Lecture Note on Photon Interactions and Cross Sections

Home , Absorption cross section, Compton scattering, Cross section (physics), Elastic scattering

KEK Internal 2000-10

November 2000

Lecture Note on

Photon interactions and Cross Sections

H. Hirayama

Lecture Note on

Photon Interactions and Cross Sections

Hideo Hirayama

KEK, High Energy Accelerator Research Organization,

1-1, Oho, Tsukuba, Ibaraki, 305-0801 Japan

Abstract

This is the lecture note used at the \Tutorials for Electron/Gamma Monte Carlo: building

blo cks and applications" as part of the International Conference on the Monte Carlo 2000,

Advanced Monte Carlo on Radiation Physics, Particle Transp ort Simulation and Applications

(Monte Carlo 2000) held at Lisb on Portugal, 23-26 Octob er, 2000.

1 Intro duction

This lecture contains descriptions of the individual atomic cross sections of photons for scat-

tering, absorption, and pair pro duction that are necessary for photon transp ort calculations,

together with related cross section data. Many parts on this lecture are based on the series of

works by J. H. Hubb ell[1, 2, 3, 4, 5, 6] in this eld.

2 Classi cation of Interaction

The interaction of photons with matter may b e classi ed according to:

1. The kind of target, e.g., electrons, atoms, or nuclei, with which the photons interact, and

2. The typ e of event, e.g., scattering, absorption, pair pro duction, etc:, whichtakes place.

Possible interactions are summarized in Table 1.[7 ] As an example, the cross sections of a

photon with Cu is shown in Fig. 1[8 ]. It is apparent from Fig. 1 that for the attenuation of

photons the most imp ortantinteractions are:

1. The photo electric e ect, ,

2. Compton scattering, , and

3. Electron-p ositron pair pro duction, ( + ).

pair tr ip

Rayleigh scattering, , is usually of minor imp ortance for the broad b eam conditions typi-

cally found in shielding, but must b e known for the interaction of attenuation co ecient data.

The photonuclear e ect, , is mostly restricted to the region of the giant resonance around

ph:n

10 to 30 MeV where, at the resonance p eak, it may amount to as much as 10 p ercent of the

total \electronic" cross section.

Elastic nuclear scattering, inelastic nuclear scattering and Delbruc k scattering are negligible

pro cesses in photon interactions. Elastic nuclear scattering is regard as a nuclear analog to very

low energy Compton scattering by an electron. In this pro cess, a photon interacts with a nucleon

in such a manner that is a photon re-emitted with the same energy.

During inelastic nuclear scattering, the nucleus is raised to an excited level by absorbing a

photon. The excited nucleus subsequently de-excites by emitting a photon of equal or lower

energy.

The phenomenon of photon scattering by the Coulomb eldofanucleus is called Delbruc k

scattering (also called nuclear p otential scattering). It can b e thought of as virtual pair pro duc-

tion in the eld of the nucleus { that is, pair pro duction followed by annihilation of the created

pair. 1 4 10 Cu

) Total 2 2

cm 10

-24 Pair production (nucleus) (10

0 10 Compton Rayleigh scattering (γ,n) Pair production (barn/atom) (electron) σ -2 10 Photoelectric Effect

Pion Quasi-deutron production disintegration 10-4 0.01 0.1 1 10 100 1000 104 105

Photon Energy (MeV)

Figure 1: Cross section of Cu for photons b etween 10 keV and 100 GeV.[8 ]. 2

Table 1. Classi cation of elementary photon interactions.

Typ e of Scattering

interaction Absorption Elastic Inelastic

Interaction (Coherent) (Incoherent)

with:

Atomic Photo electric Rayleigh Compton

electrons e ect scattering scattering

(

2 4

Z Z (L:E :)

pe C

(L:E :) Z (H:E :)

Nucleus Photonuclear Elastic Inelastic

reactions nuclear nuclear

( ,n),( ,p), scattering scattering

2 0

photo ssion, etc. ( ; ) Z ( ; )

ph:n:

(h 10MeV)

Electric eld Electron-p ositron Delbruc k

surrounding pair pro duction in scattering

charged eld of nucleus,

particles Z

pair

(h 1:02MeV)

Electron-p ositron

pair pro duction in

electron eld,

tr ip

(h 2:04MeV)

Nucleon-antinucleon

pair pro duction

(h 3 GeV)

Mesons Photomeson Mo di ed

pro duction, ( ; )

(h 150 MeV)

3 Photo electric E ect

Studies related to the photo electric e ect are reviewed historically by Hubb ell in NSRDS-NBS

29[1 ].

In the atomic photo e ect, a photon disapp ears and an electron is ejected from an atom.

The electron carries away all of the energy of the absorb ed photon, minus the energy binding

the electron to the atom. The K -shell electrons are the most tightly b ound, and are the most

imp ortant contributions to the atomic photo e ect cross-section in most cases. However, if the

photon energy drops b elow the binding energy of a given shell, an electron from that shell cannot

b e ejected. Hence, particularly for medium- and high-Z elements, a plot of versus the photon

energy exhibits the characteristic sawto oth absorption edges, since the binding energy of each

electron subshell is attained and this pro cess is p ermitted to o ccur. These feature can b e seen

well in Fig. 2, which shows the total photo electric cross section, , for Ag together with

subshell ones[9 ]. 3 6 10 L -edge Ag 3 L -edge 2 L -edge 1 Total 105 K (barn/atom) L1 L2 4 K-edge L3 10 M section

103 cross

102

Photoelectric 10 1 10 100

Photon Energy (keV)

Figure 2: Total and partial atomic photo e ect of Ag.[9 ].

The order of magnitude of the photo electric atomic-absorption cross section is

(

4 3

Z =(h ) low energy

(1)

Z =h high energy :

Dramatic resonance structures of the order of 10% just ab ove the absorption edges are

well known and can be easily observed with high-resolution sp ectrometers. Owing to their

dep endence on the temp erature, chemical binding and other variable atomic environments, this

extended X-ray absorption ne structure (EXAFS) is not included in the photon transp ort

calculation. However, EXAFS can b e a ma jor analytical to ol in X-ray di raction sp ectrometry.

3.1 Relaxation pro cesses after the photo electric e ect

Avacancy created by the ejection of an electron from an inner shells is lled by an outer electron

falling into it (de-excitation); this pro cess may b e accompanied by one of the following mo des:

1. A uorescence X-ray is emitted from the atom, with a photon energy equal to the di erence

between the vacancy-site inner-shell energy level and energy level of the particular outer

shell which happ ens to supply the electron to ll the vacancy ( uorescence yield, ! ,isthe

fraction of uorescence X-ray emission).

2. The excess energy ejects an outer-shell electron from the atom. This electron is known as

an Auger electron (Auger yield, a, is the fraction of Auger emission).

3. A vacancy is lled by an electron in a higher subshell, like from the L subshell to the

L subshell. This pro cess is called Coster-Kronig. As a result, a new vacancy is created,

in which is lled by one of the mo des (Coster-Kronig yield, f , is the fraction of Coster-

Kronig). 4

The sum of uorescence yield, ! , Auger yield, a, and Coster-Kronig yield, f , is unity:

! + a + f =1 (2)

Exp erimental and theoretical uorescence-yield information have b een reviewed by Fink et

al.[10 ], Bambynek et al.[11 ], Krause[12 ] and Hubb ell[2].

The yields of uorescence, Auger electrons and Coster-Kronig after the K-, L -, L - or L -

1 2 3

photo electric e ect[13 ] are shown in Fig. 3.

1 L1-Fluorescence 1.2 L2-L1 Coster-Kronig 0.8 L3-L1 Coster-Kronig L1 Auger 1 0.6 0.8

0.6 K-Fluorescence Yields 0.4 K-Auger Yields 0.4 0.2 0.2

0 0 0 20406080100 0 20406080100 Atomic Number, Z Atomic Number, Z

1 1

0.8 0.8

0.6 0.6 L3 Fluorescence L2 Fluorescence L3 Auger

Yields 0.4 L3-L2 Coster-Kronig Yields 0.4 L2 Auger

0.2 0.2

0 0 0 20406080100 0 20406080100

Atomic Number, Z Atomic Number, Z

Figure 3: Yields of uorescence, Auger electrons and Coster-Kronig after the K-, L -, L - or

1 2

L -photo electric e ect[13 ].

The uorescence X-ray intensities are imp ortant for low energy photon transp ort. Sco eld

published K and L uorescence X-ray intensities calculated with the relativistic Hatree-Slater

theory for elements with Z=5 to 104[14 ]. His data are cited in a Table of Isotop es Eighth

Edition[13], but are slightly di erent with the exp erimental results, as presented by Salem et

al[15 ]. 5

4 Scattering

4.1 Compton scattering, Klein-Nishina Formula

In Compton scattering, a photon collides with an electron, loses some of its energy and is

de ected from its original direction of travel (Fig. 4). The basic theory of this e ect, assuming

the electron to b e initially free and at rest, is that of Klein and Nishina[16].

hν θ hν0 O X φ

Figure 4: Compton scattering with a free electron at rest.

The relation b etween photon de ection and energy loss for Compton scattering is determined

from the conservation of momentum and energy b etween the photon and the recoiling electron.

This relation can b e expressed as:

h = ; (3)

1+ (1 cos )

m c

2 2

2(h ) cos

E = h h = m c ; (4)

0 e

2 2 2 2

(h + m c ) (h ) cos

0 e 0

; (5) cot tan =

m c

where h is the energy of incident photon, h is the energy of scattered photon, E is the energy

of recoil electron, m is the rest mass of an electron, and c is the sp eed of light.

For unp olarized photons, the Klein-Nishina angular distribution function per steradian of

solid angle is

( )

2 KN 2 2

1 + cos d 1 h (1 cos )

( ) = r 1+

2 2

d 2 [1 + h (1 cos )] (1 + cos )[1 + h (1 cos )]

k k k 1

2 2 2 1 1

+ r sin (cm sr el ectr on ); (6) =

2 k k k

0 0

h h

k = ; k =

2 2

m c m c

e e 6 90

80 120 60 70 60 -1 50 30 40

electron 30 -1 20 θ deg. str. 2 10 cm 0 -27 0 10

, Ω Ω Ω Ω /d C σ σ σ σ d

210 330

0.01MeV 0.016MeV 240 0.2MeV 300 0.5MeV 1.2MeV270

3.0MeV

Figure 5: Di erential cross section of Compton scattering (original gure from [17 ]).

where r is the classical electron radius. This angular distribution is shown in Fig. 5.

Integration of Eq. 6over all angles gives the total Klein-Nishina cross section,

1+k 2(1 + k ) ln(1 + 2k ) ln(1 + 2k ) 1+3k

2 1 KN 2

+ (cm el ectr on ): =2r

C 0

2 2

k 1+2k k 2k (1 + 2k )

(7)

4.1.1 Electron binding correction in Compton scattering

The Klein-Nishina formula for Compton scattering from atomic electrons assume that the elec-

trons are free and at rest, which is a go o d approximation for photons of the order of 1 MeV or

higher, particularly for low-Z target materials.

Binding corrections have usually b een treated by the impulse approximation or by Waller-

Hatree theory[18 ], taking account not only of the K-shell, but all of the atomic electrons. This

involves applying a multiplicative correction, the so-called \incoherent scattering function",

S (q; X ), to the di erential Klein-Nishina formula,

BD KN

d d ( )

C C

( )=S (q; Z ) : (8)

d d

The momentum transfer, q , is related to the photon energies and de ection angle, , according

q = k + k 2k k cos ; (9)

2 2

where q is in m c units and k and k are the photon energies (in m c units) b efore and after

e 0 e

de ection.

Incoherent scattering functions have b een surveyed and studied by Hubb elle et al.[3,4]and

are presented for all elements Z=1 to 100, for photon energies 100 eV to 100MeV.

KN BD

for photon energies of 1, 10, and 100keV to Fig. 6 shows the ratios of

C C 7 1 100 keV

10 keV KN C σ σ σ σ / 0.1 BD C σ σ σ σ 1 keV

0.01 0 20406080100

Atomic Number, Z

Figure 6: Ratio of the b ound-electron Compton scattering cross section, [9], to that for free

electrons, ,evaluated from the Klein-Nishina formula.

4.1.2 Doppler broadening

In the energy region where the binding correction is necessary, the motion of the atomic electrons

around the atomic nucleus gives rise to a Doppler broadening of the apparent energy of the

incident photon, resulting in a corresp onding broadening of the Compton \mo di ed line" for

a given de ection angle of the outgoing scattered photon. The shap e of this broadened line

is called the \Compton pro le." Available data on Compton pro les have been surveyed by

Hubb ell, and are presented in Ref. [4 ].

If one is interested in the sp ectral distribution of Compton-scattered photons, and not just

the integrated cross section, Namito et al.[19 ] have made it dramatically clear by computing

the scattered sp ectrum with and without including Doppler broadening in the EGS4 co de[20 ],

b ecause such broadening must b e included in order to replicate the corresp onding measurements

using 40 keV photons from a 2.5 GeV synchrotron light source (KEK-PF) (see Fig. 7). 8 -2 10-2 10 C (a) Cu (b)

-3 -3 source Measurement

10 source 10 Measurement EGS4(CP)

per EGS4(CP)

per

EGS4(S) -1

-1 EGS4(S) 10-4 10-4 keV

keV

-1 -1 sr.

sr. -5 10 10-5 Photons

-6 Photons 10 10-6 30 32 34 36 38 40 30 32 34 36 38 40

Photon Energy, keV Photon Energy, keV

Figure 7: Comparison of the photon sp ectra scattered by C and Cu samples. The measured and

calculated values are shown in symb ols and histograms, resp ectively (k =40 keV, =90 ). The

EGS4 calculation including Doppler broadening is shown as EGS4(CP). The EGS4 calculation

without Doppler broadening is shown as EGS4(S). The p eak at 30 keV is the Ge K X-ray escap e

p eak of the Rayleigh-scattering p eak at 40 keV.

4.2 Rayleigh scattering

Rayleigh scattering is a pro cess by which photons are scattered by b ound atomic electrons and

in which the atom is neither ionized nor excited. The scattering from di erent parts of the

atomic charge distribution is thus \coherent," i.e., there are interference e ects.

This pro cess occurs mostly at low energies and for high-Z material, in the same region

where electron binding e ects in uence the Compton-scattering cross section. In practice, it

is necessary to consider the charge distribution of all electrons at once. This can be done

approximately through the use of an \atomic form factor", F (q; Z ), based on the Thomas-

Fermi, Hartree, or other mo del of the atom. The square of this form factor, [F (q; Z )] , is the

probability that the Z electrons of an atom take up the recoil momentum, q , without absorbing

any energy. In this pro cess q is well-represented by the following equation, since it is assumed

that k k =0:

q =2k sin : (10)

The di erential Rayleigh scattering cross section for unp olarized photons is

2 2 2 1 1

)[F (q; Z )] (cm sr atom ): (11) ( )= (1 + cos

d 2

The cumulative angular distributions of , based on the values of F (q; Z ) from Nelms and

Opp enheim[21], are displayed in Fig. 8 for C, Fe, and Hg. At high energies it is seen that Rayleigh

scattering is conned to a small angle (at 1 MeV more than half the photons are scattered bylee

than 5 ). At low energies, particularly for high-Z materials, the angular distribution is much

broader. 9 100 95% 75% C 6 50%

25% 10 (degree)

1 0.01 0.1 1 Photon Energy, MeV

95% 100 Fe 75% 26 50%

25% 10 (degree)

1 0.01 0.1 1 Photon Energy, MeV

95% 100 Hg 75% 80 50%

25% 10 (degree)

1 0.01 0.1 1

Photon Energy, MeV

Figure 8: Op ening half-angle, , of cone containing 25%, 50%, and 95%, resp ectively, of photons

Rayleigh scattered from C, Fe, and Hg. 10

4.3 Polarization e ects

Linear p olarization is imp ortant in low-energy photon transp ort, b ecause photons are linearly

p olarized in scattering; further, the Compton and Rayleigh scattering of linearly p olarized pho-

tons are anisotropic in their azimuthal distributions in the low-energy region.

For linearly p olarized photons with a free electron, the Klein-Nishina angular distribution

function p er steradian of solid angle is[16]

d 1 k k k

2 2

= r + 2 + 4 cos : (12)

d 4 k k k

0 0

Here, is the angle b etween the incident p olarization vector (~e ) and the scattered p olarization

vector (~e). According to Heitler[22 ], two directions are considered for ~e, i.e. ~e either in the same

plane as ~e (~e ) or p erp endicular to ~e (~e ).

0 0

The elastic scattering of a photon by one electron is called Thomson scattering. The

Thomson-scattering cross section p er electron for a linearly p olarized photon is[23]

2 2

= r cos : (13)

Linearly p olarized photon scattering for b oth Compton- and Rayleigh-scattering have been

implemented to the EGS4 co de by Namito et al[24 ]. The e ect of linear p olarization is clearly

presented in Fig. 9[25 ], whichgives a comparison b etween EGS4 calculations with and without

including linear p olarization and Doppler broadening and corresp onding measurements using 40

keV linearly p olarized photons from a KEK-PF.

10-2 10-2 Pb 40 keV Rayleigh Pb 40 keV -3 -3 Exp H 10 10 Exp H Exp V Comton Exp V EGS4 H Default EGS4 H LP+DB

EGS4 V Default -4 EGS4 V LP+DB 10-4 10 (/keV/sr./source)

(/keV/sr./source)

-5 10-5 10 Counts Counts

-6 10-6 10 32 34 36 38 40 42 32 34 36 38 40 42 Energy Deposition (keV) Energy Deposition (keV)

(a) (b)

Figure 9: Comparison of the photon sp ectra scattered by a Pb sample. Source photons are

linearly p olarized in the vertical direction. The measured and calculated values are indicated

bysymb ols and histograms, resp ectively (k =40 keV, =90 ). (a) is the comparison with the

EGS4 calculation without including linear p olarization and Doppler broadening and (b) is the

one with linear p olarization and Doppler broadening. 11

5 Electron-Positron Pair Pro duction

In this e ect, which is the most likely photon interaction at high energies, a photon disapp ears

in the eld of a charged particle, and an electron-p ositron pair app ears.

For the reaction M + M ! M + M + M + Q, it can b e shown from the conservation

1 2 3 4 5

of energy and momentum that the threshold energy for the reaction in lab oratory system is

2 2 lab

[Q 2(M c + M c )] (14) T =

1 1

2M c

when M is at rest. In a pair-pro duction reaction ( + M ! M + m + m + Q),

2 e e

M = 0;

M = M = M;

2 3

M = M = m ;

4 5 e

so that

2 CM

Q = 2m c (T ) (15)

and

2 2 2

2m c (m c + Mc )

e 2

lab

T = : (16)

thh

Thus;

1. Pair pro duction in the eld of a nucleus of mass M (M m ):

2m c

2 2 lab

(Mc )=2m c =1:022 )MeV ) (17) T '

2. Pair pro duction in the eld of an electron (M = m ):

2m c

lab 2 2 2

= T (m c + m c )=4m c =2:044 )MeV ) (18)

e e e

m c

The cross section for pair pro duction in the eld of a nucleus varies as

Z : (19)

For low photon energies,

) ln(h ): (20) (=

n n

For high energies[26 ],

; (21)

where X is the radiation length of the material. The quality of the approximation is shown in

Table 2 ( for 1000 MeV).

n 12

Table 2. Pair-pro duction cross section in the eld of a nucleus[27].

2 2 1 2 1

Material X (g cm ) (cm g ) % Di erence (cm g )

0 n

Pb 6.40 0.122 0.114 7

Cu 13,0 0.060 0.055 9

Fe 13.9 0.056 0.051 10

Al 24.3 0.032 0.028 14

C 43.3 0.018 0.014 29

H O 36.4 0.021 0.020 5

The energy distribution of an electron and p ositron pair for Pb is shown in Fig. 10[22 ].

16 Pb 14

12 hν=infinity ) - 10 1000 +E +

)(E 8 φ

)/ 50 + 6 (E pair

σ 4 ( 10 2 6 0 0 0.2 0.4 0.6 0.8 1 E /(E +E )

+ + -

Figure 10: Energy distribution of an electron-p ositron pair for Pb. E and E are the kinetic

energy of a p ositron and an electron, resp ectively. (E )dE means the di erential cross

pair + +

2 2

section which pro duces a p ositron b etween E and E + dE . = Z r =137.

+ + +

The cross section, (triplet), in the eld of one of the atomic electrons varies as Z times

tr ip

the square of the unit charge, or

Z; (22)

tr ip

and is of minor imp ortance, except for the lowest-Z materials.

This cross section is usually called the \triplet" cross section, since the atomic electron

involved in this pro cess is also ejected from the atom, giving rise to trident signature, including

the created electron and p ositron, when observed in a cloud chamber.

Finally, the \characteristic" angle b etween the direction of motion of the photon and one (or

the other) of the electrons () is given by

m c

(inradian): (23)

h 13

6 Mass Attenuation Co ecient and Energy-Absorption Co e-

cients

6.1 Mass attenuation co ecient, =

A narrow b eam of mono energetic photons with an incidentintensity I , p enetrating a layer of

material with mass thickness x and density , emerges with intensity I ,given by the exp onential

attenuation law,

I=I = exp[(=)x]: (24)

This equation can b e rewritten as

= = x ln(I =I ); (25)

from which = can b e obtained from measured values of I ; I and x. The various exp erimental

arrangements and techniques from which = can be obtained, particularly in the crystallo-

graphic photon energy/wavelength regime, have b een examined and assessed by Creagh and

Hubb ell[29, 30 ] as part of the International Union of Crystallography (IUCr) X-Ray Attenua-

tion Pro ject.

The theoretical value of = is de ned with the total cross section per atom, , whichis

tot

related to = according to

: (26) = =

tot

23 1

In this equation, N is Avogadro's number (6:022045 10 mol ), u is the atomic mass

unit (1/12 of the mass of an atom of nuclide C), A is the relative atomic mass of the target

element. Total cross section can be written as the sum over contributions from the principal

photon interactions,

= + + + + + : (27)

tot pe pair tr ip

coh incoh ph:n

Photonuclear absorption can contributed as much as 5-10% to the total photon interaction

cross section in a fairly narrow energy region, usually o ccurring somewhere between 5 and 40

MeV. This cross section has not b een included in the theoretical tabulations of the total cross

section b ecause of the diculties due to (a) the irregular dep endence of b oth the magnitude and

resonance-shap e of the cross section as a function of b oth Z and A; (b) the gaps in the available

information, muchofwhich is for separated isotop es or targets otherwise di ering from natural

isotop e mixtures; and (c) the lack of theoretical mo dels for comparable to those available

ph:n:

for calculations of the other cross sections of interest.

= without including photonuclear absorption is calculated according to

( + + + + ): (28) = =

pe pair tr ip

coh incoh

6.2 Mass energy-absorption co ecient, =

2 2

The mass energy-absorption co ecients, = (cm /g or m =kg, is a density of the medium), is

a useful parameter in computations of energy dep osited in media sub jected to photon irradiation.

= can be describ ed more clearly through the use of an intermediate quantity, the mass

energy-transfer co ecient, =.

The mass energy-transfer co ecients, =, when multiplied by the photon energy uence

( =h , where is the photon uence and h is the photon energy), gives the dosimetric 14

quantity kerma. Kerma has been de ned[28] as the sum of the kinetic energies of all those

primary charged particles released by uncharged particles (here photons) p er unit mass. Thus,

= takes into account the escap e of secondary photon radiations pro duced at the initial

photon-atom interaction site, plus, by convention, the quanta of radiation from the annihilation

of p ositrons (assumed to have come to rest) originating in the initial pair-pro duction interactions.

Hence, = is de ned as

= = (f + f + f + f ): (29)

tr pe pe inco pair pair tr ip tr ip

incoh

In this expression, coherent scattering has b een omitted b ecause of the negligible energy

transfer asso ciated with it, and the factor f represents the average fraction of the photon en-

ergy, h , that is transferred to kinetic energy of charged particles in the remaining typ es of

interactions. These energy-transfer fractions are given by

f =1 (X=h ); (30)

pe 0

where X is the average energy of uorescence radiation emitted p er absorb ed photon,

f =1 (+X )=h ; (31)

incoh

where is the average energy of the Compton-scattered photon.

Also,

f =1 2m c =h (32)

pair e 0

and

f =1 (2m c + X )=h : (33)

tr ip e 0

The mass energy-absorption co ecient involves further emission of radiation pro duced by

charged particles in traveling through the medium, and is de ned as

= =(1 g ) =: (34)

en tr

The factor g represents the average fraction of the kinetic energy of secondary charged parti-

cles (pro duced in all the typ e of interactions) that is subsequently lost in radiative energy-loss

pro cesses as the particles slow to rest in the medium. The evaluation of g is accomplished by

integrating the cross section for the radiativeprocessofinterest over the di erential track-length

distribution established by the particles in the course of slowing down.

Various calculations and compilations were p erformed for =. Most recent results were

presented by Hubb ell and Seltzer[6 ]. Fig. 11 shows a comparison between = and = for

lead presented in their pap er.

The mass energy-absorption co ecients when multiplied by the photon energy uence, ,

gives the absorb ed dose of the materials if charged-particle equilibrium exists. The mass energy-

absorption co ecients are used in cavity-chamb er theory together with the mass stopping p ower.

7 Photon Cross Section Data Base

Data on the scattering and absorbing of photons, which are explained in previous sections, are

required for many scienti c, engineering and medical applications. Available tables[1 , 6 , 31 , 32 ,

33 , 34 , 35 , 36 , 37 , 38, 39 , 40 , 41 , 42 , 43 ] usually include cross sections for many (but not all)

elements. Some tables[1, 31 , 35 , 40 ] also contain data for a limited number of comp ounds and

mixtures.

It is p ossible to generate the cross sections and attenuation co ecients r any element,

comp ound or mixtures at energies b etween 1 keV and 100 GeV with a computer program called

XCOM[45 ]. A PHOTX[9] data base was prepared using this XCOM. 15 Z=82, Lead 104

103 µ/ρ µ /ρ en

/g 10 2 cm

, ρ ρ ρ ρ 1 / 10 en µ µ µ µ

or 100 µ/ρ µ/ρ µ/ρ µ/ρ

10-1

10-2 0.001 0.01 0.1 1 10 100

Photon Energy (MeV)

Figure 11: Mass attenuation co ecient, =, and the mass energy-absorption co ecient, =,

as a function of the photon energy for lead.

References

[1] J. H. Hubb ell, \Photon Cross Sections, Attenuation Co ecients, and Energy Absorption

Co ecients From 10 keV to 100 GeV", NSDRS-NBS 29.

[2] J. H. Hubb ell, W. J. Veigele,, E. A. Briggs, R. T. Brown, D. T. Cromer, and R. .J. Hower-

ton, \Atomic Form Factors, Incoherent Scattering Functions, and Photon Scattering Cross

Sections", J. Phys. Chem. Ref. Data 4(1975)471-616.

[3] J. H. Hubb ell, \Polarization E ects in Coherent and Incoherent Photon Scattering: Survey

of Measurements and Theory Relevant to Radiation Transp ort Calculations", NISTIR

4881, 1992.

[4] J. H. Hubb ell, \Bibliography and Current Status of K, L, and higher Shell Fluores-

cence Yields for Computations of Photon Energy-Absorption Co ecients", NISTIR 89-

4144(1994).

[5] J. H. Hubb ell, \Tables of X-ray Mass Attenuation Co ecients and Mass Energy-Absorption

Co ecients 1 keV to 20 MeV for Elements Z=1 to 92 and 48 Additional Substances of

Dosimetric Interest," NISTIR 5632, 1995.

[6] J. H. Hubb ell, \An Examination and Assessment of Available Incoherent Scattering S-

Matrix Theory, also Compton Pro le Information, and Their Impact on Photon Attenua-

tion Co ecient Compilations," NISTIR 6358, 1999.

[7] U. Fano, L. V. Sp encer, M. J. Berger, in Flugge (Ed.), Encyclop edia of Physics, Vol.

XXXVI I I/2 Berlin/Gottingen/Heidelb erg: Springer 1959.

[8] E. Frytag, \Strahlenschutz an ho chenergiebschleunigern", (G. Braum, Karlsruhe, 1072). 16

[9] Radiation Shielding Information Center Data Library Package DLC-136/PHOTX, \Photon

Interaction Cross Section Library", contributed by the National Institute of Standards and

Technology.

[10] R. W. Fink, R. C. Jopson, H. Mark, and D. C. Swift, \Atomic Fluorescence Yields", Rev.

Mod. Phys 38(1966)513-540.

[11] W. Bambynek, B. Crasemann, R. W. Fink, H. -U. Freund, H. Mark, C. D. Swift,

R. E. Price, and P. V. Rao, \X-Ray Fluorescence Yields, Auger, and Coster-Kronig Tran-

sition Probability", Rev. Mod. Phys. 44(1972)716-813; erratum in bf 46(1974)853.

[12] M. O. Krause, \Atomic Radiative and Radiationles Yields for K and L shells", Phys. and

Chem. Ref. Data 8 (1979)307-327.

[13] B. Richard, B. Firestone and S. S. Virginia edited, \Table of Isotop es Eighth Edition",

John Wiley & Sons, Inc.,1996.

[14] J. H. Sco eld, \Relativistic Hartree-Slater Values for K and L X-Ray Emission Rates",

Atomic Data and Nuclear Data Tables 14(1974)121-137.

[15] S. L. Salem, S. L. Panossian, and R. A. Krause, \Exp erimental K and L Relative X-Ray

Emission Rate", Atomic Data and Nuclear Data Tables 14(1974)91-109.

[16] O. Klein and Y. Nishina, Z. Physik 52 (1929)853-868.

[17] R. D. Evans,\Compton E ect", in Handbuch der Physik XXXIV, Tlugge ed., pp2180298,

Springer-Verlag, Berlin (1958).

[18] I. Waller and D. R. Hartree, \On the intensity of total scattering of X-rays", Proc. R. Soc,

London, Ser. A 124 (1929)119-142.

[19] Y. Namito, S. Ban and H. Hirayama, \Implementation of the Doppler broadening of a

Compton-scattered photon into the EGS4 co de", Nucl. Instr. Meth. A349(1994)489-494.

[20] W. R. Nelson, H. Hirayama and D. W. O. Rogers, \The EGS4 Co de System", SLAC-265,

(1985).

[21] A. T. Nelms and L. J. Opp enheim, Rev. Nat. Bur. Stds. 55(1955)53-62.

[22] W. Heitler, \The Quantum Theory of Radiation", Oxford Univ. Press, Oxford, 1954.

[23] P.P. Kane, Phys. Rev. 140(1986)75.

[24] Y. Namito, S. Ban and H. Hirayama, \Implementation of Linearly-p olarized photon scat-

tering into the EGS4 co de", Nucl. Instr. Meth. A 332(1993)277-283.

[25] H. Hirayama, S. Ban and Y. Namito, \Current Status of a Low Energy Photon Trans-

port Benchmark", Presented at Shielding Asp ects of Accelerators, Targets and Irradiation

Facilities (SATIF5), Paris, France, July 17-21, 2000, KEK Preprint 2000-65, July 2000.

[26] B. Rossi, \High Energt Particles", Prentice=Hall, Inc., Englewo o d Cli s, New Jersey, 1952.

[27] K. R. Kase and W. R. Nelson, \Concepts of Radiation Dosimetry", Pergamon Press.

[28] \Radiation Quantities and Units", ICRU Report 33, Bethesda, MD, 1980. 17

[29] D. C. Creagh and J. H. Hubb ell, \Problems Asso ciated with the Measurement of X-Ray

Attenuation Co ecients. I. Silicon", Rep ort on the International Union of Crystallography

X-RayAttenuation Pro ject, Acta Cryst. A 43(1987)102-112.

[30] D. C. Creagh and J. H. Hubb ell, \Problems Asso ciated with the MeasurementofX-RayAt-

tenuation Co ecients. I I. Carb on", Rep ort on the International Union of Crystallography

X-RayAttenuation Pro ject, Acta Cryst. A 46(1990)402-408.

[31] J. H. Hubb ell and M. J. Berger, Section 4.1 and 4.2 in R. G. Jaeger (ed.): Engineering

Comp endium on Radiation Shielding (IAEA, Viena), Vol. 1, Ch. 4, pp. 167-202, Springer,

Berlin (1968),

[32] W. H. McMaster, N. K. DEl Grand, J. H. Malett, J. H. Hubb ell, \Compilation of X-ray

Cross Sections", Lawrence Livermore Lab., Report UCRL-50174, (1969).

[33] E. Storm and H. I. Israel, \Photon Cross Sections from 1 keV to 100MeV for Elements

Z=1 to Z=100", Nucl. Data Tables A7(1970)565-681.

[34] W. J. Weigele, \Photon Cross Sections from 0.1 keV to 1 MeV for elements Z=1 to Z=94",

Atomic Data 5(1973)51-111.

[35] J. H. Hubb ell, \Photon Mass Attenuation and Mass Energy-Absorption Co ecients for H,

C, N, O, Ar and Seven Mixtures from 0.1 keV to 20 MeV", Radiat. Res. 70(1977)58-81.

[36] J. Leroux and T. P. Thinh, \Revised Tables of X-ray mass Attenuation Co ecients",

Cororation Scienti que Classique, Queb ec (1977).

[37] J. H. Hubb ell, H. A. Gimm, and I. Overb o, \Pair, Triplet and Total Atomic Cross Sections

(and Mass Attenuation Co ecients) for 1 MeV-200 GeV Photons in Elements Z=1 to 100",

J. Phys. Chem. Ref. Data 9(1980)1023-1147 .

[38] E. F. Plechaty, D. E. Cullen, and R. J. Howerton, \Tables and Graphs of Photon-Interaction

Cross Sections from 0.1 keV to 100 MeV Derived from the LLL Evaluated-Nuclear-Data

Library", Report UCRL-50400, Vol. 6, Rev. 3 (1981).

[39] B. L. Henke, P. Lee, T. J. Tanaka, R. L. Shimabukuro, and B. K. Fujikawa, \Low Energy

X-rayInteraction Co ecients: Photoabsorption, Scattering and Re ection", Atomic Data

and Nuclear Data Tables 27(1982)1-144.

[40] J. H. Hubb ell, \Photon Mass Attenuation and Energy Absorption Co ecients from 1 keV

to 20 MeV, Int. J. Appl. Radiation & Isotopes 33(1982)1269-1290 .

[41] S. T. Perkins, et al., \Tables and Graphs of Atomic Subshell and Relaxation Data Derived

from the LLNL Evaluated Atomic Data Library (EADL), Z=1-100", Lawrence Livermore

National Lab oratory, Livermore CA, Vol. 30 (1991).

[42] D. E. Cullen, et al., \Tables and Graphs of Photon-Interaction Cross Sections Derived

from the LLNL Evaluated Atomic Data Library (EPDL), Z=1-50", Lawrence Livermore

National Lab oratory, Livermore CA, Vol. 6, Part A, Rev. 4(1991).

[43] D. E. Cullen, et al., \Tables and Graphs of Photon-Interaction Cross Sections Derived

from the LLNL Evaluated Atomic Data Library (EPDL), Z=51-100", Lawrence Livermore

National Lab oratory, Livermore CA, Vol. 6, Part B, Rev. 4(1991). 18

[44] Radiation Shielding Information Center Data Library Package DLC-179/ENDLIB-

94,\LLNL Libraries of Atomic Data, Electron Data, and Photon Data in Evaluated Nuclear

Data Library (ENDL) Typ e Format", Contributed Lawrence Livermore National Lab ora-

tory.

[45] M. J. Berger and J. H. Hubb ell, \XCOM: Photon Cross Sections on A Personal Computer",

NBSIR 87-3597, 1987. 19