DECLARE SUB PICTURE1 () DECLARE SUB PICTURE2 () DECLARE SUB PAGER () 20 PRINT CHR$(12) REM ge copeland REM Dept of Physics REM Old Dominion University REM 2/2/2001 revision for web REM Pi = 4! * ATN(1) 30 DIM THOM(200), KN(200), THETA(200) 50 PRINT "The Compton effect." 60 PRINT 70 PRINT " A. H. Compton first discovered the interaction of a photon" 80 PRINT "with a free electron. We assume the energy of the photon" 90 PRINT "Ep = hf before the interaction. After the scattering takes" 100 PRINT "place the photon energy Ep' = hf'. The photon path is dev-" 110 PRINT "iated by the angle phi and the electron leaves the event" 120 PRINT "with angle theta. The electron kinetic energy after the" 130 PRINT "collsion is E. We use the RELATIVISTIC LAWS of conservation" 140 PRINT "of momentum and total energy." 150 PRINT 160 PRINT "The diagram looks like this:" 180 CALL PAGER 190 CALL PICTURE1 200 CALL PAGER 210 PRINT "Momentum balance yields:" 220 PRINT " " 230 PRINT " 2 1/2 " 240 PRINT " hf/c = (hf'/c) cos(phi) -m b c(1-b ) cos(theta)" 250 PRINT " o " 260 PRINT " 2 1/2" 270 PRINT " (hf'/c)sin(phi)=m b c(1-b ) sin( theta)" 280 PRINT " o" 300 PRINT "Energy balance yields:" 310 PRINT " " 320 PRINT " 2 2 -1/2 " 330 PRINT " hf = hf' +m c [ (1-b ) -1]" 340 PRINT " o " 350 PRINT "where m = mo electron rest mass, b = v/c of electron." 351 PRINT " o " 352 PRINT "All 3 particles( Why 3?) lie in a plane. Why?" 353 PRINT "If we look at the wavelengths L and L', we get" 354 PRINT " L'- L = (h/mo c)(1-cos(phi))" 360 CALL PAGER 370 PRINT " First, let's look at the change in wavelength." 384 PRINT " L'- L = (h/mo c)(1-cos(phi))" 400 PRINT " Note that the wavelength change is INDEPENDENT of the " 410 PRINT "wavelength. It is instructive to look at the fractional" 420 PRINT "loss of energy of the incoming photons vs their energy." 430 PRINT "That is, (Ep-Ep')/Ep. This can be found from L'-L." 460 PRINT " Ep " 470 PRINT " Ep' = -------------------------" 490 PRINT " 1 + [Ep/mc^2][1-cos(phi)" 500 PRINT " o" 530 CALL PAGER 540 PRINT "Since the rest energy of an electron is 0.511 Mev, we will" 550 PRINT "calculate this function in those unit for phi=90 degrees." 560 PRINT 570 PRINT " Ep", "Ep'", "(Ep-Ep')/Ep" 580 PRINT "Mev", "Mev", " %" 582 FOR I = 1 TO 5 584 READ E(I) 588 DATA .01,.10,1.,10.,40 590 EA = E(I) / (1 + (E(I) / .511)) 592 PRINT E(I), EA, (E(I) - EA) * 100 / E(I) 594 NEXT I 596 PRINT "NOTICE: the fractional loss is quite large for "; 597 PRINT "ENERGETIC PHOTONS." 620 PRINT "Recall the equation for the CHANGE in wavelength." 621 PRINT " L'- L = (h/mo c)(1-cos(phi))" 640 PRINT "The term (h/mc) has units of wavelength and is called the" 650 PRINT "COMPTON wavelength of an electron. Since hc=1.24x10^4 eV-A" 660 PRINT "and mc^2=0.511MeV, the COMPTON WAVELENGTH Lc = 0.024 A." 670 PRINT "This is a small wavelength change if the incoming photon " 680 PRINT "happens to be in the visible (4000-7000A) and thus is not" 690 PRINT "noticeable. But the effect is very large for HIGH energy" 700 PRINT "photons such as the gammas emitted by nuclear transitions." 760 CALL PAGER 780 PRINT "The differential cross section for Compton scattering was" 790 PRINT "first calculated quantum mechanically by Klein and Nishina." 800 PRINT "Classically, the cross section for this process is given by" 810 PRINT "THOMPSON. The THOMPSON differential cross section will be " 820 PRINT "called d sigma/d omega. It is given by:" 830 PRINT " " 840 PRINT " _ 2 _ " 850 PRINT " d sigma 2 : 1 + cos( 0 ): 2 " 860 PRINT " ------ = r : ------------- : cm per steradian." 870 PRINT " d omega o :_ 2 _: " 880 PRINT " " 900 PRINT "Integrated over all angles, this yields the THOMPSON CROSS" 910 PRINT "SECTION 2 " 920 PRINT " sigma = (8/3) pi r " 930 PRINT " o " 940 PRINT "where" 950 PRINT " ro is the 'classical electron radius' = 2.82x10^-13 cm." 951 PRINT " The geometry is shown in this diagram." 970 CALL PAGER 972 CALL PICTURE2 973 CALL PAGER 980 PRINT " This simple cross section derived from classical argu-" 990 PRINT "ments has several defects:" 1000 PRINT "1. It does NOT depend upon frequency" 1010 PRINT "2. The electron is assumed to NOT recoil" 1020 PRINT "3. The treatment is nonrelativistic" 1030 PRINT "4. Quantum effects are not treated. " 1040 PRINT 1050 PRINT "The proper treatment was done by Klein and Nishina. Let" 1060 PRINT "g = hf/mc^2 and we have" 1080 PRINT 1090 PRINT " d sigma 2 1 + cos( 0) 1 " 1100 PRINT " ------- = r ---------- ---------------------- X " 1110 PRINT " d omega o 2 [ 1 + g(1-cos(0) )]^2 " 1120 PRINT 1130 PRINT " _ 2 2 _ " 1140 PRINT " : g (1-cos( 0) ) :" 1141 PRINT " : 1 + ------------------------------------- :" 1142 PRINT " : (1-cos^2( 0 )) [1 + g (1- cos( 0 ) )] :" 1143 PRINT " - _ " 1144 PRINT "UNITS CM2/STERADIAN" 1150 CALL PAGER 1160 PRINT " Both the THOMPSON and KLEIN-NISHINA differential cross" 1170 PRINT "sections will now be evaluated for a fixed g and varying " 1180 PRINT "angle 0. Recall that the Thompson d sigma/d omega is NOT" 1190 PRINT "a function of g = hf/mc^2. We will use g = 1.29." 1192 G = 1.29 1200 R0 = 2.8E-13 1202 R2 = R0 * R0 1210 NPTS = 0 1220 FOR ANG = 0 TO 180 STEP 5 1230 NPTS = NPTS + 1 1240 AN = ANG * Pi / 180 1250 COST = COS(AN) 1260 COS2 = COST * COST 1270 THOM(NPTS) = R2 * (1 + COS2) * .5 1280 F1 = 1 / (1 + G * (1 - COST)) ^ 2 1290 F2 = G ^ 2 * (1 - COST) ^ 2 1300 F2 = F2 / ((1 + COST ^ 2) * (1 + G * (1 - COST))) 1310 F2 = F2 + 1 1320 KN(NPTS) = THOM(NPTS) * F1 * F2 1322 THETA(NPTS) = ANG 1330 NEXT ANG 1400 CALL PAGER 1410 REM graphics file output 2060 REM TEKTRONIX OUTPUT 2070 OPEN "PLTTEK.DAT" FOR OUTPUT AS #2 2071 PRINT #2, 1 2072 PRINT #2, 2 2073 PRINT #2, NPTS 2074 PRINT #2, "CROSS SECTION CM^2/STERADIAN" 2075 PRINT #2, "SCATTERING ANGLE IN DEGREES" 2076 PRINT #2, "THOMPSON AND KLEIN-NISHINA CROSS SECTIONS" 2077 PRINT #2, "FROM COMPTON G=1.29 = hf/mc^2" 2090 FOR I = 1 TO NPTS 2100 PRINT #2, THETA(I), THOM(I) 2110 NEXT I 2111 FOR I = 1 TO NPTS 2112 PRINT #2, THETA(I), KN(I) 2113 NEXT I 2120 CLOSE #2 2130 PRINT "Now we CHAIN to convert the plotting data, so you can run WPLOT." CHAIN "TEK2WPLT.EXE" 2140 STOP 2590 END 2010 SUB PAGER PRINT , "Push RETURN to continue."; INPUT T$ PRINT CHR$(12); END SUB 2150 SUB PICTURE1 2160 PRINT " /" 2170 PRINT " / " 2180 PRINT " Ep'=hf' /" 2190 PRINT " / " 2200 PRINT " / " 2210 PRINT " / " 2220 PRINT " / " 2230 PRINT " / " 2240 PRINT " / phi " 2250 PRINT "/›/›/›->Ep= hf >---------* - - - - - - - - - - - - - - - " 2260 PRINT " › " 2270 PRINT " › theta " 2280 PRINT " › " 2282 PRINT " › " 2290 PRINT " › " 2292 PRINT " › " 2300 PRINT " › E of electron" END SUB 2320 SUB PICTURE2 2330 PRINT " ^ x axis " 2350 PRINT " | detector" 2360 PRINT " | / " 2370 PRINT " | / " 2380 PRINT " | / " 2390 PRINT " ^ | / " 2400 PRINT " | E polarization | / " 2410 PRINT " | | / theta " 2420 PRINT " | | / " 2430 PRINT "/›/›/›/›/›/›/›/›/›/›/›/›/›/›* --------------------->Z " 2440 PRINT 2450 PRINT "E is the ELECTRIC INTENSITY VECTOR. Theta is angle measured" 2460 PRINT "from the gamma ray direction of propagation to the detector." 2470 PRINT END SUB