decay hut, a st,eady state
The opposite extreme of all positive energies in the beam will be possible only
in esceptional rircumstances of extreme isolation of the line.‘0 One may expect
t,hat the energy width of t’he packet should hc larger than the line width so as
to “fill out” its shape well. In order to realize a sharp cutoff bot,h in timt and
energy we choose a, Gaussian shape (rcstrict)ed to physically accessible energies).
It is certainly possible to alter det,ails of the ensuing results by minor alt,ernatjions
in the packet shape. Specifically, the nsympt,otic behavior for infinit,c t,imc is
sensitive to such changes. That is why we shall avoid basing any argumrnt,s on
the asymptotic tail. But as in many ot’her contexts, the essrnt’ial results, if
gleaned carefully without, rwourw t#o accidental features, can be expected t,o be
independent, of the precise shape of the packet.
9 .&sfar as the first t,erm, r/co, of the outgoing wave’s peak t.ime is concerned,
the foregoing
phase argument
is exact in the limit as r - m ; it is excellent
for practical
values
of r. The
delay term, &,I&,
holvever,
must not be taken quantitatively
seriously.
It. is of yualitat,ive
significance
only. The reason
is t,hat, when dq/rlw is large in a resonance
the actual
increase
in cp is only about K so that the oscillation
does not, really
have the effect
of reducing
the
integrand
appreciably.
Furt,hermore
t,he resonance
wngniturle
of the integrand
is sharpl)
peaked
at the same point.
One must therefore
root conclude
from the phase argument
that
the observed
activity
should
be maximal
two life times after the time of flight.
The actual
maximum
will, of course,
occur near the time of flight
The real meaning
of t,he delay is contained
in the exponential
decay law. For r < 0 there would
he 110 such decay.
need not he the incident
heam; it co111ti
10 The experiment,al
realization
of the “window”
be in the acceptance.
But that, makes no difference
to the discussion.
VXSTABLEI
339
SYSTEMS
It is useful to analyze first the t,ime development of the packet’ itself as it
would be seen in t,he absence of a resonance. The typical energy integral t,hat
appears is
I = A-l 1% dlt
A
esp [-(TV
-
11’,)2/2A’ + ~‘(X.T-
TBt/h)l,
1;Ybeing the total energy and
k? = [(IQZ - ml”c4 - y&4)2
- ~m12m2?c8]/~~fnfi’c’~,
where ml and m2 are the rest masses of the emerging particles, c is the velocit,y
of light, and h, Planck’s constant divided by 2n. We may also write X-in terms
of the final kinetic encrgp
E = TV -
(m, + m&‘,
or its relative deviat,ion from the packet center EO :
Ii
= IC”{,
-1
SE,
J%l+ (ml + m2)c2>
where
s = (E - E,)l’Eo
and
Pl
=
t( 1 +
o(l)
( 1 +
1 +
oL2)
P182
+
-
01 PJ1,
p12
-
82
p2 = 11 + Lyl)(l + @)(l + 03 a!2 + p1 P2) ’
81
=
h/C,
82
=
V~‘C,
o(1 z2
(1
-
‘&y
a? = (1 - ,&3)1/2,
u1 and v2 being the velocities of the final part’icles at the center of the distribut,ion. The lower limit -4 of t,he integral is either (ml + m2)c2 or the sum of the
incident particles’ rest’ energies, (ml + m2)ine 3, whichever is the larger.
The integral 1 can be re-written
1 = CXP [i(/i#
with
- lf’~t,‘fi) -
(1 - ir)(l
- t,n)?j:!!P](l
+ iy)-“‘J
340
VEX-TON
I’
zz (1 +
y?)l/?fijA,
2% = c[(sE,,/A) - (t - 1,)/T]
+ P[(s&/A)
+ (t - t~)iy’]j
y = 2p~(AjEo)21i~r,
E = [(l + yy
- y]llZ,
l = X,r([ - 1 tp
=
r/(u1
+
u2)
p1s +
=
Pd,
Plf&~,~&
so that tp is the time of flight of the decay product,s from the center of mass to
t,he separat(ion r.
Using z as the variable of imegration we obtain
J = (1 - i) L dx PfE),
where the lower limit, of integrat’ion is
B = -y2{[(&‘/A)
+ (t - tF),h”]t
+ i[(&‘/A)
-
(t - tF)/T]e-‘1
wit,h
E,,’ =
E. = W. E()inc = Wo -
(ml + rn&?
if
ml + m2 > (ml +
(ml + mp)incC2
if
??Zl+ 7n2 < (?77l + t?l2)inty
and the integral runs over the “old rontour”
an angle
6 =
x7r
+
4;
7772)inr
of E’ig. 1, a straight line upward at
tan-l
y.
The path of integration is subsequently changed t)o t,hc “new comour” of Icig.
1, on which always Im 9 2 0. We must’ examine E on t,he dist,orted piece of the
contour.
Since .$ was chosen so that, it, is of order .s3in an expansion in powers of S, we
have 1.$ 1 5 I so long as
kor 1s 13 5 1.
Sow on t,he dist,orted port’ion of the path
1s / w (~iE,,)(t
- tp)/T
with it,s phase ranging between 0 and -?r. In order for if not t,o become large
negative anywhere on it we must, therefore demand that’
kor(A/Eo)3[(t - t,)/?‘]”
S I
t-NSTABLE
OLD
311
SYSTEMS
CONTOUR
/
--NEW
FIG.
position
1. Showing
the change
of contour
in the evnluabion
of the pole in JR when
(t - ~.P)/T > TI.
(t - tF)/T
5 (Eo,‘A),‘(kor)“3.
CONTOUR
of integrals
Z and
ZE. 20 is t,he
03)
If this inequality is seriously violated then the integral J increases rapidly. So
long as it holds J remains of order of magnitude 1, provided t)hat Im Bz 2 0.
The lat’ter condition is fulfilled so long as
(t - fp)/!T 5 &‘/A,
CC)
which is usually far less stringent t,han (B).
We have thus found t,hat so long as (B) and (C) hold, the temporal developmrnt, of / I /? is determined by the Gaussian fact,or
exp -
(t - tF)‘/T?.
Notice that for y << 1 t#he mean life of the packet, is T FZ n/A, hut, for y >> 1 it is
T x apzk~ir(A!E,)(hjl;,‘,,),
342
NEWTON
which increases with increasing A. After (B) is violated, the packet no longer
falls off as rapidly as gaussian. By t,hat t.ime it has dropped by a fack of
exp -
[(Eo/A)~/(kor)~~3]
(provided that’ E,,’ 2 Eo,~(kor)1/3). The shape is t’herefore ever really Gaussian
in time only if
&/A >> (/i~,g-)"~.
CD)
After the violation of (B) the packet decreases much less rapidly t,han gaussian,
the details of the t’ail depending on t,he experimental circumstances. The exact
asymptot,ic behavior as t -+ 30, particularly, depends on how it is cut off at,
E = 0.
We now turn to the resonance case. The relevant. energy integral appearing is
exp [i(kr -
Wt/fi)
- (W -
W,j"/2A"].
It is handled exact.ly as f. The only difference comes from the contour change.
There is now a pole at
z = z. = !+([$i(r/'A)c'
- (t - &)/!f]
- i[l,s(riA)
- c-?(t - tp)/Z']1,
which is such that
Im x0 > 0
and
Ite z,, < 0
when
(t - tp)/~ > T1 = >i(r/‘A)%-‘(1
+ y’J)“r.
(El
Consequently as soon as t’he time sat#isfies (E), the pole lies het’ween the old and
the new contour and there is a residue contribution to IR :
IR = I,,, + f*$(lT,/A) exp [i(kor -
Wet/A) -
(1 - ir)(l
- t,)“/2T”]J,
,
where
I Ftes= -ia(r/A)
exp [i(k, T -
W. t/n) + ?d(r/A)“(l
+ ir)
-
(t - t~)/2~],
JR = s, dz (x - z~)-~P?+~),
r = n/r.
Again JR becomes larger than of order of magnitude 1 only when either (13) or
(C) are violat,ed. In other words, JR is of magnit’udr 1 provided that
(t - tp)/~ 5 (r/A)(&/A)(l
+ y4)1"(ko1.)-1'3
@'I
UNST,4BLE
343
SYSTEMS
and
(t - fF)/7 5 (&‘,/A)(r/A)(l
+ -#“?.
CC’>
So long as (B’) and (C’) hold we have
lh = -ig(r/A)
rsp ki(k,,r -
W,,ti’fi) + ,l,#/A)2(1
+ iy) - (t - fF),l%](l
+ R),
whrrc
/ R ! 5 (‘exp (-l.iI;),
I’ = (1 + y’?)--l(Ajr)?[(t
- fF)‘+
-
[(t - ~F),/T] -
K(r/A)‘,
P being a constant of order of magnitude 1. The decay curve is approximat,cly
exponent,ial to t,he rxtcnt, t,o which ( R ( is small compared to 1.” We have 1’ = 16
for
(f - fF)/7 = I’? = ‘ti(r,/A)z(l
+ +)‘/?{ (1 + y”)‘/” + [y” + (f$A/r)‘l]l/z)
1 qr:/A)
+ 1*:1(I‘/A)2.
By t,hat time ( R ( is quite small (e-x E 1.3000). It, decreases further until either
(B’) or (C’) breaks down. I;rom then on I, decreases very slowly, say,
as (1 - tF)-?l. Then
1’ M (~!!,‘,,jA)~(k~r)-~‘~
+ 2n log [(t - lF),/~] -
(t - tR) !T,
which becomes negative and R dominates at t’he time when
(t - tp)j~ 2 T:, = (&/A)z(k,r)-2’3
+ -In log [(E’n./A)(kor)-“3]
(V
(assuming (D)). The number n depends on t)he cxpcrimcntal circumst,ances, but
T, is relatively insensit,ive to it.
We can now discuss the shape of the decay curve as seen at the separat,ion
distance r, provided that’ (D) holds. Since it, will he seen below that. we must
have A 2 I’, that, means we must assume that
li,f
<<
(fl(,/r)3.
(D’)
lcor (t - fF); 7 < ?‘l t’he residue term is absent and t,he shape of t,he curve is
roughly as it would be wit’hout the resonance, i.e., Gaussian. iZpproximat,ely
from T1 on, the over-all feat)ures of the decay law are beginning to be dominated
by the exponent8ial; that is, t’hc decrease of activity is exponential in its gross
features wit’hout, necessarily being very nearly so in detail. By the time T, the
decay is almost exactly exponential, to wkhin about 0.1%. It then remains indistinguishable from a pure esponent,ial until T3 , when it) can be expcckd to
flatSten out,.
But
I1 The relevant
that, makes
quantity
is, of course,
no discernible
difference
the flux
in the
density
rather
conclusions.
than
the
density
of particles.
344
NEWTON
I
TABLE
it A/I’ is the excitation width relative to the resonance width.
b 2’1 and TI are the initial times of t,he roughly and almost exact exponential curves, I’Aspectively, both measured in mean lives from the time of flight t.~
c ( IR \‘/I I, I:,, is the relative value to which the observed activity has dropped 1)s the
time Tz It was assumed t,hat y << 1.
Suppose first that A 5 I?. Table I shows the rapid unobservabilit,y of the pure
exponential when the excitat,ion widt,h (or the width of t,he accepting window)
is smaller t#han the resonance width. (When y is not small then it drops off even
more rapidly.) It is clear that, as expected, the excitation width A must be at
least about equal to the line width I’ of the decaying state in order for a clean
exponential curve to appear at, a t!ime of observable act,ivit’y.
If A > I’ and y << 1 then hhe onset of the roughly exponential region is less
t#han half a lifetime from the peak and the exponentjial is almost exact about,
(U/A) mean lives after t,he time of flight. The decay curve then remains exponential for about, (Eo’/A)2 mean lives (assuming & >> I’). It is int,eresting to
notice the dependence on the observat’ion distance from the decay center. When
t,he distance is made larger t,hen y increases, and t,he onset, of the exponential is
retarded. The smaller pz is, the larger r has t,o be in order for appreciable effects
to occur. Since
Pz =
?4
if both final particles are slow;
p2 = X(1
if both are highly relativistic;
PZ =
P(1
-
@)/Cl
-
PJ + ?4(1
-
P2!
and
+
P)(l
+
01) -
P
for
P <<
1
if one is a photon, the decay law is least sensitive t,o the observation distance
when one of the final particles is a phot#on and the ot*her suffers either very little
recoil, or else recoils estremely
rapidly.
If the distance is so large t,hat’
y W (&/r)
(/ior)-"",
UiVSTABLE
345
SYSTEMS
i.e.,
kor z (E0/A)3/“(&/2p$)3’4
,
then no part of t#he curve can be expected to be exponential any longer.
By way of an example we may consider the case of a nuclear deexcitation by
y-ray emission. In a typical instance we may have & z 200 krv and T c 1OF
sec. Then for r = 10 cm, kor = lo”, and r/E0 w 3 X 10-13. If t,he level is
filled from a broadly excited parent, level, then fi/A is t,he lifet#ime of t#he latter;
suppose that is 34~. Then (D) is excellent,ly
fulfilled
and y << 1. The decay
curve should be roughly exponential after g of a mean life from the peak, and
rxcellently
exponential
after
less than
two.
It, then
remains
exponential
for
some 1017lifetimes!
1
I
It is a pleasure to acknowledge sonle stimulating
with Professor R. Curtis and Dr. M. Wellner.
I~ECEIVEI):
discussions on the subject of this paper
January 9, 1961
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