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dσT(θ)

dΩ= r2

e

1 + cos2 θ

2

ª@qu

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dσFF

dΩ=

dσT(θ)

dΩ[f(q)]2.

ª@q¬«F

f(q)NIKtBCA@T@NQxDFX KZ=@?dBkKZDF^wLOJWxDFXe^ÛxßBCJ:KZDFXcBCA@T

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q ≡ ~k − ~k′_HlM=@?*Xe?

~kBCA@T

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qLONiyFLu`a?*AEa[#KZ=@?

Xe?*RSBkKZLODFAq = 2(E/c) sin(θ/2)

ª@q ¡alM=@?*Xe?

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xBCA@T

qLONQKZ=@?dxDFRORODÖlMLuA@y@

x =q

2h= sin(θ/2)/λ

ª@q ªalM=@?*Xe?

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λKZ=@?P@=@DCKZDFAlQB\`a?*RO?*A@yCKZ=q

WNeLOA@yiKZ=@?J*DFA@J*?*PbKZN|DCxoJ:RSBCNeNeLOJ\BCRP@=a[HNZLuJ*N*_]KZ=@?^w?\BCA@LOA@yiDCx2KZ=@?dxDFXe^æxßBCJ:KZDFXMJ\BCA%E]?j@A@T@?*XVNIKZDHDmT/Ea[J*DFA@NeLOT@?*XeLOA@yWLOAmKZ?*XIx?*Xe?*A@J*? E]?:Kl ?*?*A/KZ=@?7J*DF=@?*Xe?*AmK2lQB\`a?*N2NeJ\BkKeKZ?*Xe?*TpEa[KZ=@?7T@LONIKtXeLOE@jbKZLODCADCxT@LuXZBCJ:KZLOA@yJ*?*AaKZ?*XeNQBCXeDFj@A@TiKZ=@?WAHj@J*RO?*j@N7DCxKZ=@?|BkKZDF^q2rWNQBJ*DFA@Ne?*mj@?*A@J\?C_HKZ=@?WJ*DF^wP@RO?n°wBC^pVP@ROLuKZj@T@?*N DCxKZ=@?gNeJ\BkKeKZ?*Xe?*T"lQB\`a?*Nc^/j@NIKQE-?dBCT@T@?*T­T@DFLOA@y/NeD;Kt=@?|Xe?*RSBkKZLu`a?WP@=hBCNe?dLOAbxDFXe^4BKZLuDFA4LONXe?:KtBCLOA@?*T] E-?:xDFXe?E]?*LOA@ywNemjhBCXe?*Tqc=@?wBkKZDF^wLOJ;xDFXe^ xßBCJ:KZDFXg^4BÖ[%E-?i?n°bP@Xe?*NeNe?*T¦BCNgKZ=@?w¯hDFj@XeLO?*X|KZXZBCA@NIxDFXe^ LOA£¡FÅNePhBCJ*?ÖgDCx

KZ=@?dKZDCKtBCRT@?*A@NeLuKY[ρ(r)

DCx2E-DCj@A@T?*RO?*J:KZXeDFA@N çMäd+ ³

f(q) =

∫

ρ(r) exp(iq · r) d3r.ª@q¬³F

¯hDFXB$xXZ?:?4BkKZDF^ lMLuKZ=­BNeP@=@?*XeLOJ\BkRORu[aVYNI[H^w^w?:KZXeLOJ/?*RO?*J:KZXeDFA£T@LONIKZXeLOE@jbKZLODFAρ(r)

_KZ=@?wBCA@yFj@RSBkXLOAaKZ?*yFXZBkKZLODFA"DCxKZ=@?P@Xe?:`HLODFj@NM?*mjhBkKZLODFAXe?*Nej@RuKZNMLOA

f(q) = 4π

∫

∞

0

ρ(r)sin(qr)

qrr2dr.

ª@q |j@?7KZDMKZ=@?7NeLO^wP@ROLOJ*LzKY[gDCx@KZ=@?fxDFXe^áxßBCJ:KZDFXBCP@P@XeDaBCJt=_FBMKZXe?*^w?*A@T@DCj@N2?:DFXIK=hBCNE-?*?*A;xDmJ*j@Ne?*T

DFA£^w?:KZ=@DmT@NdxDCXdJ\BCROJ*j@RSBkKZLuA@yf(q)

_j@NeLOA@y$KZ=@?pXe?*RSBkKZLu`HLONIKZLOJ;DFXgA@DFAHVYXe?*RSBkKZLu`HLONIKZLuJ;lQB\`F?pxj@A@J:KZLODFA@NJ*DFXeXe?*NeP-DFA@T@LOA@y¦KZD#KZ=@?­BkKZDF^wLuJDFXeE@LuKtBCRON*qc=@?£xDFXe^ xßBCJ:KZDFX"J\BCAÄE-?£Xe?*P@Xe?*Ne?*AaKZ?*T¢Ea[>B#Nej@^DCxKZ?*Xe^wN

fn(q)J*DFXeXe?*NeP-DFA@T@LOA@y¦KZD¨KZ=@?J*DFAmKZXeLOE@jbKZLODFA¥DCx?\BCJt=ÄBkKZDF^wLOJ%Nej@E@Ne=@?*ROR

nqZYfBCJn=>DCx

KZ=@?*Ne?£KZ?*Xe^wN"NIKtB\[HN%BCRO^wDFNIK$J*DFA@NIKtBCAmK¨?*mjhBCRgKZD¦KZ=@?£Amj@^/E-?*X"DCx;?*RO?*J:KZXeDFA@NLOAsKZ=@?£Nej@E@Ne=@?*ROR±xDFXRODÖl`kBCROj@?*NDCx

qBCA@T£KZ=@?*A®NIKtBCXIKZNKZDT@XeDFP­DFA@J*?

qLONRSBCXeyF?*XgKt=hBkA¨B"KZ=@Xe?*Ne=@DFROT£`CBCROj@?%±KZ=@?

J*DFXeXe?*NeP-DFA@T@LOA@yKZ=@Xe?*Ne=@DFROT­`kBCROj@?iDCx1/q

LONP@XeDFP-DFXIKZLODFAhBCR2KZDKZ=@?iNeLO²*?wDCx KZ=@?wNej@E@Ne=@?*ROR±nq4c=mj@N*_xDFXe^õxßBCJ:KZDFXeNgDCx7DFjbKZ?*XdNe=@?*RORONgT@?*J*Xe?\BCNe?i?\BCXeROLO?*X*_BkKdNe^4BCRORu?*X|^wDF^w?*AaKZj@^õKZXZBCA@NIx?*XeN*_lM=@LORO?;xDFXe^xßBCJ:KZDFXeNiDCx|LOA@A@?*XiNe=@?*RORONiT@XeDFP§BkKwRSBCXeyF?*Xp^wDF^w?*AaKZj@^ KZXZBCA@NIx?*XeN% Þ©

+ ³ q#r|N4B£Xe?*Nej@RuK\_fKZ=@?Æ

ÇCÈmÉhÊmËCÌCÍdÎeÏWÐeÑÓÒ|ÔÖÕ*ÉHËkÑS×HØ/×mÙMÊCÈ]×hËm×HØÐ:ÇÉHËmËCÌCÍmÑzØ]ÚKZDCKtBCRQxDCXZ^ xßBCJ:KZDFXiLONwB^wDFA@DCKZDFA@?"xj@A@J:KZLODFA_flM=@LOJt=T@?*J*Xe?\BkNZ?:N4xXeDF^

f(0) = Z_7lM=@?*Xe?>LON

KZ=@?$BkKZDF^wLOJpAHj@^/E-?*XDCxQKZ=@?4J*DFA@NeLOT@?*Xe?*T¦BkKZDF^_9KZDf(∞) = 0

_lMLuKZ=®Ne?:`F?*XZBCR LOA[h?*J:KZLODFA­P-DFLOAaKZN߯2LOy@qbª@qunqvc=@?^4B°bLO^/j@^Å`CBCROj@?

ZLONcDFEbKtBCLOA@?*T"lM=@?*AKZ=@?NeJ\BkKeKZ?*Xe?*TlcB\`F?*N|BCXe?BCRORLOAP@=hBkNZ?kq

c=@LONMJ*DFA@T@LuKZLODFA"LONcNZBkKZLONI¤h?*TlM=@?*Aθ = 0

q

0

5

10

15

20

25

30

1e+05 1e+06 1e+07 1e+08 1e+09 1e+10

Al

Fe

f

x \^]`_ −1 a¯2LOyFj@Xe?|ª@quFcb d e4 4e/f9ghifjGkhjlf9d mnjpof9d

f q jlf9dreGd`f9gs gutv xwymzeGgeGwym q -h|jwygo4e/f9gfjGkhmvf9m4g4wym~4d# gh4j*dR d# m4Yd

x owyd Chif 4Y`twheGgHGkh)lwy Y`t9 xwhhfjvGkhjlf9d mjpof9di 9 eGlOreGgsGkhC9 xwR Y`tckf9f9g )z@e4ld# d WRlAR4/F

r|NgBiJ*DFA@Ne?*mj@?\A@J*?/DCxoKZ=@?`CBCXeLSBKtLuDFA@NMDCx2KZ=@?xDFXe^æxßBCJ:KZDFX*_dσFF/dΩ

LONMNIKtXeDFA@yCRu[P-?\BC~C?*TLOAKZ=@?|xDFXIlQBCXeT$T@LOXe?*J:KZLODFA߯2LOy@q2ª@q¬«FnqvD;KtBC~C?gLOAaKZDpBCJ*J*DFj@AaKQKZ=@?g?:?*J:KcDCx9NeDFROLOT4BCA@yFRO?C_bLuK J\BCA"BCRONeDE-? =@?*ROPbxj@RHKZDgJ*DFA@NeLOT@?*XKZ=@?QT@Lu?*Xe?*AaKZLSBCRzVåLOAHVYBCA@yFRu?7J*XeDFNeNoNe?*J:KZLODFA

dσFF/dθXZBkKZ=@?*X2KZ=hBCA

dσFF/dΩ

dσFF

dθ=

dσFF

dΩ

dΩ

dθ=

dσFF

dΩ2π sin θ.

ª@q

K"J\BCAsE-?A@DCKZ?*T¥KZ=hBkK dσFF/dθ = 0xDFX

θ = 0_MT@j@?KZD®KZ=@?P@Xe?*Ne?*A@J*?£DCx

sin θLOAKZ=@?

?n°bP@Xe?*NeNeLODFAsDCxNeDFROLOT¥BCA@yFRO?Cqác=@?yFXZBCP@=@LOJXe?*P@Xe?*Ne?*AaKnBKtLuDFAsDCxdσFF/dθ

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E-?cKtBC~C?*A4LOAaKZDBCJ*J*DFj@AaK Ea[iXe?*P@ROBCJ*LOA@ydKZ=@?cxDFXe^xßBCJ:KZDFXfEa[wBd^wDHT@Lu¤h?*TiXe?*RSBkKZLu`HLONIKZLuJ xDFXe^xßBCJ:KZDFX

©WÞ Fk

g(q) = 4π∑

i

∫

∞

0

ρi(r)sin(qr)

qr

[

mc2

Ei − V (r)

]

r2dr,ª@q Æ

lM=@?*Xe?Ei

LONdKZ=@?iKZDCKtBCRo?*A@?*XeyC[¥LOA@J*ROj@T@LOA@y"Xe?*NIK;^4BCNeN?*A@?*XeyC[@dDCx KZ=@?iVåKZ=­?*RO?*J:KZXeDFA®BCA@T

V (r)LONcKZ=@?;J*DF^w^wDFAJ*?*AmKZXZBCR9P-DCKZ?*AaKZLSBCR9BkKgBwP]DFNeLuKZLODFArT@j@?KZDwKZ=@?Amj@J*RO?*j@NgBCA@TKZ=@?DCKZ=@?*X|BkKZDF^wLOJ

?*RO?*J:KZXeDFA@N*qÆFÆ

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15

10

5

0

5

10

15

300 keV

100 keV

30 keV

10 keV

¯2LOyFj@Xe?$ª@q¬«H|f9 ddZudZhÖg) 4e/f9gfjdσFF/dΩ eGg d ghihdvh d#t e4 g zjlf9dh%o` 4Yd eGgHfj

wyg f9 d e`tid#t e4 4e/f9g q jlf9d|eGd`f9g Z = 26 q o` 6owy Y`tiwheGgHÖ-6 v gutiGkh)lwy Y`t9 xwhh

fjGkhjlf9d mUjpof9d 9 eGlOeGgGkhzt )-hÖWR¡;kh@¢£nof9m4hmvf9dZz gut¤mvf9dZjlf9d ¥> d#t ¦6h`-§a`tc¥@khgHGkhpgdí ceGgodZ`-hÖh\

0

20

40

60

80

100

0 20 40 60 80 100 120 140 160 180

10 keV

30 keV

100 keV

dσ

FF/d

θ

¨ ©ª6«¬­®

θ \°¯ a¯2LOyFj@Xe?pª@q ¡bbR d e4 4e/f9ghfj

dσFF/dθ eGg± d ghhdd#t e4 g -h¤ijwygo4e/f9g²fjzGkhch%o` 4Yd eGgH gH O eGg³tmå dZth q jlf9d´ 9Oeu-kµh%o` 4Yd eGgH£fjvwyg f9 d e`tsd#t e4 4e/f9g q jlf9d¤eGd`f9g Z = 26

qo` 6owy Y`twheGgH|Ö -) gutGkh>)lwy Y`t9 xwhhfjWGkhjlf9d m£jpof9dC 9 eGlOeGgzGkht -hÖWR

Æ

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LONQKZ=Hj@NMXe?*P@RSBCJ*?*TlMLzKt=F = g(q) + g′(ω) + ig′′(ω)

ª@q lM=@?*Xe?

g′BCA@T

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θNe?*?$\Óa9 Qf ³ n

ECo =E

1 + κ(1 − cos θ)

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eß\κ = E/mec

2.ª@quF

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0.01

0.1

1

10

0 20 40 60 80 100 120 140 160 180

30 keV

100 keV

300 keV

1 MeV

3 MeV

10 MeV

EC

o

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θ \°¯ a¯2LOyFj@Xe?Wª@q ª@>Cgdí

ECo

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dσKN(θ, E)

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2

(

ECo

E

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ECo

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E

ECo

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EComax= E − EB

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x < 30 nm−1_KZ=@?*Xe?pBCXe?;x?:lThBkKtB"BCA@T£BCT@T@LuKZLODFAhBCR9BCJ*J*j@XZBkKZ?p^w?\BCNej@Xe?*^w?*AaKZNdBCXe?

A@?*?*T@?*Tq£gBkKtBxDFX;DCKZ=@?*XKtBCXeyF?:KZN/P@Xe?*Ne?*AaKpNZLu^wLORSBkXdKZXZ?:A@T@N\q©M?*J*?*AaKZRu[F_oKZ=@?$RSBkXZyC?*X/B\`kBCLORSBCE@LORuLzKY[DCxWNI[HA@Jn=@XeDCKZXeDFA¦°mVYXZB\[¨E-?\BC^wNp=hBCNpyFLu`F?*A#A@?:l LOA@NeLOyF=aKZN*_fDFA§BCJ*J*DFj@AaKiDCxWKZ=@?*LOX/=@LOyF=¦LOAmKZ?*A@NeLuKY[F_KZj@AhBCE@LOROLzK[F_Ne^4BCRORfBCA@yFj@RSBCXT@Lu`a?*XeyF?*A@J*?$BCA@T®`CBCXeLSBCE@RO?iP-DFRSBCXeLO²\BkKZLuDFA£P@XeDFP-?*XIKZLO?*N*q A¨=@LON;Xe?:`HLO?:lDCx ?n°bLONIKZLOA@yLOAbxDFXe^4BkKZLuDFADFAKZ=@?iLOA@J*DF=@?*Xe?*AaK;NeJ\BkKeKZ?*XeLOA@yDCx P@=@DCKZDFA@N*_2çWj@E@E]?*RORvJ*DFA@J*ROj@T@?*NKZ=hBkKKZ=@?pLOA@J*DF=@?*Xe?*AaKdNZJ*BkKeKt?:XZLuA@y$xßBCJ:KZDFX*_T@?*NeP@LuKZ?pLuKZNgNe=@DFXIKZJ*DF^wLOA@yFN*_LONgNIKZLOROR2B$j@Ne?:xj@R2KZDHDFRxDFX|XZBCT@LSBVKZLODFA4KZXZBCA@NeP-DFXIKcJ\BCRuJ*j@RSBkKZLODFA@N*qvçM?BCNeNe?*XIKZNQKZ=hBkKQxDFX7xDFXIlcBCXeT$NeJ\BkKeKZ?*XeLOA@yiBCA@yFRO?*NQBCA@T"xDCXQRODÖlÄBCA@T^w?*T@LOj@^ ^4BkKZ?*XeLSBCRuN*_HKZ=@?A@DFAHVYXe?*RSBkKZLz`HLONIKZLuJ|KtBCE@j@RSBkKZLODFA@NQDCx

S(x, Z)BCXe?BCJ*J*j@XZBkKZ?dKZD/KZ=@?DFXeT@?*X

DCx ³ÓÆ DFXWE-?:KeKZ?*X*_BCA@T%KZ=hBkK|xDFXWRSBCXeyF?;BCA@yFRO?*NgBCA@T=@LOyF=­`CBCROj@?*NWKZ=@?*Ne?;KtBCE@j@RSBkKZ?*T%`kBCROj@?*Ng^4B\[E-?dKZDmDi=@LOyF=Ea[BCNc^pj@Jt=BCN|«CÆ çWj@E ak q DFA@J*?*XeA@LOA@y4KZ=@?;LO^wP@j@RuNe?;BCP@P@XeD\°bLO^4BkKZLuDFA%BCA@T%Xe?*RSBkKZLu`HLONIKZLuJLO^wP@j@RONe?BCP@P@XeD\°bLO^4BkKZLODCA_]ÞBCA@?'e4le4taÓdP@XeDFP-DFNe?*N9KZ=@?vxDFRORuDÖlMLOA@yQJ*DFA@T@LuKZLODFA@NDCxH`CBCROLOT@LzK[F_ÖEhBCNe?*TDFAT@Lu?*Xe?*AaK?n°bP]?*XeLO^w?*AaKtBCRaXe?*Nej@RuKZNBCA@TKZ=@?*DFXe?:KZLOJ\BCRJ*DFA@NeLOT@?*XZBkKZLODFA@N*

E ≥ 6EB

E − ECo ≥ EB

2EB < E ′ < E − EB

ª@qu³F

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DF^wPbKZDFAROLOA@?ECo

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Æ

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BCA@Tx?*Xe^wLOj@^ÛZ = 100

n_kDÖ`F?*XBclMLOT@?fXZBCA@yF?vDCxxq2QBCNe?*T

DFAçWj@E@E-?*RORíà NXe?*J*DF^w^w?*A@ThBkKZLODFAgKZ=@?vxDFXe^ÄxßBCJ:KZDFXeN9LOA@J*ROj@T@?*TLOAYffgÜ a BCXe?fA@DFAHVYXe?*RSBkKZLu`HLuNIKZLOJCq AKZ=@?*DFXI[F_mLuKol DFj@ROTi=hBÖ`F?ME]?*?*A4^wDFXe?QJ*DFA@NeLONIKZ?*AmKvKZDj@Ne?WXe?*ROBkKZLu`HLONIKZLOJ7xDFXe^xßBCJ:KZDFXeNolMLuKZ=wBCA@DF^4BCRODFj@NNeJ\BkKeKZ?*XeLOA@y"xßBCJ:KZDFXeNgKZDT@?:¤hA@?pKZ=@?iJ*DF=@?*Xe?*AaKNeJ\BkKeKZ?*XeLOA@yJ*XeDFNeNNe?*J:KZLODFA@N*qiçWDl ?:`F?*X*_LOAP@XZBCJ:KZLOJ*?C_?n°HKZ?*A@NeLu`F?;P@=@DkKtDCAKZXZBCA@NeP-DFXIKWJ*BCROJ*j@RSBkKZLODFA@NQ=hB\`a?d`F?*XeLu¤h?*TçWj@E@E]?*RORíà NMXe?*J*DF^w^w?:A@ThBkKZLODFA"KZ=hBkKcxDFXj@Ne?iLOAP@=@DCKZDFAKZXZBCA@NeP-DFXIKBCP@P@ROLOJ\BkKZLuDFA@N*_LuKgLONdE]?:KeKZ?*XdKZD"j@Ne?i=@LONgA@DFAHVYXe?*RSBkKZLu`HLuNIKZLOJxDFXe^ xßBCJnKtDCXZN Qç|Þ a q A¨KZ=@?4xDFRORODÖlMLuA@y%Jt=hBCPbKZ?*X*_vKZ=@?$NeLO^/j@RSBkKZLuDFA­Xe?*Nej@RuKZN/DFEbKtBCLOA@?*T®lMLuKZ=®KZ=@?$T@?:KZ?*Xe^wLOA@LONIKZLOJ4BCPHVP@XeDaBCJt=$lMLOROR]E-?gJ*DF^wPhBCXe?*T4KZD;KZ=@?|KZ=@?gXe?*Nej@RuKZNQyFLu`F?*A$Em[wKZ=@?ãDFAaKZ?QBCXeROD;J*DHT@?zõ@ö-q KcLONl DFXIKZ=A@DCKZLOA@y;KZ=hBkKQl ?dj@Ne?dKZ=@?dNZBC^i?dP@=a[HNeLOJ\BCR^wDHT@?:RON xDFXQNeJ\BkKeKZ?*XeLOA@yiBCA@T$KZ=@?dNZBC^w?gThBkKtBCEhBCNe?

BCNzõ@ö-qdc=@?*Xe?:xDFXe?C_KZ=@?/J*DF^wPhBCXeLONeDFAlMLORORE-?;xDHJ*j@Ne?*T£DFA%KZ=@?pAmj@^w?*XeLOJ\BCR2BCP@P@XeDaBCJt=DFA@Ru[T@?:KZ?*Xe^wLOA@LONIKZLOJ|`F?*XeNej@NWP@XeDFEhBCE@LOROLONIKZLuJÖnq

êWëêWëíì ð|ñoøbñfôõóHö÷oømøbùú@ûô|üÅñWÅ÷7ñ gòù¢ù ù9òùô7ø ?:xDFXe?WJ:DFA@NeLOT@?*XeLOA@yJ*DF^wP@RO?n°i^4BCJ*XeDFNeJ*DFP@LOJcNZBC^wP@RO?*N*_mLuK7LONvA@?*J*?*NeNtBCXI[iKZDNIKZj@Tb[i¤hXeNIKfKZ=@?WNeLO^wP@RO?*XJ\BCNe?dDCx9P@=@DCKZDFA"NeJ\BKZKZ?*XeLOA@y/DCB/Ne^4BCRORh`FDFROj@^w?|?*RO?*^w?*AaK±`FD\°b?*R±7RODHJ*BkKZ?*T"LOA4KZ=@?gLOA@J*LOT@?*AaK7°mVYXZB\[E-?\BC^qo<>=hBkKcxDFRORODÖlMNcT@Dm?*NMA@DCKWT@?*P-?*A@T%DCAKZ=@?dKY[HP-?DCx2NeJ\BkKeKZ?*XeLOA@yß©WB\[HRO?*LOyF="DFXW DF^wPbKZDFA]nq

DFA@NeLOT@?*XiBP@=@DCKZDFA¦E-?\BC^ ?*^wLuKeKZ?*T¦Ea[¦BNeDFj@XeJ*?"P-DFLOAaKS_fLOA@J*LOT@?:AmKpDFA#B`FD\°b?*Ríq¨Ü?:Kij@N

J\BCRORAKZ=@?%J*?*AaKZ?*X$DCxd^4BCNeNwDCxgKZ=@?`FD\°b?*RgBCA@T¥BCNeNej@^w?KZ=hBkK4KZ=@?NeLO²*?%DCxgKZ=@?`FD\°b?*R|LONw^/j@Jt=

Ne^4BCRORO?*XiKZ=hBCA¥KZ=@?T@LONIKtBCA@J*?SA

ßLzK$KZ?*A@T@N$KZD¦²*?*XeD@_MlM=@LORO?SA

Xe?*^4BCLOA@Nw¤hA@LuKZ?Ön_WNeD¨KZ=hBkK$KZ=@?P@=@DCKZDFAE-?\BC^ J*DFA@NeLONIKZN4DCxgmjhBCNeLzVYPhBCXZBCRORu?*RcXZB\[HN*q¥<®?%J\BCROR

dSKZ=@?J*XeDFNeN4Ne?*J:KZLODFA¥BCXe?\B­DCx|KZ=@?

LOA@J*LOT@?*AaKwE]?\BC^ BCA@TdNi(E)

KZ=@?KZDCKtBCRQAHj@^/E-?*XiDCxgP@=@DCKZDFA@Ni?*^wLuKeKZ?*T#Em[¨KZ=@?NeDFj@XeJ*?T@j@XeLOA@yKZ=@?i?n°bP-?*XeLO^w?*AaK$߯2LOy@qª@q nqic=@?iAHj@^/E-?*XDCx P@=@DCKZDFA@N*_NeJ\BkKeKZ?*Xe?*T­KZDÖlQBCXeT@N;B"T@?:KZ?*J:KZDFXLOAKZ=@?

A

dΩ

P

dS

¯2LOyFj@Xe?dª@q Ckf9f9gh%o` 4Yd eGgHfëUv f9xwym4/Om4gåÖF

ÎeÏ ÎeÏWÐeÑÓÒ|ÔÖÕ*ÉHËkÑS×HØp×mÙQÙÑzÍHÐ:Ë9QO×HÍ@ÌCÍdÊCÈ]×hËm×HØÐ:ÇÉHËmËCÌCÍmÑzØ]ÚT@LOXe?*J:KZLODFA

Ω = (θ, φ)BCA@TLOA"KZ=@?LOAb¤hA@LzKt?:NZLu^4BCRNeDFROLOTBCA@yFRO?

dΩ_hLONQKZ=@?P@XeDHT@j@J:KWDCx

dNi

Em[$KZ=@?P@XeDFEhBCE@LOROLzKY[

dpxDFXMBpP@=@DCKZDFAKZDiE-?NeJ\BkKeKZ?*Xe?*TLOAmKZD

dΩ

d2Ns = dNidpª@qu

lMLuKZ=dp =

dNatdσat

dSdΩdΩ

ª@qu lM=@?*Xe?

dNat

LONKZ=@?$Amj@^/E-?*X;DCxWBkKZDF^wNLOA­KZ=@?4J*DFA@NeLOT@?*Xe?*T¨`aD\°b?*RQBCA@Tdσat/dΩ

LONKZ=@?$BkKZDF^wLOJd J*DFXeXe?*NeP-DFA@T@LOA@ygKZDdNeJ\BkKeKZ?*XeLOA@ygLOAmKZD

dΩq1Yv]qFª@qu J\BCApE-?Qj@A@T@?*XeNIKZDHDHTp?\BCNeLORu[;Ea[/Xe?*^4BCXe~mLOA@yKZ=hBkK

dpLON|?*mjhBCRKZD4KZ=@?;KZDCKtBCR2BCXe?\B4DCxvKZ=@?;KtBCXeyF?:KdBkKZDF^wNWT@Lu`HLOT@?*TEa[KZ=@?/E-?\BC^ J*XeDFNeNgNe?*J:KZLODFA

BCXe?\Bbqvc=@?Amj@^/E-?*XMDCx2P@=@DCKZDFA@N*_hNeJ\BkKeKZ?*Xe?*TLOAmKZDdΩ_@J\BCAKZ=mj@NWE]?dlMXeLuKeKZ?*A%BCN

d2Ns(E′) =

dNi(E)

dSdNat

dσat

dΩdΩ = Φ(E)dNat

dσat

dΩdΩ

ª@qu Æ lM=@?*Xe?

E ′NIKtBCA@T@NxDFXKZ=@?vNeJ\BkKeKZ?*Xe?*T;P@=@DCKZDFA?*A@?*XeyC[w

E ′ = ELOAgKZ=@?fJ\BCNe?fDCx@©WB\[HRO?*LOyF=dNeJ\BkKeKZ?*XeLOA@y@_

E ′ = ECo

LOA/KZ=@?QJ\BCNe?cDCx DF^wPbKZDFApNeJ\BkKeKZ?*XeLOA@y! n2BCA@TΦ(E)

LON2KZ=@?cAmj@^/E-?*XoDCx-LOA@J*LOT@?*AaKvP@=@DCKZDFA@NP-?*XMj@A@LuK|BCXe?\Bbq ©M?*^4BCXe~kBCE@Ru[F_bKZ=@?P@Xe?:`HLODFj@NMJ\BCROJ*j@RSBkKZLODFA"DCx

d2Ns(E′)T@Dm?*NWA@DCK|T@?:P-?*A@TDFAKZ=@?

Ne=hBCP-?DCx|KZ=@?`aD\°b?*R/P@XeDÖ`HLOT@?*TLuKZNiT@LO^w?*A@NeLODCA@NiXe?*^4BCLOA#Ne^4BCROR±;E@jbKwDFA@Ru[#DCA§KZ=@?Amj@^pE]?*XwDCxBkKZDF^wNQlMLuKZ=@LOA$KZ=@?d`FDÖ°b?*R#"*qêWëêWë4S · üfñvú@ûåøbðgò ÿtñvú®øbð|ù>óHûIò ÷oøbûñfôñfÿ

1stTkñvúR|ùú£Vgð|ñoøbñfô óHö÷oømøbùú@ûô|ü

$&%(')!*,+.-,%0/!1-,%024352768-,9+:;*,27<>=0+?DNeLO^pj@RSBkKe?QP@=@DCKZDFAwNeJ\BkKeKZ?*XeLOA@yLOAiB¡C>^4BCJ*XeDFNeJ*DFP@LOJcDFEHGI?*J:K\_al ?MT@Lu`HLOT@?QKZ=@?MDFEHGI?*J:K7LOAaKZDNe^4BCRORPhBCXZBCRORO?*Ru?*P@LOP-?*T@N*_-BCNWKZ=@?:[BCXe?KZ=@?;NeLO^wP@RO?*NIKW`FDFROj@^w?*N|?*AhBCE@ROLOA@yiKZD$J*DÖ`F?*X|KZ=@?lM=@DFRO?;DFEHGI?*J:K\q AP@XZBCJ:KZLOJ*?C_fBPhBCXZBCRORO?*Ru?*P@LOP-?*T¨LOA@J*ROj@T@LOA@yKZ=@?"DFEHGI?*J:KiLON;Jn=@DFNe?*A_vKZ=@?*A#LuK/LON/T@Lu`HLOT@?*T¦LOA¨LuKZN;KZ=@Xe?*?T@LO^w?*A@NeLODFA@N*_mLOAmKZDNe^4BCRORO?*Xv`aDCROj@^w?*N =hB\`HLOA@yKZ=@?gNZBC^w?WNe=hBCP-?CqYfBCJn=$DFEbGI?:J:KcNeROLOJ*?/í«Cá^4BkKZXeLz°iDCx`FDÖ°b?*RONtn_J*DFA@NIKZLuKZjbKZ?*NglM=hBkKgl ?pJ\BCROR2B%Á `HLOXIKZjhBCRT@?:KZ?*J:KZDFX*Ãu_lM=@LOJt=£Xe?*J*?*Lu`F?*NdLOA@J*LOT@?*AaKdP@=@DCKZDFA@NdBCNBwXe?\BCRP@RSBCA@?T@?:KZ?*J:KZDFX*_?t°bJ*?*PbKdKZ=hBkK|LuKWLONWRODHJ\BkKZ?*TlMLuKZ=@LOAKZ=@?;NeJ\BkKeKZ?*XeLOA@y4DFEbGI?*J:K\q|gA@J*?`aD\°b?*RON=hB\`a?E-?*?:A%T@?:¤hA@?*T_hKZ=@?NeLO^/j@RSBkKZLuDFA"DCx

stVYDFXeT@?*XcNeJ\BkKeKZ?*XeLOA@yiJ\BCAKtBC~C?P@RSBkJ*?C_]BCNcxDFRORuDÖlMN*

FqMXZB\[HNWBCXe?J\BCNIKMxXeDF^KZ=@?NeDFj@XeJ*?P]DFLOAmKQKZDlQBCXeT@NcKZ=@?J*?*AaKZXe?DCx2?\BCJt=`FDÖ°b?*R/¹«HqcxXeDF^?\BCJt=Á¬BCJ:KZLu`F?CÃ`FD\°b?*Rg'eß\9RODHJ\BkKZ?*T­LOA@NeLOT@?pKZ=@?wDFEHGI?*J:Knn_2XZB\[HN;BCXe?wNe?*AaKKZDlQBCXeT@NKZ=@?J*?*AaKZXe?DCx2?\BCJt=P@Lz°b?*RDCxKZ=@?Xe?\BCRT@?:KZ?*J:KZDFX*q

c=@LON^w?:KZ=@DHT­LONLORORuj@NIKZXZBkKZ?*T£Ea[­¯2LOy@qª@quÖbq4 ?*NeLOT@?*NKZ=@?wNePhBkKZLSBCRvT@LONeJ*Xe?:KZLO²*BkKZLODFA£DCx KZ=@?wDFEHGI?*J:KBCA@T"T@?:KZ?*J:KZDFX*_h?*A@?:XZyk["Jt=hBCA@A@?*RONcJ\BCA"NIKZLOROR-E-?gj@Ne?*TKZDiBCJ*J*DFj@AaKcxDFX7KZ=@?dNeP-?*J:KZXZBCRT@LONIKZXeLOE@jbKZLODFA4DCxLOA@J*LOT@?*AaK\_KZXZBCA@Ne^wLuKeKZ?*T­BCA@T£NeJ\BkKeKZ?*Xe?*T®P@=@DCKZDFA@N*qpDDFEbKtBCLOA`CBCROLOTXe?*Nej@RuKZN*_KZ=@?iNeLO²*?pDCx7`aD\°b?*RON*_P@Lz°b?*RONBCA@Tá?*A@?*XeyC[>Jn=hBCA@A@?*RON=hBCNKZD§E]?®Jt=@DFNe?*AálMLuKZ=ÄJ\BCXe?C_gKZXI[HLOA@y#KZD#¤hA@T¢B#J*DF^wP@XeDF^wLONe?E-?:KYl ?*?*A­BCJ*J*j@XZBCJ:[BCA@TJ*DF^iP@jbKnBKtLuDFAKZLO^w?Cqcc=@?/J*DFAa`a?*XeyF?*A@J*?pDCxoKZ=@?;Xe?*Nej@RuKZN|DFEbKtBCLOA@?*TlM=@?*AKZ=@?gT@LONeJ*Xe?:KZLO²\BkKZLODFA$LON ^4BCT@?W¤hA@?*XQlMLOROR]E-?gT@LONeJ*j@NeNe?*TlMLuKZ=$J*DFA@J*Xe?:KZ?d?n°@BC^wP@RO?*NQLOA ?*J:KZLODFA"ª@q¬³Hq¬«Hq@BA2/24* ,+-* +-,./<-+-!/,L1 I9./<-Ðu=Ñ%./+>5 * +-u.)./<-;2/&$.4./!/#(Ñ%<- ./+>*32$ +>$ Ñ-Ñ-!/`Ô%* =$./* +p"`K<-*3&)<>+-,5 #&.)2

C Ñ-Ñ-5 !K?-!/l$(%+-* +-,9MD C * ÒÓ!/+l.W `Ô%5R24<9$ ÑÓ#2@$ !/./<78921&'+9&'* $ ?-5 FEÑ9$ !)$ 5 5 5 Ñ-* Ñ #(-2"24#&./* +92 IuÑ7L7!)$ =*3(-2MMM°ä$ !)$ 5 5 5 Ñ-* Ñ #(-2&'+92°./* ./8%./1./<-24* =Ñ-5 #2°.&)<-*3&'M

Öb

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A

P

S

uSA

uAP

¯2LOyFj@Xe?pª@quÖb¢eGmzwy 4e/f9g±fjíst¦#f9d#tmdih%o` 4Yd eGgHa´ %h¤ dZ¤o`-hjd`f9m Gkhzh%f9wyd`otf9eGg|f ¦

¥> d#t-h£`pok² fHGHfjGkhfÓJI*o q Gkhgjd`f9m `pok³po4eGC fHGH|f9¥> d#t-h­Cd ueKGHfj°khtmYof9d*L 24?M:>N-1-,%0243O276P-,9+M3QN;?R<S+T*U276V:>92-243;'W')!1--+T*,+XY2Z[1]\72^+=8-2_`1a*bX;'c15:>%d^+=2768-,9+eX+.-+)f-27*c=@?dxDFRORODÖlMLOA@y/J\BCROJ*j@RSBkKZLODFA@NcBCXe?dJ\BCXeXeLO?*TDFjbKWT@j@XeLOA@ypKZ=@?NeLO^/j@RSBkKZLuDCAFqWÜ?:K"j@N$J\BCROR

AKZ=@?J*?*AaKZXe?DCxdKZ=@?J*j@XeXe?*AaK$`FDÖ°b?*Ríq¢rQK"P-DFLOAaK

A_QKZ=@?LOA@J*LOT@?*AaK$P@=@DCKZDFA

Amj@^/E-?*XMP]?*XMj@A@LuKMNeDFROLOTBCA@yFRO?dLONcyFLu`F?*AEa[

dNA(E)

dΩ=

dNi(E)

dΩexp

(

nmat∑

m=1

−µm(E)lm

) ª@qu

lM=@?*Xe?dNi(E)/dΩ

LON|KZ=@?pAmj@^/E-?*XdDCx7LOA@J*LOT@?*AaKdP@=@DCKZDFA@NdP-?*Xdj@A@LuKdNeDFROLOTBCA@yFRO?C_?*^wLuKeKZ?*TEa[/KZ=@?WNeDFj@XeJ*?MLOApKZ=@?M?*A@?*XeyC[iJt=hBCA@A@?*R

[E−∆E/2 ;E+∆E/2]qoc=@?MNej@E@NeJ*XeLOPbK

mXe?:x?*XeN

KZDKZ=@?|T@Lu?:XZ?:AmK ^4BkKZ?*XeLSBCRuNfJ*XeDFNeNe?*TBCRODFA@ydKZ=@?|PhBkKZ=SA_bBCN7KZ=@?WNeLO^pj@RSBkKe?*TwJ*DFAb¤hyFj@XZBkKZLODFA

^4B\[$LOA@J*ROj@T@?dNe?:`F?*XZBCR9DFEbGI?*J:KZN*q«HqMrQKWP]DFLOAmK

A_@KZ=@?Amj@^pE]?*XMDCx2LOA@J*LOT@?*AaKMP@=@DCKZDFA@NMP-?*XMj@A@LuKWBCXe?\BpLON

Φ(E) =dNA(E)

dS=

dNA(E)

dΩ

dΩ

dS=

1

r2

dNA(E)

dΩ

ª@q¬«CalM=@?*Xe?

r = SALONQKZ=@?T@LONIKtBCA@J*?E-?:KYl ?*?*A%KZ=@?dNeDFj@XeJ*?P-DFLOAaK

SBCA@TKZ=@?d`FD\°b?*R9J*?*AaKZXe?

Aq

Öa«

ÎeÏ ÎeÏWÐeÑÓÒ|ÔÖÕ*ÉHËkÑS×HØp×mÙQÙÑzÍHÐ:Ë9QO×HÍ@ÌCÍdÊCÈ]×hËm×HØÐ:ÇÉHËmËCÌCÍmÑzØ]Ú¡bqMc=@?J*DFNeLOA@?DCxKZ=@?NeJ\BkKeKZ?*XeLOA@ywBCA@yFRO?dLON*

cos θ = uSA · uAP =SA · AP

‖SA‖‖AP ‖

ª@q¬«H

lM=@?*Xe?PLONfKZ=@?|J*?*AaKZXe?gDCxKZ=@?|J*j@XeXe?*AaKcP@LÓ°b?*R-DCxKZ=@?|T@?:KZ?*J:KZDFX*_

uSA

BCA@TuAP

BCXe?WKZ=@?|j@A@LuK`F?*J:KZDFXeNWJ*DFXeXe?*NeP-DFA@T@LOA@yiXe?*NeP-?*J:KZLu`F?*Ru["KZDpKZ=@?LOA@J*LOT@?*AaKWBCA@TNeJ\BkKeKZ?*Xe?*TXtB\[HN*q

ª@qMc=@??*A@?*XeyC[DCxKZ=@?; DF^wPbKZDFAROLOA@?gLONcJ\BCROJ*j@RSBkKZ?*Tj@NeLOA@yYv]qhª@quÖbq³Hq ?*?*AxXeDF^

A_bKZ=@?P@Lz°b?*RlM=@DFNe?J*?*AaKZXe?LON

PLONWBCNeNeDmJ*LSBkKZ?*TKZDpKZ=@?NeDFROLOTBCA@yFRO?

dΩP =dSP

AP 2cos(PA,n)

ª@q¬«F«FlM=@?*Xe?

dSP

LON KZ=@?dP@Lz°b?*R9BCXe?\BpBCA@TnLONcB/`F?*J:KZDFXMP-?*XeP]?*A@T@LOJ*j@RSBCXQKZDpKZ=@?dT@?:KZ?*J:KZDFXMNej@XIxßBCJ*?

BCA@T?*^w?*XeyFLOA@ypxXeDF^æLuKZNcNe?*A@NeLuKZLu`F?NeLOT@?Cq qMc=@?^wDF^w?*AmKZj@^ÅKZXZBCA@NIx?*XMPhBCXZBC^w?:KZ?*X

xJ\BCAE-?J\BCROJ*j@ROBkKZ?*Tj@NeLOA@yYv]q]ª@q ª@q

qMc=@DF^wNeDFABCA@TÞdRO?*LOAHV'WLuNe=@LOAhB;d NMBCXe?gDFEbKtBCLOA@?*T$Xe?*NeP-?*J:KZLu`F?*Ru[4xXeDF^¿Yv]q@ª@qudBCA@T"ª@qu«HqÆ qMc=@?4`kBCROj@?*N/DCxQKZ=@?wxDFXe^xßBCJ:KZDFX;BCA@T¨LOA@J*DF=@?*Xe?*AmK/NeJ\BkKeKZ?*XeLOA@y%xßBkJ:KZDFX/BCXe?4T@?:KZ?*Xe^wLOA@?*T¨Ea[LOAaKZ?*XeP-DFRSBkKZLuA@y%KZ=@?$KtBCE@j@RSBkKZ?*T¨ThBkKtB£B\`CBCLOROBCE@RO?4LOAYffgÜ a q xMKZ=@?"NeJ\BkKeKZ?*XeLOA@y^4BkKZ?*XeLSBCRJ*DFAaKtBCLOA@N7Ne?:`a?*XZBCRJt=@?*^wLOJ\BCR]?*RO?*^w?*AaKZN*_mKt=@?MxDFXe^ÛxßBCJ:KZDFX BCA@T$LOA@J*DC=@?*Xe?*AmKQNeJ\BkKeKZ?*XeLOA@yxßBCJ:KZDFXBCXe?dDFEbKtBCLOA@?*Tj@NeLOA@yiKZ=@?BkT@T@LuKZLu`HLuKY[$Xej@RO?wNe?*? ?*J:KZLODFA@NMª@q¬«Hq BCA@Tª@q ¡bq ªmn

[f(x)]2 =∑

i

αi[f(x, Zi)]2 ª@q¬«C¡a

S(x) =∑

i

αiS(x, Zi)ª@q¬«kªm

lM=@?*Xe?αi

LONQKZ=@?P-?*XeJ*?*AaKtBCyF?DCxKZ=@?BkKZDF^wN*_blM=@DFNe?;BKtDC^wLOJgAHj@^/E-?:XMLONZi

q qMc=@?AHj@^/E-?*XcDCx2P@=@DCKZDFA@NW©WBÖ[HRO?:LOyF=HVYNeJ\BkKeKZ?*Xe?*T"xXeDF^ A

KZDÖlcBCXeT@NPLON

d2NRa→P (E) = Φ(E)dNat

dσRa

dΩexp

(

nmat∑

m=1

−µm(E)lm

)

dΩP

ª@q¬«F³F

lM=@?*Xe?dσRa/dΩ = dσFF/dΩ

Ne?*?Yv]q@ª@q¬«F BCA@T$KZ=@?d?n°bP]DFA@?*AmKZLSBCR-xßBCJ:KZDFXQJ*DFXeXe?*NeP-DFA@T@NcKZDP@=@DCKZDFABkKeKZ?*AmjhBkKZLODFABCRODFA@yiKZ=@?;XZBÖ[

APqWc=@?/Amj@^/E-?*X|DCxfBkKZDF^wNWJ*DFAmKtBCLOA@?*TlMLuKZ=@LOAKZ=@?

`FD\°b?*R9LONcyFLu`F?*AEa[dNat =

ρ dV∑

iαiMi

NA

ª@q¬« lM=@?*Xe?

ρNIKtBCA@T@NgxDFX|KZ=@?p^wBkKZ?*XeLSBCRT@?*A@NeLuKY[F_

dVLON|KZ=@?;`aD\°b?*R2`aDFROj@^w?C_

Mi

LON|KZ=@?pBkKZDF^wLOJ^wDFRSBCX^4BkNZNDCxHKZ=@?

iVåKZ=?*RO?*^w?*AaK9P@Xe?*Ne?*AaKLOAgKZ=@?f^4BkKZ?*XeLSBCRCBCA@T

NA

LONr9`FDFyaBCT@XeD@Ã N9J*DFA@NIKtBCAaK\qÖF¡

ÇCÈmÉhÊmËCÌCÍdÎeÏWÐeÑÓÒ|ÔÖÕ*ÉHËkÑS×HØ/×mÙMÊCÈ]×hËm×HØÐ:ÇÉHËmËCÌCÍmÑzØ]ÚÖbq LO^wLuRSBkXeRz[F_HKZ=@?AHj@^/E-?*XcDCxv DF^wPbKZDFAHVYNeJ\BkKeKZ?*Xe?*TP@=@DCKZDFA@NMLON

d2NCo→P (ECo) = Φ(E)dNat

dσCo

dΩexp

(

nmat∑

m=1

−µm(ECo)lm

)

dΩP

ª@q¬«

lM=@?*Xe?KZ=@?BkKeKZ?*AmjhBkKZLODFA¥J*DH?¸wJ*LO?:AmKµm(ECo)

LON4DFEbKtBCLOA@?*T¥Ea[¥LOAaKZ?*XeP-DFRSBkKZLuA@y­xXeDF^ KZ=@?KtBCE@j@RSBkKZ?*TThBkKtBbq

FFqW¯2LOAhBCRORz[F_HKZ=@??*A@?*XeyC["J\BCXeXeLO?*TEm[$KZ=@?NeJ\BkKeKZ?*Xe?*TP@=@DCKZDFA@N*_@KZ=hBkKWXe?\BCJt=KZ=@?P@Lz°b?*RPLON

d2EP = d2ERa + d2ECo = d2NCo→P (ECo)ECo + d2NRa→P (E)E.ª@q¬« Æ

L 24?M:>N-1-,%0243g2768-,9+e%(?M1ah7+6i27*b?M+X<kj')!1--+T*,+Xl:>92-243;'c=@?¤hAhBCR|LO^4BCyF?DCxP@=@DCKZDFAsNeJ\BkKeKZ?*XeLOA@y¨Xe?*J*DFXeT@?*T>Ea[KZ=@?T@?:KZ?*J:KZDFX$LON$J*DF^wP@jbKZ?*Tsj@NeLOA@y®KZ=@?BCROyFDFXeLzKZ=@^P@Xe?*Ne?*AmKZ?*T%LOA¯2LOy@qbª@quFFq

'"(nm µ ¹M¼5¸n.¼1po".¶8*,"¹M3rqæ8n.»tsvu+êWë/ºcëIî »cûIò ÷oøbûñfôñfÿvV|ð|ñoøbñfô óHö÷oømøbùú@ûô|üÅñWÅ÷ |ô|û'¾|ùwfñXMù ?:xDFXe?7NeLO^/j@RSBkKZLuA@yMP@=@DCKZDFA;NeJ\BkKeKZ?*XeLOA@yWDC4¡F§^4BCJ*XeDFNeJ*DFP@LOJfDFEHGI?*J:KZN*_FLuK9LONLOA@NIKZXej@J:KZLu`F?fKZDMKZ?*NIKoDFj@XBCROyFDFXeLzKZ=@^lMLuKZ=4DFA@?c`a?*XI[iNe^4BCROR@`FD\°b?*Rí_HA@DCK7DFA@Ru[/KZD`F?*XeLux±[pKZ=@?WJ*DF=@?*Xe?*A@J*?|DCxDFj@X7P@XeDFyFXZBC^w^iLOA@yE@jbK7BCRONeDgKZDdLO^wP@XeD`F?QDFj@Xoj@A@T@?*XeNIKtBCA@T@LOA@yDCx-KZ=@?cP@XeDFP-?*XIKZLO?*NvDCx-NeJ\BkKeKZ?*XeLOA@y@q2c=@LONvj@A@T@?*XeNIKtBCA@T@LOA@ylMLORORHE-?QKZ=@?c~F?:[/xDFXvBCAhBCRu[H²*LOA@y|KZ=@?cXe?*Nej@RuKZNvDCx-^wDFXe?QJ*DF^wP@ROLOJ\BkKZ?*T/NeLO^pj@RSBKZLuDFA/J*DFAb¤hyFj@XZBkKZLODFA@N*q AKZ=@LONQNe?*J:KZLODFA_bKZ=@?NeJ\BKZKZ?*XeLOA@ypDFEHGI?*J:KMLONMB;`F?*XI["Ne^4BCRORJ*j@E@LOJ|`aDFROj@^w?C_HKZ=@?*Xe?:xDFXe?gKZ=@?BkKeKZ?*AmjhBkKZLODFADCxgP@=@DCKZDFA@NDFXiKZ=@?*LOXiP@XeDFEhBCE@LOROLzKY[¨DCxgLOAmKZ?*XZBCJ:KZLODFA];LONp`F?*XI[§RODÖlqsçW?*A@J*?%^/j@RuKZLuP@RO?"NeJ\BkKeKZ?*XeLOA@yJ\BCA­E-?iJ*DFA@NeLOT@?*Xe?*T®BCNA@?*yFROLOyFLOE@Ru?Cq A£KZ=@?*Ne?wJ*DFA@T@LuKZLODFA@N*_lQ?plMLOROR2`F?*XeLux±[LuxfKZ=@?iKZDCKtBCRoAmj@^/E-?*XeNDCxP@=@DCKZDFA@N j@A@T@?*XeyFDFLOA@yp DF^wPbKZDFA4NeJ\BkKeKZ?*XeLOA@y@_b©WB\[HRO?*LOyF=wNeJ\BkKeKZ?*XeLOA@y/BCA@T4lc=hBkKZ?:`a?*XQLOAaKZ?*XZBCJ:KZLODFABCXe?dXe?*NeP-?*J:KZLu`F?*Ru["P@XeDFP-DFXIKZLODFAhBCR]KZDpKZ=@?;7DF^wPbKZDFA_h©WBÖ[HRu?*LOyF=BCA@T"KZDCKtBCR^4BCNeNMBkKeKZ?*AmjhBkKZLODFA"J*DH?:xSV¤hJ*LO?*AaKZN*_]BCNM?n°bP]?*J:KZ?*Tqf<®?dlMLORORBCRONeDpJt=@?*Jn~$lc=@?:KZ=@?*XMKZ=@?NePhBkKZLSBCRT@LONIKZXeLOE@jbKZLODFA"DCxKZ=@?dNeJ\BkKeKZ?*Xe?*TP@=@DCKZDFA@NLONLOA­BCJ*J*DFXeThBCA@J*?ilMLuKZ=KZ=@?4BCA@yFj@RSBCXgT@?*P-?*A@ThBCA@J*?4DCx KZ=@?iT@Lu?*Xe?*AaKZLSBCRoJ*XeDFNeNNe?*J:KZLODFA@N*qc=@?T@?:KtBCLORONcDCxKZ=@?NeLO^/j@RSBkKZLuDFA"J:DFAb¤hyFj@XZBkKZLODFABCXe?P@Xe?*Ne?*AaKZ?*T%LOA¯2LOy@qbª@qu«Hqc=@?dP@XeDFEhBCE@LOROLuKY[ixDFXQB/P@=@DCKZDFA$KZDpE-?|KZXZBCA@Ne^wLuKeKZ?*T$lcLuKZ=@DFjbKcLOAaKZ?*XZBCJ:KZLODFA4KZ=@XeDFj@yF=$KZ=@?g`FD\°b?*R

LONp = exp(−µtotx) = 0.9997

_@KZ=@?*Xe?:xDFXe?dKZ=@?dLOA@LuKZLOBCR-=m[HP]DCKZ=@?*NeLONWLONGIj@NIKZLu¤h?*TlMLuKZ=B/`F?*XI["yFDHDmTBCJ*J*j@XZBCJ:[Fqc=@?w DF^wPbKZDFA£BCA@T£©WB\[HRO?*LOyF=%NeJ\BkKeKZ?*XeLOA@y"LO^4BCyF?*NgBkA@TKt=@?/J*DFXeXe?*NeP]DFA@T@LOA@y"P@XZDk¤hRO?*N*_DFEbKtBCLOA@?*T>lM=@?*AÄKZ=@?­T@?:KZ?*J:KZDFXLONRODHJ\BkKZ?*T>LOA>KZ=@?­P]DFNeLuKZLODFA

(a)_dBCXe?­T@LONeP@RSB\[F?*TÄLOAá¯2LOy@qQª@quÖ¡

BCA@TĪbqu\ª@qÛc=@?£P@XeDC¤hRO?*NBCXe?`F?*XI[>NeLO^wLOROBCXwKZD¦KZ=@?£J*j@XI`F?*NXe?*P@Xe?*Ne?*AmKZLOA@y#KZ=@?­ DF^wPbKZDFAÄBCA@T©WB\[HRO?*LOyF=%T@Lu?*Xe?*AaKZLSBCRJ*XeDFNeNgNe?*J:KZLODFA@NdBCNdBwxj@A@J:KZLODFADCx

tan θ߯2LOy@q]ª@qu³4BCA@Tª@qu nqg¯hDFXWKZ=@?*Ne?J*j@XI`F?*N*_KZ=@?/Jt=@DFLOJ*?pDCx

tan θDFAKZ=@?iBCE@NeJ*LONeNZB$LONgJ*DFAa`F?*A@LO?*AaKE-?*J\BCj@Ne?

tan θLON|P@XeDFP-DFXIKZLODFAhBCRKtD

KZ=@?iT@LONIKtBCA@J*?iE]?:Kl ?*?*A¨ByFLu`F?*A£P@Lz°b?*RvDCx7KZ=@?iT@?:KZ?*J:KZDFXBCA@T£KZ=@?iDFXIKZ=@DFyFDFAhBCRoP@XeDkGI?*J:KZLODFA£DCx7KZ=@?J*DFA@NeLOT@?*Xe?*TwNeJ\BkKeKZ?*XeLOA@yd`FD\°b?*R]DFApKZ=@?cT@?:KZ?*J:KZDFXfP@RSBCA@?Ne?*?W¯2LOy@qCª@qu nqoçWDÖl ?:`F?*X*_HDFA@?MNe=@DFj@ROTiA@DCKKZXI[%KZD"Nej@P-?*XeLO^wP-DFNe?;KZ=@?pNeLO^pj@RSBkKe?*TP@XeDC¤hRO?*NdBCA@TKZ=@?id NLOA£B$J*DF^w^wDFAP@RODCK\_E-?*J\BCj@Ne?pKZ=@?P@XeDC¤hRO?*NMLOA¯2LOy@q@ª@qu\ªpKtBC~C?LOAaKZDwBCJ*J*DFj@AaKWKZ=@?d`CBCXeLSBkKZLODFA@NQDCx2KZ=@?NeDFROLOTBCA@yFRO?BCNeNeDHJ*LSBkKZ?*TKZDw?\BCJt=P@Lz°b?*RDCxKZ=@?T@?:KZ?*J:KZDFX*_@lM=@?*Xe?\BCNMKZ=@?d NWT@DwA@DkK/±KZ=@?:[BCXe?yFLu`F?*ALOAEhBCXeA@NMP-?*XMNIKZ?*XZBCT@LSBCA]nq

ÖCª

ÎeÏ xIÏMÍHÌCÐÔÖÕ:ËCÐf×ybËtÉ@ÑzØbÌfzfÑSËCÈ|-P@ÑQ27*/?\BCJt=%NZBC^wP@RO?27*/?\BCJt=`HLOXIKZjhBCRT@?:KZ?*J:KZDFX DF^wP@jbKZ?d^4BkKZXeLz°

Dist(m, i, j)DCxKZ=@?BkKeKZ?*AHjhBkKZLODFA"RO?*A@yCKZ=@NcxDFXcLOA@J*LOT@?*AaKMP@=@DCKZDFA@N*q

¯2LORORLOAKZ=@?iE-DHDCRO?\BCA£^4BkKZXeLz°A(i, j)

qpc=@?i`CBCRuj@?4d wh4^i?\BCA@NdKZ=hBkKdKt=@?iJ:DFXeXZ?:NZP]DFA@T@LOA@y`FD\°b?*R9LONMBCJ:KZLu`F?b~:q DF^wP@jbKZ?gKZ=@?^4BkKZXeLOJ*?*N

uSA(i, j)_SA(i, j)

BCA@TSA2(i, j)

q DF^wP@jbKZ?dKZ=@?^4BkKZXeLz°

Φ(i, j, e)J*DFAaKtBCLOA@LOA@y/KZ=@?Amj@^pE]?*XMDCx2LOA@J*LOT@?*AmKMP@=@DCKZDFA@NMP-?*XMj@A@LuK

BCXe?\BbqQ27*/?\BCJt=BCJ:KZLu`F?g`aD\°b?*R DF^wP@jbKZ?QKZ=@?c^wBkKZXZLÓ°

Dist(m, ii, jj)DCx]KZ=@?MBkKeKZ?*AmjhBkKZLODFApRu?*A@yCKZ=@NoxDFXoNeJ\BkKeKZ?*Xe?*TiXZB\[HN*q

DF^wP@jbKZ?dKZ=@?^4BkKZXeLOJ*?*NuAP (ii, jj)

_AP (ii, jj)

BCA@TdΩ(ii, jj)

q27*/?\BCJt=P@Lz°b?*R

PDCxKZ=@?T@?:KZ?*J:KZDFX

DF^wP@jbKZ?cos θ, cos2 θ, dσT(θ)/dΩ

qQ27*/?\BCJt=%?*A@?*XeyC[

E DF^wP@jbKZ?ECo, x, dσKN(θ, E)/dΩ, [f(x)]2, S(x), µ(m,ECo)

_dσRa/dΩ, dσCo/dΩ, d2ECo(ii, jj)

BCA@Td2ERa(ii, jj)

q3;X6i27*

3;Xl6i27*3;X6i27*

3;Xl6i27*3;X6i27* :yc(%'./!/=* +-K<-'./<-!A$z `Ô%5*32A$&./* "×xÖ#×;K* ./<-* +$c2/&$.4./!/* +-,?,G#&.#"Ku&'8-+.|./<-+78-=>? !A I* +l./!)24#&./* +ÑÓ* +.)2u+9&'8-+l./!/#($ 5 +-,W./<-!)$`L

SAMRÞ^Ip./<-@248-!)&'Ñ * +l.

S*x2E7+-K+./?Ó8%.)24*3(%./<-?,^#&.$ +9(A./<-W+l8-=>? ! I* +l./!)24#&./* +92;*32;-(%("%./<-+./<-W#Ô%5*32$&./* M

¯2LOyFj@Xe?Mª@quFF>f9muwy) 4e/f9gHfjAGkheGmÖmjlf9d m4tzlcíst¦#f9d#tmdh%o`Ó4YdZ`tAkf9f9gh\z¡;khof9w%uO

fjieGgYåmd`h(i, j)

dZ4j*d`hzf fHGHWfjiGkhvowyd dZg|%eGd 4wR WtmYof9d q e h) gut-hjlf9dGkh4gdå okR gggwymnd| gut

(ii, jj)of9d dthf9gut-hfiueKGHfjGkh|dZ` tmYof9d*¡;khf9Gkhd9 d e4lOh

kR Cintgtm4ñWg`trudZ%e/f9whxF

S O

(a)

(b)

(c)

¯2LOyFj@Xe?iª@qu«H¢eGmzwy 4e/f9g²fjkf9f9gµh%o` /Yd eGgHfë¿wyge4lwhv fHGHuý¡;kht d# ¥WeGgHe6hgf9fh%o` O\¿¡;khHt e6h) gotjd`f9m Gkhh%f9wyd`otzf9eGg

SfµGkh fHGHotg4dZ

Oe6hSO = 100

mzmw¡;khv fHGHWe6h gûeGd`f9gýowRn tmgh eG/ ρ = 7.87

Kom3 q ¥1kf%hÖ"`tÖmchee6h

x = 1 µmGkh

tmYof9de6h|6f%o` Y`trhwoothheGCxif9g¤Gkhhe4tmh(a) q (b)

gut(c)

fj|zowRnípîî4eGm4hleuÖmdGkR gGkhc fHGH ¥eGGksGkhzh7 m4votg4dZ

OF¡;khtmYof9dze6hvt eG%e4tm`teGgf

500 × 500ueKGH h ueKGHhe

0.2 µmFóheGgH Oimvf9gf%okyd`f9m 4e/o E = 100

§aRbveGgoe4tmgCd#Ó q o` d d 9eGgHNi = 1012kf9f9gh q e6hzo`-hY`tAjd`f9m

SfO

Öa³

ÇCÈmÉhÊmËCÌCÍdÎeÏWÐeÑÓÒ|ÔÖÕ*ÉHËkÑS×HØ/×mÙMÊCÈ]×hËm×HØÐ:ÇÉHËmËCÌCÍmÑzØ]Ú

¯2LOyFj@Xe?pª@quÖ¡bkf9f9ggwymndzeGmÖmh|jlf9d>f9muf9g O4jxH gut´ 9Oeu-k d eu-kyxh%o` 4Yd eGgHa¡;khWmf9m44d e/oof9g%ñ wyd# 4e/f9gfjGkye6hheGmzwy 4e/f9ge6htmh%od e4n`tzeGg&eua-Gí q ¥WeGGkGkhitmYof9deGgvf%heG4e/f9g F

0

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1 100 200 300 400 500

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f pho

tons

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0

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¯2LOyFj@Xe?pª@qu\ª@7g4d# 1xeGgrud`fñWOh|jlf9d>f9muf9g O4jxH gut´ 9Oeu-k d eu-kyxh%o` 4Yd eGgH£GÓ¦4d#poY`tjd`f9m GkheGmÖmhfjeuau-GíÓïH

c=@?7LOAaKZ?*yFXZBkKZ?*TAmj@^pE]?*XeN9DCxbP@=@DCKZDFA@N9J*DFj@AmKZ?*TEm[gKZ=@?fT@?:KZ?*J:KZDFX9LOAdKZ=@?fP]DFNeLuKZLODFA@N(a)_(b)BCA@T

(c)BCXe?QyFLu`F?*A/LOAp¯2LOy@qª@qu Æ q2Q=@?QKZDCKtBCRHAmj@^/E-?*XeNoDCx]P@=@DCKZDFA@NoNeJ\BkKeKZ?*Xe?*TiEa[KZ=@? `aD\°b?*RbLOA

4πNIKZ?*XZBV

T@LSBCA@N BCXe?WDFEbKtBCLOA@?*TwEa[4BCT@T@LOA@yKZ=@?M`CBCROj@?*N7J*DFj@AaKZ?*T$LOAw?\BCJn=4P-DFNeLuKZLODFAwDkxKZ=@?WT@?:KZ?*J:KZDFXd±lMLuKZ=@DFjbKxDFXeyF?:KeKZLOA@yWKZDd^/j@RuKZLOP@Rz[Em[;ªgKZ=@? `CBCROj@? DFEbKtBCLOA@?*TpLOA/P]DFNeLuKZLODFA

(b)nq2c=@?c^4BCNeNvBkKeKZ?*AmjhBkKZLODFA/J*Dm?:xSV

¤hJ*LO?*AaKZNfxDFXvLOXeDFAwBkKWÖFd~F?íä/_m?n°HKZXZBCJ:KZ?*T4xXeDF^ YffgÜ a _bBCNvlQ?*ROR]BCNvKZ=@?MAHj@^/E-?*XfDCxLOAaKZ?*XZBCJ:KZLODFA@NNint

LOA4KZ=@?g`FDÖ°b?*RxDFXQ?\BCJt=P@=a[HNeLOJ\BCRP@XeDHJ*?*NeNWBCXe?gyFLu`F?*A"LOA¯2LOy@qHª@qu qoc=@?dKZDCKtBCR-AHj@^/E-?*XQDCxLOAHVKZ?*XZBCJ:KZLODFA@N*_ClM=hBkKZ?:`a?:XfP@XeDHJ*?*NeNvLONoJ*DFA@NeLOT@?*Xe?*T%LOA@J*ROj@T@LOA@y|P@=@DCKZDH?:?*J:Knn_aLONNi(1 − exp(−µtotx))

qc=@?$RSBCNIK;J*DFROj@^wA­LOA­KZ=@?4KtBCE@Ru?4?*AhBkE@RO?*N;KZD%Jt=@?*Jn~£KZ=@?4P@XeDFP-DFXIKZLODFAhBCROLzKY[E-?:KYl ?*?*A¦KZ=@?4Amj@^pE]?*XDCxgLOAaKZ?*XZBCJ:KZLODFA@NiBCA@T#KZ=@?J*DFXeXe?*NeP-DFA@T@LOA@y­^4BCNeNwBkKeKZ?*AmjhBkKZLODFA#J*DH?¸wJ*LO?*AaK\qc=@?`CBCROj@?*NiLOA#KZ=@LONJ*DFROj@^wAiNe=@DFj@ROTiE-?|BCROR@?:mjhBCRí_m=@Dl ?:`F?*X7KZ=@?*Xe?MLON7BdNeROLOyF=mKvT@LONeJ*Xe?*PhBCA@J:[iDCxKZ=@?c`CBCROj@?QxDFX7©WB\[HRO?*LOyF=NeJ\BkKeKZ?*XeLOA@y@q4rcJ:KZjhBCRORu[F_KZ=@?w©WBÖ[HRO?*LuyF=£^4BCNeN;BkKeKZ?*AmjhBkKZLODFA£J*DH?¸wJ*LO?*AaK;?n°HKZXZBCJ:KZ?*T®xXeDF^ YffgÜ aLONQEhBkNZ?:TDFA"c=@DF^wNeDFAà NMd _bxDFXe^ÅxßBCJ:KZDFXeNcBCA@TBCA@DF^4BCRODFj@N7NeJ\BkKeKZ?*XeLOA@y/xßBCJ:KZDFXeN*q A"DFj@XQJ*DHT@?C_BCA@DF^4BCRODFj@NgNeJ\BkKeKZ?*XeLOA@y"xßBCJ:KZDFXeNBCXe?iA@DCKdKtBC~F?*A­LOAaKZDBCJ*J*DFj@AaK/BCA@TlM=@?*A®LOAmKZ?*yFXZBkKZLOA@y4KZ=@?wd j@NeLOA@yYv]qª@q¬«FWDÖ`F?*X

4πNIKt?:XtBkT@LSBCA@N*_l ?pDCEbKnBkLOA

(µ/ρ)Ra = 3.736 × 10−2 cm2/gqg<>LuKZ=KZ=@LON

`CBCROj@?C_bl ?¤hA@TNint/(µ/ρ)Ra = 7.87 × 108

BCNcxDFXQKZ=@?dKYl DwDCKZ=@?*XcXZDÖlMNMDCxKZ=@?dKtBCE@RO?CqÖ

ÎeÏ xIÏMÍHÌCÐÔÖÕ:ËCÐf×ybËtÉ@ÑzØbÌfzfÑSËCÈ|-P@Ñ

0

0.5

1

1.5

2

2.5

0 0.2 0.4 0.6 0.8 1

10 MeV

3 MeV

1 MeV

300 keV

100 keV

30 keV

10 keV

dσ

Co/d

Ω

¨ ©ª«®

tan θ

¯2LOyFj@Xe?ª@qu³Hlf9mze/oµt e ë|dZg4e4 od`f%hhHhÖo4e/f9gdσCo/dΩ eGgU d gh¤hdhYd#t e4 g ±-h£

jwygo4e/f9g fjtan θ q jlf9d>f9muf9gûh%o` 4Yd eGgHfj¤wyg f9 d e`td#t e4 /e/f9g q jlf9deGd`f9g Z = 26

qo` 6owy Y`tswheGgHHÖW-Gílý guts°kh¤)lwy Y`t9 xwhhfjvGkheGgof%khdZg|h%o`Ó4Yd eGgHzjwygo4e/f9g 9 eGlOreGgsGkht )-hÖWR

0

10

20

30

40

50

60

0 0.2 0.4 0.6 0.8 1

10 keV

30 keV

100 keV

300 keV

dσ

Ra/d

Ω

¨ ©ª«®

tan θ

¯2LOyFj@Xe?ª@qu lAf9mze/o£t e ë|dZg4e4 od`f%hhHhÖo4e/f9gdσRa/dΩ eGgU d gh¤hdhYd#t e4 g ²-hµ

jwygo4e/f9gÄfjtan θ q jlf9d´ 9Oeu-kûh%o` 4Yd eGgHûfjwyg f9 d e`tµd#t e4 4e/f9g q jlf9deGd`f g Z = 26

qo` 6owy Y`twheGgH|Ö -6 gutGkh>)lwy Y`t9 xwhhfjWGkhjlf9d m£jpof9dC 9 eGlOeGgzGkht -hÖWR

Ö

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dtanθθ

dA

Hvoxel

detector

P

¯2LOyFj@Xe?dª@qu c¡;kht e6h) gotPH

n4¥ tgsGkhzotg4dtP

fjueKGH gutvGkhcf9d GkfZpf9gu yud`fBI*o¦4e/f9g

HfjzGkh fHGHCotgYd

Ae6hrud`f f9d 4e/f9gu 1f

tan θ²¡;khdZ4jlf9dZ q ¥@khgdZudZhÖg4eGgHGkh

t e ë|dZg4e4 @od`f-h`hhÖo4e/f9ghcfj>f9muf9gµ gutc´| 9Oeu-kh%o` 4YdeGgm q tan θe6hzipf%f7tokf9e/otCjlf9d

GkhA h%oe6hh7avkhgokf%f%heGgHrGkh@ d# m4Yd`hAfjheGmzwy 4e/f9g q GkhhÖ@u6f9Gh hÖteuap-Gí. guteuaK-GíT dZzwhÖ4jwp@ftHYd mzeGgiGkhihR 4e4 dth%f9xwp/e/f9gfjGk]ctmYof9dgt`tm`tf%ghwydZof9d dZo1h7 muxeGgHfjGkht e ë|dZg4e4 od`f%hhhÖo4e/f9gh\

|?:KZ?*J:KZDFXWP]DFNeLuKZLODFANRa NCo

(a) 2.619 × 107 2.478 × 107

(b) 7.097 × 105 1.498 × 107

(c) 3.650 × 105 1.769 × 1079DCKtBCRLOA4π sr 2.939 × 107 1.024 × 108

¯2LOyFj@Xe?/ª@qu Æ rð gYå d# Ytgwymnd`h¤fj|kf9f9ghvof9wygY`tHlGkhvtmYof9deGg£Gkhf%heG4e/f9gh(a) q

(b) gut

(c)NRa

gutNCo

dZrdZhho4eGCxzGkhrgwymnd`hifjkf9f9gh´ 9Oeu-k gut>f9muf9gh%o` 4YdZ`tac¡;khzof9muOYhÖfj>R d# m4Yd`hfjrGkhheGmzwy 4e/f9g dZ| eGCgHeGgeuau-Gí H

AmKZ?*XZBCJ:KZLODFA µ/ρcm2/g

Nint Nint/(µ/ρ)©WB\[HRO?*LOyF=

3.792 × 10−2 2.939 × 107 7.75 × 108 DF^wPbKZDFA1.301 × 10−1 1.024 × 108 7.87 × 108r|RORP@XeDmJ*?*NeNe?*N3.721 × 10−1 2.928 × 108 7.87 × 108

¯2LOyFj@Xe?pª@qu `-hhv 4YgwR 4e/f9g²ofÖ(ýoeígGh|jlf9dceGd`f9gû ípîaRb q GÓ4d#poY`tijd`f9m WR gutgwymndfjeGgYd#po4e/f9gh

Nint

eGgHGkh fHGHÓjlf9d/`pokky%he/o` yud`f%othh\

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Ö Æ

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tan θq

A§KZ=@?%J*DFAb¤hyFj@XZBkKZLODFA#DCx¯2LOy@qoª@q¬«Cb_7KZ=@?KZXZBCA@Ne^wLONeNeLODFA§XZBkKZ?KZ=@XeDFj@yF=B¨«CF µ ^ `aD\°b?*RWLONBCE-DFjbK ªÆ BCA@TKZ=@?;BCA@yFj@RSBCXMNIKZ?*P%xXeDF^BwP@Lz°b?*RKZDiKZ=@?;A@?n°HKgLONMNe^4BCRORO?*XQKZ=hBCA

0.2/10 = 0.02q

S I J

Detector:500pixelsSquarepixels,size200µm

Sample:ironplate50×50×5mm3

Cubicvoxels,size200µm

SJ=1000mmIJ=10mm

PointsourceIsotropicemission,1012ph/sr100keV

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µ^$ lQBCN

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Inte

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Pixel number

1 mm voxels2 mm voxels5 mm voxels

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Collimators

Detector

0

230

Source

z

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0e+00

1e+03

2e+03

3e+03

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Phot

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Position (z) of the measurement volume

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