Davisson-Germer experiment — electron diffraction: Foundations

2.3.6 · D1 · Physics › Modern Physics › Davisson-Germer experiment — electron diffraction

Is page par assume kiya gaya hai ki tumne kuch nahi dekha. Parent note ki equations ko touch karne se pehle, hum har symbol, picture, aur word ko build karte hain — ek aisi order mein jahan har naya idea sirf pehle waale ideas par tikta hai.

0. Do mental pictures jinhe hum baar baar aazmaa rahe hain

Poora experiment do tarahon se electron ko imagine karne ki ladaai hai.

Hume dono pictures ki zaroorat kyun hai? Kyunki experiment dikhata hai ki electron particle ki tarah fire hota hai (ek garam wire se beam) lekin wave ki tarah land karta hai (ek striped fan). Akela koi bhi picture kaafi nahi hai.

1. Angle — hum ek direction ki taraf kaise point karte hain

Jo kuch bhi hum measure karte hain woh ek angle hai, isliye pehle hum yeh pin down karte hain ki angle kya hota hai.

Do alag angles parent note mein aate hain aur unhe mix karna classic trap hai, isliye hum unhe carefully picture ke saath naam dete hain.

Formula

$ϕ$ aur $θ$ ke beech link derive karna — picture se Figure s02 dekho. Beam andar aati hai plane se glancing angle $θ$ banate hue, aur bahar jaati hai doosri taraf wahi glancing angle $θ$ banate hue (mirror-jaisa, "specular" scattering).

Step 1 — $ϕ$ kya hai? $ϕ$ incoming aur outgoing beam ke beech total kink hai. Kyunki har beam plane se $θ$ upar hai, aur woh ek doosre se door khulti hain, straight-through direction $18 0^{\circ}$ hogi; har beam surface se $θ$ upar jhuki hai, isliye dono beams mein $ϕ = 18 0^{\circ} - 2 θ .$ ka fark hai. Kyun? Incoming ray se outgoing ray tak jaate hue, tum $θ$ mudke plane tak pahucho aur phir $θ$ aur mudke niklo — lekin straight-ahead se deflection ke roop mein measure karo toh beams ek straight line se $2 θ$ alag hain, jo $18 0^{\circ} - 2 θ$ deta hai.
Step 2 — $θ$ ke liye solve karo. $ϕ = 18 0^{\circ} - 2 θ$ ko rearrange karo: $2 θ = 18 0^{\circ} - ϕ \Rightarrow θ = 9 0^{\circ} - \frac{ϕ}{2} .$
Numbers se check karo: observed $ϕ = 5 0^{\circ}$ deta hai $θ = 9 0^{\circ} - 2 5^{\circ} = 6 5^{\circ}$ , ek bahut grazing beam — exactly wahi jo parent note use karta hai.

2. Sine — angle ko length ratio mein badalna

Angles ko path lengths se connect karne ke liye hume ek tool chahiye: sine.

Key values jo apni pocket mein rakhna:

$sin 0^{\circ} = 0$ (koi slant nahi → koi extra path nahi)
$sin 5 0^{\circ} \approx 0.766$
$sin 6 5^{\circ} \approx 0.906$
$sin 9 0^{\circ} = 1$ (sabse bada possible)

3. Wavelength $λ$ — ek ripple ki size

Hume kyun parwah hai: diffraction — striped fan — tabhi dikhta hai jab $λ$ slits (yahan, atoms) ke beech spacing ke comparable ho. Bahut chota $λ$ aur wave seedhi nikal jaati hai bina mudi; bahut bada aur sab kuch blur ho jaata hai. Isliye experiment mein crystal chahiye, wide grating nahi.

4. Atom spacings $a$ aur $d$ — "comb teeth"

Ek crystal mein atoms perfectly regular grid mein stack hote hain. Do spacings matter karti hain.

Ab hum har spacing ko ek wavelength se connect karte hain — ek relation per picture.

Hume yeh kyun chahiye: spacings woh "slit spacing" hain jo diffraction ke liye $λ$ ke paas honi chahiye, aur har relation measured angle ko wavelength mein convert karta hai.

5. Momentum $p$ aur de Broglie link

$p$ ko $λ = h / p$ se pehle kyun introduce karein? Kyunki tum yeh claim nahi kar sakte ki ek particle ki wavelength hoti hai bina yeh bataye ki kaunsi property usse set karti hai — aur jawab hai uski momentum.

6. Charge $e$ , voltage $V$ , aur energy $e V$

Hum electron ki momentum control kaise karte hain? Usse ek voltage se push karke.

7. Diffraction order $n$ — puri wavelengths ginna

Yeh kyun exist karta hai: bright spots repeat hote hain. Waves reinforce karti hain jab bhi extra path $1, 2, 3 \dots$ full wavelengths ho — in mein se har ek apna bright angle deta hai. Section 4 ki dono bright conditions yeh counter carry karti hain: $a sin ϕ = nλ$ aur $2 d sin θ = nλ$ .

Prerequisite map

Ise top-to-bottom padho: do roads (theory via de Broglie, experiment via grating/Bragg picture) har ek ek wavelength produce karti hain, aur unka agreement hi proof hai.

Recall checks

Recall Kaun sa angle incoming beam se measure hota hai?

Scattering angle $ϕ$ ::: haan — $ϕ$ beam-to-beam hai; $θ$ atomic plane se measure hota hai.

Recall

$ϕ$ se $θ$ derive karo. $θ = 9 0^{\circ} - ϕ /2$ ::: kyunki mirror-like scatter ke liye $ϕ = 18 0^{\circ} - 2 θ$ hai; $θ$ isolate karne ke liye rearrange karo.

Recall

$sin ϕ$ hume physically kya deta hai yahan? Extra path length ::: yeh beam ki slant ko neighbouring atoms ke beech extra distance mein convert karta hai.

Recall Agar momentum

$p$ double ho jaaye, toh $λ$ ka kya hoga? Woh half ho jaayegi ::: kyunki $λ = h / p$ hai, wavelength momentum ke inversely proportional hai.

Equipment checklist

Self-test: kya tum in mein se har ek ek breath mein bata sakte ho? Check karne ke liye reveal karo.

Angle kya hota hai aur uski unit ::: do directions ke beech turning ki matra, degrees mein ( $36 0^{\circ}$ = full turn) ya radians mein ( $2 π$ = full turn).
Radian kyun matter karta hai ::: SI physics aur calculators aksar radians expect karte hain; $sin$ use karne se pehle $rad = deg \times π /180$ se convert karo.
$ϕ$ aur $θ$ mein fark, aur unka link ::: $ϕ$ incident beam se, $θ$ atomic plane se; $ϕ = 18 0^{\circ} - 2 θ$ isliye $θ = 9 0^{\circ} - ϕ /2$ .
$sin ϕ$ ki definition ::: right triangle mein opposite side over hypotenuse; yahan yeh slant ko extra path mein convert karta hai.
Wavelength $λ$ kya hai ::: wave ki crest-to-crest distance; ångström mein measure ki jaati hai, $1 \overset{˚}{A} = 1 0^{- 10}$ m.
$a$ aur $d$ mein fark ::: $a$ = surface row spacing (sideways); $d$ = interplanar spacing (layer depth).
Grating aur Bragg conditions ::: $a sin ϕ = nλ$ (surface rows) aur $2 d sin θ = nλ$ (stacked planes).
Momentum $p$ ka matlab ::: mass times velocity, "kitni motion."
de Broglie relation ::: $λ = h / p$ — momentum ek particle ki wavelength set karta hai.
KE aur $e V$ kya represent karte hain ::: KE = kinetic energy (motion ki energy, $\frac{1}{2} m v^{2}$ ); $e V$ = woh energy jo charge $e$ voltage $V$ se gain karta hai.
Poori theory-road chain ::: $e V = p^{2} /2 m \Rightarrow p = 2 m e V \Rightarrow λ = h / 2 m e V$ .
Order $n$ kya count karta hai ::: kitni whole wavelengths extra path mein fit hoti hain; $n = 1, 2, 3 \dots$