Compton (1923): primer

Document studied: Arthur H. Compton, "A Quantum Theory of the Scattering of X-Rays by Light Elements", Physical Review 21(5), 483-502 (May 1923). File: Researchers/Compton/Compton_1923.pdf.

Companion script: example.py in this folder reproduces every number in Section 5. Run it with .venv/bin/python review/Compton/example.py

Source gap (repeated from earlier steps, updated after QE). The PDF is a scan. Step 0 has since produced a full OCR text of all 20 pages, which is good for prose and numbers but garbles the equations, so equations and figures were read from page images. Between them the reviewers and the arbiter read every page as an image, later deep reads examined Fig. 7 (p.18), Fig. 1B and Eq. 1 (p.4), Eq. 23 (p.10), Fig. 4 and Table I (pp.13-14) at 200-500 dpi, and I re-read pp. 486 and 493 (scan pp. 4 and 11) for this primer, so the paper's own content is well covered [read: OCR text; page image]. What was never obtained: Compton's follow-up papers (Phys. Rev. 22, 409, 1923), the 1925 Compton-Simon and Bothe-Geiger papers, the Ishino, Hewlett and Barkla-Ayers data papers, and the Klein-Nishina paper. Statements below about those rest on web summaries or on memory and are tagged each time.

Source tags: [read] = from the paper; [web] = from a web page (URL given); [memory] = from model training, with confidence; [calc] = computed by example.py.

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1. The idea in one paragraph

By 1922 everyone agreed X-rays were light of very short wavelength, and the accepted theory of how light scatters off an electron (J. J. Thomson's) said the scattered light should come out with exactly the same wavelength it went in with. Compton measured X-rays bounced off graphite and found the scattered rays had a slightly longer wavelength (in this paper he measured X-rays at one angle, 90 degrees; the growth of the shift with angle he showed only indirectly, for gamma-rays). His explanation was bold for the time: treat the X-ray as a particle (a "quantum", today a photon) that carries both energy and momentum, and let it hit a single electron like a billiard ball. The electron recoils and carries off some energy, so the scattered photon has less energy, which means a longer wavelength. Using nothing but conservation of energy and momentum from special relativity, he got a formula for the shift that depends only on the angle, not on the X-ray's starting wavelength, and it matched his measurement to about 10%. This paper is the founding statement of what is now called the Compton effect. Its core claim is textbook physics today. Its more detailed formulas for how much light scatters in each direction were reasonable guesses that a later, exact theory (Klein-Nishina, 1928-29) replaced.

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2. What you need to know first

2.1 Photons: energy and momentum of light. Einstein (1905) proposed that light of frequency nu (Greek "nu", cycles per second) comes in lumps of energy E = h nu, where h = 6.626 x 10^-34 J s is Planck's constant. Since frequency and wavelength lambda are linked by c = nu lambda (c = speed of light), this is also E = hc/lambda. Classical electromagnetism already says a light wave carrying energy E carries momentum p = E/c (radiation pressure). Combining the two gives a photon momentum p = h nu / c = h/lambda. Compton's paper is the first to use this momentum seriously in a collision calculation. Read: any intro modern-physics text, e.g. Krane, Modern Physics, or Serway, Moses and Moyer, Modern Physics (chapters on the particle properties of light).

2.2 Special-relativity energy and momentum. For a particle of rest mass m moving at speed v, write beta = v/c and gamma = 1/sqrt(1 - beta^2) (the Lorentz factor). Then momentum p = gamma m v, total energy E = gamma m c^2, kinetic energy T = (gamma - 1) m c^2, and these are linked by E^2 = (pc)^2 + (mc^2)^2. A photon has m = 0, so E = pc exactly. These are the only relativity results you need. The electron's rest energy mc^2 is 511 keV (1 keV = 1000 electron-volts; 1 eV = 1.602 x 10^-19 J). Read: Taylor, Classical Mechanics, the special-relativity chapter; or Griffiths, Introduction to Electrodynamics (4th ed.), Ch. 12.

2.3 Elastic collisions in two dimensions. In a collision, the total momentum vector is conserved (separately in x and y), and if nothing is lost to heat, total energy is conserved. With two unknowns (here: the scattered photon's energy and the electron's speed) and two equations, you can solve. The trick that makes the algebra short is to square the energy equation so that the electron's speed drops out. You did this kind of problem in first-year mechanics with billiard balls; the only new thing is the relativistic energy formula.

2.4 Radiation from an accelerated charge (Thomson scattering). A charge q with acceleration a radiates power P = q^2 a^2 / (6 pi epsilon_0 c^3) (the Larmor formula; epsilon_0 is the permittivity of free space). An electromagnetic wave shakes a free electron at the wave's own frequency, so the electron re-radiates at that same frequency. That is the classical prediction of "no wavelength change". The scattered intensity varies with angle as (1 + cos^2 theta)/2 for unpolarised light, which is symmetric front to back. Read: Griffiths, Introduction to Electrodynamics (4th ed.), Ch. 11 (Radiation).

2.5 Cross-sections and absorption coefficients. A cross-section sigma (Greek "sigma") is an effective target area: if a beam of N photons per square metre passes an electron, sigma N photons are scattered. Per unit volume of material with n electrons per cubic metre, the fraction of the beam removed per metre of path is n sigma; X-ray physicists of 1923 called this a scattering (or absorption) coefficient. A "differential" cross-section dsigma/dOmega tells you how much goes into each direction per unit solid angle Omega (steradians); integrating it over the whole sphere gives sigma.

2.6 X-ray spectroscopy by crystal diffraction. X-ray wavelengths (about 10^-10 m = 1 angstrom, written A) are measured by reflecting them off a crystal: bright reflections occur when n lambda = 2 d sin(phi), where d is the crystal's atomic-plane spacing and phi the glancing angle (Bragg's law). Compton used a calcite crystal this way to see the scattered line shift. Read: any intro modern-physics text, section on X-ray diffraction.

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3. The textbook view

3.1 Classical (Thomson) scattering. A free electron in a wave of frequency nu_0 oscillates at nu_0 and re-radiates at nu_0. The differential cross-section, averaged over polarisation, is

dsigma_T/dOmega = (r_e^2 / 2)(1 + cos^2 theta),

where theta is the scattering angle (0 = straight on, 180 deg = straight back) and r_e = e^2/(4 pi epsilon_0 m c^2) = 2.818 x 10^-15 m is the "classical electron radius" (e = electron charge, m = electron mass). Integrating over all directions gives the Thomson cross-section

sigma_T = (8 pi / 3) r_e^2 = 6.652 x 10^-29 m^2.

Compton writes the same thing in the older Gaussian units as sigma_0 = (8 pi/3) N e^4/(m^2 c^4) per unit volume, with N electrons per unit volume [read: page image p.493]. Predictions: no wavelength change; forward and backward intensities equal; scattering independent of wavelength.

3.2 The Compton formula (textbook today). Treat the photon as a particle with energy h nu_0 and momentum h nu_0/c hitting a free electron at rest. After the collision the photon goes off at angle theta with frequency nu_theta, the electron recoils. Conservation of energy and momentum gives

lambda_theta - lambda_0 = (h / m c)(1 - cos theta) = (2h / mc) sin^2(theta/2).

The constant h/(mc) = 0.024263 A is called the Compton wavelength of the electron [calc; standard value]. Key features: the shift is independent of the incident wavelength lambda_0, is zero straight ahead, and is largest (2h/mc = 0.0485 A) straight back. It only matters when lambda_0 is not much bigger than 0.024 A, i.e. for X-rays and gamma-rays; for visible light (5000 A) the fractional shift is about 10^-5.

Compton states the same result in frequency form (his Eq. 3) [read: page image p.486]:

nu_theta = nu_0 / (1 + 2 alpha sin^2(theta/2)), alpha = h / (m c lambda_0) = h nu_0 / (m c^2).

The dimensionless number alpha (not the fine-structure constant) is the photon energy divided by the electron rest energy: small for soft X-rays, about 1 for gamma-rays.

3.3 How much scatters, exactly (Klein-Nishina). The correct quantum cross-section for a free electron came in 1928-29 from Klein and Nishina using Dirac's relativistic electron theory [web: https://en.wikipedia.org/wiki/Klein%E2%80%93Nishina_formula]. With P = nu_theta/nu_0 = 1/(1 + alpha(1 - cos theta)):

dsigma_KN/dOmega = (r_e^2 / 2) P^2 (P + 1/P - sin^2 theta).

At small alpha, P is close to 1 and this reduces to Thomson. At large alpha it falls and becomes strongly forward-peaked. You do not need to know how it is derived (that requires quantum field theory) to use it as a benchmark, which is what this primer does.

3.4 Bound electrons. The formula assumes a free electron at rest. Real electrons are bound in atoms and moving. If the recoil energy is much bigger than the binding energy, the electron acts as free and you get the shifted line (broadened a little by the electron's motion). If not, the photon scatters off the whole atom, which is so heavy that the shift is negligible: an unshifted ("coherent" or Rayleigh) line. Real spectra show both [web: https://en.wikipedia.org/wiki/Compton_scattering].

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4. Where this researcher departs

Compton is unusual among the projects in this collection: he departed from the textbook of his day, and the textbook moved to him. So the side-by-side is between 1922 physics and Compton, with modern physics as the referee.

The exact point of departure is one sentence on p.486: the momentum of the incident ray "will be h nu_0/c", and the recoil momentum of the electron must equal "the vector difference between the momenta of these two rays" [read: page image p.486]. Everything else in the first half of the paper is careful algebra from that premise. Compton's Eqs. 1 and 2 are momentum and energy conservation:

(m beta c / sqrt(1 - beta^2))^2 = (h nu_0/c)^2 + (h nu_theta/c)^2 + 2 (h nu_0/c)(h nu_theta/c) cos theta (Eq. 1 as printed) h nu_theta = h nu_0 - m c^2 (1/sqrt(1 - beta^2) - 1) (Eq. 2)

A detail worth noticing as a learner: if theta is the scattering angle, the law of cosines for a vector difference needs a minus sign in Eq. 1, and only the minus sign leads to his Eq. 3. The plus sign is clearly printed [read: page image p.4]. The arbiter suggested Fig. 1B might mark the triangle's other angle (180 deg minus theta), which would make the plus sign right. A later high-resolution read settled this: in Fig. 1B the arc marked theta sits inside the momentum triangle, at the corner between the incident momentum h nu_0/c and the scattered momentum h nu_theta/c, so it is the scattering angle itself, not 180 deg minus theta, and Fig. 1A marks the same angle [read: page image p.4, deep read; arbiter QE revision, Round 2]. The law of cosines for that angle gives a minus sign, so the printed plus is a real print error. The arbiter records it as a typo in the paper, not a fault in Compton's physics: his Eq. 3 is the minus-sign answer, and everything after it is unaffected. (A second printed slip of the same kind: the last line of Eq. 23 on p.10 has R where R^2 is needed for the units; the line above it has R^2 and the two agree once beta = alpha/(1+alpha) is put in, so nothing depends on it [read: page image p.10; arbiter Round 2].) Do the derivation yourself (Question 1 below) and you will see why.

The second, weaker departure is in the intensity calculation (pp. 488-493). To get how much light goes into each direction, Compton could not use quantum theory (it did not exist yet in usable form). He assumed that scattering from a recoiling electron looks, in intensity, like classical scattering from an electron moving forward at speed beta = alpha/(1+alpha) c, and transformed the classical pattern with the relativistic Doppler effect. He says frankly, "I have not, however, succeeded in showing rigidly" that this is valid [read: page image p.491]. He also fixes the forward intensity at the classical value (Eq. 26) by a second assumption. The resulting angular law is his Eq. 27 [read: page image p.493]:

I_theta = I (N e^4 / (2 R^2 m^2 c^4)) [1 + cos^2 theta + 2 alpha (1+alpha)(1 - cos theta)^2] / [1 + alpha (1 - cos theta)]^5,

where I is the primary beam intensity, N the number of electrons doing the scattering, and R the distance to the detector. Set alpha = 0 and you get Thomson back, which is the right low-energy limit. This is the part the later exact theory replaced.

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5. Worked example: Compton's own measurement, re-derived

All numbers below come from example.py [calc], using CODATA 2018 constants. Compton's printed values are given alongside.

Step 1. Derive the shift. Put the incident photon along x with momentum p_0. After scattering, the photon has momentum p_1 at angle theta; the electron has momentum vector p_e = (p_0 - p_1 cos theta, -p_1 sin theta), so

p_e^2 = p_0^2 + p_1^2 - 2 p_0 p_1 cos theta (note the minus sign).

Energy conservation: p_0 c + m c^2 = p_1 c + sqrt((m c^2)^2 + p_e^2 c^2). Move p_1 c to the left and square. The (m c^2)^2 and the p_0^2, p_1^2 terms cancel, leaving

m c (p_0 - p_1) = p_0 p_1 (1 - cos theta).

Divide both sides by m c p_0 p_1 and use lambda = h/p:

lambda_1 - lambda_0 = (h/mc)(1 - cos theta).

SymPy does this in example.py and prints h*(1 - cos(theta))/(c*m). Using 1 - cos theta = 2 sin^2(theta/2) gives Compton's form.

Step 2. Put in the constant. h/(mc) = (6.626 x 10^-34)/(9.109 x 10^-31 x 2.998 x 10^8) = 2.4263 x 10^-12 m = 0.024263 A. So 2h/mc = 0.04853 A. Compton printed 0.0484 A, using 1922 values of h and e/m; the 0.3% difference is in the constants, not the physics.

Step 3. Compton's experiment. Molybdenum K-alpha X-rays (lambda_0 = 0.708 A), scattered by graphite at theta = 90 deg, analysed with a calcite crystal spectrometer [read: page image p.495 per reviewers; Fig. 4].

So the prediction and measurement differ by about 9% of the shift, or about 0.3% of the wavelength. The paper gives no error bar [arbiter]. Fig. 4 is worth looking at closely: the scattered curve is one broadened peak shifted a few tenths of a degree from the primary peak, with heights normalised and no counts or error bars. No separate unshifted peak at 0.708 A is resolved, but since the shift (about 0.2 deg of crystal angle) is about the width of the peak, a small unshifted component would be hidden in the overlap; so the right wording is "not resolved in Fig. 4", not "absent" [read: page image p.13, deep read]. As evidence this is a single angle with one wavelength and modest resolution, but the sign and size of the effect are unmistakable: the classical prediction is a shift of exactly zero.

Step 4. What happens to the electron? Energy conservation gives its kinetic energy directly:

T = E_0 - E_1 = 17.51 - 16.93 keV = 0.580 keV = 580 eV.

Lorentz factor gamma = 1 + T/(mc^2) = 1 + 0.580/511 = 1.00113, so beta = sqrt(1 - 1/gamma^2) = 0.0476: the electron leaves at about 14,000 km/s, roughly 5% of light speed. Check momentum: the vector difference of the photon momenta, |p_0 - p_1| computed from (17.51, 0) and (0, 16.93) keV/c, is 24.36 keV/c; the relativistic momentum from T, pc = sqrt(T^2 + 2 T mc^2), is also 24.36 keV/c. The two agree, so both conservation laws hold. Its direction, from the standard formula cot(phi) = (1 + alpha) tan(theta/2), is phi = 44.0 deg from the beam, on the opposite side to the scattered photon. This is the recoil electron Compton predicted but, he admits, had not seen for ordinary X-rays in 1923 [arbiter, read]. Wilson cloud-chamber tracks of such recoil electrons, and the Compton-Simon and Bothe-Geiger coincidence experiments of 1925, later showed photon and electron emerging together in single events [web: https://en.wikipedia.org/wiki/Bothe%E2%80%93Geiger_coincidence_experiment].

Step 5. Units-and-size check on the "free electron" assumption (Claim 6). Is 580 eV big compared with how tightly carbon holds its electrons? Carbon's outer electrons are bound by about 11 eV (first ionisation energy 11.26 eV), its inner K-shell electrons by about 284 eV [memory: high]. So T is about 50 times the outer binding and about twice the K-shell binding. For the outer electrons the free-electron picture is excellent; for the two K-shell electrons it is marginal. This is exactly why Compton restricted his theory to light elements: in lead, K-shell binding is around 88 keV [memory: high], far more than 17.5 keV X-rays can supply, and those electrons scatter coherently with the whole atom (no shift).

Step 6. Gamma-ray check (Table I). For Compton's "primary" gamma wavelength 0.022 A at 135 deg: 0.022 + 0.04853 x sin^2(67.5 deg) = 0.022 + 0.04853 x 0.8536 = 0.0634 A. Compton's "observed" value, inferred from absorption in lead, was 0.068 A, about 7% high [read: OCR text p.14, QE-checked]. His full Table I (observed / calculated) is 0.030/0.029 at 45 deg, 0.043/0.047 at 90 deg, 0.068/0.063 at 135 deg, now confirmed from the page image (the OCR of that table is garbled) [read: page image p.14]. Observed minus calculated is +0.001, -0.004, +0.005 A: the sign changes, so there is no systematic trend. Evaluating the formula directly gives 0.0291, 0.0462, 0.0633 A, so the printed 0.047 should round to 0.046, a trivial 0.001 A difference [arbiter Round 2, deep reader's arithmetic]. Here alpha = 0.024263/0.022 = 1.10, so the shift is larger than the wavelength itself; this is the regime where the photon behaves most obviously like a particle. But note the weakness: the "observed" wavelengths are not measured directly. They follow from the photoelectric absorption coefficient tau/rho through an assumed lambda^3 law, scaled to the primary: lambda = 0.022 x ((tau/rho)/0.017)^(1/3), which reproduces .0298, .0425, .0680 A, and the 0.022 A starting value came from the same extrapolation, so this is not an independent measurement [read: page image p.14; arbiter Round 2]. The gamma beam was also a mixture of lines (Ellis), not a single wavelength. In fairness, Compton gave two partly independent cross-checks on 0.022 A: his own earlier value of 0.025 A from small-angle scattering of gamma-rays by lead (Phil. Mag. 41, 777, 1921), and Ellis's beta-ray line spectra (0.045, 0.025, 0.021, 0.020 A, with 0.020 strongest) [read: OCR text p.14]. The self-calibration criticism is softened, not removed.

Step 7. How much scatters, and where Compton's 1923 formulas go wrong. The table compares, as fractions of the classical Thomson value, Compton's Eq. 28 (energy removed from the beam), his Eq. 29 (energy truly re-radiated), and the same quantities from the exact Klein-Nishina formula, integrated numerically over all angles [calc]:

Three lessons from this table:

1. Both theories agree that scattering falls well below Thomson at short wavelength. Classical theory is plainly wrong for gamma-rays.
2. At soft X-ray energies Compton's formulas agree with the exact answer to about 0.1%; they are right to first order in alpha.
3. At gamma-ray energy (alpha about 1) Compton's numbers are about 25% low. Yet in the paper they "agree" with Ishino's 1917 gamma-ray data (0.040 calculated vs 0.045 observed for aluminium) [read]. The contrarian reviewer found that modern NIST tables for aluminium near this energy match Klein-Nishina, not Compton and Ishino [web: https://physics.nist.gov/PhysRefData/XrayMassCoef/ElemTab/z13.html]. So a formula that was too low agreed with data that were also too low. This is a classic way a wrong-in-detail theory gets apparent support.

The script also checks that integrating Compton's angular law (Eq. 27) over the sphere reproduces his Eq. 29 exactly (0.9058, 0.7453, 0.2046), confirming his algebra from Eq. 27 to Eq. 29 is internally consistent; the error lies in the assumption behind Eq. 27, not in his calculus.

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6. Evidence table

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7. What a learner should take away

What is solid. Light carries momentum h/lambda, and in a collision with an electron energy and momentum are conserved event by event. From that alone, with no adjustable constant, you get a shift (h/mc)(1 - cos theta) that does not depend on the incoming wavelength. You can derive it yourself in ten lines (Section 5, Step 1). This is one of the cleanest pieces of evidence for the photon, and it earned Compton a share of the 1927 Nobel Prize [memory: high; also stated on the Wikipedia Compton scattering page].

What was wrong, and why it is instructive. Compton's intensity formulas (Eqs. 27-30) rest on a guess he openly labelled unproven: that a recoiling electron radiates like a classical electron moving forward. The guess has the right low-energy limit and the right qualitative trend (less scattering, more forward), so it agrees with the later exact theory (Klein-Nishina) to a fraction of a percent at small alpha. The 1923 soft X-ray data themselves were only roughly matched: Hewlett's "perhaps within the experimental error" (p.16), while Barkla and Ayers's forward/back ratio (1.04) differs from his prediction "by more than their probable experimental error" (p.17) [read: OCR text pp.16-17]. At gamma energies it is about 25% low and badly wrong in angular shape. Two lessons: (1) a model can be right to first order and still be wrong; agreement in the easy regime is weak evidence. (2) Agreement with data is only as good as the data: his gamma-ray "agreement" partly came from a low formula meeting low measurements, and his gamma wavelengths were inferred by a method that also supplied his starting value. Also note the overstatement: he called his Fig. 7 comparison (gamma-rays scattered by iron, measured with an ionization chamber) "beautiful agreement" with "not a single adjustable constant" [read: p.19]. A high-resolution read of the figure shows that the eight plotted points, which have no error bars, all sit well below the classical curve (about 12 times lower at 90 deg, about 35 times at 150 deg), so the classical law is decisively rejected. But against his own Eq. 27 they sit on or slightly above the curve only near 44-60 deg and systematically 20-45% below it from 75 to 150 deg; at 90 deg the text's own number is 0.037 I_0 against the curve's about 0.07 [read: page image p.18, deep read; arbiter QE revision]. "Beautiful" is fair for the shape (steep fall, then a low plateau) and overstated for the numbers. He also admits the ionization reading measures intensity only on an unverified assumption ("It is not clear just how the ionization ... will vary with the wave-length", p.19).

A researcher who reversed himself. In a footnote (p.3) Compton says that in 1921 he had "defended the view" that the softening of scattered X-rays came from added fluorescent radiation, and that J. A. Gray (1913, 1920) and Florance (1914), who had argued for true scattering, were right: "the latter view is the correct one" [read: OCR text p.3]. In the Discussion he also drops his own earlier "large electron" hypothesis as "superfluous" (p.19). So two earlier workers were partly right before him, and the decisive step came from a researcher willing to follow his data against his own published theory. He is equally candid that the recoil electrons his theory requires had not yet been observed for ordinary X-rays (p.14) [read]. (Peter Debye derived the same shift independently in 1923; that is not in this paper [memory: medium].)

Why this project matters for the rest of the collection. Compton is the model case of a heterodox claim that won. It contradicted the consensus wave picture, was contested (by Duane experimentally, and by Bohr, Kramers and Slater theoretically [memory: medium for Duane's details]), and was settled within about two years by independent, controlled experiments designed to tell the alternatives apart. That is the standard to hold other claims in this collection to: a precise prediction with no free parameters, published with the data, and a decisive test by others.

What is genuinely open. Nothing about the physics. The open items are historical: what exactly Barkla and Ayers measured and why they got 1.04; how good Ishino's 1917 data were; the details of Duane's challenge; whether the Eq. 1 sign and Eq. 23 R slips (both now confirmed as printed errors, not misreadings of the scan) were Compton's or the typesetter's; and why the drawn Fig. 7 curve sits above the printed Eq. 27 at large angles (deep-read observation, cause unidentified). A clean copy of the journal and the cited data papers would settle these. Two questions earlier drafts left open are now closed [arbiter Round 2]: the plus sign in Eq. 1 is a real print error, and Fig. 4 does not resolve an unshifted peak at 0.708 A (nor can it rule out a small one). None of the arbiter's verdicts changed in that round.

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8. Check yourself

1. Starting from p_e^2 = p_0^2 + p_1^2 - 2 p_0 p_1 cos theta and energy conservation, derive lambda_1 - lambda_0 = (h/mc)(1 - cos theta). Then explain what Compton's printed plus sign in Eq. 1 would require the angle to be.
2. Visible green light has lambda = 5000 A. What is the largest possible Compton shift, and what fraction of the wavelength is it? Why did nobody notice the effect with visible light?
3. A 0.022 A gamma ray scatters straight back (theta = 180 deg) from a free electron. Find the scattered wavelength and the recoil electron's kinetic energy in keV.
4. Compton's formula for energy removed, sigma_0/(1+2 alpha), and the exact Klein-Nishina result agree at 0.708 A but differ by 25% at 0.022 A. Why does agreement with soft X-ray data say little about whether a theory is right at gamma-ray energies?
5. Why does Compton's theory apply to graphite but not to the K-shell electrons of lead, for 17.5 keV X-rays?

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9. Further reading

The researcher's own work - A. H. Compton, "A Quantum Theory of the Scattering of X-Rays by Light Elements", Phys. Rev. 21, 483 (1923): the paper reviewed here. - A. H. Compton, "The Spectrum of Scattered X-Rays", Phys. Rev. 22, 409 (1923): the promised experimental paper [memory: medium; not obtained]. - A. H. Compton and A. W. Simon, cloud-chamber test of recoil electrons, Phys. Rev. 26 (1925) [memory: medium; not obtained]. - Compton's Nobel lecture, 1927, "X-rays as a branch of optics" [memory: medium], at nobelprize.org.

Mainstream sources - Griffiths, Introduction to Electrodynamics (4th ed.): Ch. 11 (radiation, Larmor formula), Ch. 12 (relativistic energy and momentum). - Krane, Modern Physics, or Eisberg and Resnick, Quantum Physics of Atoms, Molecules, Solids, Nuclei, and Particles: the photon chapter covers the Compton effect with worked problems. - Wikipedia, "Compton scattering" (shifted and unshifted lines, history): https://en.wikipedia.org/wiki/Compton_scattering - Wikipedia, "Klein-Nishina formula": https://en.wikipedia.org/wiki/Klein%E2%80%93Nishina_formula - Wikipedia, "Bothe-Geiger coincidence experiment": https://en.wikipedia.org/wiki/Bothe%E2%80%93Geiger_coincidence_experiment - NIST X-ray mass attenuation coefficients (to compare theory with modern data yourself): https://physics.nist.gov/PhysRefData/XrayMassCoef/ElemTab/z13.html - Roger H. Stuewer, The Compton Effect: Turning Point in Physics (1975), the standard history [memory: high].

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Answers

1. Squaring p_0 c + mc^2 - p_1 c = sqrt((mc^2)^2 + p_e^2 c^2) gives (p_0 - p_1)^2 c^2 + 2 (p_0 - p_1) m c^3 + m^2 c^4 = m^2 c^4 + (p_0^2 + p_1^2 - 2 p_0 p_1 cos theta) c^2. The m^2 c^4 cancels; (p_0 - p_1)^2 = p_0^2 + p_1^2 - 2 p_0 p_1, so what is left is 2 (p_0 - p_1) m c = 2 p_0 p_1 (1 - cos theta). Divide by m c p_0 p_1: 1/p_1 - 1/p_0 = (1 - cos theta)/(mc). Multiply by h. The plus sign: the law of cosines gives a plus only if the angle used is 180 deg minus the scattering angle, since cos(180 - theta) = -cos theta. With theta as the scattering angle (which is how the text defines it, and how Figs. 1A and 1B mark it, confirmed from a high-resolution page image), the sign must be minus, so the printed plus is a print error. It does not matter for the result: Compton's Eq. 3 is the minus-sign answer.

2. Maximum shift is 2h/mc = 0.0485 A at theta = 180 deg. Fraction = 0.0485/5000 = 9.7 x 10^-6, about one part in 100,000. That is far below what an optical spectrometer of 1923 could see against the natural width of spectral lines, and in any case visible light scatters mostly from bound electrons as whole atoms.

3. lambda_1 = 0.022 + 2 x 0.024263 = 0.0705 A. E_0 = 12.398/0.022 = 563.6 keV; E_1 = 12.398/0.0705 = 175.8 keV; T = 387.8 keV. The electron takes about 69% of the photon's energy. (Here alpha = 1.10, which is why the effect is so large.)

4. Every reasonable theory must reduce to Thomson when alpha goes to 0, and any two such theories agree to zeroth order and often to first order in alpha. At 0.708 A, alpha = 0.034, so the theories can only differ at order alpha^2, about 0.1%, which is below the precision of 1920s data. Only at alpha near 1 do the higher-order terms differ enough to tell the theories apart.

5. The recoil kinetic energy for 17.5 keV X-rays at 90 deg is about 0.58 keV. Graphite's outer electrons are bound by about 11 eV, so they act as free. Lead's K-shell electrons are bound by about 88 keV, far more than the whole photon energy; the photon cannot knock them loose by scattering, so it scatters off the whole atom, whose large mass makes the shift negligible (the unshifted line).