7/31/2019 Alpha Project

1/2

Joseph Berg and Opeoluwa Matthews

Physics 411, Quantum Mechanics: The problem of Alpha Decay

Wednesday, November 23, 2011

Abstract:

The purpose of this project is to determine the alpha decay half-life for a given mass number Thorium.

This is determined by approximating the probability of emission through a series of potential barriers,

akin to a Riemann sum. The half-lives are then determined using a model of the alpha particle rattling

classically inside its well. The half-life obtained using this method is then compared to that obtained by

calculating the transmission probability using an integral as an infinite set of barriers.

Experimental Procedure:

Calculation of Expected Time using Reimann Approximation:

At the core of this experiment is that the probability of transmission, T, from a rectangular barrier is given

by:

where , ,

, and r = . Z1 and Z2 are simply the number of protons of thealpha particle and the number of protons of Thorium after emitting the alpha particles. E is the

energy of the alpha particle emitted.

Vo is given as -30MeV.

a is the barrier width, given by r=R+na, a=(b-R)/100.

R is given as 1.2(AThorium-A)1/3

+ 1.2(A)1/3

b is the is the radius at which the Coulomb potential energy and the alpha particle energy are

equal. It is given as b = k (ZThorium-2)Z e2/(E) = 176*1.44MeV fm/ E.

We will simply refer to E as E for the remaining of the report.

It can be shown that, when q*a is large, , approximately. This approximation was

used in the calculations and is derived below.

For large qa, sinh(qa) --> eqa/2

7/31/2019 Alpha Project

2/2

Hence sinh2(qa) --> (eqa/2)2 = e2qa/4

Also, >>

Hence,

The calculations were made for Transmission probabilities through 100 barriers. The transmission of the

alpha particle through each of the 100 barriers is considered an independent event. Hence, the probability

of transmission through all 100 of them would be the product of those independent probabilities. Of note

is the fact that to calculate V(r), the center of the rectangle was used so as to get more accurate

calculations.

After calculating the Transmission probability T, the expected number of attempts for one transmission to

occur is gotten from NA = 1/T. Next, the velocity of the alpha particle can be gotten from its energy E =

mv2. Hence, v = (2E/m)

1/2.

Assuming a width of R (discussed above) for the well, the number of attempts per second N A/s =v/R. Hence, the expected time for one emission is given as t = NA/NA/s.

Calculation of Expected Time using Integral:

( )

Where e=E, f=176*1.44MeV fm.

Results/ConclusionsThis probabilities gotten from this integral is usually so small, however, that zero probability is reported,

which would mean an artificially infinite amount of time for alpha decay emmission. It is likely that this

is attributable to a rounding error in the calculation of these integrals. This is consistent with the results

for all the isotopes of Thorium.

The probability gotten from the excel spreadsheet using Reiman Sums also yielded an artificially small

probability of 1.82655E-58. Again, this could be as a result of rounding error. This is consistent with the

results for all the isotopes of Thorium.