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