Foundations — Compton scattering — wavelength shift derivation

This page assumes you have seen none of the symbols in the parent note. We build every one from the ground up, in an order where each rests on the last. By the end you should be able to read the parent derivation without pausing on any letter.

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0. The scene we are describing

Look at the figure. One photon (the lavender arrow) flies in from the left and hits an electron (the mint dot) sitting still. After the collision two things fly out: the photon (now coral, bent by an angle $\theta$) and the electron (slate arrow, bent the other way by angle $\varphi$). Everything below is just naming the pieces of this picture.

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1. Wavelength $\lambda$ — the "length of one ripple"

The picture: imagine the wiggly line of the sea. The gap from one wave-top to the next wave-top is $\lambda$. See the top strip of the figure below — the tight ripple is small $\lambda$, the loose ripple is big $\lambda$.

Why the topic needs it: the whole result of Compton scattering is that the wavelength grows after the bounce. We write the "before" wavelength as $\lambda$ and the "after" as ${\lambda }^{\prime }$ (read "lambda-prime" — the little tick just means the new one). The quantity everyone cares about is the increase:

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2. Frequency $f$ and the light-speed link $c=f\lambda$

Light always travels at the same speed $c\approx 3\times 10^8$ m/s (the speed of light). In one second the wave advances a distance $c$; if $f$ crests go by and each is $\lambda$ long, then $c=f\lambda$. So frequency and wavelength are two ways of saying the same thing:

$f=\frac{c}{\lambda }$

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3. Planck's constant $h$ and the photon energy $E=hf$

The revolutionary idea (from the Photoelectric effect): light of frequency $f$ can only be handed over in whole packets, each carrying energy

$E=hf=\frac{hc}{\lambda }$

The last step just swapped $f$ for $c/\lambda$. So a short wavelength means a big energy — that's why X-rays, not lamplight, are used.

Why the topic needs it: conservation of energy in the parent note is written entirely with $hc/\lambda$. See Photon momentum and energy.

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4. Momentum $p$ — the "push" a moving thing carries

The picture: a rolling bowling ball has lots of momentum (heavy, moving); a drifting feather has almost none. In a collision the total arrow-sum of momentum before equals the total after (see Conservation of momentum).

For a photon there is no mass, yet it still carries a push. The right rule (justified next) is:

Notice again: short $\lambda$ → big push. A photon is a little ball with energy $hc/\lambda$ and momentum $h/\lambda$.

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5. Why $E=pc$ for a photon — the relativistic energy relation

Ordinary mechanics says $E=\frac{1}{2}m{v}^2$, but a photon has no mass and moves at $c$, so that formula is useless here. Einstein's replacement (see Relativistic energy-momentum relation) is the master equation for any object:

Read this as a right triangle (see the figure): the total energy $E$ is the long side (hypotenuse), and the two shorter sides are the "motion part" $pc$ and the "existence part" $m{c}^2$. Pythagoras stitches them together.

• Photon ($m=0$): the "existence" side vanishes, so $E=pc$, i.e. $p=E/c$. That's exactly the box in Section 4.
• Electron ($m=m_e\ne 0$): both sides survive, so $E_e=\sqrt{(p_{e}c{)}^2+(m_e{c}^2{)}^2}$. This is why the electron must be treated relativistically — its recoil can be fast, and only this triangle stays correct.

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6. The electron mass $m_e$ and its rest energy $m_e{c}^2$

Why the topic needs it: the final shift formula divides by $m_{e}c$. Using the whole atom's mass instead (a common trap) would make the shift almost vanish — that actually explains the extra "unshifted peak" Compton saw, but the moving-electron peak needs $m_e$.

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7. The angles $\theta$ and $\varphi$

The picture: in Section 0's figure, $\theta$ opens between the incoming direction and the coral (out-going photon) arrow; $\varphi$ opens below the line to the electron's slate arrow. We keep them separate because the two objects go different ways.

The reason $\varphi$ never appears in the final answer: we only measure the photon. The parent derivation squares-and-adds the two momentum equations so that ${\cos }^2\varphi +{\sin }^2\varphi =1$ makes $\varphi$ disappear.

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8. Cosine, and the special angles we will need

$\cos \theta$ (cosine) answers: how much of the outgoing arrow still points forward? At $\theta =0$ the photon didn't turn, so it fully points forward, $\cos 0=1$. Turned sideways, $\cos 90^{\circ }=0$. Turned all the way back, $\cos 180^{\circ }=-1$.

That is all the trigonometry the topic needs, and it explains the three headline cases in one glance:

angle $\theta$	$\cos \theta$	$1-\cos \theta$	meaning
$0^{\circ }$	$1$	$0$	no turn → no shift
$90^{\circ }$	$0$	$1$	sideways → shift $={\lambda }_C$
$180^{\circ }$	$-1$	$2$	backscatter → maximum shift $=2{\lambda }_C$

Every case in between lands smoothly between these, because $\cos \theta$ slides smoothly from $1$ to $-1$.

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9. Putting the vocabulary together

Now every symbol in the parent's final box is defined:

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Prerequisite map

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Equipment checklist

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Connections

• Photon momentum and energy — where $E=hf$ and $p=h/\lambda$ come from.
• Relativistic energy-momentum relation — the triangle $E^2=(pc{)}^2+(m{c}^2{)}^2$.
• de Broglie wavelength — the reverse idea: matter has a wavelength too.
• Conservation of energy and Conservation of momentum — the pool-table rules.
• Photoelectric effect — the first proof that light comes in packets.
• X-ray production and Bremsstrahlung — where the incoming X-rays are made.
• Parent: Compton scattering derivation.