Electron Diffraction Experiment by Eric Cotner (presenting) and Yukun Zhang

Apparatus Electron gun Voltage supply Multimeter Carbon diffraction grating Phosphorescent screen

Electronics Electron gun accelerating voltage controlled by power supply Accelerating voltage determines kinetic energy (and wavelength) of electrons Current is limited to avoid burning out the filament

Theory De Broglie hypothesis: matter can actually be described with wavelike properties (p=h/λ, E=hf) From λ= ℎ /(2mE) 1/2, the wavelength of an electron is inversely proportional to the square root of its kinetic energy We expect that accelerated electrons will be diffracted from a sufficiently small spacing, such as the crystals in a carbon lattice

Derivation of Results Electrons are diffracted by carbon lattice; 2 different spacings will create 2 concentric rings obeying the law nλ=dγ where γ=2θ, n=1, and d is the crystal spacing Using E=hf and p=h/λ, we can calculate λ in terms of accelerating voltage in the following way: – E = p 2 /2m = (h/λ) 2 /2m – λ = h/(2m e eV) 1/2 – λ = (1.23 V -1/2 ) nm Using small angle approximation, γ=D’/2L Setting λ = dγ = (1.23 V -1/2 ) nm allows us to calculate d (in nm) from the slope of V -1/2 = d(γ/1.23) by plotting γ vs. V -1/2

Calculation of D’ Find D’ from ratios of similar triangles using the radius of curvature of the phosphorescent screen Equation to use:

Diffraction Ring Data V (V) D 1 (mm) D 2 (mm) V (V) D 1 (mm) D 2 (mm) measurement for 2 different voltages: 1 measurement for 8 different voltages:

Derived Quantities V (V) D’ 1 (mm) D’ 2 (mm) λ (nm) γ1γ γ2γ V (V) D’ 1 (mm) D’ 2 (mm) λ (nm) γ1γ γ2γ

D 1 ’ d 1 /(1.23x L) vs. V -1/2 d 1 = nm

D 2 ’ d 2 /(1.23x L) vs. V -1/2 d 2 = nm

Error Analysis Used the above error propagation formulae For d 1, evaluated at D 1 ’=57.5 mm, V=2000 V, δD 1 ’=1.73 mm, δV=50 V – δd 1 = nm For d 2, evaluated at D 2 ’=30 mm, V=2000 V, δD 2 ’=1.55 mm, δV=50 V – δd 2 = nm

Comparison to accepted values Accepted values: – d 1 =0.123 nm and d 2 =0.213 Derived values: – d 1 =0.128 ± nm and d 2 =0.220 ± Accepted values fall within uncertainties of derived values, strong support for validity of accepted values

Conclusions 4% error for spacing of d 1 3.3% error for spacing of d 2 Strong support for de Broglie wave theory of moving particles Strong support for accepted crystal structure of carbon