Particle pool

1 Particle pool Adolf Cusmariu [email protected] In the celebrated Compton Effect, a photon bouncing at angle q off a stationary electron undergoes a wavelength increase unexpected from classical electrodynamics, the impact also recoiling the electron at angle f ; see graphic below (photon in red, electron in blue). Compton first reported this effect in 1922 [1] - after several years of careful experiments - and only provided a spectral image as evidence (below), stating in a footnote ‘It is hoped to publish in the near future a full account of the experiments in the X-ray region’. This he did and much more, shortly thereafter [2]; the positive change in wavelength Dl was expressed mathematically as Dl = l(q) - l 0 = (h/mec)(2sin2( q/2 )) where q = Photon bounce angle h = Planck’s constant c = Speed of Light me = Electron rest-mass q f l(q) l0 2 Compton with his famous formula; ‘vers’ means ‘1 - cos’; the angle notation is reversed. To recover the usual form for the Effect, from trigonometry cos(q) = cos(q/2 + q/2) = cos2(q/2) - sin2(q/2) hence 1 - cos(q) = 1 - cos2(q/2) + sin2(q/2) = 2sin2(q/2) which yields the justly famous expression Compton used Molybdenum K a x-rays with incoming wavelength l0 = 0.0709 nm ≈ 17KeV. Now, three Effects compete for attention during X-ray photon/electron interaction - in energy as a function of emitter Atomic Number: The Photoelectric, Compton, and Electron/Positron Pair production; an on-line source displays their regional dominance (see below). Mo(Z) = 42 Compton (17 KeV) Dl = l(q) - l 0 = (h/mec)(1 - cos( q )) = l c(1 - cos( q )) l c = 0.002426 nm 3 The energy region for the Effect overlaps the Photoelectric Effect region at Molybdenum’s Atomic Number of 42; small wonder it took Compton years of careful experiments to detect (only) his Effect using the chosen energy level. C.T.R. Wilson and his Cloud Chamber subsequently discovered the recoiled electrons (see below), an equally significant and predicted phenomenon, strongly emphasized by Compton. Compton and Wilson shared the 1927 Nobel Prize, nominated by Einstein himself, among others. Einstein and Compton during an event at the University of Chicago in 1940. Compton’s experimental set-up and numerical measurements - from an on-line source - are shown below. 4 Set-up and spectral results for 0o, 45o, 90o, 135o Thus l0 = 0.0709 nm = 17.487 KeV l (0o) = l0 l (45o) = 0.0715 nm = 17.340 KeV l (90o) = 0.0731 nm = 16.961 KeV l (135o) = 0.0749 nm = 16.553 KeV A rewrite of Compton’s formula yields (in principle) the rest-mass of the electron: me = (h/c)(1 - cos( q ))/( l(q) - l0 ) an astonishing result on the face of it: electron rest-mass estimation from photon recoil angles! Set A( q ) = (1 - cos( q ))/( l(q) - l0 ) = 1/ l c = 412.201 A( q ) should be a constant independent of the photon recoil angle q and the wavelength l(q ); but how really so, for the measured (angle, wavelength) sets above?; we get A(45o) = 488.155 A(90o) = 454.545 A(135o) = 426.776 5 Not a particularly impressive agreement; why the difference? Either the (single?) recoiled electron wasn’t really stationary - contrary to the assumptions - or measured photon recoiling wavelengths, or angles, (or both), were a bit off as reported. The incoming x-ray wavelength of 0.0709 nm will be assumed accurate. Still, the pairing (130o, 0.0749 nm) ⟹ A(130o) = 410.696 is very close indeed to the expected value of 412.201; an angular error of 5o is perhaps acceptable. Similarly, the pairing (85o, 0.0731 nm) ⟹ A(85o) = 414.929 is again close to the ideal, showing a numerically similar angular error. Finally, the pairing (41o, 0.0715 nm) ⟹ A(41o) = 408.817 also fits. More exact pairings were estimated as: (41.177, 0.0715 nm) ⟹ A(41.177o) = 412.201 (84.6547o, 0.0731 nm) ⟹ A(84.6547o) = 412.201 (130.415o, 0.0749 nm) ⟹ A(130.415o) = 412.201 What if instead the recoil wavelength l(q) measurements were a bit off? The pairings below work now. (45o, 0.07161056 nm) ⟹ A = 412.201 (90o, 0.073326 nm) ⟹ A = 412.210 (135o, 0.07504144 nm) ⟹ A = 412.201 By contrast, only (laboriously) minute changes in the scattering wavelengths - while keeping the angular measurements as reported - brought agreement; perhaps the Bragg spectrometer was a bit off. The bounce angle f of the recoiling electron satisfies [2] cot( f ) = (1 + l c/ l0 )tan( q /2) Thus f = arccot{(1 + l c/ l0 )tan( q /2)} Note the recoiling electron’s explicit angular independence of the recoiling photon’s wavelength l(q) ; after all, the electron is a true particle in this setting. Angularly, though, the two particles do seem to ‘know’ each other! In fact, some algebra will wrap it all up neatly in one package; first, from the above expression cot( f )/tan( q /2) = (1 + l c/ l0 ) while from Compton’s formula l(q) / l0 = 1 + ( l c/ l0 )(1 - cos( q )) = 1 + ( l c/ l0 ) - ( l c/ l0 )cos( q ) 6 hence l(q) / l0 + ( l c/ l0 )cos( q ) = 1 + ( l c/ l0 ) Altogether then To verify experimentally this particular form of the Effect, angular data on the recoiling electron would also have been needed! The ‘corrected’ photon recoil angles of {41.177o, 84.6547o, 130.415o} would correspond to electron recoils of {68.0861o, 45.7141o, 23.3317o}. For 0 < q < p /2 these two recoil angles are plotted below; (photon in red, electron in blue) They actually meet around 600 (58.86080 more precisely) where the photon wavelength is ≈#0.07207 nm. Compton playing banjo for students at Washington University in St. Louis, 1949. References [1] A. H. Compton, ‘Secondary radiations produced by X-rays’, Bulletin of the National Research Council, Vol. 4, Part 2, Number 20, October, 1922, P. 16. [2] A. H. Compton, ‘A Quantum Theory of the Scattering of X-Rays by Light Elements’, Phys. Rev, 21, 1923. [3] Wikipedia article on the Compton Effect. l(f,q) = lo cot( f )/tan( q /2) - l ccos( q )