Download The Radial Equation

e
{lj) ¡aro uIt"c -l
178
Quantum
-
{}
uamJuJ1tV
(})
Physics
totation, which is consistent with the faer that for a perfectly
such a totation would be unobservable.
10. Express V
equation obeyed by
in terms of %e
and O/O<jJ. Write
defined In Eq. 10-49.
down
smooth
11
chapter
cylinder
the differential
Glm
[Hint. Use the variable z
tion for 1 = m.
=
cos e). Show that (sin e)l is the solution
of the equa-
The Radial Equation
References
This is standard material found in any of the books listed on page 501. For a
deeper look into the consequences
of invanance
under totation
see especiaIIy
K. Gottfried,
Quantum
Mechanics,
Vol. 1, W. A. Benjamin,
M. E. Rose, Elementary Theory o/ Angular
1957.
Momentum,
)ohn
The radial Schrodinger
equation
(10-64)may
1ne., 1966.
+- -2r -dd)
r
Wiley and Sons, 1nc.,
R"lm(r) -
-h2
2¡.¿
[
be written
+ l(l+2¡.¿r
V(r)
as
8
h2J R"lm(r)
Z
1)
2¡.¿E
+ r;: R"lm(r)
e we ha ve replaced
the label E by n in the subscripts
=
O
of the eigenfunerion
,1"1.;~"",(r). We wiII examine
the solutions
to this equation
for a variety of po'!lU\tials restriered by the condition
thar they go to zeto ar infinity faster than
l/I'"...except for the important special case of rhe Coulomb porentia!' We wil!
~ro
il$SUme rhat the porentials
are not as singular
Lim r2V(r)
r-·O
Jt ts sometimes
convenient
to introduce
U"lm(r)
dr2
(_d2
i¡
follows
barrier,
r2
ar rhe origin,
O
so that
(11-2)
the function
=
rR"lm(r)
r
u"I,,1,':.2
(11-3)
= ~r ~dr2
(11-4 )
u ,,1m(r)
rhat
-d-z
r d2unlm(r)
111is looks
r dr
+ -~~)
as 1/
very much
(a) the potential
+ --r..-2
2¡.¿ [
.
f6
E - Ver) - l(!
~---2+2¡.¿r1) h2J tinlm(r)
like a one-dimensional
equation,
V(r) is altered by the addition
V(r) ---. V(r)
+ l(l +2¡.¿r21)
h.z
except
=
O
(11-5)
rhat
of a repulsive
centrifugal
( 11-6)
179
(i)
.",0
Quanrum
for large r we can clrop the potentia!
Ver)
t
1=
f
d:'rl
= foCO
f(r)
fo""
=
12
foCO drIUnlm(r)
so that the wave function
I
J
should
the asymplOric
in radial equarion
amI rhe finireness
for
11
=
rR(r)
when rhe
of rhe wave funcrion
1
(11-13)
-a
If E
<
O,
so that
(11-14)
2
1I(r) ~
If E > O, we have solurions
in Chapter 4). Wirh
(11-15)
e-ar
that are only normalizable
in a box (see discussion
2¡.¡E
k2
=
If2u
IU + 1)
---'---u~O
dr2
(11-8)
1'2
because rhe potential cloes not
sarisfiecl. If we make rhe Ansatz
contri bu re for smal! enough
u(r)
we fincl rhar rhe equation
~
wil! be sarisfied,
J(J -
= -1.
1) -
IU
The solurion
l'
when
(11-2)
is
(11-9)
1'"
the solurion wil! be a linear combination
of eikr ancl e-ikr, the'proper
combination
being cleterminecl by the requirement
that the asymplOtic solurion tie on continuously
lO the solution that is regular at rhe origino We now consicler some
examples.
A. The Free Particle
In this example V(r) = O, but there is stil! a centrifugal
The raclial equarion (11-1) takes rhe form
proviclecl rhar
+
1)
=
O
rhar sarisfies rhe conclirion
[~ + 2.r ~dr
_
dr2
(11-10)
If we introcluce
u(O)
=
O,
rhar ¡s, rhe solurion rhat behaves like rH-1 is calJecl rhe regular Jolution; rhe solurion rhar behaves like 1'-1 is rhe irregular Jotution.
(11-16)
1'2
1.
(11-7)
O
+
1 or J
12
ar rhe
which makes ir more like rhe one-dimensional
problem for which V =
ro in
rhe lefr-hand region (Fig. 11.1).
Firsr we consider rhe radial equarion near rhe origin, clropping aJ] subscriprs for convenience.
As l' -'> O, rhe leacling rerms in our equarion are
= 1+
Y1m((),¡p)
is
rhar
U"I",(O)
that ¡s, J
1
solution
=
12
vanish at infinity.
-=
acring
11"",,(1')
Rn1m(r)
(11-12:
2¡.¡E
h2
the c!efinirion of
dal
r2drf
r2drl Rn1m(r)i2
Vol,)
I
I
Fig. 11-1.
Effecrive potenrial
real pare mi al is a square \Vell.
rhat
1I
~
I
________
Implies
chat is,
I
V.,,(r)
conclition
(11-11
2
2Mr2
require
becomes
2¡.¡E
d211
The square integrabiliry
(lo)
rerms, ancl rhe equation
--+--u~O
dr2
lF
1(1 + 1)11
origin
18
The Radial Equarion
Physics
the variable
p
=
tU ~r
l)J
R(r)
+
kr, we get
+
-+------R+R=O
cPR
2
dR
dp2
p
dp
IU
1)
pO
barrier
O
k2R(r)
8
presento
e
'""'~''"'"-~''-''''"--''''''''_''''
rf1Jwtft
182
Quanrum Physics
__
'''''''''''_\o\'''!;<'''~
rP
.,1,U ~
e i t:.-; ;- f"b ~JÚ-fI ) i J~(je¡~)?e. (~J e-)
and
= 2. L 47T ¿ ~J~
/h) , (k) '([ ~~1e Radial Equation
tr\
R.
This equation can actually be solved in tetms of simple functions. The sOlutions
are known as spherieal Bessel functions. The regular sOlution is
= (-
jz(p)
(Id
--;
p)Z
d; ) ¡ (sin-p- p)
=
-;;
(Id
-(-p)¡
=
Slll
P
(11-26)
p
»
1,
we have the asymptotic expressions
jl(p)
c::-::
1
-;;
sin
B
-2
l'lr )
( p -
and
cos
=
no(p)
P
(21-1)
...
p1tl
(11-20)
-P-p)
) ¡ (cos
dp
The first few functions are listed below.
jo(p)
n¡(p)c::-::_1+5'
For
n¡(p)
183
.t.
(11-19)
and the irregular one is
®
p
n¡(p)
p
c::-::
1
-
-;;
cos
:2
( p -
(11-28)
l'lr )
so that
Slll
}I. (p) =_E
cos
---
2
P
p
cos p
nl(p)
P
p2
Slll
P
I
--P
h(I)( p)
c::-::
h(p)
(~ p3 -~)
The combinations
[unetions
-
p -
sin
p
= - \~
13p" -- -p1)
cos
p2
cos p -
P
hP) (p)
=
-p23 sin p
+ in¡(p)
j¡(p)
h¡2)
=
(p)
[h;!)
*
(p)]
(11-29)
8
The solution that is regular at the <¡rigin is
(11-21)
that will be of interest for large pare the spherieal Hankel
and
¡ e/(p-I~/2)
.
p
R¡(r)
n2(p)
-
3
(11-22)
(11-23)
=
j¡(kr)
Its asymptotic form is, using (11-27)
,Z~"céJ
¡It.{;'€.-
R zrc::-::----:-e
( )
2¡k1 r [ -;(k,.-I~/2)
-e
-
;(kr-'~/2)1.
We describe this as a sum of an "incoming" and an "outgoing" spherjcal wave.
The nomenclature is arrived at in the fol!owing way. The generalization of the
one-dimensional flux is
h
Again the first few spherical Hankel functions are
j = -. [¡f*(r) V¡f(r) - Vo/*(r) ¡f(r)]
27¡;,
eip
(11-32)
h¿I)(p)
We shal! see that it is only the flux in the radial direction that js of interest for
large r. Thus the radial flux, integrated over al! angles, is
¡P
h1(1) (p)
W)(p)
= - -;;
eip
= -i peip
(
1
1
+ ~i)
+ -- - 3i
P
(
p2
3 )
f dQ¡,·j(r)
(11-24)
= 2i¡;,
Tif
(o0/* -¡;; ¡f -
~00/*) ¡f
(11-33)
For a solution of the form
Of special interest are
e±ikr
(a) the behavior near the origin: for
p
«
1,
¡f(r)
it turns Out that
= 1+5·
."
e _.r
YZm(IJ,cP)
(11-34)
with
p¡
j¡(p)
dQ
(21
+ 1)
(11-25)
f
dQI
YZm(IJ,cP)[2
= 1
(11-35)
(f)
184
Quanrum
!
The Radial Equation
Physics
we get
dfJ/r
= -.
I
ej
2
and at large distances, such a flux, when multiplied
by the are a factor r2dQ, stil!
term in the radial flux. This is the
vanishes as 1/ r relative to the dominant
±
-
2'fl
f¿
ik _
[e'F rikr (
_
_
e±rikr
_
e±r2ikr)
justification
complex
hk I el
=
r2dQjr
For our solution
(11-31),
(independem
- hk
fl
1
-
r2
I
i
-2k
r2
Consider
that emerges
since the flux
of r)
the potential
the radial equation
d2R
dr2
(11-38)
+ ~r
In general,
flux conservation
demands
solutionsfollowing
for which
Ver) be
'7' O--whose
mems
(11.16)J
Rl(r)
~
-
~k
2, r
this includes
-
Sl(k)
12
=
o
r> a
6
SI (k) /(kr-I"./2)J
1
dR
dr
_
t(l ~r
+ ~
1) R
t(l
dR
r
We look for bound
dr
r2
state solutions,
2¡.t
-
2fl
(11-41)
e2i01(k)
The real function o¡(k) is called the phaJe Jhijt beca use the radial function
asymptotic
regio n (11-39) may be rewritten as
Aside from the phase
jz(kr),
Oz(k).
whose asymptotic
~
eiól(k) sin [kr -
lrr/2
+ o¡(k)J
kr
--
in the
(11-42)
factor in from, this dilfers from the free particle
form is [sin (kr -
/¡¡/2)J/
The solution
for r
<
10
direction
The solution
(11-44)
for r
>
a must
is just the equation
.
j
= 2'fl
---:- f* -r1 -08O f f¿
(
complex
(11-44)
<
E
=
o. We write
K2
= -a2
E
=
(11-45)
at the origin,
is
vanish
as r
----+
for the spherical
for r > a. The two solutions
This leads to the condition
=
( 11-46)
AjZ(Kr)
replaced by ia. The solution that behaves
al!y falling one, that is, we have
Bh¡t)
must match
ro. The second
Bessel funetion,
like
of the equation,
except
now becomes
eikr
k
i,
the exponenti
(11-47
(iar)
at r
that
=
a and so must the derivatives
sOlution
kr, only by the shift in phase,
K.
P~K"
_.
la
(11-48
(1)
[dh¡t)(p)/dp]
h, (p)
p=iaa
involves
This is a very complicated
10
O
r> a
a, which must be regular
[djZ(p)/dp]
Jl(p)
We note that rhe flux in the
r<a
=
h2
R(r)
R¡(r)
R
ñ,2
+ E)
(vo
R(r)
=
¡.tE
for which
fi2
+ E)
(Vo
(11-40)
as otherwise the Outgoing flux wouJd dilfer from the incoming one. A function
whose absolute square is unity can always be written in the form
S¡(k)
(11-43)
has the form
form for r very large must [by the argu-
requlres
i
dr2
flux. The net flux is therefore
of flux.
that any sOlution-and
[e-i(kr-I"./2)
r<a
2
+ 1)
2
-+------R+-R=O
d2R
and this is equal in magnitude
to the Outgoing
zero, as it should be, since there are no sources
-Vo
Ver)
1/r2,
= - --hk
fl -4k2
12
flux at large distances.
(11-37)
flux is, aside from
eiZ,c/2
al! but the radial
B. The Square Well, Bound States
(11-36)
Then
the incoming
-
2
fl
The
signs describe outgoing/incoming
flux. The factor 1/
from our calculation
is actually necessary for flux conservation,
going through the spherical surface <tt radius r is
!
for ignoring
conjugateJ
± --
=
±
185
conjugate
)
~
r31
(... )
t
=
transcendental
equation
it simplifies greatly if one uses the function
is obtained by matchi~g A sin Kr and Be-ar at r
O
involving
=
=
t,
Vo,
and E. Fa
The eigenvalu
a. The details are left as a
u(r)
rR(r).
~-,....--.
~
(§)
chapter
24
Collision Theory
Atomic and molecular structure was largely explored through spectroscopy. When it comes to trying to understand nuclear forces and the laws that
. govern the interactions of elementary particles, the only technique available is
that of scattering a variety of partic1es by a variety of targets. In some sense,
spectroscopy is also a form of "scattering." The atom in the ground state is
,excited by some projectile (it may be electrons in a discharge tu be or collisions
. with other target particles, as in heating up of the gas), and then an outgoing
photon is observed, with the atom going ¡nto the ground state again, or possibly
'another excited state. We do not usually describe these processes as "col1ision
J, 'processes"
beca use the atom has very wel1-defined energy levels, in which it
i'/~ays foe times that are enormously long compared to col1ision times,l so that
,~'iI;is possible to separate the "decay" from the excitation process. In particular,
,,~ characteristics of the decay are not sensitive to the particular mode of
C:xcitation. In nuc1ei and also in elementary partic1es, there exist levels, but
quently the lifetime is not sufficiently long to warrant a separation into
citation and decay, especial1y since accompanying the "resonant" scattering
re is also nonresonant "background"
scattering, and the disentangling of
two is sometimes complicated. In this chapter we wil1 therefore discuss the
ocess as a whole.
A. Collision Cross Section
The ideal way to talk about scattering is to formulate equations that
ibe exactly what happens: an incident partic1e, described by a wave packet,
oaches the target. The wave packet must be spatially large, so that it does
spread appreciably during the experiment, and it must be large compared
1
Recall rhar rhe liferime of a 2p hydrogen srare is 1.6 x 10-9 sec, which is large comrime ao/ Ol e"'" 2 x 10-17 sec.
lO rhe characterisric
379
380
Quamum
Physics
Collision Theory
with the target
particle,
but small compared
tory, that is, it must not simultaneously
lateral dimensions
are, in fact, determined
with the dimensions
overIap the target and detector. The
by the beam size in the accelerator.
There follows an interaction with the target, and finally we see two wave packets:
one continues
in the forward direction,
describing
the unscattered
part of the
beam, and the other flies off at some angle and describes the scattered particles.
The number of particles scattered into a given solid angle per unit time and unit
incident flux is defined to be the diflerential scattering cross section. We will not
follow this approach
directly,2 buc will instead use some of the material developed in Chapter 11 to obtain the differential cross section. We will, however,
keep the wave-packet
treatment in mind
In Out discussion of the continuum
as we interpret our formal results.
solutions of the Schrodinger equation
in Chapter 11 we concluded thar: (a) A solution of the Schrodinger
equation in
the absence of a potential is the plane wave form /k.r, which describes a flux
•
J
h
=
2im (tf¡*Vtf¡ -
tf¡Vtf¡*)
When the target is much more massive than the projectile,
tinction between the laboratory
angle and the center-of-mass
matics
are easily worked
out using the material
..
i
:
.
-1-:"
/k.r
:.:
...
i.'·¡?MkJ..e.
/i'l(J)yt. R
¡i
¿
00
2k 1=0
=}.!...
(21
(b) The conservation
a radial potential
tf¡(r)
=} -
¿
2ki I~o
00
subject
+ 1) il
of particles
[
j =
2im
h
f[
l /k. r
+ [(O)
SV /k. r + [(8) S-. ikr] *
[ikr]
where we have defined
the gradient
-.-
-f(kr-I"ff/2)
can only alter this to a function,
(21
+ 1) il [e
--
h ,.
l(kr-I"ff/2)]
I
p¡(cos
O) ~24-<cJ/} 1':V'd"D
that the presence
whose asymptotic
)
of
r
-i(kr-I"ff/2)
S¡(k)
-~-
r
/(kr-I"ff/2)]
P¡(ms
-ik.
+ tr[(O) (k
form is
-.h [ ik
21m
-
e
r
1=0 ~/+l)fi~)~~w~
[SI(k)1
+ ik[*(O)
hk
tf¡(r)
=}
e .
ik r
+ [ '0
(21 +
Pl(cos O) I~o
Sl(k)21k- 1
] r
wave + an outgoing spherical wave.3 Note
L.....
1)
(k) -
lJ/2ik
(24-8)
e---ikr(l-cosO)
r
1
[-k
e
ik. r
complex
+ _ 1r
LO
_
0[(8)
00 /krr
1
conjugate
+ iktr[(O)
+ to 0[(0)
--00 ----
r2
/kr(l-cosO)
-
+ iktrl[(8) [2_1
-/kr(l-coso)
__
r
r2
complex
j = -m
hk
+ -m
conJugate
.
k. r =
1
kr cos O, in
1
trl[(8)12-r2
eikr
point
r, where,
presumably
the counter
See Eg. 11-36, which explains why this is called an outgoing spherical
+
2m
r
hk
.2.-
2m
+ !!!.
[[*(0)
e -ikr(l-cosO)
+ [(8)
/kr(l-COSO)]
[[*(0)
e -ikr(l-cosO)
+ [(O)
/kr(l-COSO)]
r
.!!...
will be set up '.'
'This is done very nicely in R. Hobbie, American Journal o/ PhYJicJ,30, 857 (1962),
the leve] of mathematics
that we use in this book.
3
[SI
.
corresponding
to aplane
that we areworking with the effective one-particle
Schrodinger
equation,
so that m is the
reduced mass and O is the center of mas s angle between the direction of k (the·
z-axis) and the asymptotic
=
(24-7)
where we ha ve left out 1/ r3 terms, and where we ha ve used
the exponential
factors_ Thus the flux is
= 1
-
21m r2
~
~ [[(O) /kr(l-cosO) -
+
2im r2
_h_
~
wave.
1_
gives
e:r - e~:)]-
tr/(O) ---
conjugate
~:~
¿
e -ikr]
r2
/kr(l-cosO)
J.
l.
wave.
~
+ [*(8) -- r
O)
to
complex
00
[(8)
J -
r
r
e
_ e
forces us to the conclusion
section
solucion that describes the scattering is the one involving
the ourgoing
Let us calculate the flux for the asymptotic
solution
(24-5).
. - 21m
h [[
¡~
of the Special Topics
+
(24-3) that could
fi(k)
wave
is no disThe kine-
of COutse have set up a solution that has the asymptotic
an incoming spherical wave since it is the first term in
be modified by a coefficient satisfying
(24-4). However the
with
= hk
m
there
angle.
Note also that we could,
form of aplane wave
Calculating
i)
381
of the labora-
00
[0[(0)
/kr(l-cosO)
[*(8)
_
e -ikr(l-COSO)]
08
0[*(0) e -ikr(l-COSO)]
(24-9)
®
382
Quantum Physics
Collision Theory
This rather involved expression simplifies considerably when we consider that
8 r!' O, since one never does a scattering experiment directly in the forward
direction,4 and that in a measurement one always integrates the flux over a small
but finite solid angle. Thus in the last four terms of this expression we should
replace /kr(l -cosO) by
~D""'''
Outgoingwave
scattered
J sin 8 d8dcP g(8,cP)
fík
_ 1/(8)
m
r2
12
~ J ] ]
1/(8)12
r2
8
Fig. 24-1. Schematic layout for scattering experimento The scattering angle is the
laboratory angle.
The total cross section is given by
error
dA
1/(8)
fík
= - m . -- r2
The differential cross section is this number,
fík/ rn, that is,
der
= 1/(8)
12
r2dn
--
\
erro,
(~
divided by the incident flux,
one cel1 scattered
1/(8,cPW
froro unscattered
/(8)
= k t:o
=f
dn
(21
+ 1)
sin
/o,(k)
f¡¡(k) PI(COS
(24 ..17)
8)
D
f
D~
(21
+ 1)
+ 1)
(2/'
eio,(k)
e-iol'(k)
sin
D¡(k) PI(COS
8) J
sin o¡,(k) Pl,(cos 8)J
and using
dn
der
" How could
(24-16)
dn dn
der
rhen
G
47T'
If the potential has spin dependence, there may be an azimuthal depen
that more generally,
dn
=J
00
~
~)
[2
(k)
If we now use /(8) as expressed in terms of S¡(k), and express the latter in terms of
rhe phase shifr (d. 11-41) Sl(k) = e2io,(k), so rhar
Thus the num ber of particles crossing the area that subtends a solid angle dn at
the origin (the target) is
j . ir
J+ J4- } "O~::~"od
~
V:J~~~V
] ] ] 1]
1
= !!:.!
m .
a I,b
..../
beam
In the absence of a potential, only the first term is there: it represents the incident
flux. In a wave-packet treatment, fík/ll} would be multiplied by a function that
defines the lateral dimensions of the beam. Thus, if we ask for the radial flux,
ir' j, then that term gives a contribution fík- ir/ m = fík cos 8/ m, but only within
a finite regio n of the z-axis (see Fig. 24-1). Since the counter is put outside of
that region, this first term does not contribute to the radial flux in the asymptotic
region, so that
j . ir
~
~
_____
(24-10)
/kr(l-cosO)
where g(8,cP) is some sort of smooth, localized acceptance function for the
counter. Now, as r ~ ro, we have an integral over a product of a smooth function and an extremely rapidly varying one, and this vanishes faster than any
power of l/r. This is what is known in the mathematical literature as the
Riemann-Lesbegue lemma, and the reader can convince himself that this is
indeed so by working out an example, with a gaussian acceptance funcrion, sayo
Thus, only the first two terms remain, so that
.
fík
J=-+-Ir-m
383
particles?
J dnp¡(cos
8) Pl,(cos
8)
21
+1
Ol!'
(24-18)
we ger
47T'
erro,
=
-k2
00
L
1=0 (21
+ 1) sin2D¡(k)
(24-19)
384
Quanrum Physics
385
Collision Theory
IS
It is an interesting fact that
1 '"
Imf(O)
(21
+ 1) 1m
-k 1=0 (21
+ 1) sin2
k
t:o
sin
[/ól(k)
1 '"
o¡(k)]
(ñ~) [(~2 ]
p¡(1)
k
L
O¡(k)
(24-20)
= -47f (Tror
This relation is known as the optical theorem and it is true even when inelastic
processes can occur, as chey do in nuclear and pareicle physics scaccering processes. Ir is a very useful relacion and in wave language ic follows from che fact
thac che rocal cross seccion represencs che removal of flux from the incidenc
beam. Such a removal can only occur as a resulc of descructive incerference, and
the laccer can only occur becween che incidenc wave and che elascically scaccered
wave in che foeward direcrion. This explains why f(o) appears linearly. A more
decailed examinacion shows why che
¡ imaginary pare is involved.5
The requiremenc chac I S¡(k) I = 1 followed from conservacion of flux.
Accually, in many scaccering experimencs there is absorption of the incidenc beam;
the cargec may merely gec exciced, or change ics scace, or anocher pareicle may
emerge. Vnder chese circumstances our discussion is unchanged except that
=
5¡(k)
11¡(k) /iÓI(k)
(d. Eq. 11-36 and the fact that YlO = p¡(cos 8)/V 47f). The outward radial flux
is (ñk/m)(15¡(k)[2
47f/4k2),
so that the net flux lost is (ñk/m)(7f/k2)[1
- 11¡2(k)]
foe each I-value. Hence, dividing by the incident flux, we get
(Tror
because we are dealing with absorption.
15 now
f¡(k)
=
S¡(k)
-
2ik
1
+
(Te!
11¡(k)
e2iÓ/(k)
1
-
= ~k2
L
(21
+ 1)
(1
L
(21
+ 1)
(1 -
1
11¡ sin 20¡
+ i1
-
11¡ ~os 20¡
(24-23)
=
47f
=
47f
L
L
(21
+ 1) lji(k)
(21
+
1
1
1) 1
+
12
11¡2 -
211¡ cos 20¡
4k2
=
-k
1
2
It also follows from
i
e
-ikr
2k ·r~
(24-25 )
'1¡2(k)]
+ 11¡2 -
211¡ cos 2 o¡
+1
-
11¡2)
11¡ cos 2 o¡)
(24-26)
thac
(24-23)
=
L
(21
+ 1) 1m ji(k)
=
L
(21
+ 1)
1
1
1 -
11¡ cos 20¡
2k
=
k
47f
(Tror
(24-27)
so thac the optical theorem is indeed sacisfied.
If 11¡(k) = 1, we have no absorpcion, and the inelascic cross seccion vanishes.
When 11¡(k) = O we have total absorpcion. Nevereheless chere is still elastic
scactering in chat pareial wave. This becomes evident in scattering by a black disco
The black disc is described as follows: (a) ic has a well-defined edge and (b) it is
totally absorbing. Since we will consider scartering for shore wavelengths,
thac is, large k-values, condicion (a) specifies chac we only consider pareial
wa ves I ;S L, where
(24-24)
There is also a cross section for the inelastic processes. Since we do not specify
whac che inelascic processes consisc of, we can only talk about the total inelastic
cross section, which describes the loss of flux. If we look at a particular term in
(24-3),
the inward radial flux carried by
+ 1) [1 -
27f
The pareial wave scattering amplitude
=
(21
(Tinel
(24-22)
and the total elastic cross section is
(Te¡
=
(24-21)
1
.z;=
Thus the total cross section is
Imf(O)
.::;
= ;
(Tinel
is to be used, with
o .::; 11¡(k)
?
L
= ka
and a is che radius of che disco Condition
relevane values of I .::;L. Thus
(24-28)
(b) specifies chat
11¡(k)
o for che
L
(T'
me
1
= - 7f
"
(21
k2 1=0
L....J
+
1)
= - 7f
k2
L2
=
7fa2
(24-29)
and
p¡(cos
8)
7f
• See 1. 1. Schiff, Prog. Theo. PhYJ., (Kyoto),
11, 288 (1954).
(Te!
=
-2
k
L
L
/=0
(21
+ 1)
=
7fa2
(24-30)
386
Quantum Physics
®
102
Optical
point
lO'
x
.....
x
x-« x
10°
region
eE
Fig. 24-2.
:3
'"
~;::
.gl~
.!:1
x
10-1
10-2
10-3
,~
Black dise scarrering and the shadow effect.
r
so that the total cross section is
(Ttot
~
=
(Te!
+
(Tinel
=
27ra2
(24-31)
The result looks peculiar; on purely classical grounds we might perhaps expect
thar the total cross section cannor exceed the area presented by the disc; we
might also expect to see no elastic scattering when there is total absorption.
This is wrong; the absorptive disc takes flux proportional to 7ra2 out qf the
incident beam (Fig. 24-2), and rhis leads to a shadow behind the disc. Far away,
however, the shadow gets filled in-far enough away you cannor "see" the
disc-and the only way in which this can happen is through the diffraction of .
some of the incident wave at the edge of the disc. The amount of incident wave
that must be diffracted is the same amount as was taken out of the beam to
f¿
ap
1
,\)
10-4
./"-::-1'
\
f1 Ifl1NI
10-5
l.
make the shadow. Thus the elastically scattered flux must also be proportionaI '.~
to 7ra2. The elastic scattering that accompanies absorption is called shado!
scattering for rhe above reason. It is strongly peaked forward. The angle to which
it is confined can be esrimated from rhe uncertainty principIe: an uncertainty in
rhe lateral direcrion of magnitude a will be accompanied by an uncontrolled
lateral momentum transfer of magnitude p1. "'-' hj a. This, however, is equal to
P8, so that
()",-,-",-,-
'I\
10-6
o
20
22
24
26
28
Fig. 24- 3. Angular disrriburion of 1000 MeV (1 BeV) protons scarrered by
160 nuclei. The angular disrriburion shows rhe dips rhat characrerize diffraction
scattering. The departures fram rhe shape of Frauenhofer scattering in oprics is due
to rhe fact rhar nucIei are nor sharp, nor are rhey totally absorbing. The curve is
the result of a rheorerical calculation rhar rakes rhese effecrs into aCCOunt.(Fram H.
Palevsky et al., Phys. Rev. Letters, 18, 1200 (1967), by permission.)
ak
This agrees with the optical result 8 "'-' 'Aja. These features are observed both in
nuclear scattering and in particle scattering at high energies, since the central
region of nuclei and of prorons is strongly absorprive, and the edges of these
objects are moderately sharp. (See Fig. 24-3.)
387