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s(oth)328 1437 y(passage)k(from)f Fl(x)c(<)f(x)1153 1452
y Fi(0)1226 1437 y Fp(to)32 b Fl(x)c(>)g(x)1587 1452
y Fi(1)1626 1437 y Fp(.)328 1557 y(In)38 b(order)g(to)f(parametrize)f
(this)i(in)f(a)g(simple)f(w)m(a)m(y)j(for)e(the)h(repla)m(y)g(co)s(de,)
h(a)e(general)328 1677 y(parametrization)i(for)i(all)f(materials)f(w)m
(as)k(c)m(hosen:)63 b Fl(x)2402 1692 y Fi(0)2486 1677
y Fp(=)43 b(0,)h Fl(x)2780 1692 y Fi(1)2863 1677 y Fp(=)f(3,)h
Fl(k)i Fp(=)d(3)f(and)328 1798 y Fl(a)28 b Fp(=)f Fj(\000)p
Fl(C)657 1813 y Fi(0)697 1798 y Fl(=)p Fp(27.)328 2178
y Fd(2.1.2)112 b(Comparison)37 b(of)h(the)f(Energy)h(Loss)g
(Calculations)e(for)h(Hadrons)428 2483 y Fp(The)c(di\013eren)m(t)f
(results)h(of)f(Equations)g(1)g(and)h(2)f(with)f(di\013eren)m(t)i
(parametrizations)328 2603 y(of)45 b(the)i(densit)m(y)f(correction)g
(are)g(sho)m(wn)h(in)e(Fig.)f(1)i(for)f(four)h(di\013eren)m(t)g
(materials.)328 2724 y(Most)i(signi\014can)m(t)g(are)f(the)i(e\013ect)f
(of)g(the)g(densit)m(y)h(correction)f(at)f(high)g(energies.)328
2844 y(Fig.)31 b(2)h(sho)m(ws)i(what)f(errors)g(are)f(made)g(b)m(y)i
(either)e(omitting)e(the)j(densit)m(y)g(correction)328
2965 y(altogether)38 b(\(b)s(ottom\))f(or)h(b)m(y)i(putting)e(in)f(the)
j(high)d(energy)j(densit)m(y)g(correction)e(at)328 3085
y(lo)m(w)d(energies)g(\(i.e.)51 b(without)34 b(using)h(Sternheimer's)g
(parametrization\).)49 b(The)36 b(result)328 3205 y(of)31
b(this)g(exercise)i(is)e(that)g(the)h(energy)g(loss)f(calculation)e(in)
i(the)h(co)s(de)f(in)g(its)g(old)f(state)328 3326 y(\(7/98\))g(do)s(es)
h(a)g(reasonable)g(job)g(\()p Fj(\006)p Fp(10\045\))g(for)f(protons)i
(up)f(to)f(ab)s(out)h(10)g(GeV/c)g(and)328 3446 y(for)42
b(pions)h(up)g(to)g(1.5)g(GeV/c.)75 b(In)43 b(order)g(to)g(do)g(a)f(b)s
(etter)i(job)f(for)f(pions,)j(and)f(in)328 3566 y(the)e(ligh)m(t)d(of)i
(the)h(up)s(coming)e(upgrade)h(of)g(the)h(CEBAF)g(accelerator,)h(the)f
(densit)m(y)328 3687 y(correction)37 b(in)g(the)h(Sternheimer)f
(parametrization)e(clearly)i(should)h(b)s(e)f(co)s(ded)i(in)m(to)328
3807 y(the)33 b(engine.)328 3928 y(The)40 b(error)f(made)f(b)m(y)i
(making)e(a)g(general)h(parametrization)d(for)i(the)i
Fl(x)3070 3943 y Fi(0)3148 3928 y Fl(<)f(x)g(<)f(x)3526
3943 y Fi(1)328 4048 y Fp(part)30 b(will)e(b)s(e)i(at)g(least)g(an)g
(order)g(of)g(magnitude)f(lo)m(w)m(er)h(than)g(the)h(densit)m(y)g
(correction)328 4168 y(itself)g(and)i(can)g(therefore)g(b)s(e)g
(tolerated.)1922 5280 y(3)p eop
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%%EndDocument
 @endspecial 328 4012 a Fp(Figure)29 b(1:)41 b(Energy)31
b(loss)f(of)f(hadrons)h(in)f(LH)1984 4027 y Fi(2)2054
4012 y Fp(\(upp)s(er)h(left\),)f(graphite)g(\(upp)s(er)i(righ)m(t\),)
328 4133 y(alumin)m(um)22 b(\(lo)m(w)m(er)j(left\))g(and)g(copp)s(er)g
(\(lo)m(w)m(er)g(righ)m(t\).)41 b(The)26 b(energy)g(loss)e(as)i
(calculated)328 4253 y(b)m(y)36 b(the)f(standard)g(engine)g
(\(dashed-dotted)g(line\))f(is)g(compared)h(with)f(the)h(standard)328
4373 y(Bethe-Blo)s(c)m(h)c(form)m(ula,)f(with)g(t)m(w)m(o)i
(parametrizations)d(for)i(the)g(densit)m(y)h(correction)f
Fl(\016)328 4494 y Fp(\(solid)e(and)h(dashed)h(lines\).)42
b(See)31 b(text)g(for)f(explanations)f(on)h(the)h(densit)m(y)g
(correction.)328 4614 y(The)45 b(momen)m(tum)d(scale)i(is)g
Fl(\014)6 b(\015)52 b Fp(=)47 b Fl(p=)-5 b(M)10 b(c)p
Fp(,)47 b(whic)m(h)d(is)g(the)g(particle)f(momen)m(tum)g(in)328
4735 y(terms)32 b(of)h(the)g(particle)e(mass.)1922 5280
y(4)p eop
%%Page: 5 5
5 4 bop 360 3809 a @beginspecial 103 @llx 54 @lly 557
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4494 y(made)29 b(b)m(y)h(the)g(standard)g(repla)m(y)f(engine)h(form)m
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4614 y(corrections.)48 b(BB)34 b(stands)i(for)d(Bethe-Blo)s(c)m(h)h
(with)g(Sternheimer's)g(parametrization)328 4735 y(of)e(the)h(densit)m
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%%Page: 6 6
6 5 bop 328 31 a Fm(2.2)135 b(Electrons)328 216 y Fd(2.2.1)112
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521 y Fp(F)-8 b(or)31 b(electrons,)i(the)g(matter)f(is)g(more)g
(complicated)f(than)i(for)f(hadrons)h([2]:)572 749 y(\\While)23
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b(for)24 b(hea)m(vy)i(c)m(harged)572 869 y(particles)h(is)g(also)f(v)-5
b(alid)26 b(for)h(electrons,)i(the)f(Bethe-Blo)s(c)m(h)g(form)m(ula)e
(m)m(ust)572 990 y(b)s(e)40 b(mo)s(di\014ed)f(somewhat)h(for)g(t)m(w)m
(o)h(reasons.)67 b(One,)42 b([..],)h(is)d(their)f(small)572
1110 y(mass.)88 b(The)48 b(assumption)f(that)g(the)h(inciden)m(t)f
(particle)f(remains)h(un-)572 1231 y(de\015ected)g(during)d(the)i
(collision)c(pro)s(cess)k(is)f(therefore)g(in)m(v)-5
b(alid.)79 b(The)572 1351 y(second)44 b(is)e(that)g(for)g(electrons)h
(the)g(collisions)c(are)k(b)s(et)m(w)m(een)i(iden)m(tical)572
1471 y(particles,)k(so)e(that)f(the)h(calculation)d(m)m(ust)i(tak)m(e)h
(in)m(to)f(accoun)m(t)h(their)572 1592 y(indistinguishabilit)m(y)-8
b(.)84 b(These)49 b(considerations)e(c)m(hange)h(a)f(n)m(um)m(b)s(er)h
(of)572 1712 y(terms)37 b(in)g(the)g(form)m(ula,)g(in)g(particular,)g
(the)g(maxim)m(um)e(allo)m(w)m(able)h(en-)572 1833 y(ergy)25
b(transfer)f(b)s(ecomes)h Fl(W)1602 1848 y Fk(max)1773
1833 y Fp(=)j Fl(T)1934 1848 y Fk(e)1971 1833 y Fl(=)p
Fp(2)c(where)h Fl(T)2423 1848 y Fk(e)2485 1833 y Fp(is)e(the)i(kinetic)
f(energy)572 1953 y(of)32 b(the)h(inciden)m(t)f(electron)h(or)f(p)s
(ositron.)42 b([..]")428 2181 y(The)36 b(Bethe-Blo)s(c)m(h)f(equation)f
(\(Eq.)i(2\))f(calculates)g(a)f(mean)h(energy)h(loss.)51
b(A)35 b(rela-)328 2302 y(tivistic)26 b(correction)g(for)h(electrons)h
(to)e(the)i(Bethe-Blo)s(c)m(h)f(equation)g(has)h(b)s(een)g(devised)328
2422 y(b)m(y)j(Bethe)h([2,)f(5)o(].)43 b(This)31 b(not)g(only)f(tak)m
(es)i(in)m(to)d(accoun)m(t)j(the)f(c)m(hanged)h Fl(T)3024
2437 y Fk(max)3168 2422 y Fp(,)f(but)g(also)328 2542
y(the)h(e\013ect)g(that)f(a)g(collision)d(b)s(et)m(w)m(een)33
b(electrons)f(can)g(result)f(in)f(a)h(big)g(c)m(hange)h(in)e(the)328
2663 y(direction)40 b(of)g(the)i(particle)d(momen)m(tum.)67
b(With,)43 b(among)d(others,)j(this)e(correction,)328
2783 y(the)33 b(form)m(ula)e(reads)429 3148 y Fj(\000)516
3081 y Fl(dE)p 516 3125 129 4 v 527 3216 a(dx)683 3148
y Fp(=)83 b Fl(K)942 3109 y Fk(Z)p 942 3125 53 4 v 942
3182 a(A)1036 3109 y Fi(1)p 1015 3125 78 4 v 1015 3182
a Fk(\014)1058 3163 y Fe(2)1124 3148 y Fj(\001)1174 3027
y Fh(\024)1218 3148 y Fp(ln)1326 3105 y Fk(m)1388 3113
y Ff(e)1422 3105 y Fk(c)1453 3082 y Fe(2)1487 3105 y
Fk(\014)1530 3082 y Fe(2)1564 3105 y Fk(\015)1604 3082
y Fe(2)1639 3105 y Fk(E)1691 3113 y Ff(e)p 1326 3125
399 4 v 1472 3182 a Fi(2)p Fk(I)1543 3163 y Fe(2)1757
3148 y Fp(+)22 b(\(1)f Fj(\000)i Fl(\014)2124 3112 y
Fi(2)2163 3148 y Fp(\))f Fj(\000)2333 3105 y Fi(2)p Fk(\015)t
Fg(\000)p Fi(1)p 2333 3125 166 4 v 2378 3182 a Fk(\015)2418
3163 y Fe(2)2525 3148 y Fp(ln)16 b(2)22 b(+)2802 3109
y Fi(1)p 2802 3125 36 4 v 2802 3182 a(8)2864 3052 y Fh(\020)2923
3105 y Fk(\015)t Fg(\000)p Fi(1)p 2923 3125 131 4 v 2968
3182 a Fk(\015)3064 3052 y Fh(\021)3113 3075 y Fi(2)3175
3148 y Fj(\000)g Fl(\016)3321 3027 y Fh(\025)3441 3148
y Fp(\(4\))683 3405 y(=)251 b Fl(K)1110 3366 y Fk(Z)p
1110 3382 53 4 v 1110 3440 a(A)1204 3366 y Fi(1)p 1183
3382 78 4 v 1183 3440 a Fk(\014)1226 3421 y Fe(2)1292
3405 y Fj(\001)1342 3259 y Fh(")1390 3405 y Fp(ln)1548
3359 y Fk(\034)1587 3335 y Fe(2)1622 3359 y Fi(\()p Fk(\034)8
b Fi(+2\))p 1498 3382 357 4 v 1498 3440 a(2\()p Fk(I)d(=m)1693
3448 y Ff(e)1728 3440 y Fk(c)1759 3421 y Fe(2)1793 3440
y Fi(\))1820 3421 y Fe(2)1887 3405 y Fp(+)22 b(\(1)g
Fj(\000)h Fl(\014)2255 3369 y Fi(2)2294 3405 y Fp(\))f(+)2472
3322 y Ff(\034)2507 3301 y Fe(2)p 2472 3334 70 4 v 2491
3376 a(8)2551 3349 y Fg(\000)p Fi(\(2)p Fk(\034)8 b Fi(+1\))13
b(ln)f(2)p 2462 3382 482 4 v 2593 3440 a(\()p Fk(\034)c
Fi(+1\))2776 3421 y Fe(2)2975 3405 y Fj(\000)23 b Fl(\016)3122
3259 y Fh(#)3187 3405 y Fl(;)227 b Fp(\(5\))428 3682
y(with)34 b(K)g(is)g(de\014ned)h(as)g(in)e(Eq.)j(2)e(and)g
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(energy)h(of)f(the)328 3803 y(electron,)g Fl(m)810 3818
y Fk(e)847 3803 y Fl(c)889 3766 y Fi(2)929 3803 y Fp(\()p
Fl(\015)27 b Fj(\000)d Fp(1\),)33 b(and)h Fl(\034)41
b Fp(=)30 b Fl(\015)e Fj(\000)23 b Fp(1)33 b(is)h(the)g(kinetic)f
(energy)i(of)e(the)h(electron)g(in)328 3923 y(terms)k(of)f(the)h
(electron)g(mass.)59 b Fl(\016)41 b Fp(is)d(the)g(densit)m(y)h
(correction)e(discussed)j(in)d(section)328 4043 y(2.1.1.)j(In)25
b(the)f(case)h(of)f(electrons)g(at)g(CEBAF)h(energies)g(\(b)s(et)m(w)m
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4187 y Fi(10)732 4164 y Fl(\014)6 b(\015)29 b Fp(is)23
b(big)g(enough)i(\(2.3-3.9\))d(that)i(the)h(densit)m(y)g(correction)e
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(parametrization.)49 b(The)36 b(error)f(made)g(here)h(is)e(0.1-0.2\045)
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%%Page: 7 7
7 6 bop 328 31 a Fd(2.2.2)112 b(Most)38 b(Probable)f(v)m(ersus)h(Av)m
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(O'Brien)e Fn(et)k(al.)42 b Fp(for)31 b(50)f(and)i(100)e(MeV)i
(electrons)f(on)g(a)g(n)m(um)m(b)s(er)g(of)328 577 y(solid)g(targets)i
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541 y Fi(2)2855 577 y Fp([6]:)1474 855 y Fl(E)1546 870
y Fk(p)1614 855 y Fp(=)d Fl(A)1791 814 y Fg(0)1814 855
y Fl(t)1866 709 y Fh( )1931 855 y Fl(K)i Fp(+)22 b(ln)2257
788 y Fl(t)p 2250 832 51 4 v 2250 923 a(\032)2310 709
y Fh(!)2392 855 y Fl(:)1022 b Fp(\(6\))428 1096 y(Here,)38
b Fl(A)763 1060 y Fg(0)821 1096 y Fp(=)d(0)p Fl(:)p Fp(1536)p
Fl(Z)r(=)-5 b(A)34 b Fp(MeV)k(cm)1792 1060 y Fi(2)1831
1096 y Fp(/g.,)f Fl(K)42 b Fp(=)35 b(19)p Fl(:)p Fp(26,)h
Fl(t)h Fp(is)f(the)i(target)e(thic)m(kness)328 1216 y(in)30
b(\(g/cm)701 1180 y Fi(2)740 1216 y Fp(\))g(and)h Fl(\032)g
Fp(is)f(the)i(target)e(densit)m(y)i(in)e(\(g/cm)2333
1180 y Fi(3)2371 1216 y Fp(\).)43 b(This)31 b(equation)g(is)f(based)i
(on)328 1337 y(\\Landau)24 b(straggling".)39 b(In)25
b(a)f(Landau)h(straggling)d(distribution,)j(the)g(a)m(v)m(erage)h
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1577 y(do)s(es)35 b(not)f(tak)m(e)i(in)m(to)d(accoun)m(t)j(the)f
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1698 y(relativistic)30 b(e\013ects)k(in)e(the)h(medium.)328
1818 y(A)f(problem)e(with)h(using)g(the)h(most)f(probable)g(energy)i
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b(con)m(v)m(olution)d(of)h(a)328 2179 y(narro)m(w)36
b(Landau)h(distribution)d(and)i(a)g(wide)g(gaussian)g(distribution)f
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(energy)h(loss)e(than)h(the)g(most)f(probable)g(energy)h(loss)3233
2384 y Fi(1)3305 2420 y Fp([7].)328 2680 y Fd(2.2.3)112
b(Comparison)37 b(of)h(the)f(eloss)g(calculations)f(for)i(electrons)428
2985 y Fp(Fig.)31 b(3)h(sho)m(ws)i(the)f(di\013erence)g(of)f(the)h(n)m
(umerical)e(outcome)i(of)f(energy)h(loss)g(calcu-)328
3105 y(lations)e(after)i(Bethe-Blo)s(c)m(h,)f(Bethe)i(and)f(O'Brien)f
(for)h(a)f(n)m(um)m(b)s(er)h(of)g(targets.)44 b(Both)328
3226 y(the)39 b(Bethe-Blo)s(c)m(h)f(calculation)e(and)i(the)h(Bethe)g
(calculation)d(with)i(the)h(relativistic)328 3346 y(correction)32
b(include)h(the)g(densit)m(y)h(correction.)44 b(The)34
b(dashed-dotted)g(line)d(represen)m(ts)328 3466 y(the)k(O'Brien)f(form)
m(ula)e(for)i(the)h(most)e(probable)h(energy)i(loss)e(\(Eq.)h(6\).)48
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 @endspecial 328 4130 a Fp(Figure)29 b(4:)42 b(Ratio)29
b(of)h(the)h(calculations)e(of)h(electron)g(energy)h(losses)g(in)f(LH)
3116 4145 y Fi(2)3156 4130 y Fp(,)h(graphite)328 4250
y(and)36 b(copp)s(er.)54 b(T)-8 b(op)36 b(left:)50 b(ratio)34
b(for)i(Bethe-Blo)s(c)m(h)f(form)m(ula)f(and)j(O'Briens)f(form)m(ula;)
328 4371 y(top)47 b(righ)m(t:)73 b(ratio)46 b(for)h(Bethe)h(with)g
(relativistic)d(corrections)i(and)h(O'Brien;)55 b(b)s(ot-)328
4491 y(tom:)40 b(ratio)26 b(b)s(et)m(w)m(een)k(Bethe)f(with)e
(relativistic)e(corrections)j(and)g(Bethe-Blo)s(c)m(h.)42
b(The)328 4611 y(Bethe-Blo)s(c)m(h)32 b(calculation)f(is)h(the)h
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%%Page: 16 16
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%%Page: 17 17
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%%Page: 18 18
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y(write\(6,'\(4A10\)'\))i('velocity','beta','pmass',)q('den)q(sco)q
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%%Trailer
end
userdict /end-hook known{end-hook}if
%%EOF