Davisson-Germer experiment — electron diffraction

2.3.6 · Physics › Modern Physics

KYA claim kiya ja raha hai?

Experiment ki genius: yeh $λ$ ke do raaste deta hai jo agree karne chahiye.

Theory road: electrons ko voltage $V$ se accelerate karo, $λ = h / p$ compute karo.
Experiment road: diffraction angle measure karo, crystal geometry use karke $λ$ nikalo.

Agar dono same number dete hain, toh matter waves real hain.

Apparatus KAISE kaam karta hai

Figure — Davisson-Germer experiment — electron diffraction

DERIVATION 1 — "Theory road" wavelength

DERIVATION 2 — "Experiment road" wavelength

Formula Bragg cross-check (CORRECT Ni spacing use karke)

Atomic planes se constructive interference ke liye Bragg condition: $2 d sin θ = nλ$ jahan $d$ = atomic planes ki spacing, $θ$ = glancing angle (plane se).

Geometry link: scattering angle $ϕ$ (incident aur scattered beam ke beech) glancing angle se relate karta hai $θ = 9 0^{\circ} - \frac{ϕ}{2}$ Kyun? Incident aur diffracted rays plane normal ke baare mein symmetric hoti hain; $ϕ$ angle ko bisect karke $9 0^{\circ}$ se subtract karne par glancing angle milta hai.

Ni(111) planes ke liye correct interplanar spacing $d = 2.03$ Å hai. $ϕ = 5 0^{\circ} \Rightarrow θ = 6 5^{\circ}$ , first order $n = 1$ ke saath: $λ = 2 d sin θ = 2 (2.03) sin 6 5^{\circ} = 3.68 \overset{˚}{A}$ Yeh higher-order reflection se correspond karta hai ( $n = 2$ : $λ = 1.84$ Å) — baat yeh hai ki upar wali grating analysis, jo surface row spacing $a = 2.15$ Å use karti hai, wahi seedha $1.65$ Å deti hai. (Purane textbooks $d = 0.91$ Å quote karte hain; woh value Ni(111) plane spacing nahi hai aur use nahi karni chahiye.)

Forecast-then-Verify

Recall Answer padhne se pehle forecast karo

Q: Agar tum accelerating voltage $V$ badhao, toh diffraction peak bade ya chote detector angle $ϕ$ par jaayegi?

Verify: Higher $V$ → bada $p$ → chota $λ$ (kyunki $λ = h / p$ ). Surface-grating view ( $a sin ϕ = nλ$ ): chota $λ$ → chota $sin ϕ$ → peak chote $ϕ$ par shift hoti hai. Bragg view ( $2 d sin θ = nλ$ ): chota $λ$ → chota $θ$ ; lekin $ϕ = 18 0^{\circ} - 2 θ$ hone se, chota $θ$ bada $ϕ$ deta hai. ⚠️ Yeh opposite lagte hain kyunki $ϕ$ dono pictures mein alag define hai (grating $ϕ$ incident beam se surface convention ke along outgoing ray tak hai; Bragg $ϕ = 18 0^{\circ} - 2 θ$ poora beam-to-beam deflection hai). Physically observed result, D–G applied standard surface-grating relation use karke, yeh hai ki $V$ badhne par peak chote $ϕ$ ki taraf shift hoti hai. Peak disappear bhi ho sakti hai agar $λ$ us order ke liye bahut chota ho jaaye. Lesson: shift direction predict karne se pehle hamesha batao ki tumhara $ϕ$ kis geometric convention ko refer karta hai.

Common mistakes (Steel-manned)

Worked Example (poore numbers)

Flashcards

Davisson–Germer experiment ne kya prove kiya?

Ki electrons (matter) wave behaviour exhibit karte hain — de Broglie matter waves ki direct confirmation.

Kaunsa target use kiya gaya aur kyun?

Nickel ka single crystal, kyunki uska atomic spacing (~0.1 nm) electron wavelengths se match karta hai, diffraction enable karta hai.

Voltage V se accelerate kiye electron ka wavelength (derive)?

λ = h / 2 m e V = 12.27/ V

Å.

Kis voltage aur scattering angle par famous peak observe hui?

V = 54

V, scattering angle

ϕ = 5 0^{\circ}

Data se seedha

λ = 1.65

Å konsa relation deta hai?

Surface-grating relation

a sin ϕ = nλ

a \approx 2.15

Å,

ϕ = 5 0^{\circ}

n = 1

ke saath.

Actual Ni(111) interplanar spacing kya hai (aur kya use NAHI karna)?

d \approx 2.03

Å (0.203 nm); purana

0.91

Å value mat use karo.

Detector angle

ϕ

aur Bragg glancing angle

θ

mein relation?

θ = 9 0^{\circ} - ϕ /2

, equivalently

ϕ = 18 0^{\circ} - 2 θ

54 V par non-relativistic treatment kyun valid hai?

KE = 54 eV

≪

511 keV rest energy, toh

v ≪ c

Surface-grating convention use karke,

V

badhne par peak

ϕ

kaise shift hoti hai?

λ

ghatta hai,

sin ϕ = nλ / a

ghatta hai, toh peak chote $ϕ$ ki taraf jaati hai.

Recall Feynman: ek 12-saal ke bacche ko samjhao

Socho tum tiny balls ek comb ke evenly spaced teeth par phek rahe ho. Agar balls sirf balls hote, toh woh everywhere randomly scatter hote. Lekin yeh special balls (electrons) pond par ripples ki tarah act karti hain — aur evenly spaced teeth se bounce hoti ripples ek special direction mein line up aur reinforce hoti hain, ek bright "loud" spot banaati hain. Davisson aur Germer ne woh bright spot dekha. Balls aisa nahi karti; waves karti hain. Toh electrons secretly wavy hain. Aur comb teeth ki spacing ne unhe exactly bataya ki electron kitna "wavy" tha — aur woh de Broglie ke formula se perfectly match kiya.

Connections

de Broglie hypothesis — woh prediction jise is experiment ne confirm kiya.
Bragg's law — angle se $λ$ cross-check karne ke liye use ki gayi geometry.
Wave-particle duality — D–G matter-wave half hai; photoelectric effect wave→particle half hai.
Photoelectric effect — complementary evidence.
Electron microscope — electron waves ka practical application (chota $λ$ → high resolution).
Heisenberg uncertainty principle — particles ko wave packets ki tarah treat karne se follow karta hai.