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