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All Rights Reserved) readonly def /FullName (CMTI10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMTI10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 46 /period put dup 97 /a put dup 101 /e put dup 108 /l put dup 116 /t put readonly def /FontBBox{-163 -250 1146 969}readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMBX10 %!PS-AdobeFont-1.1: CMBX10 1.00B %%CreationDate: 1992 Feb 19 19:54:06 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMBX10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Bold) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMBX10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 56 /eight put readonly def /FontBBox{-301 -250 1164 946}readonly def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 9098D5FEE67660E69A7AB91B58F29A4D79E57022F783EB0FBBB6D4F4EC35014F D2DECBA99459A4C59DF0C6EBA150284454E707DC2100C15B76B4C19B84363758 469A6C558785B226332152109871A9883487DD7710949204DDCF837E6A8708B8 2BDBF16FBC7512FAA308A093FE5F00F963068B8B731A88D7740B0DDAED1B3F82 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMBX12 %!PS-AdobeFont-1.1: CMBX12 1.0 %%CreationDate: 1991 Aug 20 16:34:54 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMBX12) readonly def /FamilyName (Computer Modern) readonly def /Weight (Bold) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMBX12 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 82 /R put dup 99 /c put dup 101 /e put dup 102 /f put dup 110 /n put dup 114 /r put dup 115 /s put readonly def /FontBBox{-53 -251 1139 750}readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMMI6 %!PS-AdobeFont-1.1: CMMI6 1.100 %%CreationDate: 1996 Jul 23 07:53:52 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI6) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI6 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 105 /i put readonly def /FontBBox{11 -250 1241 750}readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMSY8 %!PS-AdobeFont-1.1: CMSY8 1.0 %%CreationDate: 1991 Aug 15 07:22:10 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY8 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /minus put readonly def /FontBBox{-30 -955 1185 779}readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMSY10 %!PS-AdobeFont-1.1: CMSY10 1.0 %%CreationDate: 1991 Aug 15 07:20:57 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /minus put dup 1 /periodcentered put dup 25 /approxequal put readonly def /FontBBox{-29 -960 1116 775}readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMR8 %!PS-AdobeFont-1.1: CMR8 1.0 %%CreationDate: 1991 Aug 20 16:39:40 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR8 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 11 /ff put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 54 /six put dup 56 /eight put dup 57 /nine put dup 61 /equal put dup 77 /M put dup 83 /S put dup 99 /c put dup 104 /h put dup 105 /i put readonly def /FontBBox{-36 -250 1070 750}readonly def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 9098D5FEE67660E69A7AB91B58F29A4D79E57022F783EB0FBBB6D4F4EC35014F D2DECBA99459A4C59DF0C6EBA150284454E707DC2100C15B76B4C19B84363758 469A6C558785B226332152109871A9883487DD7710949204DDCF837E6A8708B8 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMEX10 %!PS-AdobeFont-1.1: CMEX10 1.00 %%CreationDate: 1992 Jul 23 21:22:48 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMEX10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMEX10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 18 /parenleftbigg put dup 19 /parenrightbigg put dup 48 /parenlefttp put dup 49 /parenrighttp put dup 50 /bracketlefttp put dup 51 /bracketrighttp put dup 52 /bracketleftbt put dup 53 /bracketrightbt put dup 64 /parenleftbt put dup 65 /parenrightbt put dup 66 /parenleftex put dup 67 /parenrightex put dup 80 /summationtext put dup 88 /summationdisplay put dup 90 /integraldisplay put dup 104 /bracketleftBig put dup 105 /bracketrightBig put readonly def /FontBBox{-24 -2960 1454 772}readonly def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 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cleartomark %%EndFont %%BeginFont: CMMI10 %!PS-AdobeFont-1.1: CMMI10 1.100 %%CreationDate: 1996 Jul 23 07:53:57 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 11 /alpha put dup 13 /gamma put dup 21 /lambda put dup 27 /sigma put dup 30 /phi put dup 58 /period put dup 59 /comma put dup 61 /slash put dup 69 /E put dup 78 /N put dup 97 /a put dup 98 /b put dup 100 /d put dup 105 /i put dup 106 /j put dup 107 /k put dup 110 /n put dup 121 /y put readonly def /FontBBox{-32 -250 1048 750}readonly def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA0529731C99A784CCBE85B4993B2EEBDE 3B12D472B7CF54651EF21185116A69AB1096ED4BAD2F646635E019B6417CC77B 532F85D811C70D1429A19A5307EF63EB5C5E02C89FC6C20F6D9D89E7D91FE470 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR10 %!PS-AdobeFont-1.1: CMR10 1.00B %%CreationDate: 1992 Feb 19 19:54:52 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 1 /Delta put dup 11 /ff put dup 12 /fi put dup 13 /fl put dup 39 /quoteright put dup 40 /parenleft put dup 41 /parenright put dup 43 /plus put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 58 /colon put dup 61 /equal put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 81 /Q put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 89 /Y put dup 91 /bracketleft put dup 93 /bracketright put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put readonly def /FontBBox{-251 -250 1009 969}readonly def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 9098D5FEE67660E69A7AB91B58F29A4D79E57022F783EB0FBBB6D4F4EC35014F D2DECBA99459A4C59DF0C6EBA150284454E707DC2100C15B76B4C19B84363758 469A6C558785B226332152109871A9883487DD7710949204DDCF837E6A8708B8 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All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR12) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR12 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 44 /comma put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 66 /B put dup 67 /C put dup 74 /J put dup 97 /a put dup 98 /b put dup 101 /e put dup 104 /h put dup 105 /i put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 114 /r put dup 116 /t put dup 117 /u put readonly def /FontBBox{-34 -251 988 750}readonly def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 9098D5FEE67660E69A7AB91B58F29A4D79E57022F783EB0FBBB6D4F4EC35014F 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR17 %!PS-AdobeFont-1.1: CMR17 1.0 %%CreationDate: 1991 Aug 20 16:38:24 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR17) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR17 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 45 /hyphen put dup 76 /L put dup 80 /P put dup 85 /U put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 105 /i put dup 108 /l put dup 110 /n put dup 111 /o put dup 115 /s put readonly def /FontBBox{-33 -250 945 749}readonly def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 9098D5FEE67660E69A7AB91B58F29A4D79E57022F783EB0FBBB6D4F4EC35014F D2DECBA99459A4C59DF0C6EBA150284454E707DC2100C15B76B4C19B84363758 469A6C558785B226332152109871A9883487DD7710949204DDCF837E6A8708B8 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b(of)h(the)f(electron) i(b)s(eam)e(and)g(the)h(energy)g(loss)f(of)h(the)g(electron)h(b)s(eam)e (in)g(the)448 4118 y(radiator.)51 b(Findla)m(y's)35 b(mo)s (di\014cation)f(w)m(as)g(successfully)f(demonstrated)h(to)g(pro)s(duce) 448 4231 y(correct)e(cross)e(sections)i(in)e(\([5)q(],)h(see)g(Figure)g (2,)g([6]\).)589 4344 y(These)38 b(da)m(ys,)i(there)e(are)g(v)m(ery)g (accurate)h(Mon)m(te-Carlo)h(sim)m(ulations,)h Fj(N)3203 4358 y Fh(ij)3301 4344 y Fk(can)448 4457 y(b)s(e)31 b(calculated)i(for) e(eac)m(h)i(sp)s(eci\014c)e(exp)s(erimen)m(tal)h(conditions)g(without)g (the)f(need)g(to)448 4570 y(use)h(theoretical)j(form)m(ula.)46 b(This)31 b(remo)m(v)m(es)j(problems)d(in)h(the)g(unfolding)f(related)i (to)448 4683 y(the)e(kno)m(wledge)g(of)g Fj(N)1221 4697 y Fh(ij)1281 4683 y Fk(.)589 4795 y(Ho)m(w)m(ev)m(er,)40 b(this)c(is)g(not)h(the)f(only)g(reason)g(that)h(ma)m(y)f(cause)h(P)m (enfold-Leiss)g(un-)448 4908 y(folding)23 b(to)h(fail.)39 b(Careful)22 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1 X 2 Y 1 X -2 Y 1 X 1 Y 1 X -1 Y 1 X 1 Y 1 X -2 Y 5 X -1 Y 2 X 2 Y 2 X -1 Y 1 X -2 Y 1 X 1 Y 1 X -1 Y 1 X 2 Y 1 X -3 Y 1 X 1 Y 2 X 1 Y 2 X -1 Y 1 X -1 Y 1 X 1 Y 1 X -2 Y 1 X 1 Y 1 X -1 Y 2 X 1 Y 1 X -1 Y 2 X -1 Y 1 X 1 Y 1 X -2 Y 2 X 2 Y 1 X -1 Y 1 X -1 Y 1 X 1 Y 2 X -1 Y 2 X -1 Y 1 X 1 Y 1 X -1 Y 1 X 1 Y 1 X -2 Y 1 X 2 Y 3 X -2 Y 1 X -1 Y 1 X 2 Y 1 X -1 Y 1 X -2 Y 1 X 1 Y 1 X -1 Y 1 X 2 Y 1 X -4 Y 1 X 2 Y 5 X 1 Y 1 X 1 Y 1 X -2 Y 1 X -1 Y 1 X 2 Y 1 X -3 Y 1 X 3 Y 1 X -2 Y 1 X -1 Y 3 X 1 Y 2 X -1 Y 2 X -1 Y 2 X 2 Y 1 X -1 Y 1 X -1 Y 2 X 1 Y 1 X -2 Y 1 X 1 Y 4 X -1 Y 1 X -3 Y 1 X 4 Y 1 X -4 Y 1 X 2 Y 1 X -2 Y 1 X 2 Y 1 X -1 Y 3 X 1 Y 1 X -2 Y 1 X 2 Y 2 X -3 Y 1 X 3 Y 1 X -3 Y 1 X 3 Y 1 X -1 Y 1 X -3 Y 1 X 2 Y 1 X -2 Y 2 X 3 Y 1 X -3 Y 1 X 1 Y 1 X -1 Y 2 X 2 Y 1 X -3 Y 2 X 1 Y 2 X -2 Y 1 X 1 Y 1 X 1 Y 1 X -2 Y 1 X 2 Y 2 X -1 Y 1 X -1 Y 3 X -1 Y 1 X 1 Y 2 X -1 Y 2 X 1 Y 2 X -1 Y 3 X 2 Y 1 X -2 Y 2 X 1 Y 1 X -1 Y 1 X -3 Y 3 X 2 Y 1 X -1 Y 2 X -1 Y 1 X -1 Y 1 X 2 Y 1 X -2 Y 1 X 3 Y 2 X -3 Y 3 X 1 Y 1 X -2 Y 2 X 1 Y 2 X -3 Y 1 X 3 Y 1 X -1 Y 2 X -2 Y 2 X 2 Y 2 X -2 Y 1 X -1 Y 1 X 1 Y 1 X 2 Y 1 X -2 Y 1 X 1 Y 1 X -1 Y 2 X -3 Y 1 X 2 Y 1 X -2 Y 2 X 2 Y 1 X 1 Y 1 X -1 Y 1 X -1 Y 1 X 1 Y 1 X -1 Y 1 X -1 Y 1 X 2 Y 2 X -2 Y 1 X 1 Y 5 X -3 Y 1 X -2 Y 1 X 2 Y 1 X 2 Y 1 X -2 Y 1 X 2 Y 1 X -1 Y 1 X -1 Y 1 X 1 Y 2 X -1 Y 4 X -2 Y 1 X 3 Y 1 X -2 Y 1 X -1 Y 3 X -1 Y 1 X 1 Y 2 X 1 Y 1 X -4 Y 1 X 3 Y 3 X -1 Y 1 X 1 Y 1 X -1 Y 1 X -1 Y 1 X -1 Y 1 X 1 Y 2 X -1 Y 1 X 2 Y 1 X -3 Y 1 X 2 Y 1 X -1 Y 2 X -1 Y 1 X 1 Y 1 X -3 Y 1 X 3 Y 1 X -2 Y 1 X 2 Y 1 X -2 Y 1 X 1 Y 1 X -1 Y 1 X -1 Y 2 X -2 Y 1 X 3 Y 2 X -2 Y 1 X 1 Y 1 X -1 Y 1 X 1 Y 1 X -2 Y 2 X 2 Y 1 X -1 Y 1 X -2 Y 1 X -2 Y 1 X 3 Y 1 X 1 Y 2 X -3 Y 2 X -1 Y 1 X 4 Y 1 X -2 Y 1 X 1 Y 1 X -3 Y 1 X 1 Y 1 X -1 Y 3 X -1 Y 1 X 1 Y 2 X -1 Y 1 X -1 Y 1 X -1 Y 1 X 1 Y 1 X 1 Y 3 X -2 Y 2 X -1 Y 1 X 2 Y 1 X -3 Y 1 X 1 Y 4 X -1 Y 1 X -2 Y 1 X 2 Y 1 X -2 Y 1 X 2 Y 1 X -3 Y 1 X 1 Y 1 X 2 Y 1 X -3 Y 2 X 1 Y 2 X -2 Y 1 X -4 Y 1 X 4 Y 4 X -2 Y 1 X 3 Y 1 X -1 Y 1 X -1 Y 1 X 1 Y 1 X -4 Y 1 X -2 Y 2 X 4 Y 1 X -2 Y 4 X -3 Y 2 X 4 Y 1 X -3 Y 2 X -1 Y 2 X -3 Y 2 X 3 Y 1 X -1 Y 1 X 1 Y 2 X -1 Y 1 X 2 Y 1 X -3 Y 2 X -5 Y 1 X 6 Y 1 X -4 Y 1 X -3 Y 1 X 8 Y 2 X -8 Y 1 X 1 Y 1 X -3 Y 1 X 3 Y 1 X -1 Y 1 X -3 Y 1 X 3 Y 1 X -1 Y 2 X -2 Y 2 X 1 Y 1 X -1 Y 3 X -2 Y 1 X 1 Y 2 X -1 Y 1 X -2 Y 1 X -1 Y 1 X -1 Y 1 X -2 Y 1 X 6 Y 1 X -6 Y 2 X 1 Y 2 X -1 Y 1 X -2 Y 1 X 1 Y 1 X -4 Y 1 X 4 Y 2 X -3 Y 1 X -2 Y 1 X -4 Y 1 X 2 Y 1 X -1 Y 1 X 2 Y 1 X -3 Y 4 X -1 Y 1 X -2 Y 1 X -2 Y 1 X 1 Y 1 X 1 Y 1 X -5 Y 2 X 2 Y 1 X -6 Y 1 X 3 Y 1 X -5 Y 2 X -1 Y 1 X 2 Y 1 X -5 Y 1 X 4 Y 1 X -1 Y 1 X -9 Y 1 X 3 Y 1 X -4 Y 1 X 3 Y 1 X -3 Y 1 X -5 Y 1 X 1 Y 1 X -13 Y 2 X -49 Y 1 X -235 Y 53 X s black[ ] 0 sd 3 lw 159 138 m 861 X s gsave 1134 769 0 0 C 384.469 39.0986 t 0 r /Helvetica findfont 42.3568 sf 0 0 m (Photon Energy \(MeV\)) show NC gr 159 156 m -18 Y s 180 147 m -9 Y s 202 147 m -9 Y s 223 147 m -9 Y s 245 147 m -9 Y s 266 156 m -18 Y s 288 147 m -9 Y s 310 147 m -9 Y s 331 147 m -9 Y s 353 147 m -9 Y s 374 156 m -18 Y s 396 147 m -9 Y s 417 147 m -9 Y s 439 147 m -9 Y s 460 147 m -9 Y s 482 156 m -18 Y s 503 147 m -9 Y s 525 147 m -9 Y s 547 147 m -9 Y s 568 147 m -9 Y s 590 156 m -18 Y s 611 147 m -9 Y s 633 147 m -9 Y s 654 147 m -9 Y s 676 147 m -9 Y s 697 156 m -18 Y s 719 147 m -9 Y s 740 147 m -9 Y s 762 147 m -9 Y s 783 147 m -9 Y s 805 156 m -18 Y s 827 147 m -9 Y s 848 147 m -9 Y s 870 147 m -9 Y s 891 147 m -9 Y s 913 156 m -18 Y s 934 147 m -9 Y s 956 147 m -9 Y s 977 147 m -9 Y s 999 147 m -9 Y s 1020 156 m -18 Y s gsave 1134 769 0 0 C 151.507 110.779 t 0 r /Helvetica findfont 29.3239 sf 0 0 m (1) show NC gr gsave 1134 769 0 0 C 257.399 110.779 t 0 r /Helvetica findfont 29.3239 sf 0 0 m (2) show NC gr gsave 1134 769 0 0 C 364.92 110.779 t 0 r /Helvetica findfont 29.3239 sf 0 0 m (3) show NC gr gsave 1134 769 0 0 C 472.441 110.779 t 0 r /Helvetica findfont 29.3239 sf 0 0 m (4) show NC gr gsave 1134 769 0 0 C 579.962 110.779 t 0 r /Helvetica findfont 29.3239 sf 0 0 m (5) show NC gr gsave 1134 769 0 0 C 689.112 110.779 t 0 r /Helvetica findfont 29.3239 sf 0 0 m (6) show NC gr gsave 1134 769 0 0 C 796.633 110.779 t 0 r /Helvetica findfont 29.3239 sf 0 0 m (7) show NC gr gsave 1134 769 0 0 C 904.154 110.779 t 0 r /Helvetica findfont 29.3239 sf 0 0 m (8) show NC gr gsave 1134 769 0 0 C 1011.68 110.779 t 0 r /Helvetica findfont 29.3239 sf 0 0 m (9) show NC gr 159 138 m 554 Y s gsave 1134 769 0 0 C 57.0187 317.676 t 90 r /Helvetica findfont 42.3568 sf 0 0 m ( /\(0.01 MeV\)) show NC gr gsave 1134 769 0 0 C 66.7934 303.014 t 90 r /Symbol findfont 29.3239 sf 0 0 m (g) show NC gr gsave 1134 769 0 0 C 57.0187 281.835 t 90 r /Helvetica findfont 42.3568 sf 0 0 m (n) show NC gr 171 172 m -12 X s 171 191 m -12 X s 171 205 m -12 X s 171 216 m -12 X s 171 225 m -12 X s 171 232 m -12 X s 171 238 m -12 X s 171 244 m -12 X s 183 249 m -24 X s gsave 1134 769 0 0 C 136.845 249.253 t 0 r /Helvetica findfont 19.5493 sf 0 0 m (6) show NC gr gsave 1134 769 0 0 C 104.263 236.22 t 0 r /Helvetica findfont 29.3239 sf 0 0 m (10) show NC gr 171 282 m -12 X s 171 302 m -12 X s 171 316 m -12 X s 171 327 m -12 X s 171 335 m -12 X s 171 343 m -12 X s 171 349 m -12 X s 171 355 m -12 X s 183 360 m -24 X s gsave 1134 769 0 0 C 136.845 358.403 t 0 r /Helvetica findfont 19.5493 sf 0 0 m (7) show NC gr gsave 1134 769 0 0 C 104.263 347 t 0 r /Helvetica findfont 29.3239 sf 0 0 m (10) show NC gr 171 393 m -12 X s 171 413 m -12 X s 171 427 m -12 X s 171 437 m -12 X s 171 446 m -12 X s 171 453 m -12 X s 171 460 m -12 X s 171 466 m -12 X s 183 471 m -24 X s gsave 1134 769 0 0 C 136.845 470.812 t 0 r /Helvetica findfont 19.5493 sf 0 0 m (8) show NC gr gsave 1134 769 0 0 C 104.263 457.779 t 0 r /Helvetica findfont 29.3239 sf 0 0 m (10) show NC gr 171 504 m -12 X s 171 523 m -12 X s 171 537 m -12 X s 171 548 m -12 X s 171 557 m -12 X s 171 564 m -12 X s 171 571 m -12 X s 171 576 m -12 X s 183 581 m -24 X s gsave 1134 769 0 0 C 136.845 581.591 t 0 r /Helvetica findfont 19.5493 sf 0 0 m (9) show NC gr gsave 1134 769 0 0 C 104.263 568.558 t 0 r /Helvetica findfont 29.3239 sf 0 0 m (10) show NC gr 171 615 m -12 X s 171 634 m -12 X s 171 648 m -12 X s 171 659 m -12 X s 171 667 m -12 X s 171 675 m -12 X s 171 681 m -12 X s 171 687 m -12 X s 183 692 m -24 X s gsave 1134 769 0 0 C 127.07 692.37 t 0 r /Helvetica findfont 19.5493 sf 0 0 m (10) show NC gr gsave 1134 769 0 0 C 94.4882 679.337 t 0 r /Helvetica findfont 29.3239 sf 0 0 m (10) show NC gr 0 1 0 c[ ] 0 sd 6 lw 1 1 1 c black 0 1 0 c 159 658 m 1 X -1 Y 2 X -1 Y 1 X -1 Y 2 X -1 Y 2 X -1 Y 3 X -1 Y 3 X -1 Y 1 X -1 Y 2 X -1 Y 2 X -1 Y 2 X -1 Y 2 X -1 Y 4 X -1 Y 2 X -1 Y 2 X -1 Y 2 X -1 Y 2 X -1 Y 4 X -1 Y 2 X -1 Y 2 X -1 Y 2 X -1 Y 3 X -1 Y 2 X -1 Y 4 X -1 Y 1 X -1 Y 4 X -1 Y 2 X -1 Y 3 X -1 Y 3 X -1 Y 1 X 1 Y 1 X -1 Y 1 X -1 Y 2 X -1 Y 4 X -1 Y 2 X -1 Y 4 X -1 Y 4 X -1 Y 2 X -1 Y 2 X -1 Y 2 X -1 Y 6 X -1 Y 3 X -1 Y 2 X -1 Y 3 X -1 Y 5 X -1 Y 3 X -1 Y 3 X -1 Y 3 X -1 Y 5 X -1 Y 2 X -1 Y 4 X -1 Y 4 X -1 Y 4 X -1 Y 4 X -1 Y 2 X -1 Y 4 X -1 Y 3 X -1 Y 1 X 1 Y 1 X -1 Y 4 X -1 Y 3 X -1 Y 5 X -1 Y 3 X -1 Y 4 X -1 Y 4 X -1 Y 5 X -1 Y 3 X -1 Y 1 X 1 Y 1 X -1 Y 2 X -1 Y 6 X -1 Y 2 X -1 Y 6 X -1 Y 5 X -1 Y 1 X -1 Y 1 X 1 Y 1 X -1 Y 4 X -1 Y 4 X -1 Y 6 X -1 Y 4 X -1 Y 6 X -1 Y 3 X -1 Y 1 X 1 Y 1 X -1 Y 1 X 1 Y 1 X -1 Y 1 X -1 Y 6 X -1 Y 4 X -1 Y 5 X -1 Y 3 X -1 Y 3 X 1 Y 1 X -1 Y 2 X -1 Y 2 X 1 Y 1 X -1 Y 2 X -1 Y 2 X 1 Y 1 X -1 Y 3 X -1 Y 4 X -1 Y 6 X -1 Y 5 X -1 Y 8 X -1 Y 4 X -1 Y 1 X 1 Y 1 X -1 Y 3 X -1 Y 4 X -1 Y 6 X -1 Y 1 X 1 Y 1 X -1 Y 4 X -1 Y 6 X -1 Y 5 X -1 Y 5 X -1 Y 8 X -1 Y 4 X -1 Y 1 X -1 Y 1 X 1 Y 4 X -1 Y 5 X -1 Y 3 X 1 Y 1 X -1 Y 2 X -1 Y 6 X -1 Y 3 X -1 Y 1 X 1 Y 2 X -1 Y 3 X -1 Y 2 X 1 Y 1 X -1 Y 3 X -1 Y 1 X 1 Y 1 X -1 Y 1 X 1 Y 1 X -1 Y 4 X -1 Y 3 X 1 Y 1 X -1 Y 2 X -1 Y 3 X -1 Y 4 X 1 Y 1 X -2 Y 2 X 1 Y 2 X -1 Y 1 X 1 Y 1 X -1 Y 1 X -1 Y 2 X 1 Y 1 X -1 Y 1 X 1 Y 1 X -1 Y 2 X -1 Y 3 X 1 Y 1 X -1 Y 4 X -1 Y 4 X -1 Y 5 X -1 Y 3 X 1 Y 1 X -1 Y 5 X -1 Y 6 X -1 Y 2 X 1 Y 1 X -2 Y 1 X 1 Y 2 X -1 Y 7 X -1 Y 3 X -1 Y 6 X -1 Y 1 X 1 Y 1 X -1 Y 2 X 1 Y 1 X -1 Y 3 X -1 Y 2 X 1 Y 1 X -1 Y 3 X -1 Y 7 X -1 Y 8 X -2 Y 1 X 1 Y 4 X -1 Y 1 X -1 Y 1 X 1 Y 4 X -1 Y 6 X -1 Y 2 X -1 Y 1 X 1 Y 4 X -1 Y 1 X -1 Y 1 X 1 Y 2 X -1 Y 3 X -1 Y 1 X 1 Y 3 X -1 Y 3 X -1 Y 1 X 1 Y 2 X -1 Y 1 X 1 Y 1 X -1 Y 5 X -1 Y 4 X -1 Y 2 X 1 Y 1 X -1 Y 5 X -1 Y 1 X 1 Y 1 X -1 Y 2 X -1 Y 2 X -1 Y 1 X 1 Y 2 X -1 Y 2 X 1 Y 2 X -1 Y 3 X -1 Y 2 X -1 Y 1 X 2 Y 1 X -1 Y 2 X -1 Y 7 X -1 Y 2 X -1 Y 2 X 1 Y 2 X -2 Y 1 X 2 Y 1 X -3 Y 1 X 3 Y 1 X -2 Y 5 X -1 Y 1 X 1 Y 1 X -1 Y 1 X 1 Y 1 X -1 Y 1 X -1 Y 1 X 1 Y 1 X -1 Y 5 X -1 Y 3 X -1 Y 4 X 1 Y 1 X -1 Y 1 X -1 Y 5 X -1 Y 5 X -1 Y 3 X -1 Y 1 X 1 Y 1 X -1 Y 2 X 1 Y 1 X -1 Y 1 X -1 Y 1 X 1 Y 1 X -1 Y 1 X 2 Y 1 X -3 Y 3 X -1 Y 1 X 1 Y 1 X -1 Y 3 X -1 Y 3 X 1 Y 1 X -2 Y 2 X 2 Y 1 X -3 Y 1 X 2 Y 1 X -2 Y 2 X 1 Y 1 X -2 Y 4 X 1 Y 1 X -2 Y 4 X -1 Y 3 X -1 Y 2 X -1 Y 3 X -1 Y 1 X 2 Y 1 X -2 Y 1 X 1 Y 1 X -1 Y 1 X -2 Y 1 X 1 Y 2 X -1 Y 1 X 1 Y 1 X -1 Y 1 X 1 Y 2 X -1 Y 3 X -2 Y 2 X -1 Y 2 X 1 Y 2 X -3 Y 1 X 1 Y 1 X 1 Y 1 X -2 Y 2 X -1 Y 3 X -1 Y 1 X 1 Y 1 X -2 Y 1 X 1 Y 1 X -2 Y 2 X -1 Y 1 X 1 Y 1 X -1 Y 1 X -1 Y 2 X 1 Y 1 X -2 Y 2 X -1 Y 1 X -2 Y 1 X 1 Y 4 X -1 Y 1 X -1 Y 1 X -1 Y 3 X 1 Y 1 X -3 Y 1 X -1 Y 1 X 1 Y 1 X -1 Y 1 X -1 Y 2 X -1 Y 1 X -1 Y 2 X -1 Y 2 X -1 Y 3 X -3 Y 1 X -1 Y 1 X 1 Y 2 X -1 Y 1 X -2 Y 2 X -1 Y 1 X -1 Y 1 X -2 Y 1 X -1 Y 1 X 1 Y 2 X -1 Y 1 X -3 Y 2 X -2 Y 2 X 1 Y 1 X -2 Y 1 X -3 Y 1 X -2 Y 2 X -1 Y 1 X -2 Y 1 X -1 Y 2 X -3 Y 1 X -3 Y 2 X -3 Y 1 X -3 Y 1 X -7 Y 1 X -5 Y 1 X -4 Y 1 X -9 Y 1 X -12 Y 1 X -12 Y 1 X -17 Y 1 X -26 Y 2 X -41 Y 1 X -162 Y 53 X s 1 0 0 c 1 1 1 c black 1 0 0 c 159 653 m 1 X -1 Y 2 X -1 Y 2 X -1 Y 1 X -1 Y 2 X -1 Y 3 X -1 Y 2 X -1 Y 2 X -1 Y 2 X -1 Y 2 X -1 Y 2 X -1 Y 2 X -1 Y 3 X -1 Y 2 X -1 Y 2 X -1 Y 2 X -1 Y 2 X -1 Y 3 X -1 Y 3 X -1 Y 2 X -1 Y 3 X -1 Y 2 X -1 Y 2 X -1 Y 2 X -1 Y 4 X -1 Y 2 X -1 Y 3 X -1 Y 2 X -1 Y 3 X -1 Y 4 X -1 Y 2 X -1 Y 2 X -1 Y 3 X -1 Y 3 X -1 Y 4 X -1 Y 2 X -1 Y 3 X -1 Y 3 X -1 Y 4 X -1 Y 3 X -1 Y 3 X -1 Y 3 X -1 Y 3 X -1 Y 4 X -1 Y 2 X -1 Y 4 X -1 Y 3 X -1 Y 4 X -1 Y 3 X -1 Y 3 X -1 Y 4 X -1 Y 4 X -1 Y 3 X -1 Y 4 X -1 Y 3 X -1 Y 5 X -1 Y 3 X -1 Y 4 X -1 Y 3 X -1 Y 4 X -1 Y 4 X -1 Y 4 X -1 Y 5 X -1 Y 3 X -1 Y 4 X -1 Y 4 X -1 Y 4 X -1 Y 5 X -1 Y 3 X -1 Y 5 X -1 Y 4 X -1 Y 3 X -1 Y 6 X -1 Y 4 X -1 Y 4 X -1 Y 4 X -1 Y 5 X -1 Y 5 X -1 Y 4 X -1 Y 5 X -1 Y 4 X -1 Y 5 X -1 Y 5 X -1 Y 5 X -1 Y 3 X -1 Y 7 X -1 Y 4 X -1 Y 6 X -1 Y 3 X -1 Y 7 X -1 Y 5 X -1 Y 4 X -1 Y 5 X -1 Y 5 X -1 Y 5 X -1 Y 5 X -1 Y 6 X -1 Y 4 X -1 Y 1 X 1 Y 1 X -1 Y 5 X -1 Y 5 X -1 Y 5 X -1 Y 4 X -1 Y 7 X -1 Y 3 X -1 Y 7 X -1 Y 5 X -1 Y 6 X -1 Y 4 X -1 Y 6 X -1 Y 6 X -1 Y 6 X -1 Y 5 X -1 Y 5 X -1 Y 6 X -1 Y 5 X -1 Y 6 X -1 Y 4 X -1 Y 6 X -1 Y 7 X -1 Y 4 X -1 Y 6 X -1 Y 5 X -1 Y 6 X -1 Y 6 X -1 Y 6 X -1 Y 6 X -1 Y 3 X -1 Y 7 X -1 Y 4 X -1 Y 4 X -1 Y 7 X -1 Y 4 X -1 Y 6 X -1 Y 4 X -1 Y 1 X 1 Y 1 X -1 Y 3 X -1 Y 6 X -1 Y 6 X -1 Y 3 X -1 Y 6 X -1 Y 4 X -1 Y 5 X -1 Y 4 X -1 Y 5 X -1 Y 3 X -1 Y 1 X 1 Y 1 X -1 Y 5 X -1 Y 3 X -1 Y 4 X -1 Y 5 X -1 Y 4 X -1 Y 2 X -1 Y 5 X -1 Y 4 X -1 Y 2 X -1 Y 5 X -1 Y 3 X -1 Y 4 X -1 Y 3 X -1 Y 3 X -1 Y 3 X -1 Y 3 X -1 Y 3 X -1 Y 3 X -1 Y 4 X -1 Y 3 X -1 Y 2 X -1 Y 1 X 1 Y 1 X -1 Y 1 X -1 Y 2 X -1 Y 3 X -1 Y 1 X -1 Y 5 X -2 Y 3 X -1 Y 3 X -1 Y 1 X -1 Y 3 X -1 Y 3 X -1 Y 1 X -1 Y 3 X -1 Y 1 X -1 Y 2 X -1 Y 1 X -1 Y 2 X -1 Y 1 X -1 Y 3 X -1 Y 1 X -1 Y 2 X -1 Y 2 X -1 Y 1 X -1 Y 2 X -1 Y 1 X -1 Y 1 X -1 Y 3 X -1 Y 2 X -2 Y 2 X -1 Y 1 X -1 Y 1 X -1 Y 1 X -1 Y 3 X -2 Y 2 X -2 Y 3 X -1 Y 1 X -2 Y 2 X -1 Y 1 X -1 Y 1 X -1 Y 1 X -1 Y 1 X -1 Y 1 X -1 Y 2 X -2 Y 1 X -1 Y 2 X -1 Y 2 X -2 Y 1 X -1 Y 1 X -1 Y 1 X -1 Y 1 X -1 Y 1 X -1 Y 1 X -2 Y 1 X -1 Y 2 X -2 Y 1 X -1 Y 2 X -1 Y 1 X -1 Y 1 X -2 Y 1 X -1 Y 1 X -1 Y 1 X -1 Y 1 X -2 Y 1 X -2 Y 1 X -1 Y 1 X -2 Y 1 X -1 Y 1 X -1 Y 1 X -2 Y 2 X -3 Y 2 X -3 Y 2 X -2 Y 1 X -2 Y 1 X -2 Y 1 X -1 Y 1 X -2 Y 1 X -3 Y 1 X -2 Y 1 X -1 Y 1 X -3 Y 2 X -3 Y 1 X -2 Y 1 X -1 Y 1 X -3 Y 1 X -3 Y 1 X -2 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\date{June 11, 2013} \maketitle %% When using the continuous spectrum of Bremsstrahlung photon beam to study photo-nuclear reactions, the measured quantity is a yield. The yield (number of interactions) is a convolution of the cross section with the Bremsstrahlung spectrum: \begin{equation} y(E) = \int_{Threshold}^{E} N_\gamma (E,k) \sigma(k) dk , \end{equation} \noindent where $E$ is the electron beam kinetic energy, $N_\gamma(E,k)$ is the number of gammas per energy unit which depends on the electron energy and the gamma energy. The continuous range of photon energies means that the cross section is not measured directly, instead it must be unfolded from the measured yields. An integral equation of this form is known as Volterra Integral Equation of the First Kind. Mathematically the problem is one of numerical solution of the yield integral equation and $\sigma(k)$ is the function to be solved for. One way to solve this equation is to use the Method of Quadratures (a method for constructing an approximate solution of an integral equation based on the replacement of integrals by finite sums). First the yields are measured at $E = E_1, E_2, \ldots, E_n$ where $E_i - E_{i-1} = \Delta$, $i=2, \ldots, n$. Then, \begin{equation} \label{eq_approx} y(E_i) = \int_{Threshold}^{E_i} N_\gamma (E_i,k) \sigma(k) dk \approx \sum_{j=1}^{i} N_\gamma (E_i, \Delta,k_j) \sigma(k_j) , \end{equation} \noindent where $N_\gamma (E_i, \Delta,k_j)$ is the number of gammas in the energy bin of width $\Delta$. Equation \ref{eq_approx} is a set of linear equations which can be written in the matrix form: \begin{equation} \begin{pmatrix} y_1 \\ y_2 \\ \vdots \\ y_n \end{pmatrix} = \begin{pmatrix} N_{11} & 0 & \cdots & 0 \\ N_{21} & N_{22} & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ N_{n1} & N_{n2} & \cdots & N_{nn} \end{pmatrix} \begin{pmatrix} \sigma_1 \\ \sigma_2 \\ \vdots \\ \sigma_n \end{pmatrix} . \end{equation} This matrix equation can be solved with matrix inversion. Equivalently, the solution to Equation \ref{eq_approx} can be written: \begin{equation} \label{eq_sol} \sigma_i = \frac{1}{N_{ii}} \left[ y_i - \sum_{j=1}^{i-1} (N_{ij} \sigma_j) \right] . \end{equation} The error propagation of Equation \ref{eq_sol} is given by: \begin{equation} \label{eq_error} \left( \frac{\mathrm d \sigma_i}{\sigma_i} \right)^2 = \frac{\left[ (\mathrm d y_i)^2 + \sum_{j=1}^{i-1}(N_{ij} \mathrm d \sigma_j)^2 \right]} {\left[ y_i - \sum_{j=1}^{i-1}(N_{ij} \sigma_j) \right]^2} . \end{equation} For mono-chromatic photon beam, Equation \ref{eq_error} reduces to: \begin{equation} \left( \frac{\mathrm d \sigma_i}{\sigma_i} \right)^2 = \left( \frac{\mathrm d y_i}{y_i} \right)^2 = \frac{1}{y_i}. \end{equation} \begin{figure}[h!] \centering \includegraphics[width=1.0\textwidth]{Sund_141Pr.eps} \caption{Comparison of the $^{141}\mathrm{Pr}(\gamma{},n)$ cross sections. Cook and Cannington were done with Bremsstrahlung beams. The correct cross section by Bramblett (dashed line) was done with a monochromatic gamma beam from positron annihilation in flight.}. \label{fig:Fig_Sund} \end{figure} Initially, the above unfolding method known as Penfold-Leiss unfolding (\cite{ref_PL}) (aka the Inverse-Matrix Method) gave unreliable results (see for example \cite{ref_Sund} and Figure \ref{fig:Fig_Sund}) because (in the sixties and seventies) the unfolding procedures have been often considered in isolation from the photon energy spectrum of the bremsstrahlung beam used experimentally. At that time, experimentalists used the Schiff theoretical formula (\cite{ref_Schiff}) to calculate $N_{ij} = \Delta N_{\mathrm{Schiff}}(E_i, k_j-\Delta/2)$. Findlay proposed (\cite{ref_Fin1}) that a simple modification to $N_{ij}$ prevents the generation of spurious results. He replaced $k - \Delta/2$ by $k - \lambda \Delta$ where $\lambda$ is a parameter determined by considering the energy spread of the electron beam and the energy loss of the electron beam in the radiator. Findlay's modification was successfully demonstrated to produce correct cross sections in (\cite{ref_Fin2}, see Figure \ref{fig:Fig_Findlay}, \cite{ref_Watson}). \begin{figure}[h!] \centering \includegraphics[width=0.7\textwidth]{Findlay_232Th.eps} \caption{Photofission of $^{232}\mathrm{Th}$ near threshold. Solid circles present data from \cite{ref_Fin2} with Bremsstrahlung beam and lines data with tagged photon beam.} \label{fig:Fig_Findlay} \end{figure} These days, there are very accurate Monte-Carlo simulations, $N_{ij}$ can be calculated for each specific experimental conditions without the need to use theoretical formula. This removes problems in the unfolding related to the knowledge of $N_{ij}$. However, this is not the only reason that may cause Penfold-Leiss unfolding to fail. Careful inspection of Equation \ref{eq_error} reveals that statistical errors of the measured yields play a role in two ways. First, the statistical errors add up as can be seen in the numerator of the right hand side of Equation \ref{eq_error}. Although $\sigma_1$ and probably $\sigma_2$ will be very closed to their real values, the remaining cross section data points will start to oscillate. Second, the denominator of the difference of two large numbers and thus will enhance the error in the cross section since the difference will be a smaller number. Indeed, having a very steep cross section is an advantage here, since it reduces the second term in the denominator and give a denominator with large number. To determine the required statistical error for each yield measurement, the steepness of the cross section must be taken into account. A relatively flat cross section requires very accurate yield measurements to be able to successfully unfold the cross section. Indeed, the $^{16}\mathrm{O}(\gamma{},\alpha)^{12}\mathrm{C}$ cross section is very steep (shown in Figure \ref{fig:Fig_csO16}) and only photons near the endpoint contribute to the yield for each beam energy, see Figure \ref{fig:Fig_Beam_Peak}. Similar arguments show that it is beneficial to maximize the number of gammas near the endpoint relative to the number of gammas at lower gamma energies, $N_{ii}/N_{ij}$, $j=1, \ldots, i-1$. Figure \ref{fig:Fig_Schiff} shows the Schiff Bremsstrahlung cross section for 8.5 MeV elctron beam kinetic energy. This is one reason, among many others, why we choose to run with a very thin Bremsstrahlung radiator. Figure \ref{fig:Fig_RadThick} shows the Bremsstrahlung yield for three different radiator thicknesses. \begin{figure}[h!] \centering \includegraphics[width=1.0\textwidth]{csO16.eps} \caption{The cross section of $^{16}\mathrm{O}(\gamma{},\alpha)^{12}\mathrm{C}$.} \label{fig:Fig_csO16} \end{figure} \begin{figure}[h!] \centering \includegraphics[width=1.0\textwidth]{Brem_Peak.eps} \caption{The probability of Bremsstrahlung photons to undergo the interaction $^{16}\mathrm{O}(\gamma{},\alpha)^{12}\mathrm{C}$.} \label{fig:Fig_Beam_Peak} \end{figure} \begin{figure}[h!] \centering \includegraphics[width=1.0\textwidth]{SchiffBrem.eps} \caption{Schiff Bremsstrahlung cross section for a Cu radiator.} \label{fig:Fig_Schiff} \end{figure} \begin{figure}[h!] \centering \includegraphics[width=1.0\textwidth]{Geant4_RadThick.eps} \caption{Bremsstrahlung yield for three different radiator thicknesses.} \label{fig:Fig_RadThick} \end{figure} %% As was discussed above, poor statistics will cause the unfolded cross section to oscillate as a function of photon energy especially at energies above the giant resonance energy where the cross section is flat or decreasing. These cross sections are unacceptable physically and smoothing must be used. Under the assumption of the cross section smoothness, the deconvolution method is known as the Regularization Method. There are several kinds of regularization methods such as Cook's Least Structure Method \cite{ref_Cook}, the Second Difference Method \cite{ref_Geller} and Tikhonov Regularization \cite{ref_Tikhonov}. %% Another deconvolution method is called the Photon Difference Method \cite{ref_Camp}. In this method a nearly mono-energetic photon spectrum can be constructed artificially by taking an algebraic sum of three Bremsstrahlung spectra with consecutive endpoint energies: \begin{equation} \phi_\mathrm{M}(E_i) = \phi_\mathrm{Schiff}(E_i) - a \; \phi_\mathrm{Schiff}(E_{i-1}) +b \; \phi_\mathrm{Schiff}(E_{i-2}), \end{equation} \noindent where $\phi_\mathrm{Schiff}(E_i)$ is the Schiff Bremsstrahlung spectrum with endpoint energy $E_i$ and the parameters $a$ and $b$ are both positive and chosen such that $\phi_\mathrm{M}(E_i)$ represent a mono-energetic photon spectrum. An example is shown in Figure \ref{fig:Fig_Schiff_PDM} where: \begin{equation} \phi_\mathrm{M}(8.5) = \phi_\mathrm{Schiff}(8.5) - 1.35 \; \phi_\mathrm{Schiff}(8.4) +0.30 \; \phi_\mathrm{Schiff}(8.3). \end{equation} \begin{figure}[h!] \centering \includegraphics[width=1.0\textwidth]{SchiffPDM.eps} \caption{Mono-energetic photon spectrum constructed artificially by taking an algebraic sum of three Bremsstrahlung spectra with consecutive endpoint energies.} \label{fig:Fig_Schiff_PDM} \end{figure} \noindent A differential yield spectrum can be constructed artificially by using the same linear combination of the corresponding yields. Then, the photo-nuclear cross section is simply the ratio of this differential yield to the corresponding mono-energetic photon flux. %% %% References: \begin{thebibliography}{10} \bibitem{ref_PL} A.~S.~Penfold and J.~E.~Leiss, Analysis of photonuclear cross sections. Phys. Rev. {\bf 114}, 1332 (1959). \bibitem{ref_Sund} R.~E.~Sund {\it et al.}, $^{141}\mathrm{Pr}(\gamma{},n)$ Cross Section from Threshold to 24 MeV. Phys. Rev. C {\bf 2}, 1129 (1970). \bibitem{ref_Schiff} L.~I.~Schiff, Energy-Angle Distribution of Thin Target Bremsstrahlung. Phys. Rev. {\bf 83}, 252 (1951). \bibitem{ref_Fin1} D.~J.~S.~Findlay, A Modification to the Penfold-Leiss Method of Cross Section Unfolding. Nucl. Inst. Meth. {\bf 213}, 353 (1983). \bibitem{ref_Fin2} D.~J.~S.~Findlay {\it et al.}, Photofission of $^{232}\mathrm{Th}$ near threshold. Nucl. Phys. A {\bf 458}, 217 (1986). \bibitem{ref_Watson} S.~J.~Watson {\it et al.}, Photofission and photoneutron measurements on $^{241}\mathrm{Am}$ between 5 and 10 MeV. Nucl. Phys. A {\bf 548}, 365 (1992). \bibitem{ref_Cook} B.~C.~Cook, Least Structure Solution of Photonuclear Yield Functions. Nucl. Inst. Meth. {\bf 24}, 256 (1963). \bibitem{ref_Geller} K.~N.~Geller and E.~G.~Muirhead, High Resolution Second difference Analysis of Photonuclear Yield Curves. Nucl. Inst. Meth. {\bf 26}, 274 (1964). \bibitem{ref_Tikhonov} A.~N.~Tikhonov {\it et al.}, About the determination of photonuclear reaction cross-sections. Vestn. Mosk. Univ., Ser. III. Fiz. Astron. 11: No. {\bf 2}, 208 (1970). \bibitem{ref_Camp} E.~Van Camp {\it et al.}, Experimental determination of the proton escape width in the giant dipole resonance $^{89}\mathrm{Y}$. Phys. Rev. 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