X Ray Spectra Electrons bombarding a high-Z target Incident electron KE: Ko = eVo Wavelength distribution of X-Rays ”Lines”: knock atomic e’s out of metal anode (see spectral lines in a few slides)Smooth continuum: Deceleration of incident e’s in anode radiate X rays Distribution of energies: bremsstrahlung Xrays: E = hν = hc/λ F o sλ min = hc/Ko or Ko = 1240/ λ min (eV nm) r a given incident KE, largest los = smallest λ Example: cutoff wavelength of .035 nm corresponds to acceleration voltage V of Ko = eV = 1240 eV nm / .035 nm, so V = 3550 Volts . Note: hc is very flexible: coordinated change of units in energy and wavelength hc = 1240 eV nm nano = 1240 keV pm pico = 1240 MeV fm femtoCompton Effect (c=1) E cons: p + m = p’ + √(q2 + m2) (1) Px cons: p = p’ cosθ + q cosφ (2) Py cons: 0 = p’ sinθ + q sinφ (3) Eliminate φ: add squares of (2), (3) (p-p’ cosθ)2 + (p’ sinθ)2 = q2 = p2 + p’2 – 2pp’ cosθ Or (Δp = p – p’) q2 = (Δp)2 + 2pp’ (1-cosθ) (4) Eliminate q with square of (1): (Δp)2 + 2 m Δp + m2 = q2 + m2 ; combine with (4) 2 m Δp = 2pp’ (1-cosθ) (5) 1/p’ – 1/p = (1-cosθ)/m (6) Restore c’s: 1/p’c – 1/pc = (1-cosθ)/mc2 Use pc = hc/λ (only NOW use λ at all!) 1/ λ’ – 1/ λ = (h/mc) (1-cosθ) 4 vector rules (“not on the exam”—but slick!) To denote 4-vector I use P (caps); personally, I use p (script) P = (e,p) 2nd slot a vector (underscore or arrow); p is vector or p magnitude depending on context P.P’ = EE’ – (p.p’) mixed-sign dot product (invariant!) P2 = e2 – p2 = M2 special case of dot product compare: (Δct)2 – (Δx)2 Compton Effect with 4‐vector conservation: P=(p,p) and m=(m,0) to P’, Q P + (m,0) = P’ + Q so, using ΔP = P – P’ Q = ΔP + m (1) Squaring: Q2 = m2 = ΔP2 + 2m.ΔP+ m2 For a photon, P = (p,p) and P2 = 0 0 = 0 + 0 -2p.p’ + 2mΔp which is again (5) COM frame M2 for fixed target: (e,p) and (m,0) M2 = (P+m)2 =P2 + m2 + 2P.m = 2m2 + 2em = 2m(e+m) M = √2m(k+2m) ~ √2mk COM frame M2 for collider: (e,p) and (e,‐p) M2(cm) =(P+P’)2 = 2 m2 + 2P.P’ = 2m2 + 2( e2 –p.(-p) ) = 4e2 M = 2e = 2(k+m) ~
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