Introduction
Wavelength
DiffractionOrder Graphite Grating Tasks
Introduction
Wavelength
DiffractionFor high energy electrons, the non-relativistic calculation using $\lambda_{\text{de Broglie}} =\frac{h}{p}=\frac {h}{\sqrt{2\cdot m_\text e \cdot e\cdot V_{\text a}}}$ is no longer accurate. The effects of relativity must be taken into account. The new calculation is $$\bbox[8px,border:2px solid red]{\lambda_{\text{de Broglie}} = \frac {h}{p}=\frac{h\cdot c}{\sqrt{\left(e\cdot V_\text{a}\right)^2+2\cdot e\cdot V_\text{a}\cdot m_\text{e}\cdot c^2}}}$$
For comparison, the tabel shows de Broglie wavelengths calculated using both the claccical and relativistic formulas:
($h=6{.}6\cdot 10^{-34}\,\text J\cdot \text s$, $c=3\cdot 10^8\,\rm{\frac{m}{s}}$, $ m_{\text e}=9{.}1\cdot 10^{-31}\,\text{kg}$, $e=1{.}6\cdot 10^{-19}\,{\text C}$)
| Acceleration voltage Va | $\lambda_{\text {de Broglie}}$ (classic) | $\lambda_{\text {de Broglie}}$ (relativistic) | relative error |
|---|---|---|---|
| 1000 V | $3.8677\cdot 10^{-11}\,\text m $ | $3.8658\cdot 10^{-11}\,\text m $ | 0.05% |
| 10000 V | $1.2231\cdot 10^{-11}\,\text m $ | $1.2171\cdot 10^{-11}\,\text m $ | 0.49% |
| 50000 V | $5.4697\cdot 10^{-12}\,\text m $ | $5.3408\cdot 10^{-12}\,\text m $ | 2.41% |
| V | ${}$ | ${}$ |
The following graph compares the two calculations and shows the relative errors as a function of the acceleration voltage. Due to the small error of approximately 0.58% with an acceleration voltage of 12 kV, this experiment can still be calculated non-relativistic.
