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= /(2mE)ℎ 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 = p2/2m = (h/λ)2/2m– λ = h/(2meeV)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) 1400 1400 1400 1400 2500 2500 2500 2500

D1 (mm) 68.55 66.84 66.98 67.29 52.50 52.67 53.10 52.74

D2 (mm) 39.72 42.36 41.06 41.28 33.60 31.19 32.27 30.67

V (V) 1600 1700 1800 1900 2000 2100 2300 2700

D1 (mm) 63.99 62.60 61.10 57.75 58.63 56.33 55.19 50.65

D2 (mm) 38.32 38.72 36.99 35.19 33.57 32.58 32.02 29.86

4 measurement for 2 different voltages:

1 measurement for 8 different voltages:

Derived QuantitiesV (V) 1400 1400 1400 1400 2500 2500 2500 2500

D’1 (mm) 73.40 71.30 71.47 71.85 54.55 54.74 55.22 54.82

D’2 (mm) 40.58 43.41 42.01 42.25 34.11 31.60 32.72 31.08

λ (nm) .0329 .0329 .0329 .0329 .0246 .0246 .0246 .0246

γ1 .242 .236 .237 .238 .186 .186 .188 .186

γ2 .140 .150 .145 .146 .119 .110 .114 .108

V (V) 1600 1700 1800 1900 2000 2100 2300 2700

D’1 (mm) 67.86 66.20 64.43 60.53 61.54 58.89 57.59 52.48

D’2 (mm) 39.09 39.51 37.68 35.78 34.08 33.05 32.46 30.22

λ (nm) .0308 .0298 .0290 .0282 .0275 .0268 .0256 .0237

γ1 .226 .221 .216 .204 .207 .199 .195 .179

γ2 .135 .137 .131 .124 .119 .115 .113 .106

D1’ d1/(1.23x10-6 2L) vs. V-1/2

d1 = 0.128 nm

D2’ d2/(1.23x10-6 2L) vs. V-1/2

d2 = 0.220 nm

Error Analysis

• Used the above error propagation formulae• For d1, evaluated at D1’=57.5 mm, V=2000 V,

δD1’=1.73 mm, δV=50 V– δd1=0.0044 nm

• For d2, evaluated at D2’=30 mm, V=2000 V, δD2’=1.55 mm, δV=50 V– δd2=0.0138 nm

Comparison to accepted values

• Accepted values:– d1=0.123 nm and d2=0.213

• Derived values:– d1=0.128 ± 0.004 nm and d2=0.220 ± 0.014

• Accepted values fall within uncertainties of derived values, strong support for validity of accepted values

Conclusions

• 4% error for spacing of d1

• 3.3% error for spacing of d2

• Strong support for de Broglie wave theory of moving particles

• Strong support for accepted crystal structure of carbon