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y Fd(2)2348 4702 y Fc(\000)h Fh(\()p Ff(E)2558 4717 y Fe(\015)2602 4702 y Fh(\))2640 4661 y Fd(2)1611 4848 y Fh(=)83 b Ff(m)1855 4806 y Fd(2)1855 4872 y Fe(e)1895 4848 y Ff(c)1937 4806 y Fd(4)1998 4848 y Fh(+)22 b Ff(E)2174 4806 y Fd(2)2168 4872 y Fe(\015)2236 4848 y Fh(+)g(2)p Ff(E)2455 4863 y Fe(\015)2499 4848 y Ff(m)2584 4863 y Fe(e)2622 4848 y Ff(c)2664 4806 y Fd(2)2725 4848 y Fc(\000)h Ff(E)2903 4806 y Fd(2)2897 4872 y Fe(\015)1611 5002 y Fh(=)83 b Ff(m)1855 4961 y Fd(2)1855 5027 y Fe(e)1895 5002 y Ff(c)1937 4961 y Fd(4)1998 5002 y Fh(+)22 b(2)p Ff(E)2217 5017 y Fe(\015)2262 5002 y Ff(m)2347 5017 y Fe(e)2384 5002 y Ff(c)2426 4961 y Fd(2)411 5222 y Fh(Therefore,)40 b(for)c(this)h(to)g(b)s(e)g(satis\014ed)h(requires)g(2)p Ff(E)2367 5237 y Fe(\015)2411 5222 y Ff(m)2496 5237 y Fe(e)2533 5222 y Ff(c)2575 5186 y Fd(2)2650 5222 y Fh(=)d(0)i(whic)m(h) h(is)e(only)h(realizable)411 5343 y(for)32 b Ff(E)632 5358 y Fe(\015)705 5343 y Fh(=)27 b(0,)33 b(i.e.,)f(there)h(is)f(no)h (photon.)233 5505 y(\(b\))48 b(In)34 b(the)g(Compton)f(e\013ect)h(the)g (\014nal)e(state)i(con)m(tains)f(b)s(oth)g(the)h(photon)g(and)f(the)h (electron.)411 5625 y(No)m(w)40 b(some)e(of)g(the)h(excess)i(momen)m (tum)d(can)g(b)s(e)h(carried)g(o\013)f(b)m(y)h(the)g(photon)g(so)g (energy)411 5745 y(and)33 b(momen)m(tum)e(can)i(b)s(e)g(conserv)m(ed.)p eop %%Page: 2 2 2 1 bop 72 481 a Fh(2.)49 b Fg(\(6)37 b(marks\))197 601 y Fh(A)31 b(photon)h(of)f(frequency)i Ff(\027)38 b Fh(collides)30 b(elastically)f(with)i(an)g(electron)g(at)g(rest.)44 b(The)32 b(photon)g(loses)197 722 y(half)38 b(its)h(energy)i(in)e(the)h (collision)d(and)i(its)g(\014nal)g(path)h(mak)m(es)g(a)f(60)2820 686 y Fb(\016)2899 722 y Fh(angle)g(with)g(its)g(original)197 842 y(direction.)j(What)33 b(is)f Ff(\027)6 b Fh(?)44 b(What)32 b(sort)h(of)f(photon)h(is)f(this?)98 1046 y Fc(\017)49 b Fh(The)33 b(expression)h(for)e(Compton)g(scattering)h (from)e(an)i(electron)f(is:)1512 1309 y Ff(\025)1569 1268 y Fb(0)1614 1309 y Fc(\000)23 b Ff(\025)28 b Fh(=)1966 1242 y Ff(h)p 1912 1286 165 4 v 1912 1378 a(m)1997 1393 y Fe(e)2034 1378 y Ff(c)2086 1309 y Fh(\(1)22 b Fc(\000)h Fh(cos)17 b Ff(\036)p Fh(\))1190 b(\(1\))197 1575 y(Here)33 b(w)m(e)h(are)e(giv)m(en)h(that)f Ff(E)1277 1539 y Fb(0)1328 1575 y Fh(=)c Ff(E)6 b(=)p Fh(2)32 b(so)1428 1845 y Ff(\025)1485 1804 y Fb(0)1536 1845 y Fh(=)1651 1777 y Ff(hc)p 1649 1822 102 4 v 1649 1913 a(E)1727 1884 y Fb(0)1788 1845 y Fh(=)1940 1777 y Ff(hc)p 1902 1822 176 4 v 1902 1913 a(E)6 b(=)p Fh(2)2115 1845 y(=)27 b(2)2277 1777 y Ff(hc)p 2277 1822 99 4 v 2287 1913 a(E)2413 1845 y Fh(=)g(2)p Ff(\025)197 2121 y Fh(Plugging)k(this)h(in)m(to)g(\(1\))g(giv)m(es:) 1135 2390 y Ff(\025)1192 2349 y Fb(0)1238 2390 y Fc(\000)22 b Ff(\025)83 b Fh(=)1700 2323 y Ff(h)p 1646 2367 165 4 v 1646 2458 a(m)1731 2473 y Fe(e)1768 2458 y Ff(c)1820 2390 y Fh(\(1)22 b Fc(\000)h Fh(cos)17 b Ff(\036)p Fh(\))1110 2643 y(2)p Ff(\025)22 b Fc(\000)g Ff(\025)83 b Fh(=)1699 2576 y Ff(hc)p 1646 2620 204 4 v 1646 2711 a(m)1731 2726 y Fe(e)1768 2711 y Ff(c)1810 2683 y Fd(2)1860 2643 y Fh(\(1)21 b Fc(\000)i Fh(cos)17 b Ff(\036)p Fh(\))1337 2902 y Ff(\025)83 b Fh(=)1646 2834 y(\(1239)p Ff(:)p Fh(8)31 b(k)m(eV)25 b Fc(\001)d Fh(pm)o(\))p 1646 2878 754 4 v 1812 2970 a(\(511)32 b(k)m(eV)r(\))2409 2902 y(\(1)22 b Fc(\000)h Fh(cos)17 b(60)2863 2861 y Fb(\016)2902 2902 y Fh(\))1477 3112 y(=)83 b(1)p Ff(:)p Fh(21)32 b(pm)60
b(=)g(1)p Ff(:)p Fh(21)21 b Fc(\002)i Fh(10)2567 3071 y Fb(\000)p Fd(12)2729 3112 y Fh(m)197 3332 y(And)33 b(the)g(frequency)h(of)f(the)g(incoming)d(photon)i(is)g Ff(\027)j Fh(=)27 b Ff(c=\025)h Fh(=)f(2)p Ff(:)p Fh(47)22 b Fc(\002)g Fh(10)3003 3296 y Fd(20)3110 3332 y Fh(Hz)33 b(corresp)s(onding)197 3453 y(to)f(an)h(energy)g(of)f(1.022)g(MeV.)h (This)g(is)f(a)h(gamma)d(ra)m(y)j(photon.)72 3656 y(3.)49 b Fg(\(8)37 b(marks\))32 b Fh(A)h(photon)f(collides)f(with)h(a)h(free)g (electron)f(at)g(rest.)238 3859 y(\(a\))48 b(\(4)32 b(marks\))f(Sho)m (w)h(that)g(the)f(kinetic)g(energy)i(of)e(the)h(electron)f(after)h(the) g(collision)c(is)j(equal)411 3980 y(to)i(the)g(energy)g(lost)f(b)m(y)i (the)f(photon.)313 4142 y Fc(\017)48 b Fh(The)31 b(initial)26 b(state)k(consists)g(of)f(a)g(photon)h(of)f(energy)i Ff(E)2483 4157 y Fe(\015)2557 4142 y Fh(and)f(an)f(electron)h(at)f (rest.)43 b(So)29 b(the)411 4262 y(total)g(energy)i(in)f(the)g(initial) d(state)j(is)g Ff(E)1912 4277 y Fe(i)1971 4262 y Fh(=)g Ff(E)2149 4277 y Fe(\015)2223 4262 y Fh(+)g Ff(m)2414 4277 y Fe(e)2452 4262 y Ff(c)2494 4226 y Fd(2)2533 4262 y Fh(.)43 b(The)31 b(\014nal)e(state)i(consists)g(of)e(the)411 4382 y(scattered)35 b(photon)e(and)h(scattered)g(electron)g(ha)m(ving)f (energies)h Ff(E)2903 4346 y Fb(0)2897 4407 y Fe(\015)2975 4382 y Fh(and)f Ff(E)3237 4397 y Fe(e)3274 4382 y Fh(,)h(resp)s(ectiv)m (ely)-8 b(.)411 4503 y(Therefore)34 b Ff(E)925 4518 y Fe(f)1003 4503 y Fh(=)f Ff(E)1190 4467 y Fb(0)1184 4527 y Fe(\015)1261 4503 y Fh(+)f Ff(E)1441 4518 y Fe(e)1478 4503 y Fh(.)44 b(Conserv)-5 b(ation)32 b(of)g(total)f(energy)j Ff(E)2865 4518 y Fe(f)2938 4503 y Fh(=)28 b Ff(E)3114 4518 y Fe(i)3175 4503 y Fh(giv)m(es:)1673 4732 y Ff(E)1751 4691 y Fb(0)1745 4757 y Fe(\015)1812 4732 y Fh(+)22 b Ff(E)1982 4747 y Fe(e)2047 4732 y Fh(=)28 b Ff(E)2223 4747 y Fe(\015)2289 4732 y Fh(+)22 b Ff(m)2472 4747 y Fe(e)2510 4732 y Ff(c)2552 4691 y Fd(2)411 4952 y Fh(Therefore,)34 b(the)f(energy)h(lost)e(b)m(y)h(the)g(photon)g(is:)1546 5172 y Ff(E)1618 5187 y Fe(\015)1685 5172 y Fc(\000)23 b Ff(E)1863 5131 y Fb(0)1857 5197 y Fe(\015)1929 5172 y Fh(=)28 b Ff(E)2105 5187 y Fe(e)2164 5172 y Fc(\000)22 b Ff(m)2348 5187 y Fe(e)2386 5172 y Ff(c)2428 5131 y Fd(2)2495 5172 y Fh(=)27 b Ff(K)2681 5187 y Fe(e)p eop %%Page: 3 3 3 2 bop 233 481 a Fh(\(b\))48 b(\(4)37 b(marks\))g(If)h(the)f(photon)h (had)f Ff(\027)42 b Fh(=)36 b(2)25 b Fc(\002)h Fh(10)2150 445 y Fd(20)2261 481 y Fh(Hz)38 b(and)f(the)h(scattering)f(angle)g(w)m (as)h(54)3787 445 y Fb(\016)3826 481 y Fh(,)411 601 y(what)33 b(w)m(as)h(the)f(kinetic)f(energy)h(of)g(the)g(electron?)313 755 y Fc(\017)48 b Fh(Again)32 b(using)g(the)h(Compton)f(form)m(ula)f (for)h(photon-electron)g(collisions:)785 989 y Ff(\025)842 947 y Fb(0)887 989 y Fc(\000)23 b Ff(\025)83 b Fh(=)1350 921 y Ff(h)p 1296 965 165 4 v 1296 1057 a(m)1381 1072 y Fe(e)1418 1057 y Ff(c)1470 989 y Fh(\(1)22 b Fc(\000)g Fh(cos)17 b Ff(\036)p Fh(\))964 1242 y Ff(\025)1021 1201 y Fb(0)1127 1242 y Fh(=)83 b Ff(\025)22 b Fh(+)1525 1174 y Ff(hc)p 1473 1218 204 4 v 1473 1310 a(m)1558 1325 y Fe(e)1595 1310 y Ff(c)1637 1281 y Fd(2)1686 1242 y Fh(\(1)g Fc(\000)h Fh(cos)17 b Ff(\036)p Fh(\))1127 1495 y(=)1302 1427 y Ff(c)p 1296 1472 55 4 v 1296 1563 a(\027)1382 1495 y Fh(+)1543 1427 y Ff(hc)p 1490 1472 204 4 v 1490 1563 a(m)1575 1578 y Fe(e)1612 1563 y Ff(c)1654 1534 y Fd(2)1704 1495 y Fh(\(1)22 b Fc(\000)g Fh(cos)17 b Ff(\036)p Fh(\))1127 1758 y(=)1296 1690 y(\(3)k Fc(\002)i
Fh(10)1602 1654 y Fd(20)1709 1690 y Fh(pm)p Ff(=)p Fh(s\))p 1296 1734 674 4 v 1349 1826 a(\(2)f Fc(\002)g Fh(10)1655 1797 y Fd(20)1762 1826 y Fh(Hz\))2002 1758 y(+)2110 1690 y(\(1239)p Ff(:)p Fh(8)31 b(k)m(eV)25 b Fc(\001)d Fh(pm)o(\))p 2110 1734 754 4 v 2276 1826 a(\(511)31 b(k)m(eV)s(\))2873 1758 y(\(1)22 b Fc(\000)g Fh(cos)c(\(54)3365 1716 y Fb(\016)3404 1758 y Fh(\)\))1127 1968 y(=)83 b(2)p Ff(:)p Fh(5)32 b(pm)59 b(=)i(2)p Ff(:)p Fh(5)21 b Fc(\002)i Fh(10)2119 1927 y Fb(\000)p Fd(12)2281 1968 y Fh(m)411 2158 y(Therefore,)34 b(the)f(kinetic)f(energy)i(of)e(the)h(electron)f(is:)742 2348 y Ff(K)825 2363 y Fe(e)945 2348 y Fh(=)82 b Ff(E)1175 2363 y Fe(\015)1242 2348 y Fc(\000)23 b Ff(E)1420 2307 y Fb(0)1414 2373 y Fe(\015)945 2555 y Fh(=)82 b Ff(h\027)29 b Fc(\000)23 b Ff(h\027)1446 2514 y Fb(0)1530 2555 y Fh(=)60 b Ff(h\027)29 b Fc(\000)1908 2488 y Ff(hc)p 1908 2532 99 4 v 1917 2624 a(\025)1974 2595 y Fb(0)945 2799 y Fh(=)82 b(\(4)p Ff(:)p Fh(136)22 b Fc(\002)g Fh(10)1583 2758 y Fb(\000)p Fd(18)1745 2799 y Fh(k)m(eV)j Fc(\001)d Fh(s\)\(2)g Fc(\002)g Fh(10)2366 2758 y Fd(20)2473 2799 y Fh(Hz)q(\))g Fc(\000)2759 2732 y Fh(\(1239)p Ff(:)p Fh(8)32 b(k)m(eV)25 b Fc(\001)c Fh(pm\))p 2759 2776 754 4 v 2952 2867 a(\(2)p Ff(:)p Fh(5)32 b(pm)o(\))945 2995 y(=)82 b(331)32 b(k)m(eV)72 3201 y(4.)49 b Fg(\(6)37 b(marks\))32 b Fh([V)-8 b(ariation)30 b(on)j(T)-8 b(ext)33 b(Problem)f(37,)g(page)h(191])197 3322 y(A)40 b(m)m(uon)h(is)f(a)g (particle)f(m)m(uc)m(h)j(lik)m(e)d(an)i(electron)f(except)j(that)d Ff(m)2736 3337 y Fe(\026)2783 3322 y Ff(=m)2917 3337 y Fe(e)2995 3322 y Fh(=)h(206)p Ff(:)p Fh(77.)67 b(Find)39 b(the)197 3442 y(maxim)m(um)25 b(w)m(a)m(v)m(elength)k(shift)e(for)g(a) g(Compton)g(collision)e(b)s(et)m(w)m(een)k(a)f(photon)f(and)h(a)f(free) h(m)m(uon.)98 3637 y Fc(\017)49 b Fh(In)23 b(the)h(Compton)e (scattering)h(of)g(a)g(photon)g(o\013)f(an)h(electron,)i(the)f (expression)g(for)f(the)g(w)m(a)m(v)m(elength)197 3757 y(shift)32 b(\001)p Ff(\025)h Fh(v)m(ersus)h(the)f(angle)f(of)g(the)h (scattered)h(photon)f Ff(\036)f Fh(is:)1223 3991 y(\001)p Ff(\025)c Fh(=)1557 3924 y Ff(h)p 1502 3968 165 4 v 1502 4060 a(m)1587 4075 y Fe(e)1625 4060 y Ff(c)1677 3991 y Fh(\(1)22 b Fc(\000)g Fh(cos)17 b Ff(\036)p Fh(\))28 b(=)f Ff(\025)2316 4006 y Fe(C)2375 3991 y Fh(\(1)22 b Fc(\000)h Fh(cos)17 b Ff(\036)p Fh(\))901 b(\(2\))197 4231 y(where)44 b(the)f(Compton)g(w)m(a)m(v)m(elength)h Ff(\025)1677 4246 y Fe(C)1781 4231 y Fc(\021)i Fh(\()p Ff(h=m)2132 4246 y Fe(e)2169 4231 y Ff(c)p Fh(\))c(=)h(2.43)f(pm)h(=)f (2)p Ff(:)p Fh(43)29 b Fc(\002)g Fh(10)3330 4195 y Fb(\000)p Fd(12)3502 4231 y Fh(m.)74 b(The)197 4352 y(maxim)m(um)28 b(w)m(a)m(v)m(elength)k(shift)e(will)f(clearly)g(b)s(e)i(for)f Ff(\036)e Fh(=)f(180)2476 4316 y Fb(\016)2545 4352 y Fh(giving)i(the)i(expression)h(\001)p Ff(\025)3605 4367 y Fe(max)3777 4352 y Fh(=)197 4472 y(2)p Ff(\025)303 4487 y Fe(C)362 4472 y Fh(.)43 b(Equation)31 b(\(3\))h(w)m(as)g(deriv)m (ed)h(for)e(scattering)h(o\013)f(a)h(free)g(electron.)43 b(Clearly)-8 b(,)31 b(replacing)f(the)197 4593 y(electron)i(with)h(a)f (m)m(uon)g(will)e(giv)m(e)j(the)g(expression:)1568 4827 y(\001)p Ff(\025)27 b Fh(=)1906 4759 y Ff(h)p 1847 4803 174 4 v 1847 4895 a(m)1932 4910 y Fe(\026)1979 4895 y Ff(c)2031 4827 y Fh(\(1)22 b Fc(\000)g Fh(cos)c Ff(\036)p Fh(\))197 5070 y(resulting)31 b(in)h(a)g(maxim)m(um)f(w)m(a)m(v)m (elength)j(shift)e(of:)1159 5317 y(\001)p Ff(\025)1297 5332 y Fe(max)1524 5317 y Fh(=)83 b(2)1749 5171 y Fa( )1883
5249 y Ff(h)p 1824 5293 V 1824 5385 a(m)1909 5400 y Fe(\026)1956 5385 y Ff(c)2008 5171 y Fa(!)2151 5317 y Fh(=)60 b(2)2353 5171 y Fa( )2433 5249 y Ff(m)2518 5264 y Fe(e)p 2428 5293 132 4 v 2428 5385 a Ff(m)2513 5400 y Fe(\026)2570 5171 y Fa(!)2652 5317 y Ff(\025)2709 5332 y Fe(C)1524 5581 y Fh(=)83 b(2)1749 5460 y Fa(\022)1931 5514 y Fh(1)p 1820 5558 271 4 v 1820 5650 a(206)p Ff(:)p Fh(77)2101 5460 y Fa(\023)2178 5581 y Fh(\(2)p Ff(:)p Fh(43)22 b Fc(\002)g Fh(10)2609 5540 y Fb(\000)p Fd(12)2771 5581 y Fh(m\))1524 5773 y(=)83 b(2)p Ff(:)p Fh(34)22 b Fc(\002)g Fh(10)2076 5732 y Fb(\000)p Fd(14)2238 5773 y Fh(m)60 b(=)g(23)p Ff(:)p Fh(4)32 b(fm)p eop %%Page: 4 4 4 3 bop 72 481 a Fh(5.)49 b Fg(\(8)37 b(marks\))32 b Fh([P)m(art)h(\(a\))f(of)g(this)g(question)h(is)f(the)h(text)h(Problem) d(16)h(\(page)h(264\).])197 601 y(A)22 b(b)s(eam)g(of)h(alpha)e (particles,)j(of)e(kinetic)g(energy)i(5.30)e(MeV)h(and)g(in)m(tensit)m (y)g(1)p Ff(:)p Fh(0)r Fc(\002)r Fh(10)3319 565 y Fd(4)3380 601 y Fh(particles/s,)197 722 y(falls)34 b(normally)g(on)i(a)g(lead)g (foil)d(of)j(atomic)e(n)m(um)m(b)s(er)j(82,)g(densit)m(y)g(11.4)e(g/cm) 3150 686 y Fd(3)3189 722 y Fh(,)i(atomic)e(w)m(eigh)m(t)197 842 y(207)30 b(g/mol,)e(and)j(thic)m(kness)h(1.0)e Ff(\026)p Fh(m.)42 b(A)30 b(detector)h(of)f(area)h(1.0)f(cm)2734 806 y Fd(2)2803 842 y Fh(is)g(placed)g(at)g(a)h(distance)f(of)197 962 y(10)i(cm)g(from)f(the)i(foil.)42 b(What)32 b(is)g(the)h(coun)m (ting)g(rate)f(for)g(a)h(scattering)f(angle)238 1161 y(\(a\))48 b(\(4)33 b(marks\))65 b(of)32 b(10)1133 1125 y Fb(\016)313 1318 y Fc(\017)48 b Fh(In)26 b(the)g(text)h(\(pg)e(236\)) g(w)m(e)i(are)e(giv)m(en)h(the)g(probabilit)m(y)e(an)h Ff(\013)i Fh(particle)d(is)h(scattered)i(through)411 1439 y(an)32 b(angle)e Ff(\036)p Fh(.)43 b(F)-8 b(rom)30 b(this)h(w)m(e)i(can)e(calculate)g Ff(N)10 b Fh(\()p Ff(\036)p Fh(\),)31 b(the)h(n)m(um)m(b)s(er)g(of)f(scatters)i(observ)m (ed)g(at)411 1559 y(an)g(angle)f Ff(\036)p Fh(,)g(for)g Ff(N)43 b Fh(input)32 b Ff(\013)h Fh(particles)f(p)s(er)h(unit)f(time.) 42 b(Sp)s(eci\014cally)-8 b(,)1463 1762 y Ff(N)10 b Fh(\()p Ff(\036)p Fh(\))83 b(=)g Ff(N)10 b(P)k Fh(\()p Ff(\036)p Fh(\))1768 1986 y(=)83 b Ff(N)2032 1840 y Fa( )2162 1919 y Ff(e)2207 1883 y Fd(2)p 2107 1963 193 4 v 2107 2055 a Fh(4)p Ff(\031)t(")2261 2070 y Fd(0)2310 1840 y Fa(!)2376 1863 y Fd(2)2432 1840 y Fa( )2515 1919 y Fc(A)p Ff(\032tZ)2754 1883 y Fd(2)p 2508 1963 293 4 v 2508 2055 a Fh(4)p Ff(R)2632 2026 y Fd(2)2671 2055 y Ff(K)2761 2026 y Fd(2)2810 1840 y Fa(!)3082 1919 y Fh(1)p 2903 1963 407 4 v 2903 2056 a(sin)3023 2016 y Fd(4)3079 2056 y Fh(\()p Ff(\036=)p Fh(2\))1315 2256 y(and)162 b Ff(\032)84 b Fh(=)1937 2188 y Ff(dN)2066 2203 y Fe(A)p 1937 2232 186 4 v 1993 2324 a Ff(A)945 2507 y Fh(where)1341 2361 y Fa( )1470 2439 y Ff(e)1515 2403 y Fd(2)p 1416 2483 193 4 v 1416 2575 a Fh(4)p Ff(\031)t(")1570 2590 y Fd(0)1619 2361 y Fa(!)1768 2507 y Fh(=)f(2)p Ff(:)p Fh(307)21 b Fc(\002)i Fh(10)2369 2466 y Fb(\000)p Fd(28)2530 2507 y Fh(J)g Fc(\001)e Fh(m)1230 2725 y(and)163 b Ff(N)1628 2740 y Fe(A)1768 2725 y Fh(=)83 b(6)p Ff(:)p Fh(02)21 b Fc(\002)i Fh(10)2320 2684 y Fd(23)2427 2725 y Fh(mol)2584 2682 y Fb(\000)p Fd(1)411 2928 y Fh(W)-8 b(e)33 b(are)g(giv)m(en)g(the)g(follo)m(wing:)1584 3132 y Ff(N)94 b Fh(=)83 b(10)2013 3091 y Fd(4)2084 3132 y Ff(\013)34 b Fh(particl...