n p + e e e e + ne * ne + n c + e e pu u + 20 10 20 10 13 7 13 6 236 94 232 92...
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n p + ee
ee +
Ne* Ne +
N C + e e
Pu U +
20 10
20 10
13 7
13 6
236 94
232 92
Fundamental particle decays
Nuclear decays
Some observed decays
The transition rate, W (the “Golden Rule”) of initialfinalis also invoked to understand
ab+c (+ )decays
How do you calculate an “overlap” between ???nep e |
It almost seems a self-evident statement:
Any decay that’s possible will happen!
What makes it possible?What sort of conditions must be satisfied?
initialtotal mm Total charge q conserved.
J conserved.
NdtdN teNtN )0()(
tet )(Pprobability of surviving
through at least time t
mean lifetime = 1/
For any free particle (separation of space-time components)
/0)0()( tiEet
Such an expression CANNOT describe an unstable particle
since2//22 )0()0()( 00 tiEtiE eet
Instead mathematically introduce the exponential factor:
2//0)0()( ttiE eet
/22 )0()( tet
2//0)0()( ttiE eet
then
a decaying probabilityof surviving Note: =ħ
/)2/(0)0()(
tiEiet
Also notice: effectively introduces an imaginary part to E
/)2/(0)0()(
tiEiet
Applying a Fourier transform:
0
/)()( dtetEg iEt
0
]/)(2
[
0
/)2/(/
0
0
)0(
)0()(
dte
dteEg
tEEi
tiEiiEt
)](2
[0
EEit
e
2/)()(
0
iEEEg
still complex!
What’s this represent?
E distribution ofthe unstable state
4/)(
4/)(
220
2
max
EE
E
Breit-Wigner Resonance Curve
Expect
4/)(
4/)()(*~)(
220
2
max
EEEgEgE
some constant
Eo E
1.0
0.5
MAX
= FWHM
When SPIN of the resonant state is included:
4/)(
4/
)12)(12(
)12()(
220
2
max
EEss
JE
ba
130-eV neutron resonancesscattering from 59Co
Transmission
-ray yield for neutron radiative
capture
+p elastic scattering cross-section in the region of the Δ++ resonance.
The central mass is 1232 MeV with a width =120 MeV
Cross-section for the reaction
e+e anything
near the Z0 resonanceplotted against
cms energy
Cross section for the reaction B10 + N14* versus energy.The resonances indicate levels in the compound nucleus N14*.[Talbott and Heydenburg, Physical Review, 90, 186 (1953).]
Spectrum of protons scattered from Na14 indicating its energy levels.[Bockelman et al., Physical Review, 92, 665 (1953).]
Resonances observed in the radiative proton capture by 23Na.[P.W.m. Glaudemans and P.M. Endt, Nucl. Phys. 30, 30 (1962).]
In general: cross sections for free body decays (not resonances)are built exactly the same way as scattering cross sections.
DECAYS (2-body example) (2-body) SCATTERING
except for how the “flux” factor has to be defined
pE
ofcons
space
phasefluxd
,42
M
pE
ofcons
space
phasefluxd
,42
M
)()2(
2)2(2)2(2
1
32144
33
33
23
322
1
ppp
E
pcd
E
pcd
m
M
)()2(
2)2(2)2()(4
432144
33
34
23
332
121
2
pppp
E
pcd
E
pcd
pEE
M
in C.O.M.
in Lab frame:
cpm 12
2
4
enforces conservationof energy/momentum
when integratingover final states
Now the relativisticinvariant phase space
of both recoilingtarget and
scattered projectile
Number scatteredper unit time = (FLUX) × N × total
)()2(2)2(2)2()(4 4321
44
33
34
23
332
121
2
ppppE
pcd
E
pcd
pEEd
M
(a rate)/cm2·sec
A concentrationfocused into a small spot and
small time interval
densityof targets size of
eachtarget
Notice: is a
function of flux!
X
Y
Z
Rotations
Y´
X´
= Z´
Changes in frame of reference
or point of view involve transformations
of coordinate axes (or, more generally, basis set)
X
Y
Z
Rotations
Y´
X´
= Z´
x
y
x´
y´
sincos yx'x
cossin yx'y z'z
sincos yx'x
cossin yx'y z'z
R =cos sin 0-sin cos 0 0 0 1
v´ = R v
X
Z
Y
r
aX´
Y´
Z´
r´
Translationsparallel translation
(no rotation) of axes
iii ax'x
r´ = r a
iii ax'xT :
)()(: arfrfT
Vectors (and functions) are translated in the “opposite direction” as the coordinate system.
How can we possibly express an operator like this as a matrix?
The trick involves using
0
432
!4!3!21
!n
nx xxx
xn
xe
to cast matrix operators as exponentials
0
432
!4!3!21
!
)(
n
niH xH
iH
iHn
iHe
where H is an operator…or matrix the unit matrix
1 0 0 0 ···0 1 0 00 0 1 0
nn
axn
afa''faxa'faxafxf )(
!
)()()(
!2
1)()()()(
)(2
Taylor Series (in 1-dimension)
and we’ll make that connection through
…and this useful limitx
N
eN
xim
1
N
For an infinitesimal translation f (x0+δx) f (x0) + δx f x
3
1
3
1
1)()()(i i
ii i
i xarf
xarfarf
3
i=1
Ok…but how can any matrix represent this?
Imagine dividing the entire translation a intoδax=
δay=
δaz=
ax
N ay
N az
N
3
1
1)(i i
i xaarf N
f (r)
f (r)
)(1)(3
1
rfxN
aarf
N
i i
i
)(1)( rfN
zayaxaarf
N
zyx
x=x0
and applying this little step N times
making this a continuous smooth translation
)(1 rfN
aN
lim
N∞)(rfe a
)()( / rfearf pai
)( rfe a i
ħ(-iħ )
)(1)( rfN
zayaxaarf
N
zyx
)(1 rfN
a N
For homework you will be asked to do the same thing for rotationsi.e., show you can cast in the same form.R=
cos sin 0-sin cos 0 0 0 1
You should start from: R= 1 0 - 1 0 0 0 1
Later we will generalize this result to:
)()( / rer Ji
)()( / rer Ji
)()( / rer Ji
Rotation of coordinate axes by about any arbitrary axis ̂
Rotation of the physical system within fixed coordinate axes
)0,(),( / retr iHt Recall, even more fundamentally, the QM relation:
Time evolution of an initial state,generated by the Hamiltonian
“Generator”OperatorAmount of
transformationNature of the
transformation
/paie
/ Jie
/iHte
p
J
H
a
t
Translation:moving linearlythrough space
rotatingthrough space
translationthrough time
The Silver Surfer, Marvel Comics Group, 1969
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