The following needs to be demonstrated for proof of de Broglies assumption:
For various acceleration voltages \(V_{\rm{a}}\), the radius of the inner circle must be determined on the screen of the electron diffraction tube. Here are three example measurements from the experiment:
$V_\text a$ | 5 kV | 7 kV | 10 kV |
---|---|---|---|
Radius rinner | 1.05 cm | 0.9 cm | 0.75 cm |
$V_\text a$ | $$\lambda=\frac{h}{\sqrt{2\cdot m_{\rm e}\cdot e\cdot V_{\rm a} }}$$ | $$\lambda_{\text {Ex}}=d\cdot \sin\left(\frac{1}{2}\tan^{-1}\left(\frac{r}{L-R+\sqrt{R^2-r^2}}\right)\right)$$ | relative error | |
---|---|---|---|---|
\(5\,\rm{kV}\) | $1{,}73\cdot 10^{-11} \,\text m$ | \(1.05\,\rm{cm}\) | $1.77\cdot 10^{-11} \,\text m$ | 2.31 % |
\(7\,\rm{kV}\) | $1.46\cdot 10^{-11} \,\text m$ | \(0.90\,\rm{cm}\) | $1.51\cdot 10^{-11} \,\text m$ | 3.42 % |
\(10\,\rm{kV}\) | $1.22\cdot 10^{-11} \,\text m$ | \(0.75\,\rm{cm}\) | $1.26\cdot 10^{-11} \,\text m$ | 3.28 % |
The table, along with the small relative deviation between the calculated wavelengths, shows that the de Broglie hypothesis is actually true. Particles with mass also show wave-like properties, and their wavelength is described by $\lambda_{\text {de Broglie}}=\frac{h}{p_{\text e}}$.