Alpha Decay II

[Sec. 7.1/7.2/8.2/8.3 Dunlap]

TPR

The one-body model of α-decay assumes that the α-particle is preformed in the nucleus, and confined to the nuclear interior by the Coulomb potential barrier. In the classical picture, if the kinetic energy of the -particle is less than the potential energy represented by the barrier height, the α-particle cannot leave the nucleus.

In the quantum-mechanical picture, however, there is a finite probability that the -particle will tunnel through the barrier and leave the nucleus.

The α-decay constant is then a product of the frequency of collisions with the barrier, or ``knocking frequency'‘ (vα/2R), and the barrier penetration probability PT.

vα

r=br=R

Qα

How high and wide the barrier?

rcZ

rZerV 1..2)4(

2)(0

2

The height of the barrier is:

RcZE ..2

max

The width of the barrier is

2 . .w b Z cR RQ

w

Lets calculate these for taking R0=1.2F, we have U23592 FR 4.7)235(x2.1 3/1

MeVFFMeVE 36

4.7x137.197x92x2

max FFMeV

FMeV 494.768.4x137

.197x92x2w

30MeV

The 1D square potential tunneling problem

This problem is quite algebraically difficult to solve exactly (although it can be done) because one has to match the sinusoidal wavefunctions and the gradient of the wavefunction at barrier entry (A) and output (B).

The 1D square potential tunneling problem

The 3D tunneling problem

So taking the case when l=0, we have to solve the 1D SE for the potential

The WKB approximation

How do we apply this result to the case of the barrier with a height V, that is not constant? As with the finite barrier really exact solution is not possible, but a good approximation exists – it is called the WKB approximation- after physicists (Wentzel, Kramers and Brillouin).

The WKB approximation

G is known as the Gamow factor. It is this Gamow factor that it is now our task to calculate.

which in the limit of r becoming small becomes:

(Q in MeV and R in fm)

The WKB approximation

The WKB approximation

The formula for decay rateSo finally we have to put everything together to get the decay rate of an alpha unstable nucleus. The decay rate can be considered as a compounding of three probabilities.(i) The probability Pα of the α-particle forming (or being in existence) at a specific time(ii) The probability fα of the α-particle hitting the “wall” of the nucleus each second(iii) The probability T of the α-particle transiting to the “freedom distance” r=b

The formula for decay rate

where the Gamow factor G can be either expressed by (8.17) or by (1) depending on preference.

The range energy relationship for alphas is so that the Geiger-

Nuttal law gives . The barrier penetration theory gives

3/ 2R T ln lnQ ln 1/ Q

In the range 4<Q<7MeV which covers most alpha emission – a quantity linear in

is close to be linear in so that the Geiger Nuttal law is vindicated.

lnQ

1/ Q

1/ 21/ 2

2

3.95 2.97( )

0

2( ). G

Z ZRQ

Q UcP eR mc

e

Where we have ignored the small pre-exponential dependence on Q. We find on taking logarithms

1/ 2

1/ 20ln ln 2.97( ) 3.95 ZZR

Q

Explaining the Geiger-Nuttal Law