Diffraction and the wave nature of electrons flashcards

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Diffraction
The spreading out of a wave as it passes through a gap or past the edge of an obstacle.
Huygens' construction
Every point on a wavefront is a source of secondary wavelets spreading at the wave speed; the new wavefront is the envelope of the wavelets.
Diffraction grating
A plate with a large number of equally spaced, parallel, very narrow slits; it gives sharp maxima at angles set by nλ = d sin θ.
Slit spacing d of a grating
Distance between the centres of adjacent slits: d = 1 ÷ (number of lines per metre).
Order of a maximum (n)
The whole number of wavelengths in the path difference between adjacent slits; n = 0 is straight through.
De Broglie wavelength
The wavelength associated with a moving particle: λ = h/p, where p is its momentum.
Diffraction grating equation
nλ = d sin θ (θ measured from the straight-through direction). (Yes)
De Broglie equation
λ = h/p, with p = mv. (Yes)
Highest order from a grating
n_max = whole-number part of d/λ (from sin θ ≤ 1). (No — derive it)
Electron accelerated from rest through p.d. V
eV = ½mv² (from W = QV and Ek = ½mv²). (No — derive it)
De Broglie wavelength of an electron accelerated through V
λ = h/√(2meV), so λ ∝ 1/√V. (No — derive it)
Angle from positions on a screen
tan θ = x/D (x from the central maximum, D from grating to screen). (No — derive it)
Wavefronts after a gap about one wavelength wide
Semicircular wavefronts spreading into the whole region beyond; same spacing (λ) as before the gap.
Wavefronts after a gap many wavelengths wide
Straight across most of the gap width, curving only slightly at the edges; little spreading.
Path difference between adjacent grating slits
Drop a perpendicular from one slit onto the ray from its neighbour: the extra path is d sin θ.
Electron diffraction tube screen
Bright central spot with concentric bright rings; the rings shrink as the accelerating p.d. increases.
Graph of electron λ against accelerating p.d.
A falling curve, λ ∝ 1/√V: four times the p.d. halves the wavelength.
What does diffraction change, and what does it not change?
It changes the direction and shape of the wavefronts; wavelength, frequency and speed are unchanged.
When is diffraction most noticeable?
When the gap (or obstacle) is similar in size to the wavelength.
Why does sound diffract round a doorway but light does not noticeably?
Sound wavelengths are similar to the doorway width; light wavelengths are millions of times smaller.
Why are grating maxima sharp and bright?
With many slits, waves only all arrive in phase at exactly nλ path difference; slightly off that angle they cancel. Many slits pass more light.
Why does electron diffraction show that electrons behave as waves?
Diffraction is a wave property; the ring pattern needs a wavelength similar to the atomic spacing, and ring size changes with p.d. as λ = h/p predicts.
Why are rings, not spots, seen with a graphite film?
Graphite contains many tiny crystals at random orientations, so each diffraction angle occurs in every direction round the beam.
Core Practical 6: measurements and instruments
D (grating to screen) with a tape measure; distance 2x between same-order maxima either side with a metre rule; repeat and take means.
Core Practical 6: laser safety
Never look into the beam or its reflections; low-power laser; beam below eye level; remove reflective objects.
Slit spacing of a grating with 300 lines per mm?
d = 1 ÷ 3.00 × 10⁵ = 3.33 × 10⁻⁶ m.
532 nm light, grating of 200 lines per mm. First-order angle?
d = 5.00 × 10⁻⁶ m; sin θ = 532 × 10⁻⁹ ÷ 5.00 × 10⁻⁶ = 0.106; θ = 6.11°.
590 nm light, grating of 600 lines per mm. Highest order and number of maxima?
d/λ = 2.82, so highest order 2; maxima 0, ±1, ±2 = 5 in total.
150 lines per mm; first-order maximum 0.120 m from centre on a screen 1.20 m away. Wavelength?
tan θ = 0.100, θ = 5.71°; λ = 6.67 × 10⁻⁶ × sin 5.71° = 6.63 × 10⁻⁷ m.
De Broglie wavelength of an electron at 5.0 × 10⁶ m s⁻¹?
p = 9.11 × 10⁻³¹ × 5.0 × 10⁶ = 4.6 × 10⁻²⁴ kg m s⁻¹; λ = h/p = 1.5 × 10⁻¹⁰ m.
Electron accelerated from rest through 500 V. Speed and de Broglie wavelength?
eV = ½mv² gives v = 1.33 × 10⁷ m s⁻¹; λ = h/(mv) = 5.49 × 10⁻¹¹ m.
Speed of a proton with de Broglie wavelength 1.0 × 10⁻¹⁰ m?
v = h/(mλ) = 6.63 × 10⁻³⁴ ÷ (1.67 × 10⁻²⁷ × 1.0 × 10⁻¹⁰) = 4.0 × 10³ m s⁻¹.
Lines per mm in the grating equation
Convert to lines per metre (× 10³) and invert to get d in metres before substituting.
Highest order: d/λ = 3.9
Round down: the highest order is 3. A fourth order would need sin θ > 1.
Angle measured between the two nth-order maxima
θ in nλ = d sin θ is from the centre, so halve the angle (or the distance) first.
Is the second-order angle twice the first-order angle?
No: sin θ doubles, not θ. Calculate each angle from sin θ = nλ/d.
Two particles with the same de Broglie wavelength
They have the same momentum, not the same speed or kinetic energy.
Explaining electron diffraction rings
Use wavelength, diffraction and superposition — not collisions or repulsion by atoms.

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