Verifying the Relativistic Dispersion Relation

Wenjun Li, Douglas Richmond
Department of Physics, University of Wisconsin-Parkside
900 Wood Road, Kenosha, Wisconsin 53141 USA

February 21, 2006

Abstract

By recording the energies of electrons excited by Compton scattering, one can verify the relativistic dispersion law numerically. Classical energy determination does not give the correct results.

1 Introduction

Using previously sampled photopeak and Compton edge data, we will verify that the relativistic dispersion relation holds true for determining the energy of the scattered electrons.

2 Procedure

We had previously obtained the data below.


Element	E_{γ, 1} (KeV )	E_{γ, 2} (KeV )	E_ce,exp (KeV )	mc² (KeV )

²²Na	522.36	1268.13	349.48	516.812

⁶⁰Co	1173.21	1332.47	1118.08	510.998

¹³³Ba	81.685	369.82	261.346	306.994

¹³⁷Cs	668.125	-	478.295	530.343

By using

--hf----- Ece = mc2- 1 + 2hf

we were able to determine the Compton edge energy from the photopeak energy.
By graphing the inverse of the photopeak energy vs. the inverse edge energy, and performing a least squares ﬁt of the data we were able to see how well our data agrees with a relativistic dispersion relation.

The value for the Compton edge energy for ¹³³Ba was occluded by the photopeak, therefore we omitted it’s contribution for the purposes of this graph.
The curve of best ﬁt has equation:

2 y = - 0.00196037 + 1.01268x + 0.253547x

1 1 x = ----,y = ---- hf Ece

3 Analysis and conclusions

By comparing our numerical equation with the theoretical relationship

2 -1-- 1--- mc------1--- E = hf + 2 (hf )2 ce

we obtained a numerical value of mc² = .5071 which only differs from the theoretical by .76%.