HIS CIRCULAR
,
AD-A278 139
4,
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X-ray Attenuation Coefficients
From 10 kev to 100 Mev
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APR n 4 1994
94-10025
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Z1UNITED STATES DEPARTMENT OF COMMERCE
NATIONAL BUREAU OF STANDARDS
94 .4
1 096
Data on Radiation Physics
Graphs of the Compton Energy-Angle Relationship and
the Klein-Nishina Formula from 10 Key to 500 Mev
NBS Circular 542
The Compton energy versus angle relationship and the differential and integral Klein-Nishina cross sections are presented
graphically as functions of the energy and direction of the scattered photon and of the recoil electron. These graphs are intended
to serve the purpose of tables. Unpolarized primary gamma rays
in an energy range from 10 Key to 500 Mev are considered. The
accuracy of.allcufrves is estimated at 1 percent. The advantage
,of this form of presentation is the convenience and accuracy of
two-way interpojation. In general, interpolated values may be
obtained with an accuracy of 2 percent.
National Bureau of Standards Circular 542, 89 pages, 81 graphs.
Available by purchase from the Superintendent of Docujments,
Government Printing Office, Washington 25, D. C., price ba cents.
Energy Lofs and Range of Electrons and Positrons
Tabulations of the mean energy loss due to ionization and excitation and .the rarge derived from this quantity are given for
electrons and positrons in several materials.
National Bureau of Standards Circular 577,30 pages, 10 graphs.
Available by purchase from the Superintendent of Documents,
Government Printing Office, Washington 25, D. C., price 30 cents.
UNITED STATES DEPARTMENT OF COMMERCE
NATIONAL BUREAU OF STANDARDS
•
Sinclair Weeks, Secretary
, A. V. Astin, Dirvcew
X-ray Attenuation Coefficients
From 10 key to 100 Mev
Gladys White Grodstein
Accesion For
I .,.
Ia. -
-•
NTIS CRA&
FOTIC TAB
Unannournced
D
JustifiCation ..............................
BY*......... ........
Dit. ibtion I
.......
-.
Availabiilty Codes
Dist
Avai# and/or
Special
National Bureau of Standards Circular 583
Issuw1 April 30, 1957
Fo
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ent 'rindg
S- Go
unns
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/
CONTENTS
Page
1. Introduction -------------------------------------------------1.1. Narrow-beam attenuation ------------------------------1.2. Absorption and scattering processes ----------------------1.3. Corrections to narrow-beam measurements ----------------1.4. Combination of attenuation coefficients -------------------1.5. Energy absorption -------------------------------------2. Probability of processes ---------------------------------------2.1. Photoelectric effect ------------------------------------2.2. Scattering by atomic electrons ---------------------------2.3. Pair production
-----------------------------------2.4. Nuclear absorption and scattering -----------------------3. Calculation of attenuation coefficients and comparison with experiment ----------------------------------------------------3.1. Photoelectric effect ------------------------------------3.2. Scattering by atomic electrons ----------------------3.3. Pair production ---------------------------------------3.4. Total attenuation coefficient -----------------------------4. Figures and tables --------------------------------------5. Appendix ----------------------------------------------------6. References ---------------------------------------------------
iii
1
1
1
2
2
3
3
3
5
7
8
9
9
10
11
12
13
51
54
X-ray Attenuation Coefficients From 10 key to 100 Mev*
Gladys White Grodstein
A tabulation of attenuation coefficients of X-rays and gamma rays from 0.01 to 100
Mev for 29 materials is presented. A summary of information orr the probability of the
basic interaction processes of photons with matter and a detailed analysis of experimental
and theoretical evidence are included. Present information on the basic processes is adequate for many applications; however, improved theory and additional experimental data
are needed in certain areas. A comparison of calculated and experimental coefficients
points up this need.
1. Introduction
1.1. Narrow-Beam Attenuation
The attenuation coefficients tabulated here are
narrow-beam, as opposed to broad-beam, coefficients. The total probability that a photon of
given energy interacts with matter may be studied
experimentally with a well-collimated beam of
homogeneous X-rays incident upon an absorber
(fig. 1). A well-shielded detector measures the
intensity of the trinsmitted beam, and any photon
absorbed or dof_.- ed appreciably does not reach
the detector, if .4he. detector is sufficiently collimated and far from the absorber. The attenuation of the intensity received by the detector as
the absorber thickness is increased measures the
total probability of the interaction processes.
The usual semflogarithmic plot of transmitted
intensity, I, versus thickness of absorber, t, follows
a straight line, indicating exponential decay of the
intensity according to 1(t)=I(O)exp.(--). The
slope, 1, of the straight line represents*the total
attenuation coefficient, namely, the probability
that a photon be removed from the incident beamh
per unit thickness of material traversed. A layer
of matter absorbs according to the quantity of
matter it contains, which is the thickness traversed
times. the density of the material. Therefore
absorber thicknesses are conveniently expressed
on a mass basis, in grams per square centimeter.
Accordingly, the attenuation coefficient is often
expressed in (g/cm2)-i=cm2/g and called the
mass-absorption coefficient.
1.2. Absorption and Scattering Processes
Photons may be absorbed or scattered as the
result of interaction with a material. Absorption
is characterized by the disappearance of a photon.
Scattered photons are deflected from the original
direction with or without a decrease in energy.
The total probability that a process takes place
per unit thickness of absorber is the sum of the
probabilities of occurrence of the various absorption and scattering processes [1].1.2 To each kind
of absorption process corresponds a process of
-This ofsurvey
has been carried out with
Branch
the Atomic
Energy Commission.
the support of the Biophysics
C¥1tmr in bracket
indicate the literature references at the end of this
cirusphoteonls
3RIerence m1]contains a classification and a qualitative description of the
abeorption and scattering processes,
scattering; the scattering may be regar(ded as at
combination of absorption and emission of a
photon, the emission taking place in a new direction.
The most important process at low photon
energy is the photoelectric effect, defined as the
absorption of a photon with subsequent ejection
of an atomic electron. Electrons in the K and L
shells account for most of the absorption by this
process at frequencies greater than the K-edge
frequency; the K electrons contribute more than
80 percent of the total absorption at these frequencies. Photons with energy very much in
excess of that required to eject an electron are
unlikely to be absorbed. Consequently, the absorption coefficient for the photoelectric effect
decreases rapidly as the photon energy increases.
Scattering of photons by atomic electrons makes
a large contribution to the total attenuation coefficient in the middle energy range (0.5 to 5 Mev).
Most of the scattering is incoherent, Compton
scattering; a photon is deflected with a reduction
in energy and an atomic electron recoils out of the
atom. The probability of this process may be
calculated approximately as though the atomic
electrons were free. Incoherent radiation consists of a spectrum of frequencies smaller than the
primary frequency. The intensity scattered in
any direction is simply the sum of the intensities
scattered by the individual electrons.
Some of the scattering by an atomic system is
coherent, Rayleigh scattering; a photon may be
deflected with no loss in energy, and the atomic
system recoils as a whole under the impact. The
probability of this process is large only for photons
with low energy; that is, in the region where
photoelectric ab'sorption gives the main contribution to the total attenuation coefficient.
A photon with energy greater than 1 Mev may
be absorbed in the neighborhood of an atomic
nucleus or an atomic electron and produce an
electron-positron pair. The probability for this
process increases rapidly with photon energy
above the threshold but levels off at higher
energies. The positron of the pair is eventually
annihilated
with of
production
new X-rays. The
lret
fraction
the new of
radiation
consists of
with energy me' emitted in pairs in
lar
opposite directions.
S~1
Absorption of a photon by the atomic nucleus
[2] occurs with subsequent emission of nuclear
particles, mostly neutrons, and little gamma radiation. The probability of this photonuclear process
has a maximum around 15 to 25 Mev, depending
upon the atomic number of the absorber. In a
There is also an appreciable amount of coherent
scattering at small angles. The intensity of this
radiation scattered within a cone of aperture
0.,,, can be obtained by integrating numerical data
on the differential cross section for this purpose.
This was done by Colgate [51, using the numerical
narrow energy interval about the maximum it may
give a contrib'ution of 5 to 10 percent to the total
attenuation coefficient.
Scattering of photons by atomic nuclei occurs in
a manner analogous to the scattering by atomic
electrons. Scattering by nuclei may be either
elastic or inelastic. The probability of nuclear
scattering is generally small compared to the
probability of scattering by the atomic electrons.
Its contribution to the total attenuation coefficient
is negligible, except as noted at the end of section
2.4; it is less than 0.1 percent in the 15- to 20-Mev
range for heavy elements.
Even though the contribution of these nuclear
effects to the total attenuation is quite appreciable
in small regions, and even though information on
these effects begins to be abundant and reasonably
accurate, these data do not yet constitute a body
of knowledge comparable to the knowledge for
electronic effects. Therefore, the main tables of
this Circular include only the effects of electronic
processes. Information on nuclear effects is discussed briefly in section 2.4, and some data on the
nuclear contribution to attenuation are given,
data of Debye [6] and the equations of Franz [7] ;1
see also Moon's discussion of the Franz equations
[8].
The need for these theoretical corrections to the
attenuation of the incident beam can be eliminated
if one follows the extrapolation procedure to
Om---0 suggested by Colgate [5]. This procedure
eliminates only the effect, of Compton scattering,
unless measurements are actually taken down to
the very small values of 6 ,,, at which coherent
scattering is important.
Fluorescent radiation originating in an absorber
as a result of photoelectric absorption can also
reach the detector. However, the intensity ilitercepted by the detector in the usual narrow-beam
experiment is quite small. For example, for Pb
exposed to 100-key radiation, the intensity of
fluorescence per steradian is roughly 6 percent
10.95(76/100)(1/4r)] of the radiation absorbed
photoelectrically. (The fluorescent yield is 0.95,
and K. radiation is isotropic with 76-key energy.)
Assuming for the detector aperture a solid angle
of 0.01 steradian, which is rather large for this type
of experiment, the measured intensity of the 76Key radiation is roughly 0.06 percent of the
intensity absorbed photoelectrically from the
1.3. Corrections to Narrow-Beam Measure-
ments
Some radiation scattered in an absorber will
always reach the detector, as seen in figure 1.
The effect of receiving this scattered radiation is
to increase the intensity of the transmitted beam.
The intensity of singly scattered radiation can be
easily calculated. If the maximum angle (Omax)
through which radiation is scattered into the
detector is small, and if the experimental arrangement has cylindrical symmetry, the intensity of
the transmitted beam is increased by the amount
of scattering within a cone of aperture 0.a..
incident 100-kev radiation.
The number of annihilation photons from the
absorber that reach the detector will b3 similarly
small in the usual narrou--beam experiment. Assuming that all radiation emitted is from 2 quanta
annihilation and is isotropic, the number of
photons per steradian will be approximately 16
percent of the number of pairs produced. 'The
number of photons detected in a solid angle of 0.01
steradian is only 0.16 percent of the number of
electron-positron pairs produced in the absorber
by the incident radiation.
The
intensity of radiation scattered within this cone
can be subtracted from the measured intensity to
1.4. Combination of Attenuation Coefficients
small 0m.. the intensity of Compton scattering
within the cone according to the Klhin-Nishina
formula is given by 3
X-ray photon with different atoms of an absorber
add up without mutual disturbance, in general.
The effect of chemical binding on the interaction
of X-ravs with valence electrons is exceedingly
weak. 'H-owever, the orderly arrangeme t of
give the attenuation of the incident beam.
cxZ.
9
For
012
[1--12 (9a-4)]
where
x=the thickness of the absorber, in g/cm2 ,
a=the incident energy, in me2 units, and
•.
C-Nirr0 Z
-=0.150Z
A
a cn
e
IA
similer
lculation was made by Davisson and Evans [31 and by
[41, but the Tarrant paper contains an erroneous re~alt.
Tarrant
2
The probabilities of interaction processes of an
atoms next to one another does influence thi total
probability of interaction processes to an extent
that is quite considerable, expecially in Bragg
reflection by crystal lattices, when the momentum
transfer from photon to matter is of the order
of the Planck constant divided by the spacing
of adjacent atoms. Special situations of this
kind are disregarded in the present Circular.
Within this approximation, the mass-attenuation
4 The total cross section of Franz is too small by a factor of 2 owing to an
analytical error.
coefficient of a chemical compound or mixture is
an average of the mass attenuation coefficients of
the constituent elements, weighted in proportion
to the abundance of each element by weight. For
example, for water (1 part H, 8 parts 0), we have
1A2 o:-(1/ 9 )jH+ (8 /9 )Ao, provided the g's are expressed as mass-attenuation coefficients.
1.5. Energy Absorption
Most of the energy transferred from X-rays and
gamma rays to a material is given to electrons or
positrons and then dissipated along the path of
these particles. Part of this energy is absorbed
by inelastic collisions with other atomic electrons
and some is released to photons of lower energy.
Thus the energy of the incident photon is not
entirely absorbed at the point of its interaction in
the material. In fact, at energies greater than a
few million electron volts, electrons may travel
distances comparable to the mean free path of
photons of the same energy. There are problems,
as in dosimetry and in medical or biological studies,
that require a calculation of the probable energy
transfer to a material by a beam of X-rays. The
fraction of energy dissipated locally by a narrow
beam of X-rays is given by the product of the
probabilityfraction
of eachof interaction
probable
the photon process
energy and
thatthe
is
tha
eer gy
probate loctin the phor
dissipated locally in the absorber as a result of the
process. The definition of the term "locally" is
not unique; it will depend on the energy of the
incident radiation, on the material of th.- absorber,
and further on the purpose of a particular measurement and the viewpoint of the observer.
2. Probability of Processes
Theoretical methods for calculating the probability of the basic interaction processes of photons
with matter are well established. However, systematic calculations are complicated. Various
kinds of approximations can be utilized, but their
proper application requires some care. Substantial uncertainty still exists regarding many details
of the approximation procedures.
Nevertheless, theory has progressed to the point
where the present tabulation of data has been
derived primarily from theory, with experimental
data providing the necessary checks and some
additional fitting.
2.1. Photoelectric Effect
effect5T
decrease
exhibits, as main features, a very rapid
as the frequency of the incident X-ray increases
and a rapid increase as the atomic number of the
material increases. This behavior appears natural because an electron can resonate under the
driving action of a high-frequency disturbance
only if it is held by a very strong force such as
obtains in the space immediately surrounding an
atomic nucleus. This portion of the atomic
volume, where the force is adequate, is a decreasing function of the dAving frequency and an
increasing function of the magnitude of the
nuclear charge. When the photon energy hi
exceeds mc most of the momentum of the ejected
electron is imparted directly by the incident
photon. The attraction by the nucleus need
supply only a momentum of the order of mc, no
matter how large is the energy hG. Accordingly,
the probability of the photoelectric effect decreases more slowly as the energy hp keeps
increasing in the relativistic range.
Simplifying assumptions. The main approximnations that are usually considered in any theoI See Sommerfeld (9] and Hall 1101for reviews of the theory of the photoThe probability of tihe photoelectric
electric egrect.
retical analysis of the photoelectric effect involve
one or more of the following features:
(a) Schematic* treatment of the. interaction
among atomic electrons, in the form of "screening
effects," which permits the use of hydrogen-like
wave functions for the atomic electrons.
(b) Treatment of the electron motion according
to nonrelativistic quantum mechanics (valid for
hvl/mc 2<
, (Z/137)2«1).
(c) Disregard of the attraction exerted by the
nucleus on the electron as it leaves the atom
(Born approximation valid for Z/(137v/c) «1,
where v is the speed of the ejected electron).
(d) Disregard of the possibility that the ejected
electron may receive from the radiation an
angularapproximation).
momentum larger
than
h/2w (dipole
transition
This
assumption
is justifled if the X-ray wavelength is much larger than
the initial wavelength of the atomic electron.
(e) Treatment of the electron motion by the
Sommerfeld-Maue-Furry approximation (angular
momentum quantum number j»Z/137). This
approximation is useful when the conditions are
opposite to (d), that is, when (at very high ener.es) most of the photoelectric effect.is contributed
by high order multipoles. 6
The interaction of radiation with the atomic
electron is normally treated as. "weak." Higherorder electrodynamic effects require corrections of
the order of 1/137 or smaller.
As a further approximation, one often assumes
that the probability ratio of photoelectric ejection
of different electrons is energy independent in the
range of interest. Because this approximation is
reasonable, and as K electrons have the largest
chance of being ejected by X- 'ays above the K
edge, most data in 1 :"tereture deal with the
photoelectric effect in ie K shell.
The principal calct'lations which hb ve been
8 Bethe and Maximon ill] used this approximation in the calculation of the
differential crosm section for brem-.strahlung and pair production.
3
carried out in detail are listed below, with an
indication of the pertinent approximations.
Simple Born calculation (approximations a,
b, c). 7he cross section for photoelectric effect in
the K shell of an atom with atomic number Z
for a photon of energy hv is [12, p. 207]
vjr
2" 4
Z5 .
=8
h--4/k,
)
1T37'
(1)
2
electron in units of mc2 . The effect of screening
3
Sauter formula (approximationsa, c). TI - corresponding relativistic calculation was made. by
Sauter [131. The assumption (b) is thereby
eliminated.
Stobbeformukas (approximationsa, b, d). A basic
calculation, using exact nonrelativistic hydrogenlike wave functions, was made by Sto'bbe [14]
for electrons of the K, L, and
shells. Its results
can be expressed by a factor f [12], which represents the ratio of the "non-Born" cross section to
the "Born" cross section eq (1).
Sauter-Stobbe combined formula. The SauterBorn approximation cross section may be corrected to a considerable extent by multiplying it
with the factorf derived by Stobbe under a nonrelativistic approximation. "The combined formula
becomes
2
ýmcl5y2-1
3ý4 Zý5
3(
-= o(137
"'Y(7-i-2)
14
'Nagasaka formula (approximation8 a and e).
Nagasaka [171 developed a high-energy formula,
using the Sommerfeld-Maue function for the-final
state and the exact Dirac wave function for the
initial state of the K-electron. The SommerfeldMaue function may be used for the final state of
the electron in the photoelectric effect so long as
(Z/137) 2(e-1 In e)<<1, where e is the energy orthe
was completely neglected'in this calculation, which
is justified by the remark in footnote 7.
Nagasaka's cross section has the form
aK-
3
Z" mI
00
13
o
G 010L
- 2-
Ie--
ZZ
0.832 -±1.476--'
1376 ,
137
hp e-1
there
elec1ro(n(--i)/ncuis the total energy of
the ejected electron (including its rest, mass) in
units of me 2 , G is a factor discussed below, and
2-11'f4j(I)f12
____In_
( 2- 1)3/2-r4E(2e•
00(1)4[3e+±1
92e
E
/_,E2_LlljJ
+Y)2-
-ý,hP+-+ •~,13
(•-)Lt
(4)
(5)
Notice that eq (4) reduces to the Sauter formula
Y+±Ve
1
I1-n
y
- --4.F.,c
2•r|I_
I
t
I
2
(2)
where
hp
and/=(Z--.3Y
_'__i,
(&is
o-o are, disregarded;
terms
(2) if the
taken
as 1,
and following
I is disregarded
in the definition
of e, so that E='Y. In the high-energy limit, more
specifically for
/e2«<
1, Nagasaka
finds
G=-exp [--Z/137+2(7/137) 2-- (7/137) 211n (7,/137)].
y
Hulme calculation (approximation a). A calculation using exact relativistic hydrogen-like
wave functions was made by Huline [151. The
results are given numerically for a few values of
the
atomic number
and of the photon energy,
Interpolation
is possible
to a considerable extent.
The corresponding factor in Hall's formula (3)
has an additional factor of 2 in front of the loga-
Approximation (d) is set aside, but the requirement
to
carryof out
the
calculations
for many successive
terms
dipole,
quadrupole
sequence
makes
thethe
procedure
prohibitiveiy.... laborious
at
rithm in the exponent. The Hall and Nagasaka
calculations differ in formal procedure but utilize
in fact the same approximation. Part of the
difference, between the results (3) and (4) has
d
een t
e results (3 al
(4)tas
been
traced
by Nagasaka to an algebraic mistake
in Most
Hall's
calculation.
calculations in the literature deal with
h
the photoelectric absorption in the K shell, which
Intrpoatin
i posibe t
a
onsderbleextnt.
mc2>> 1.
Hall formula (approximation a, wlmc'>>1).
Hall [16, 10] developed a high-energy formula that
does not rely on the Born approximation, like the
Sauter formula, or on a separate evaluation of the
dipole, quadrupole, . . . sequence, like the Hulme
Hall gives
calculations.
3
aK=3 *0
Z5 me2
13-74Z
-mh
Re-"a+2a2 -2
where
R=1+[4(1-a)ý--5/3]1
4
n
&
nucleus by centrifugal action, and therefore, expe-
rience less attraction than 1=0 electrons. The
relative probability of photoelectric effect in the
L,, M,, . . . subshells should be approximately
2K,
-,
(3)
greatly exceeds the absorption in other shells for
X-ray energies above the K edge. For energies
well above the K edge, absorption in the L, A,
.. subshells reatly exceeds the absorption in
the L, L,,,, M.. . . . subshells, because electrons with azimuthal quantum number 1=1, 2,
. . . are kept away from the proximity of the
and a=Z/137.
independent of the photon energy at high energies, according to elementary theory. These prob-
abilities should be approximately in the same ratios, 1, 1/23, 1 /3 s, . . . as the probabilities that
K, L,, M,, . . . electrons be near the nucleus,
The Stobbe formulas indicate a slight decrease of
the ratios of L to K, M to K, . . . as Av increases,
An application of the Hall formula to a calculation of ajL/
at 2.62 Mev for Pb yields 0.20,
which is considerably more than one-eighth.
Limited experimental evidence has indicated (18],
as an approximate rule, that the total probability
of photoelectric effect at high energies equals 5/4
of the probability for the K shell alone. Notice
that 5/4 is a little larger than the sum 1±1/21+
1/33 = 1.16.
A simplified treatment of the interaction among
atomic electrons (approximation a) may suffice for
the photoelectric action on K shell electrons for
which nuclear attraction greatly predominates over
other attractions. The portion of the electronic
cloud that lies nearer to the nucleus than the
photoelectron effectively offsets, or "screens,"
the nuclear charge to some extent. This effect
may be taken into account by attributing to the
nucleus an "effective atomic number" 7-s. The
number 8, called the "inner screening number,"
was evaluated semiempirically for the electrons of
the various shells by Slater [19]; values of s are
.
given in table 1.
The portion of the electronic cloud, that is farther away from the nucleus than the initial position of the photoelectron affects this electron like
an "outer screening," that is, like an external
electrically charged shell. This shell does not
exert any electric force upon a charge inside,
where the photoelectron is, but establishes a negative potential difference of V0 volts between the
interior of the shell and external points at infinite
distance. The effect of this potential energy becomes apparent when the electron escapes from
the atom. As soon as the electron reaches the
drogenlike wave functions of the electron within
the atom before and after absorption of the photon
correspond to energy levels evaluated as though
the outer screening were absent. (The energy of
the ejected electron may become negative when
reduced by eVe, if ht, is only a little above thQ
absorption edge. This circumstance introduces
no real difficulty because formulas for the hydrogenlike approximation carry over to negative
values of the energy.)
2.2. Scattering by Atomic Electrons
The main contribution of scattering to the total
attentuation coefficient arises fPom simple Compton effect processes' in which the bonds of the
atomic electrons within the material can be dis'regarded. More complex scattering conditions
obtain at the lower photon energies where photoelectric absorption predominates over the attenuation due to scattering. Therefore, these more
complex effects, which include coherent Rayleigh
scattering, do not influence the over-all attenuation very greatly.
The Compton scattering by "free" electrons is
described to a very good approximation by the
theoretical Klein-Nishina law. Corrections arising from higher-order electrodynamic effects have
been calculated and amount to about 1 percent
only. Experimental evidence agrees well with
the Klein-Nishina value of the scattering cross
section by free electrons, in the energy region
where Compton scattering gives the main contribution to total attenuation. The differential
cross section for scattering of a photon of frequency v, with a deflection 0 into a solid angle
dS, is
1
1
_1Ocose
+
da(8)=-- 1
2l1+(1-cos
outside of this shell the charge of the bhell exerts
a repulsive force and thus helps the escape from
the nuclear attraction. The effective value of Vo
may be determined by observing that the experimental value of the initial binding energy of the
photoelectron is eV 0 ev smaller than the energy
pertaining to a hydrogenlike wave function with
effective atomic number Z-s.
On this basis, the absorption of a photon with
energy hy by an atomic electron appears to take
place inside the outer screening shell under the infuence of attraction by a nulear charge (Z-s)e.
The outer screening does not influence the process
of absorption or the probability of the subsequent
ejection of the electron from t atom.8 The hy' Bethe has pointed out (in a private discussion) that when photoelectric
effect takes place near the nucleus, well inside the Kshell, the inner screening
effect should vanish. Accordingly, it may be inappropriate to utilize an
2(l
This probability would be influenced only ifthe outer screenigrpotential
varied rapidly from point to potnt• which is not the cas (see.M.E. Rose,
Phys. Rev. 48, 727 (1036)). The calculation by Hall 1I0,
p. 383 ofa correction
to the cros section of the photoelectric effect srising from outer screeng
appears to be inconsistent with his application of aWKB approximation,
0)2
f
1 a(1-cos 0)
whc.e r 2
d(
di,
(7)
(e2/mc2)2 =7.94X O-28 cm 2 , a=h/mc
h, m, and c have the usual meaning.
cross section is
2,
f l±
0X-N-=o2i
-
-2(l±a)-la n(1
L-.2a
, and
The integral
Jl),-
1 +3a
(8)
(I+2a)2
I
[21], the following formula is con-
In (1+2a)
2a
For a«1
venient,
XN= 00[1 -
inner screening number s>0 whenever the photon energy Isgreatly in excess
of the Kabsorption edge. This remark robab explains why the probabilities of photoelectric effect calculated with s-0.for low-Z elements are suband h to
stantlally
lower
than indicated
by as
experimental
modified by
an empirical
correction,
discussed inevidence
section 3.1..
-COS
1_a_(_-_Cos 0)
2a+ 5.2a--13.3a 3 +32.7a 4 ...
1.
(9)
The integral cross section (8) is tabulated in
table 2.
____
'For a fuller discussion of Compton seatterlne and extensive tabulations
of the Klein-Nishina formula, see Nelms 1201.
5
The assumption of free electrons that underlies
the Klein-Nishina formula holds only if the
momentum transferred to the electron greatly
exceeds the initial momentum of the electron's
motion within an atom or molecule. In terms
of the initial wavelength of the radiation (X) and
of the atomic electron X., this condition reads
2 sin (0/2)
and obtains less frequently that one may be
inclined to expect.
When this condition does not obtain, Compton
scattering is complicated by the bonds that hold
the atomic electrons and becomes less frequent
than predicted by the Klein-Nishina law. The
decrease of incoherent (inelastic) Compton scattering is accompanied by an increase of coherent
scattering 10 in which the photon loses no energy.
As a result of constructive interference of the
radiation scautc1 ed coherently by different electrons, the total cross section for scattering of
lower-energy
photons formula.
grows larger than predicted
by the Klein-,Nishina
In an approximate calculation, one may regard
the probability of Compton scattering by an
atomic electron as the produtct of two factors.
The first factor concerns the probability that the
Photon be deflected by a certain angle and transers to the electron a corresponding amount of
momentum q as though the electron were free.
The momentum transfer is given by q'-,(h,/c)X
2 . The second
2sin(0/2) for hy(--cos 0)«<mc
factor concerns the probability that the electron,
having received a momentum q, will actually
absorb energy and thereby become excited or
leave the atom. This analysis of probability
into two factors derives from the impulsive
character of the scattering process.'
For the first factor one may take the KleinNishina cross section (8) for free electrons. For
the second factor one may take the incoherent
scattering function S(q,Z) which is discussed in
some detail in the appendix. If q is much smaller
than the root mean square momentum of the
electron before the scattering, the s,ýond factor
S becomes very small in proportion to 1 and any
actual energy transfer is comparable to the binding
energy of the atomic electron. If q is much larger
than the initial rms momentum, S equals approximately 1, and the actual energy transfers are in a
narrow band about qt /2m. Thus incoherently
scattered radiation disappears at very low energies
and approaches the value given by the KleinNishina formula at high energies. The total cross
10
Forafullerdiscussionotfcoherent
datasee
Nelms and Oppenheim (22. scatteringand tabulations ofform factor
The momentum trnsfer takes place, in the main, an
a time short as com-
pared to the reaction time of the mechanism that binds the electron in the
atom. Thus the determination of momentum transfer and %ngulardeflection
occurs in a much shorter time than the determination of the energy transfer.
Th
6
eletron-atton, the snd on t
section for incoherent scattering with deflection
9 by the Z electrons of an atom equals approxi-
mately
dainh=(1I2)Zr2[1-(hv/mc 2)(1-cos
)]-2{
+cos 20
(hY/mc 2 )
)df2. (10)
cos o)I}S(q,Z SX(1-
-(hv/mcI) 2(1 -cos e)2/[l
To calculate the probability of coherent Rayleigh
scattering one must combine the amplitudes rather
than the intensities corresponding to scattering
with a given momentum transfer to the different
electrons. Here again the probability results as
the sum of two factors. The first factor follows
from the Klein-Nishina formula (7) by deleting
(a) the last term in the braces, which corresponds
to a flipping of an electron spin and is inconsistent
with coherent scattering, and (b) the factor
[1+(hv/mc 2)(1 -cos 8)]-2, which arises from the
ratio of the incident and scatterel frequencies and
must equal 1 for coherent scattering. The second
factor of the coherent scattering cross section is
somewhat complementary to the incoherent scattering function S, in that. it represents the probability, IF(q,Z)12, that the Z electrons of an atom
take up a recoil momentum, q, without absorbing
any energy. The function F(q,Z) is called the
form factor. The cross section for coherent
scattering equals:
2 d12.
d0o'~h-(1/2)'i(l+cos 2 0)JF(q,Z)1
(11)
The form factor F and the dross section (11)
are usually calculated separately for each kind of
atom in a material. This procedure was indicated
in section 1.4 as generally adequate, with exceptions. Additional scattering may actually arise
from interference among the X-rays scattered
coherently by electrons of different atoms. This
effect depends on the state of aggregation of
adjacent atoms. Its order of magnitude may be
lower than or comparable to the effect of interference of electrons from the same atom for
polyatomic gases, liquids, or amorphous solids.
It becomes extremely large for crystalline solids
under conditions of Bragg reflection. To calculate this effect one must define and evaluate a
suitable scattering factor F. which depends on the
arrangement of atoms of the material.
The cross sections (10) and (11) are derived
under the restrictive assumption that the X-ray
frequency is much larger than the proper oscillation frequencies of atomic electrons, i. e., that the
photon energy greatly exceeds the energies at
which photoelectric absorption is intense. Insofar
as this assumption is not fulfilled, the coherent
scattering cross section depends more critically on
the
ratiofrequencies
between the
frequency(effect
and the
proper
of X-ray
the electrons
of
anomalous dispersion). However, the assumption
fails seriously just at those energies where the
photoelectric cross section is much larger than
the scattering cross section. Therefore, an accurate knowledge of scattering is not required for
acceptable accuracy on the total probability of
interactions. The progress towards improved
calculations of coherent scattering is discussed in
1221. The Ravleigh scattering by electrons cornines coherently with other processes of elastic
scattering, such as Delbrtkck scattering and elastic
nuclear scattering; however, this interference effect
is of importance only for large photon energies
and scattering angles at which all of these processes
together yield a negligible contribution to the total
attenuation coefficient.
2.3. Pair Production
The production of an electron-positron pair bythe absorption of a photon may be regarded as a
phtoelabsrtic effect with thotoejectn oelectrd
s
photoelectric effect with the ejection of electrons
from negative energy states. Calculation of the
pair production probability is, therefore, analogous
to the photoelectric calculation. The pertinent
approximations are of the same types except for
two main differences; (a) pair production occurs
only at relativistic energies (approximation (b) of
section 2.1 is never valid) and (b) the initial state
of the electron belongs to a continuum for pair
production and to a discrete spectrum for the
photoelectric effect.
The principal calculations that have been carried
out in detail for the production of an electronositron pair in the field of the nucleus are listed
•elow with an indication of the pertinent approximations.
Born calculation (approximations a and c). The
differential cross section was calculated for pair
production in the Coulomb field of the nucleus by
Bethe and Heitler [23] and concurrently by Sauter
[24] and Racah [25].1 The effect of screening of
the nuclear field by the atomic electrons was
studied by Bethe [26]. In Born approximation
the screening effect consists of a destructive interference of the field of the atomic electrons with
the nuclear field. This interference reduces the
cross section by a factor [1-F(q,Z)], 2 where
F(q,Z) is the same atomic-form factor that
describes coherent scattering of X-rays; that is,
the probability amplitude that the atomic electrons
absorb a momentum q without absorbing any
energy. Bethe and Heitler calculated screening
functions for a Fermi-Thomas distribution of
electrons. Analytical integration over the possible
values of recoil momentum given to the atom is
possible only for the limiting cases of complete or
no screening; numerical integration must be
performed for the cases of incomplete screening.
The necessary formulas and numerical data are
given, e. g., [2, p. 260].
Wheeler and Lamb [27] calculated screening
functions for hydrogen using atomic wave func12For a discuqslon of the angular distributions In electron-posltron pair
and bremsstrahlung production see, H. Brysk (informal communication).
412901-57-2
tions. A comparison of the cross sections for
atomic hydrogen in the case of complete screening
shows the Wheeler-Lamb value to be approximately 2 percent above the Fermi-Thomas result.
Non,-Born calculationfor low energy (approximation a). A calculation using exact relativistic
wave functions for an electron in an unscreened
nuclear field was made by Jaeger and Hulme [28]
and Jaeger [29]. They obtained numerical results
for photon energies of 3 and 5.2 mc 2 and for a few
elements; some interpolation of their results is
possible. For Pb at 3 inc2 the Born approximation
value is lower by a factor of about 2 than the
Jaeger-Hulme value; the difference is much smaller
at higher photon energy and lower atomic number.
Non-Born calculation for high energy (approximations a and e). The cross section for specified
energy and direction of each particle of the pair
was calculated by Bethe and Maximon [11] without the use of forn approximation for energies
large compared to ree 2 The total cross section
was obtained by analytical integration by Davies,
bana[30].itecrrtion
btandand 'Maximon
to tve
the
[301. The correction to
Bethe,
Born approximation calculation is important
only for large momentum transfer to the atom
where screening is not important; therefore, this
correction ma
my be applied equall, to the cases of
correti
complete, incomplete, or no screening. For the
practical cases of incomplete screening a correction (calculated in reference 11 and approximately
propor-ional to Z") may simply be subtracted from
the screened Born approximation calculation to
give the total cross section. For photon energy
eoh,/mc2 the main residual error in the calcuiations
of reference 30 is known to be of the form
(a 2 log E)/e, where a2 caii be determined by fitting
to the experimental data for each element.
Pair production in the electron field. Pair production necessarily imparts a recoil momentum
to the electric field in which it takes place. The
calculations indicated above pertain to the case
where the recoil is absorbed by an atom as a whole;
the electrons remain rigidly attached to the
nucleus so that their fields combine coherently
with the nuclear field to yield a screening effect.
In addition, the recoil may be absorbed by a single
atomic electron which is thereby ejected from tie
atom. The total cross section for this process
results as the sum of the incoherent contributions
from all electrons. The recoiling electron can
take up a substantial fraction of the energy of the
incident photon but, this occurs mainly for photon
energies near the threshold; the threshold here is
4 mc2 instead of 2 mc 2.
Calculation without exchange (approximation c).
This calculation was made by Borsellino [31],
assuming the electron to be free from atomic
bonds. The cross section was integrated analytically over the energies and directions of the pair
particles for photon energies from 4 to 100 mc2.
aIn this calculation
the total cross section for the
daprchsttofalucend
electron field appraches that of an unscreened
H atom as the photon energy becomes much
larger than meo.
rn order to take into account the bonds of electrons within atoms, the cross section for pair
production with a given recoil of a free electron
must be multiplied by the probability that this
recoil actually ejects an electron from i's atom.
This probability is the same incoherent scattering
function S(q,Z) that appears in (10) and is discussed in the appendix. The cross section thus
reduced must then be integrated over all possible
values of the recoil momentum.
A calculation of this type was made by Wheeler
and Lamb [271, uqing the incoherent scattering
function derived from the Thomas-Fermi model.
The total cross section was obtained by integrating
only over momentum transfers up to mc for inci(lent photons with energies large compared to
me2 . The Thomas-Fermi model gives an erroneously large probability of incoherent scattering
75 percent of the latter at a photon energy of
100 Mev.
A calculation by Rohrlich and Jobeph [331 for
atomic hydrogen in the limit of photon energy
very large compared to me' shows that the exchange effect modifies the result of the WheelerLamb hydrogen calculation by about 19 percent."
Exchange weights the momentum transfer distribution toward smaller momenta and therefore
decreases the cross section for pair production
with electron recoil. (The cross s :-.tion for pair
production with nuclear recoil is increased slightly.)
Other calculations of the cross section in the
electron field were made by Nemirovsky [35] and
Watson [36]. Nemiro sky was coicerned only
with a photon energy near the threshold, and his
numerical result is essentially in agreement with
Vortruba. \Watson obtained a cross section that
approaches twice that of an uiniscreened H atom
as the photon energy becomes very large comlpared
for small values of the recoil momentum.
with
This
fact is borne out in the comparison Wheeler and
Lamb made between a calculation using the
Thomas-Fermi model and a similar calculation
using atomic wave functions for hydrogen; for
photon energies very large compared to me2 the
two calculations differ by approximately 12
percent.
Calculation with exchange (approximation c).
This calculation was made by Vo)rtruha [32] for
an electron free of atomic bonds. An integral
cross section was obtained only for the limiting
cases of photon energy near the'threshold or large
compared to me'2 and yielded the approximate
formulas 13
r=2,
O*,,i,ý5.6X<1-"j-.1-•r°
10, -
(c2_4
r,2 /2 8,I
2h Y_-11.3 ±-0.5)
13j,- -_ n m22
for 0_<
2
e-C
.
2.4. Nuclear Absorption and Scattering
The absorption of a photon with subsequent
emission of nuclear particles (nuclear photoeffect)
makes a contribution to the total attenuation coefficient that is usually of the order of 5 percent or
less and confined mainly to an energy interval of
less than 10 Mev, but is occasionally substantially
larger. No data on nuclear absorption are ginfen
in the main tables of this Circular, but soni infermation on the process is given and is utilized for
the analysis of experimental data in the region
Where this process is comparatively important.
The probability of the nuclear photoeffect has
the following main trend. It increases rapidly
with energy above the threshold for emission of
-- 4<1
nuclear particles, reaches a maximum and then
decreases rapidly as the energy of the incident
(12)
for hv >>
photon incr.eases' further. The position of maximual cross section varies from about 13 Mev in
mc2
(13)
uranium to about 23 Mev in carbon. The width
rof the absorption curve appears to show no svswith
tematic variation but varies from 5 to 8 Mev.
Values of the cross section for neutron emission in
The exchange effect (due to the identity of
this broad maximum vary with atomic weight
the recoil and pair electron) is very large near
from about 10 millibarns in carbon to 1 barn in
the threshold energy; a factor of 4.5 between the
uranium [37, 38]. A cross section of the same
4--7.94X10- 28
(13)
cm 2 .
results of Borsellino and Vortruba is attributed
order of magnitude is estimated for proton as for
to this effect. The effect'of exchange decreases
freatly when the recoil electron takes up very
litle of thew havailable energy. This situation predominato me2w
Therpoor e nis
l
arg e m
pared
Therefore, it was believed that the
Borsellinorein.Hwee,
calculationh would
be adequate in this
dtildreacuaion
neutron emission from low-Z nuclei.
Cross sections for photoneutron emission are
given in table 3 for comparison with total cross
An experiment by Bernstein and Panofsky 1341
indicates that exchange
ffects are not negligible at very high photon eneries in the production of
bremnsstrahiun,
which
closely
to pair
of
235.Mev photons
productIon.
byIs
500andrelated
5'i0-Mev
electrons
in liquidTheproducton
hydrogen wac
measured and compared with the Wheeler and
Lamb calculations; the
Rohrhic of Vortruba
and and
Joeht3
between the cross sections
Borsel-
nof
measured
wasIn2.4-2.8
percent below the calculated value.
3result
percent
the between
measured
dueabout
to interference
effects
lino is quite substantial; the former is only about
change
tn the
calJculation.
The magnitude
of this
be
inferred effects
from the
calculations
of Rohriich
and Joseph,
who decrease
find for can
atomic
An increas
the
individual
nuclei
value
and electrons in the
above the
calculation
is
expected
hydrogen molecule, wherees a decrease Is expected because of neglect of ex-
___
"An
b eactevauatin
Ti~hrlch
nd oseh 33 oftheconsantin his
equation gives 1i.7h instead of 11.34-.5.
8o
Stfor
pair production
hydlrogen
in the very
highofenergy
the total
(sum
valueslimit
in thethat
nuclear
ad
integral
cross field)
section
the eletron
Is
i
I
decreased by about 9 percent by the exchange effect.
I th v
hilh e r
li
t
tli
sections for non-nuclear processes tabulated in
this circular. The data for the photonuclear effect
from Katz 13 et al. were plotted and values were
read from a smooth curve over the interval for
which the cross section is about I percent or more
of the total absorption cross section. The interval
covers 5 to 8 Mev, and the photoneutron cross section at maximum is 4 or 5 percent of the total
are negligible compared to the total attenuation
coefficient.
X-rays can also be absorbed or scattered by
nuclei with high probability, if their frequency lies
within certain narrow resonance lines. These
lines lie at lower energy 'han the main continuous
absorption spectrum, mostly near to or below the
threshold for particle disintegration. The photon
absorption cross section.
Data on the elastic scattering of X-rays (391
associated with the nuclear photoeffect sho(w the
same general features as the neutron yield data.
The maximum cross section varies from 0.12
millibarn for Na to 15 millibarns for Pb. These
cross sections are about 100 times smaller than the
corresponding values from neutron yield data and
energy corresponding to individual lines is known
only in few instances. The width of typical lines
is of the order of I ev. The aggregate absorption
of the line spectrum from a continuous spectrum
of X-rays is negligible, but for X-raxs within the
line width the cross section probably often approaches a iheoretical limit of the order of 100
barns.
3. Calculation of Attenuation Coefficients and Comparison With Experiment
The data tabulated in tables 12 to 40 were
derived primarily from theoretical calculations,
Experimental data served primarily as a check,
but also as a guide in settling dubious questions
and providing empirical corrections.
The three sets of values so obtained were then
plotted together and graphical adjustment was
made by drawing a smooth curve which represents
the final photoeffect cross section. Figures 2 and
3 illustrate the procedure followed and show comparisons with both theoretical and experimental
3.1. Photoelectric Effect
data. Only a limited revision of the analysis
based on Hall's calculation was required by
Nagasaka's results indicated on p. 4.
Discussion of datafor Yv>1 Mer. Comparison of
The cross section for the photoeffect in the K
shell was calculated by the Sauter-Stobbe formula
(2) in the low-energy range. Correction factors
in table 4 were applied to the Sauter-Stobbe formula at energies from 10 to 100 key (see discussion below). In the energy range between 0.34
and 1.1 Mev interpolated data from the Hulme
calculations (see p. 4) were utilized.
In the
high-energy region the Hall formula for hv>>rc 2
(3) was used. An effective nuclear charge of
Z-0.3 was used throughout to correct for screening in the K shell (see p. 5).
Cross sections for the L and M shells were
calculated by the Stobbe formulas.' 6 Above the
K edge lengthy calculations for the L and M
shells were avoided by a procedure that relies on
the slowness of variation of the ratios among the
cross sections for different shells. The ratios given
by the Stobbe formulas were calculated at the K
edge and at an energy of 340 key. These ratios
7
are given in table 5 for a number of elements."
The Sauter-Stobbe calculations, which serve as
a zero approximation to the K shell cioss section
throughout the interval from the K edge to 340
key, were corrected for the effect of the L and M
shells on the basis of the ratio at the K edge.
The other two calculations, from Hulme and from
the Hall formula, were corrected initially on the
basis of the ratio at 340 key.
"i
The data for Ph, 1. and Cu iRfrom L. Katz et al. for the natural elements
(private communication with E. 0. Fuller). The dkta for C Is from L. Katz
and A. 0. W. Cameron, Can. J. Phyn. 2% 518 (1951).
For tabulations of the oi'llator strength for photoeffect on the K. L,
and",a
M shels as calclatedtfrom Stobbe iirm uhs, see Loewls 140],
17Notice that the ratios In table Aare substantially lower than the standard
ratio U4 whichis often utilited in the literature (seethediscussiononp. 5).
This dikernce Is reflected In the difference between the photo effect crow
sections given in this Circular and in the tables by Davieson and Evans (3).
calculations by the Hall formula (3) and the
Nagasaka formula (4) is shown in figure 2. The
calculations agree within 1 percent at hv=2.6
Mev, which is approximately the crossover point
of the two calculations. The Hall data at 2.6
Mev was used to interpolate from the Hulme data
at 1.1 Mev into the high-energy region. The
errors in Hall's formula will affect the photoeffect
cross sections tabulated in this report above about
3 Mev for Pb; the resulting uncertainty in the
total cross section is not significant.
Latyshev [18] made the only direct measurement
of photoeffect cross section in the high-energy
region. Other data shown in figure 2 was obtained from measurements of the total cross
section by subtracting the scattering (coherent
and incoherent) and pair production cross sections.
The errors indicated on each point. correspond to
the error quoted by the author for the total cross
section. The only data showing significant (leviation from the calculated curves are the values
at 2.62 inev (X-0.2) and at 5.3 Mbev (X-0.1).
i
Discqev. ion of the data for hybetween 0.1 and
1ev. The experimental data shown in figure 3
were obtained by subtracting the scattering cross
section (coherent and incoherent) from the mcasurement of total attenuation coefficient. The
data of Jones [41] for Pb and Sn are generally
higher than the calculated curves. The data of
Cuykendall [42] for Al agree with the calculated
curve within the experimental error. Experimetl aant
e rieginofthe Hulrne calculamental data in the region Of
tions are within the error estimated for the calcul i
(4%
(1) in Pb; about 8% in Sn). Although
9
measurements of total cross section at energies
of 0.411, 0.511, and 0.655 Mev were made with
high accuracy an error of a few percent may result
in the photoeffect cross section due to uncertainty
in the cross section used for coherent scattering
(see section 3.2). A direct measurement of the
.photoeffect cross section for 0.511 Mev 7-rays on
"the K shell of Pb by Seeman (43] gives a value
7 percent above the Hulme calculation. The
result is within the combined error of the calculation and the experiment (3%). Additional direct
measurements of the photoeffect cross section
in this energy region would be very desirable.
Discussion of data for hv<O.1 Mev. The data
in this energy range are fairly numerous for
low-Z materials but only moderately accurate
(-10%). Exceptions are the data from Cuykendall, Hubbell," and French, 19 with errors of 2
to 5 percent. Data used for this comparison
were assembled from Allen (44], Grosskurth [45]
Cuykendall and Hubbell, as well as empiricai
data from Victoreen [46] and a British group.20
Even though there is considerable variation among
the data and obvious errors in spots, there is a
general trend toward valubs for the experimental
data higher than calculated from the SauterStobbe formula. Empirical correction factors
were obtained by this comparison and are given in
table 4; the data presented in the main table are
obtained by applying these corrections to the
Sauter-Stobbe calculations. A theoretical interoretation of these corrections is indicated in
ootnote 7. The measurements by French (which
were not available for the above comparison)
suggest that for Al no corrections might actually
hitve been required.
The estimated uncertainty in the calculated
cross sections tabulated in this report for the
photoelectric effect varies from 5 to 15 percent.
Great improvement could be made in the lowenergy region by a systematic study (either
theoretical or experimental) especially for low-Z
elements.
3.2. Scattering by Atomic Electrons
Column 2 of tables 12 to 40 gives the cross secand incoherent scattering
tions for coherent
data
by
e
various atms.Thee
atoms. These dtareprethe electrons
theeletros
oof vaiou
scatterbecause
sections
cross
scattering
sent total
other than atomic electrons conparticles
in.oby
tributes to the total scattering cross section an
amount smaller than the estimated error of the
within the
data. The binding of the electronsmethods
disbv
account bymetods
into
atoms was taken
atom ito
wa takn
acout
iscussed in section 2.2. To obtain'the total cross
section, the numerical values of the separate differential cross sections (10) and (11) were calculated
numerically for a number of values of the scatter-
"is
1. Hubbell (private communication),
"it .
L. French (Private communication, measurements for Al and Cu.).
20Th data are from Hospital Physicists Associatiin, C.3.1. 3. 1 and A.
1.2. (1-4) %Mr. F. S. Stewart, Mount Vernon Hspitl, Northwood,
1.
Middlesex,
10
ngland.
ing angle 0, then added, and finally integrated
numerically over all directions of scattering.
The contribution of either the coherent or the
incoherent process was neglected at any 0 where it
amounts to less than 0.5 percent of the other.
Thereby each of the cross sections was omitted
just in the range of variables where its accuracy
is lowest, that is, incoherent scattering was omitted
where the momentum transfer to the atomic
electrons is small (low photon energy, small
scattering angle) and coherent scattering where
the momentum transfer is large.
The incoherent scattering function S(qZ) that
was entered in (10) is derived from the ThomasFermi theory of atomic structure. The numerical
values utilized are given in the second column of
table 41 and are discussed in the appendix. The
form factor F(q,Z) to be entered in (11) consists
of an integral over the density distribution of
atomic electrons [221. The density distribution
given by the Thomas-Fermi model was utilized
at Z>26 for all values of q, and at Z_<26 only
for large values of q. For Z<_ 26 and for small q,
the values of F(q,Z) tabulated by James and
Brindley [47] and by Compton and Allison [481
served as a basis. These values utilize electron
distributions derived from Hartree wave functions.
Further corrections were made on the values of
F(q,Z) for C, N, 0, utilizing more recent data
of [22].21
For the high-Z elements, where photoelectric
absorption edges occur the cross section for coherent scattering departs substantially from the form
(11) as indicated in section 2.2. and is no longer a
smooth function of energy. H6nl [49] "2investigated the variation of the coherent scattering
cross section in regions of anomalous dispersion
and. calculated particularly its decrease in the
region of the K absorption edge. A rough calculation indicates an error of 10 to 20 percent in
the cross section for elements from U to Mo,
giving an error of less than 3 percent in the total
cross section at an energy just below that of the
K edge.
Experimental measurements of the cross section
for coherent scattering consist mainly of data for
very small or very large [51] momentum changes
for small q have
been
The
data
qof the photon.
reviewed
[221. The
only
measurements
of direct
of
lculats
ea
i
cmaion
for
interes
interest for comparison with the calculations of
this Circular were made by Storruste (52] and
Mann [53]. At 0.411 Mev both sets of data
show good agreement with the Thomas-Fermi
photon
form factor calculations for Pb. Attpoo
energiesatrcluain
of 0.662 Mev for Sno and bPb and of 1.33
for Pb, Mann's data still show satisfactory agreemnent at angles of scattering in the range os the
raege ofper.
the
calcuation s of
paper.
present
the
of
calculations
21A survey was made to estimate the sensitivity of the total cross sections
to further Improvements in the values of F(q,Z), which could be Introduced
on the basis of [22]. These improvements would modify the total scattering
cross sections by no more than 5 percent and the total absorptibn coefficient
by no more than 1percent.
1 See Compton and Allison [48, p. 315] and also Parrattand Hempstead [501.
Tabulation excluding coherent scattering. Coherent scattering has usually a minor influence on
the penetration of X-rays under condition, other
than "narrow beam", because it is accompanied
by no energy loss and by a deflection that is
most frequently negligible. Therefore, this process has been disregarded in many studies, and
it becomes desirable to provide data on absorption
coefficients that do not include any contribution
from coherent scattering. Column 3 of tables
12 to 40 gives a scattering cross section that is
simply the Klein-Nishina cross section of one
electron, as given in table 2, multiplied by the
number of electrons 7.
3.3. Pair Production
The Bethe-Heitler Born approximation calculation was used as a zero approximation to the
pair production cross section in the field of the
all mc,
Z's.screening.efinal
nucleus for
<10sf
For hv_'10 mcl, screening effects are negligible
and the cross section for an unscreened nucleus
was obtained from the formula of Hough [541,
which fits numerically the Bethe-Heitler results to
within 0.1 percent. For hvi>10 mce, the BetheHeitler formula given in [2, p. 260] was utilized
and was integrated numerically over the energy
distribution between the pair particles. Interpolation in Z is easily accomplished as the cross
section in units of r4ZI/137 is a smooth and
slowly varying function of Z, particularly in the
low-energy region. For h,>30 Mev the interpolation is further helped by plotting F=op,,l
(r1Z2/137)+ (28/27)1nZ against f= l00mc 2 /hZ.Z'
Table 6 indicates the dependability of this procedure by showing that the relationships between
F and "v for Al and Pb are almost identical for
ht'>15 Mev.
Corrections to the Born approximation values
were applied at all energies. These corrections
depend p. "marily upon the theoretical calculations
of Jaeger and Hulme [28, 29] at low photon energy
(Yv<10 me2 ) and of Davies, Bethe, and Maximon [30] at high photon energy (hv>»10 =2).
The calculations of Jaeger and Hulme have been
verified in several experiments including those of
Dayton [55], Hahn et al. [56], and Schmid and
Huber [57]. These authors measured relative pair
production cross sections at hv_<2.62 Mev and
fitted their data by Z.dependent formulas of the
form oapr-a--o(1-aZ2 ), assuming that the Born
approximation is correct in the limit of low Z.
Schmid [58] measured the absolute pair cross secThese calculations
tion for Pb with Co' and Na.
served as a basis for graphical interpolation, as
illustrated by the plot of the ratios
ffalr/ftorn
for
Pb on the left side of figure 4.
Following the work of Davies, Bethe, and
Maximon [30] see section 2.3) a correction to the
2 A theoretical interpretation of this procedure is given in (2).
Born approximation for h6>5 Mev was obtained
by fitting a semiempirical formula
a
arBnm--Aa-om,
i,+a2n
e(4
(14)
where t is the photon ener,
i inc 2 , Osor is the
Bethe-Heitler cross section 'r a screened nucleus,
Aa, is the Coulomb correction calculated in reference [30] for the high-energy limit, and a' is a
constant to be determined from experimental data.
Values of Aa, and a2 for a few 7 are given in table 7.
The values of a2 were determined primarily from
the data of Paul [59] and Colgate [5] at 6.13 Mev,
except at low Z.
the fitting of
a2 the
much
lIt was given to
theInrequirement
that
plotwei
of(13)exrplt
extrapolate
data
smoothle to the experimental
to the
and
resultsa
calculation
smoothulme
tis resultsmen low energy
me) c
Jaeger-(h2
(h,<2.6 Mev).
This requirement caused the
estimates of PApir to fall 4 to 5 percent
below the estimates drawn from experimental
data for Al and C at higher energies (hi'>6 Mev).
This descrepancv does not appear serious because
the main experimental evidence is derived from
measurements of total attenuation coefficients
from which one must subtract the contributions of
other processes. For low-Z elements the contribution from triplet formation is considerable.
This contribution has to be estimated theoretically
and deducted from the measured absorption coefficient to obtain the experimental values of Uar.,.
The use of Vortruba's calculation rather than
those of Borsellino or Wheeler and Lamb makes up
to 8 percent difference in the estimate of apjI in C,
butgmuch less (1 to 2%) for Al around 17 to 20
Met'. The contribution from photonuclear processes is likewise relatively more important for
low-Z elements. Differences of about 5 percent
in the estimate of 0pi for Z<29 are caused by
assuming an uncertainty of 100 percent in the
cross section for the production of neutrons. The
values of amuear used to reduce the data entered
in figure 4 were taken from various sources of
experimental data on the photonuclear processes.
For Z<:13 it was assumed that the probability
for production of protons equals the probability
for production of neutrons.
The data at 6.13 Mev were given much weight
in fitting a2 for Z>29 to minimize the uncertainties in unraveling the pair production cross
section from total attenuation coefficients. This
energy lies below the threshold of the main photonuclear processes. With regard to the photoelectric effect, at 6.13 Mev its contribution to
the total absorption is small even in Pb. On the
other hand, fitting a' at large energies, above the
range of large photonuclear cross sections, would
yield low accuracy because the value of e-1 In e
uecomes quite small.
A complete curve of the ratio 0rpair/Ofo, is
given in figure 4 for Pb, with all the relevant
11
experimental data. The curves thus obtained
by fitting a2 in (4) agree with all experimental
data satisfactorily except for the data of Rosenblum et al. [601 at 5.13 and 10.3 Mev, where they
are well outside the experimental error stated by
the authors,
The curve for iodine in figure 5 shows a comparison of calculated ratios of up~,/Ionu by West
161], using experimental data derived from various
sources including his own measurement of absolute and relative pair production cross sections in
sodium iodide using sources of Co6° and Na14 .
This comparison is especially interesting since
most of the data for iodine was calculated using
Z-dependence formulas determined by the various
authors. Agreement is mostly within the experimental errors, excluding the data of Rosenblum
et al. at 5.13 and 10.3 Mev. The data of West at
1.17 Mev are the only ones available at this low
energy and cannot be compared directly with
other experimental or theoretical data. They
indicate an increase in the ratio Oftir/9Bor, as the
threshold energy is approached more rapidly than
expected by the extrapolation carried out in the
figure.
Generally, experimental data fit the calculated
curves within experimental errors of a few percent.
The estimated error in the. pair cross sections
given in the main table is about 3 percent except
at the lowest energies (<3 Mev) and in the region
where absorption by the nuclear photoeffect is
important (10 to 30 Mev).
Pair Production in the electron field. Calculations of the pair cross section in the field of
electrons were made by using .the formula of
Vortruba (12) and (13). Graphical interpolation
was made in the energy region where the two
formulas were not valid. This was accomplished
by assuming the validity of the formulas to be less
restricted than indicated and also by using the
calculations of Borsellino [31] (see 2.3) as a guide
to the shape of a curve of aEleOtron/GProton.
It is difficult to assign an error to this estimate. There are no direct measurements of the
cross section for pair production with electron
recoil (triplet production). Some evidence is
obtained indirectly from measurements of the
total absorption coefficient in hydrocarbons [62]
12
and also from measurements on the related process
of bremstrahlung (341 (see footnote 13). However
tile weight of this evidence is diluted in the
process of extracting information on the triplet
process, and the resulting accuracy is not adequate
to improve tile theoretical estimates.
3.4. Total Attenuation Coefficient
Total cross sections were obtained by sumnming
the cross sections for the individual absorption and
scattering processes discussed (3.1, 3.2, and 3.3).
Cross sections for nuclear processes are not
included for the reasons indicated in 2.4. -The
results are given in tables 12 to 40. Cross sections
for the individual processes are expressed in hbarns
(10-24 cm 2 ), and the total absorption coefficient is
given as a mass coefficient in square centimeters
per gram. Conversion factors from barns to
square centimeters per gram are tabulated for
each Z. Attenuation coefficients with and without the contribution of coherent scattering (see
section 3.2) are given separately. The purpose
for which the data are used will determine the
choice between the two sets of data.
In general, data are tabulated with a number of
digits such that the uncertainty in the last digit
amounts to a very few units. However, the total
attenuation data are given throughout with three
digits, for purpose of smoothness, even when the
last digit may be in substantial error.
The estimated errors have been discussed in
some detail in the preceding sections. A comparison of the tabulated total absorption coeffici ents with experimental data is shown in tables
8 to 11. As an over-all estimate, the errors may
easily approach 10 percent below 50 key, especially for light elements, but probably do not
exceed 3 to 5 percent above 100 key.
The author thanks the large number of persons
who assisted in the preparation of this Circular by
contributing generously of their time and information in discussions and by correspondence. The
cooperation of U. Fano in the preparation of the
manuscript is greatly appreciated.
4. Figures and Tables
ABSORBER
41max
SOURCE
-•ýDETENTO
COLLIMATOR
Fiouxuz 1. Experimental arrangement in measuring "narrow-beam" attenuation
coefficients.
.4
i
.3-.-------
NT
--
.2
I
N
0
.1
.2
.3
.4
.5
.6
X ,COMPTON WAVELENGTH
Fiounz 2. Data analysis and interpolation for the photoelectric cross section of Pb at energies
>1 Mev.
Calcwedat:
0, Hubme [l1'].
A,al 11, 0}K shellony
I.D,Ne•alam t17]0J.
Erperhnesma dat&.
V, lats•hev [IS) direct observation.
~
,Pal
[J [60]i Total absorption coefficient less an estitnate of other absorption and scattering Proceses
ORosenblum
,Cvalues used.
13
i-A
10
0
0
b
an a
I It Ion-coe.....ts
I
01
2
n
fr
45'
140
X, GOMPTON WAVELENGTH
FIGURE 3.
Data analysis and interpolation for the photoelectric cross section for lead, tin,
and aluminum from 0. 1 to 1 Mev.
TEJnhe•
solid curves are drawn through the SanteroStobbe points at large values ofA. and through the Hulme calculations
edus= .t•the
co~ntribution offthe L and M shells) In the, indicated region. Values used in the present circular were
taken
he solid curves. The dashed curves show the departure of the Santer-Stobbe from the Huhne values.
Experimental date were obtained by subtracting scattering (ohbersut+incoherent) from the measured total attenuation coefficients.
OJones-Cuykendall [41, 42]; O, Colgate [5]; A, Seeman [43]
14
2.O
,
Cr4-
I-
b---
i
t
--
b01.4
00"
0"
1.0
ENERGY, Mev
FIGURE 4.
The ratio ffPaIM/WBon for lead.
Cacidated data: 0. Kaeger [28, 29]; S Davies et al. [30].
EBperimaental data.
c i, [.
Pau1 [69].
Dsytont•55.
V:,Bean[.
Walker [L8].
Hahn
et al. [MI.
9,Colgate [84].
(), Lawson [68.
A, Roeenblum [60].
, Dewlre [67].
-. values used.
The significance of the experimental data' plotted in this figure Is discussed in section 3.3.
_
1.6
_
iV
T
8
9
---
1.4
C
b
12
A:
1.0
2
3
4
5
6
7
t
ENE'RGY, Mev
FIGUJRE
5.
The ratio o"a,.iHso, for iodine.
= =Wafmntl adat
2 LO,
et[1;V,
Dayton 155]%, Colgate [51; A, Rosenblum [601.
•
, values wLed
The significance of the experimental date plotted in this figure Is discussed In section 3.3.
412901--57--s
15
u(,tin V)
H, EXACT
Li AND O,HARTREE
C AND Pb,WENTZEL MODEL
T-TF, THOMAS- FERMI
.68
0 1.
g
___________Pb
.001
.-
0.1
V
1.0
-4
HARTREE, No AND A
WALLER HARTREE, No AND A
u(-2AnV)
0
-2
A
2
4
oe
(AA
.0l
0.
FieuRz 7.
16
I
Incoherent scattering function 8(v), for H, Li, C, 0, and Pb.
-6
-8
-
I
.01
FIGURE 6.
---
HH
F1111
0
1.0
Li
/C
.1)
V
Incoherent scatteringfunction S(v) for Ne and A.
1J
50
TABLE
1.
Slater screening constant for different electron
groups
Electron groups
Atomic
number
s
2
3
4
5
6
7
28 or 2p
3a or 3p
3
4 or 4p
0.30
1.70
105
140
178
& 10
8
3.45
&380
9
11
10
TABLE
2.
Cross section calculated from the Klein-Niahina
formula
4.15
&so
9.15
9.50
9.88
12
13
14
15
Photon energy
10. 20
16
Square centimeters per elec-
tron
10.88____________
17
18
0.30 for all Z
20
21
2
23
____
10.90
11.25
19
4.15
18.00
1&38
ra11Z
,1&.70
25
26
27
28
29
30
I
19.40
11.25for
aZ
19.40
19.78
20.10
20.48
21.15
Mer
16. so
17.18
1&00
1&85
19.70
21.08
0.010
.015
.020
.030
.040
21.40
U925
23.10
23.98
28.30
2 65
21.15
all zfor
O.640X 1O"0
.629
.618
.597
.578
3.
Square centimeters per elec-
tron
______
Me.
1.0
1.5
2.0
3.0
4.0
0. 2112X 10-24
.1716
.1464
.1151
.0960
.050
.060
.080
.10
.15
.20
.562
.46
.517
.492
4436
.4066
.0
6.0
&.0
10
15
20
.0828
.0732
.030
.05100
.037
.30
.40
.50
.60
.38M
.3167
.2892
30
40
50
.02199
.01746
.0106•
.2675
6o
.01254
.0
.2380
80
100
.00988
.00120
1
TABLE
Photon energy
.03024
Comparison of the sum of the scatteringcross section, including coherent, the photoelectric, and the pair cross sections
with the photoneutron cross section, in barns
Energy (Mev)
Pb--------------------
10
12
13.2
14
16
18
16.8
17.7
18.38
18.6
0
--
19.5
20.3
P--------10.2
-....
-------------------Cu------C,
SA cros
15.2
19.2
20
0.8
0.84
0.8
---
0.4
0.2
&66
0.16
----.-.
9.02
0.36
9.24
0.47
9.37
0.41
9.68
0.31
------
9.96
0.19
3.32
0.08
----
3
&38
3..
0.09
3.45
3.49
0.12
3.52
-----
0.11
0.11
21
22
23
24
25.
26
0.297
.013
0.294
.009
0.292
.005
0.289
.003
---- 3.95
----
{0.0D2
------------------------------------------------------------------------ {0:302
0.03
0.300
.009
section of the same order of magnitude is expected for projon emission as for neutron emission.
17
__t _____• i
TABLE
|
,
4. Correction factors applied to the Sauter-8*obbe formula at photon energy of 10 to 100 key
Element
K
Ca
Fe
Cu
1.0
1.10
1.08
1.07
i..06
1.05
1.00
1.06
1.07
1.06
1.05
1.08
1.07
1.06
1.05
1.04
1.08
1.07
10
1.05
104
1,08
1.07
1,06
1.05
10.4
1:06
1.05
1.04
1.03
1.03
1.05
1.04
1.04
1.03
1.02
1.04
1.03
1.02
1.02
1.04
1.03
1.02
1.02
1.04
1.03
1.02
1.0
404
1.03
1.02
1.02
1.03
1.03
1.02
1.02
1.03
1.03
1.02
1.02
1.02
1.02
1.01
1.01
1.02
1.01
1.01
1.01
TABLE
6.
81
112
111
1.09
1.10
1.06
1.08
1
.06 0
1.05
1.05
1.11
1.00
1.08
1.00
1.05
1.10
1.09
1.08
1.e
1.06
1.10
1.06
1.07
1.04
1.03
1.03
1.02
1.04
1.03
1.03
1.02
1.04
1.03
1.03
1.02
N
0
10
15
30
so
40
1.14
1.11
1.00
1.06
1.06
1.13
1.11
1.09
1.06
1.05
1.13
1.10
1.06
1.06
1.06
s0
60
s0
100
1.04
1.03
1.03
1.02
1.04
1.03
1.03
1.02
1.04
1.03
1.03
1.02
TABLE
A
Al
C
Na
a
Mg
U.
1.04
1.03
1.03
1.02
5. The ratio vX+L+Mu/aj calculated from the Stobbe
z1o.
formulas
I
The function F(-y) for Al and Pb
_
Pb
-AlI
7
K edge
340 kev
6
8
13
18
26
1.02
1.03
1.05
1.07
1.10
1.01
1.02
1.03
1.04
1.06
29
42
,50
74
82
1.11
1.13
1.14
1.141
1.164
1.07
1.09
1.10
1.124
1.131
92
1.167
1.138
A,,
TABLE
I
MW
7.
0.1
.15
.2
.4
.6
.8
1.0
1.5
2.0
2.5
217
147
107
54.3
36.2
27.2
21.7
14.5
10.7
8.69
The Coulomb correction Av, and the energy de-
pendent term a2 of the empirical correction
13
29
42
50
53
74
78
82
92
18
F(,r)
Barun
0.0M2
.079
.333
.649
.809
2.78
3.36
4.02
6.03
Barns
0.0159
.380
1.61
3.14
3.72
11.8
14.0
1&8
24.7
13.8
13.1
12.5
10.9
9.82
9.0o
8.41
7.29
6.49
S.
'i
,
F(
Afev
118
78.4
58.8
29.4
19.6
14.7
11.8
7.84
13. 8
13.15
12.5
10.9
9.82
9.08 1
8.51
7.50
588
4.70
6. 83
6.34
1
Id
4L
I
C.s
.0
_
_
_
_
_
_
_
_
..-O ONA
ZSCM
C5~
I~ ~
_
__C3
galz~d s
1-
.1-
II
faa
Go
I"
I
E-Hi
I
i
.~i
.i
..i
c eii ic-;-.i
20
i
c!:..
::
I,,,
C
.i
C
&
i
... ..
iii
_
_
_
i
0
_
I
i
', :I i
+ +
-
*1
Sal
HgI Mb2s
21
ie:0 Ga~199i29imi~go
S. . . . . . . . . . . . . . . . . . . . . . . . . ..
. . m
, + -,
,. ••
0I
44
boo
S
cS
ci
.
_,
_
_
.
.
. .
.
.
o
.
-----
.". .
_.R,
... .. . .. . . . . . .= . . .
.
,--
.
.
.
. .
.
.
.
.
.
94_
.. . ... .. . .. .. . ..
c
TILBL) 12. IR'drogen
Settr~
Photon
fr prodiatio
htotoelectrie
Tota
without
without
coherent
MY
Barns/atom
0.01
.0o5
.02
0.640
.629
.618
.03
.597
z~ab
2icoherent
earns/aton
Nucleus
Ilectron
Barns/atom
earra/atom
o.0046
.0o01
om2/g
0.385
.377
.369
.397
.345
.04
.05
.561
.o6
.0o
.5h6
.517
.326
.309
.10
.15
.
..
2Z
.265
.444.1
.30
.407
.i54
.40
.50
.60
.80
.317
.289
.268
.235
.20
.335
.2143
.212
.189
.173
.260
.240
1.0
1.5
2.0
3.0
.211
.1726
.1464
.:.51
o.oooo14
.00018
.00051
o.00001
4.0
5.0
6.0
8.0
.0960
.00082
.00005
.0001
.0002
.0004
.0502
.016
.0371
15
20
30
.0510
.0377
.0302
.0220
.0021
.0028
.0033
.0040
.0006
.0031
.0015
.0021
.0321
.0249
.0209
.0o68
40
50
60
80
.o1746
.011456
.01254
.00988
.005
.0048
.0051
.0056
.oo26
.0029
.0033
.0038
.01•7
.0333
.0125
.0125
100
.00820
.0059
=42
4209
10
.08B
.0732
.o599
-0011.
.00
.0018
.126
.103
.0876
.0691
.0979
a
Total scattering for %ydrogen is given by the flein-Nishina fornala for free electrons.
b
Brns/ato-•
4121Hfl -57--- 4
0.5M- a'
'23
?AMI 23.
Soattering
Ploton
SI~
with
coherent
NOv
]Photoelectric
K and L sh*U.•
Without
coherent
3.54
3.01
2.77
2.565.5.2
2.52
2.1
.03
2.53
.014
Pair productio
Nucleus
uoherent
lamn/aton Barn/atom
0.0
.015
.02
Uft13iumk
Dams/atolu
Total
lectron
With
Without
coherent
OR2/g
Darn. /atou Darm/aton
ou 2 /g
1.39
0.52
0.5.
.2914
.220
2.39
.23
.178
.168
2.38
2.31
.052
.163
.158
.06
.06
2.21
2.10
2.18
2.07
.010
.1W.
.1110
.1146
.138
.20
.25
.20
.30
1.99
1.78
1.63
1.972
1.774
1.626
.11,
.133
.119
.109
.132
.119
-W9
.0945
0.533
.261
.200
t14
1.197
1..070
am81
.o7'73
.073S
.80
0.940
.0628
.60
1.0
1.5
2.0
3.0.1460
.845
.686
.586
4.0
5.0
.384
.331
0.00071
.0028
.0081
0.00005
.0565
.01W9
.0394
.0323
.013
.018
.0002
.0004
.0266
.o234
6.0
.293
.022
.0008
.021
8.0
.240
.028
.002
.0180
10
15
20
30
.204
.3509
.1210
.0880
.034
.044
.052
.063
.003
.004
.006
.008
.o16
.0233
.0120,
.0106
40
.0698
.070
.010
.oioo
50
60
80
.0M82
.0502
.395
.076
.081
.087
.012
.023
.015
.009r7
.00964
.009•6
100
.0328
.093
.017
.00955
a Data in the first colum is given by the sun of coherent scattering and of incoherent
scattering from the Klein-Niehina for=Lla corrected for binding effects. In the second colum
inooherent scattering is given by the Klein-Nishina forzmla for free electrons.
b Barn/atom x 0.06684 -cu
*
24
2
/g
Mergy region in which dipole absorption attains a msaxim cross section.
TABLE 114.
Photon
energy
Soatteringa
With
coherent
Total
Fair production
Photoelectric
K and L shells
With
Without
erent
Barns/atom Barns/atom
HOT
Carbon
Nucleus
Barns/atom
Barns/atom
without
Electroohn
Earns/ato,
38.6
10.2
3.91
0.99
c%2/g
on 2 /g
2.26
0.7n
.429
.252
2.13
0.701
.382
.229
0.01
.015
.02
.03
6.88
5.)0
4.64
4.0o4
3.84
3.77
3.71
3.58
.04
.05
.06
.08
3.71
3.50
3.37
3.18
3.47
3.37
3.28
3.10
.38
.18
.096
.037
.205
.185
.174
.161
.193
.178
.169
.257
.10
3.02
2.69
2.1.6
2.33
2.96
2.66
2.44
2.12
.017
.0040
.152
.135
.123
.107
.149
.134
.122
.206
.15
.20
.30
.40
.50
.6o
.8c
1.900
1.735
1.605
1.410
.0953
.0&70
.0805
.0707
0.0016
.0063
.018
0.00007
.0636
.0518
.0444
.0356
.5'76
.497
.030
.040
.0003
.0007
.0304
.0270
6.o
.439
.048
.001
.o245
8.0
.359
.063
.002
.0223
10
15
20
30
.306
.226
.1814
.1319
.X76
.099
.n6
.140
.004
.006
.009
.012
.0194
.0166
.-054*
.0142
40
50
60
80
.1048
.0874
.0752
.o593
.157
.170
.180
.195
.o15
.018
.020
.023
.0139
.0138
.0138
.0o39
100
.0492
.207
x02
.0141
1.0
1.5
2.0
3.0
1.267
1.030
0.878
.691
4.0
5.0
a Data in the first column is given by the sum of coherent scattering and of incoherent
scattering from the nlein-Nishina formula corrected for binding effects. In the second column
incoherent scattering is given by the Klein-Nishina formula for free electrons.
b Barns/atom x 0.05016 - cm2 /g
* Energy region in which dipole absorption attains a mariuam cross section.
25
TABIZ 15.
Nitroven
Sat~erina
Pair production
Thoton
energy
Photoelectric
With
Without
coherent
coherent
K and L shells
0.01
.015
.02
.03
Nucleus
Electron
N
Barns/atom Barns/atom
Mev
Totalb
Earns/atom
With
Without
coherent
coherent
Barns/atom Earns/atom
€M2/g
c, 2 /g
3.61
8.96
4.48
79.4
6.72
24.0
3.80
21.2
1.20
5.73
4.84
4.33
4.18
1.10
8.21.
2.15
0.600
.301
0.539
.272
14.45
14
4.05
o.81
3.93
.06
.08
3.98
3.73
.38
3.82
3.62
.226
.1914
.209
.14k
.21
.082
.180
.164
.173
.59
.10
.15
3.54
3.25
3.45
2.85
2.7
.154
.136
.150
.134
.20
.30
2.87
2.48
.0o1
.010
.123
.107
.123
.106
.0
.05
.40
.50
.60
.80
1.0
1.5
3.1
.095
2.22
2.02
1.872
1.645
.0869
.08m%
.0707
1.478
2.0
1.201
1.025
3.0
0.806
4.0
5.0
6.0
8.0
.0636
.o517
.0445
0.0022
.0086
.025
0.00009
.0357
.672
.580
.040
.054
.0003
.0008
.512
.119
.0306
.0273
.066
.086
.001
.003
.0249
.0218
.3V7
.103
.oo4
.0200
.158
.010
.190
.015
.0163,
.1222
.213
.018
.0152
80
.0878
.0692
.244
.264
.023
.026
.0153
.0154
100
.0574
.280
.029
.0158
10
.264
15
20
.334
.212
30
.1539
40
50
60
.1019
.231
.008
.020
a Data in the first column is given by the sum of coherent scattering and of incoherent
scattering from the Klein-Nishina formula corrected for binding effects. In the second colum
incoherent scattering is given by the Klein-Nishina formula for free electrons.
b
#
"26
.185
,arns/atomx 0.0o4301
C-2/g
Energy region in which dipole absorption attains a maximum cross section.
.o175
.0154
.0152
TABLE 16.
Scatteringa
Thoton
With
Without
coherent
ooherent
Barns/atom Barns/atom
0.01
.015
.02
.03
.04
Photoelectric
Totalb
Fair production
K and L shells
energy
Mev
0Wgen
n1.5
8.2
6.95
5.77
5.12
5.03
14.914
4.78
5.18
4.80
14.61
4.30
14.62
.05
.06
.08
.15
.20
.30
1.0
4.06
3.61
3.29
2.84
3.914
3.55
3.25
2.83
.40
2.54
2.53
14.149
,4.3?
4.14
.50
2.31
.60
.80
2.14
1.880
With
Without
coherent
coherent
o12/i
o.2/6
14.o9
5.93
1.80
0.842
.Y71
5.69
1.68
0.766
.3314
1.55
0.73
.40
.15
.253
.208
.189
.168
.232
.197
.1
.a62
.3.56
.237
.124
.107
.151
.134
.123
.107
.o056
.0953
Nucleus
Barns/atom
Electron
Barns/atom Darnsseton
116
39.6
5.4
.071
.020
.010
.0870
.086
.0708
1.0
1.5
2.0
3.0
1.69o
1.373
1.171
0.921
0.0028
.on.
.032
0.0001
.0636
.0528
.01445
.0359
4.0
.768
.663
.586
.479
.053
.070
.086
.112
.000o4
.0009
.002
.003
.0309
.0276
.0254
.0224
.408
.302
.242
.1759
.134
.175
.206
.248
.005
.009
.012
.017
.o026
.03.83
.0173*
.0166
.1397
.X65
.IM00
.278
.300
.317
.021
.023
.26
.o065
.0165
.0167
5.0
6.0
8.0
10
15
20
30
40
50
60
80
.O790
.344
.030
.0171
200
.0656
.36h
.o314
.0175
a Data in the first cohi:i is given by the sum of coherent scattering and of incoherent
scattering from the Klein-Nishina formi2a corrected for binding effects. In the second coluwm
inOoherent scattering is given by the Klein-Nishira• formula for free electrons.
b gmw/aton z o.o3765 - =2'/g
* Umrgy region in whieh dipole absorption attains a maxmm aress seeetin.
27
TABIS 17.
Sodium
Scatterinog
Fair production
Photoh
eneSrg
Photoelectric
K,L and H shells
With
coherent
"loy
0.01
.015
.02
.03
Totalb
Without
coherent
Nucleus
Barns/atom Barna/atom
20
14
11.2
8.8
Barns/atom
7.0o
6.92
6.80
6.5.
Electron
Barns/atom Barns/atom
588
169
67.5
18.1
7.0
3.3
1.87
0.74T
With
coherent
Without
coherent
= 2 /g
cm2 /g
15.9
4.8o
2.o6
0.705
15.6
4.61
1.95
0.646
.388
.273
.224
.179
.350
.2W
.206
.168
.04
.05
.06
.08
7.8
7.1
6.67
6.08
6.36
6.17
6.01
5.69
.10
.15
.20
5.7o
5.M01
14.514
5.42
14.88
Up.17
.35
..091
.040
.159
.a34
.120
.a5
.2"
.118
.30
3.92
3.89
.010
.103
.102
.40
.50
.60
.80
3.5o
3.19
3.48
3.18
2.94
2.58
.0917
.0836
.0912
.0833
.0770
.0676
1.0
1.5
2.0
3.0
2.32
1.088
1.61o
1.M66
o.0054
.021
.061
0.0001
4,.o
5.o
6.0
8.0
i.o56
0.91n
.805
.659
.100
.233
.163
.211
.0005
.001
.002
.oo0
.0305
.0274
.0254
.0229
.561
.252
.007
.0215
.415
.330
.012
.0198
10
15
20
30
.333
.242
40
50
60
8o
.1921
.1602
.3379
.1087
100
.0901
.387
.•65
.016
.023
.013
.0191
.521
.562
.595
.615
.028
.032
.036
.oAi
.0194
.0198
.0201
.0208
.68o
.oA,6
.0214
a hta in the first column is given by the sum of coherent scattering and of incoherent
scattering from the Kigin-Nishina formula corrected for binding effects. In the second column
incoherent scattering is given by the Ylein-Nishina formula for free electrons.
b Vam/atom z 0.02620 - =2 /g
25•
mmmmmm
mulm
mmmunun
u
ii
.0608
.0496
.A27
.0348
TABLE 18.
Magnesium
Scatteringa
Photon
energ
With
coherent
HOT
0.01
.015
.02
.03
Pair production
To talb
Potoelectric
Without
coherent
Barns/atom Barns/atom
25
17
13
10.2
7.68
K,L and M shells
Barns/atom
Nucleus
Electron
BaRs/atom
Barns/atom
With
coherent
Without
coherent
oR2 /g
oG2 /g
7.42
7.16
8L7
246
99.7
27.2
21.6
6.51
2.79
0.926
21.2
6.28
2.65
0.851
i.0.6
.478
.434
7.55
.04
8.7
6.94
.05
.06
.08
7.9
7.4
6.66
6.73
6.55
6.2o
5.1
2.8
1. U
.322
.253
.192
.293
.232
.181
.10
6.24
5.148
5.91
5.32
4.88
4.214
0.53
.15
.14s
.060
.168
.339
.125
.160
.135
.122
.020
.107
.106
.010
.0949
.0862
.0944
.0860
.079
.0699
.20
.30
.4O
.50
.60
.80
1.0
1.5
2.0
3.
4.0
5.0
6.0
8.0
4.97
4.28
3.82
3.48
3.80
3.10
3.21
2.82
2.53
2.06
1.757
1.381
o.0064
.o26
.073
1.152
.219
.878
.719
.194
.251
o.994
.59
0.0001
.062
.x512
.01
.0360
.0006
.0315
.002
.005
.266
.0242
.001
.026
10
15
20
30
.612
.453
.363
.264
.300
.393
.459
.553
.007
.013
.018
.025
.0228
.O23
.0208
.0209
40
50
60
80
.210
.1747
.1505
.3-185
.619
.667
.707
.765
.031
.o35
.039
.045
.0213
.0217
.0222
.0230
100
.0983
.807
.05o
.0237
a Data in the first coluim is given by the sum of coherent scattering and of incoherent scattering from the Klein-Nishina formula corrected for binding effects. In the second colum incoherent
scattering is given by the Klein-Nishina formula for free electrons.
b
Barns/atom x 0.0217M7 - o/g
29
TABLZ 19.
Alnminm
Photoelectric
,L and N shells
Sr
With
coherent
0.01
.015
.02
.03
.29
19
e
Barns/atom
11.5
8.32
8.18
8.03
7.76
.04
9.8
8.8
8.1
7.26
7.51
7.29
7.10
6.72
.10
.35;
2D
.30
6.79
5.96
53
6.1.1
5.77
5.29
43.6Z
4.60
0.78
.a
.080
.020
43.12
.010
.40
.50
.60
.80
1.0
1.5
2.0
3.0
4.0
5.0
6.0
8.0
10
15
20
30
Without
coherent
Barns/atom Darns/atom
)"
.05
.06
.08
Totalb
Pair production
Scatterig
Raatca
15
4.14
3.78
3.49
3.76
3.48
3.06
Barns/atom Barns/atom
With
coherent
Without
coherent
cm2/g
cm2 /g
26.8
8.08
3.48
1.13
26.3
7.814
3.33
1.043
1170
343
141
39.0
15.2
7.3
4.0
1.61
0.558
.360
.270
.198
0.507
.326
.248
.186
.169
.138
.122
.104
.161
.134,
.120
.103
.0927
.0844
.0779
.0922
.o84o
.- 77
.0683
2.75
2.23
1.903
1.496
0.00,76
.030
.086
0.0002
.0614
.0500
.0432
.o353
1.247
1.07
0.952
.778
.140
.186
.227
.295
.0006
.001
.002
.005
.0310
.0282
.0264
.0211
.663
.353
.008
.0229
.393
.286
.539
.647
.019
.027
.0212*
.0214
.49o
.o60
.014
50
60
80
40
.227
.1893
.1630
.284
.726
.782
.828
.896
.033
.038
.0A2
.0o9
.0220
.0225
.0231
.0240
2M
.1065
.944
.055
.0247
a Data in the first colum is given by the sum of coherent scattering and ol" incoherent
scattering from the Klein-Nishina formula corrected for binding effects. In the second column
incoherent scattering is given by the Klein-Kishina formula for free electrons.
b hans/atom x 0.02233 - =/
Snergy region in which dipole absorption attains a m•dxm cross section.
80
.0225
TABLE 20.
Scatteringa
Photon
energy
Photoeleotrio
Totalb
Pair production
KL and M shells
With
coherent
Without
coherent
earns/atom Barns/atom
HOT
Silioon
Nucleus
earns/atom
Electron
With
coherent
Earns/atom Earns/atom
Without
coherent
2
CM
/g
cm2 /g
34.6
1O.6
4.53
1.44
34.1
10.3
4.35
1.35
0.01
.015
33
22
8.96
8.81
.02
17
8.65
8.36
1580
470
194
54.4
.04
.05
10.8
9.7
8.09
7.85
21.4
10.3
0.691
.429
.06
.08
8.9
8.0
7.64
7.24
5.8
2.3
.315
.221
.288
.205
.10
.15
.20
.30
7.38
6.90
1.fl
.182
.172
5.69
.12
.127
.125
.40
.50
4.46
4.07
4.43
.020
.0961
.0954
.60
.80
3.75
3.30
.03
1.0
12.8
6.144
5.82
5.o0
6.21
4.95
4.05
3.74
3.29
2.96
0.29
0.633
.389
.144
.o0o
.108
.0873
.0804
.0708
.139
.107
.0869
.0802
.0706
.0635
.0517
0.0088
.035
.100
0.0002
1.343
.162
.0007
.0323
1.160
1.025
0.838
.216
.264
.342
.002
.003
.006
.0296
.02o7
.0254
10
15
.714
.408
.oo9
.0243
.528
.533
.015
.oz31
40
50
60
.244
.204
.1756
.838
.904
.957
.036
.o0
.046
.0240
.0246
.0253
80
.M383
1.03
.053
.0262
M1417
1.09
.059
.0271
1.5
2.0
3.0
2.40
2.05
1.611
4.0
5.0
6.0
8.0
20
30
.4z3
.308
100
.623
.749
.021
.029
.04W
.03Y7
.0229*
.0233
a Data in the first
column is given by the sum of coherent scattering and of incoherent
scattering from the Klein-Nishina formula corrected for binding effects.
In the second column
incoherent scattering is given by the Flein-Nishina forMUl for free electrons.
b Barns/atom x 0.0216 . =2/g
*
Energy region in which dipole absorption attains a mazimmm cross section.
31
?ADIS 21.
Rasphorum
Scatterina
Pair production
Photon
energy
0.01
Without
coherent
Kuleus
Dars/atom Drns/atom
38
Darns/atom
9.60
9.44
.025
.02
.03
25
19
14.3
9.27
8.96
.04
.05
12.0
lo.6
8.67
.06
btap
KL and N shells
With
coherent
NOv
T
Photoelectric
Electron
Darns/atom BDa•n/ato
• om2/g
2D90
J4.1
619
12.5
259
5.41
74.3
1.72
0.794
2.84
8.42
8.19
7.76
With
coherent
.475
13.8
7.8
3.1
Without
coherent
O,2/g
o40.8
12.2
5.22
1.62
0.729
.31.0
.228
.432
.331
.211
.08
9.7
8.6
.10
.15
7.98
6.93
7.39
1.55
0.40
.185
.174
6.65
.30
5.37
5O3o
*17
.05
.11w
.125
.1o5
.Ox7
.122
.02
.0936
.0928
.0782
.0687
.0780
.0685
.20
.140
.5D
.60
.80
1.0
6.26
479
4.36
4.02
3.53
6.i0
4.75
4.34
4.01
.01
.0850
3.52
.1o4
.086
1.5
2.0
3.0
3.17
2.57
2.20
1.726
0.010
.040
.J
0.0002
.0617
.o5o2
.0436
.0358
4.0
1.439
.186
.0007
.0326
1.243
.218
.J02
.393
.002
.003
.006
.0290
10
25
20
.765
.566
.454
.169
.61o
.71)4
.009
.016
.022
.o242
.0232
.0231
40
50
.262
.218
.961
1.03
.038
.0o4
.0245
.0251
60
.1881
1.10
.049
.0260
.1282
1.19
.056
.0271
.1229
1.25
.063
.0279
5.0
6.0
8.0
1.098
0.898
30
.330
80
100
.858
.091
.0273
.0252
.0237
a Data in the first column is given by the sum of coherent scattering and of incoherent scattering from the Kloin-Mishinm form.la corrected for binding effects. In the second columm Incoherent
scattering is given by the Knein-Nishina formula for free electrors.
b
arns/atom x 0.01916
/
TABIZ 22.
Sulphur
Scatteringa
Pbatom
eI
NOT
With
coherent
Without
coherent
Barns/atom
Barns/atom
44
0.01
29
22
.035
.02
.03
15.9
.04
13.2
u.6
.05
Pair productio
Photoelectric
KL and M shells
10.24
I0.06
9.89
9.55
9.25
8.96
Barns/atom
Total
Nucleus
Electron
With
coherent
Without
coherent
Barns/atom
Barns/atom
cm2/g
2
=m
/g
51.6
16.o
270
820
344
5o.9
15.6
98.7
6.88
2.15
38.5
0.971
0.897
.567
.528
18.6
6.65
2.03
10.7
8.74
1o.6
.400
.363
.06
9.3
8.27
4.2
.254
.234
.10
.•5
.20
8.6
6.69
7.89
7.10
6.51
5.66
2.1
0.57
.a
.070
.201
.310
.1"
.109
.288
.J14
.127
5.12
5.07
.030
4.30
3.77
4.28
3.76
.0968
.0679
.so10
.0708
.0958
.0&74
.0806
.O707
.0637
.0635
.0519
.Ow
.06
7.43
5.74.
.30
.40
.50
4.66
.60
.80
3.39
1.0
.5
2.0
4.63
3.38
2.75
2.34
1.842
3.0
4.0
5.0
1.535
6.o
8.0
0.958
1.325
1.171
10
33
20
30
.13
0.0002
.21
.0008
.002
.032
.0302
.003
.006
.o0k
.0266
.010
.017
.0o
.033
.0255
.0o1
.0265
.28
.34
.45
.53
.604
.484
.69
.81
S.279
8o
.233
.201
.1580
100
.a311
60
0.012
.046
.816
.352
50
.02D
.010
.We
.98
1.09
1.18
1.24
.0371
.02W
.0247
.0256
1.34
.ol7
.052
.060
.0o27
.0281
.o293
1.42
.067
.030k
a Data in the first column is given by the sum of coherent scattering and of incoherent
scattering from the flein-Nishina formila corrected for binding effects. In the second colum
incoherent scattering is given by the Klein-Nishina foriwla for free electrons.
b
Barns/atom x 0.01879 - cM2/g
83
TABJiZ 23.
$Natterine
Win
Pheom~
.04
.05
.06
.08
.10
.35
.20
.30
.40
.50
.60
.80
1.0
1.5
2.0
3.0
K,L and M shells
with
Without
coherent
coherent
Barns/atom Barns/atom
0.01
.015
.02
.03
Totalb
production
P
cir
Totoeleotric
*o~raF
Nov
Argon
36
28
19
56
3.52
11.32
o.12
10.75
15.8
13.6
it.,
20.8
10.40
10.10
9.83
9.31
9.85
8.143
7.57
6.48
8.87
7.98
7.32
6.36
5.76
5.24
5.70
5.21
14.82
4.814
4.24
3.81
4.23
3.80
3.09
2.614
2.07
With
N
Barns/atom
Elctr
Barns/atom
Barns/atom
coherent
coherent
cm2/g
65.14
20.5
8.88
2.76
4280
132D
561
1614
Without
2
14
/g
64.7
20.1
8.63
2.64
1.21
0.682
.459
.271
64.5
31.6
18.0
7.2
.203
.14~2
.120
.0995
.188
a135
.117
.0977
.0876
.0795
.0733
.o6o
.086'7
.0790
.0730
.0638
.0575
0.0002
.0573
.0o68
.017
.0338
.0301
.0279
.0266
.o248
3.6
0.98
.41
.12
.050
.030
.020
0.015
.058
.17
1.727
1.491
1.318
1.078
.27
.36
.56
.0009
.002
.003
.007
10
3,
20
30
0.918
6-79
.544
.396
.67
.87
1.02
1.23
.011
.019
.026
.037
.o2.1
.02W7
.o14o
.o21
40
5D
80
.314
.262
.226
.1778
1.37
1.48
1.57
1.69
.o46
.053
.o59
.068
.0261
.0271
.0280
.0292
100
.11475
1.78
.076
.0302
4.0
5.0
6.0
8.0
60
.44
a Data in the first coluti is given by the sum of coherent scattering and of incoherent
scattering from the Klein-Nishira formula corrected for binding effects. In the second colum
incoherent scattering is given by the Klein-Nishina fornula for free electrons.
b BarnB/atom x 0.02508 =-cm2/
34
1.23
0.629
.1420
.2149
TAM•
SooScatterize
lenea
otaasiu
24.
Photoelectric
IKL and M shellh
With
Vthu
WithoutWth
coherent
NOT
0.01
.03o
.02
.03
.04
.05
Bar/ato.
63
140
31
21.
Barns/atom
12.16
Barns/atom
n.34
1.52
0.844
.559
1.43
n1.6
.10
10.5
9.37
1.0
2.0
2.0
3.0
4.0
5.0
6.o
8.0
25.6
10.9
3.35
23.0
9.2
.80
81.2
3.5o
1037
9.82
.60
82.0
206
13.3
6.09
nt
o2/g
nl.2
.06
.08
.40
h
o. 2 /g
26.0
81.5
4o.1
.50
arns/atom Darns/atom
1650
696
10.96
io.66
8.95
8.02
6.85
ren
coherent
5260
U.95
11.71
17.1
14.7
.15
.20
.30
Totaib
Fair production
4.6
.321
0.782
.514
.293
.233
.225
.35
.197
.)32
.AD
6.02
.070
.0949
.020
.0791
.0786
.010
.0692
.0689
.0619
.0618
8.43
7.73
6.72
5.53
5.1.9
5"n
5.06
4.15
4.46
4.02
14.01
1.27
0.52
.0658
.040
0.017
3.26
.149
.127
.106
.0938
.0852
.0505
.065
.28
0.0002
.0438
.0365
.30
.40
.1.
.0009
.002
.004
.0327
.0305
.0289
.75
.575
.97
.012
.020
.0267
.0263
.0269
.0262
2.78
2.19
1.823
1.5714
1.391
1.)38
.63
.008
.o274
10
15
20
30
0.969
.717
1.14
.025
.1
1.37
.040
40
1.53
1.65
1.74
.o04
.056
.062
.0294
60
.332
.2n
.Me8
8o
.i8
1.88
.072
.0330
200
.. 597
1.96
.060
.o34.
.0
.0306
.o034
a Data in the first colum is given by the sam of coherent scattering and of incoherent
scattering from the noin-Niehina forxz3a corrected for binding effects. In the second columa
for free electrons.
inohoerent scattering is given by the Klein-Niushina formnm
b Barns/atom x 0.015o41
om2 /g
35
TABIZ 25.
Calcium
s te a
i
PhotScattering
enety
With
0.01
•03
O44
.02
.03
.04
.05
.06
,06
.10
Electron
With
Without
coherent
coherent
2
o, 2/g
hrns/at,
Brus/atom
P=
69
12.80
12.5a
12.36
6380
2010
859
2514
96.9
30.9
13.14
14.16
U.56
102
1.81
1.71
0.998
.359
0.929
.597
.330
.257
.167
.237
.112
.238
.158
.132
.lo9
33
2331.914
18.5
15.8
14.3
12.3
Barns/atom
11.22
10.92
10.34
Barns/atom Barns/ton
",
50.6
28.8
11.6
/g
96.1
30.14
13.1
.6A
14.oo
.20
.30
.4O
6.142
.090
.0979
.0965
.60
.80
5.38
6.33
5.78
14.72
4.70
.030
.010
.0813
.073
.0809
.0708
14.24
14.22
3.43
2.9
2.30
.0637
0.018
.072
.20
0.0002
.06314
.0518
.0451
.376
1.919
1.657
.33
.A4
.0009
.002
.0338
.m326
1.1,98
.69
.008
.x285
.83
1.08
1.26
.012
.022
.029
.0280
.0279
.o25*
02
.0299
.03114
1.0
1.5
2.0
3.0
14.0
5.0
6.0
8.0
20
30
Nucleus
coherent
9.86
8.87
8.23
7.07
•50
15
Without
U.1
9.18
8.1g
7.23
.15
10
Ttlb
KL and X shells
coherent
V
rdtoa
PhotoelectriTot
5.814
6.0
1.63
0.67
.20
.050
5.35
1.464
.0885
.54
1.020
0.755
.605
.004
1.51
.4W
.o76
.0302
40
.349
1.69
.051
60
.291
.251
1.82
1.93
.059
.065
80
.198
2.08
x75
x354
.2639
2.19
.0814
.0366
50
100
.033
.0338
a Data in the first coluim is given by the sum of coherent scattering and of incoherent
scattering from the flein-Nishina formula corrected for binding effects. In the second colum
incoherent scattering is given by the Xlein-Nishina formula for free electrons.
b Darno/atom x 0.02503 - o 2/g
9 Enrgy region in which dipole absorption attains a gmalsm cross section.
86
inimmanlit
iill
U•
U
IB
IMMMMMO
TABLE 26.
*mz
With
coherent
0.01
-.
015
.02
.03
.04
.05
.06
.08
.1o
.15
.20
.30
Without
coherent
Barns/atom Barns/atom
120
75
55
K,L and Mshells
Nucleus
Barns/atom
16.64
16.35
16.07
37
15.52
16500
5380
2380
729
29
21
2D.7
17.2
15.03
AS8
14.2o
13.44
91
38
15.4
12.82
3.153
10o.7
9.19
19.1
5.h
2.3
0.66
12.8
1L.3
9.5o
14.59
8.42
7.63
7.03
6.15
8.23
7.52
6.96
6.n
.29
.16
.10
.05
5.52
5.49
.03
1.5
2.0
3.0
4.0
5.A
6.0
8.0
10
15
20
30
10
Barns/atom Barns/atom
Without
coherent
cm2/g
QA2 /g
179
58.8
26.3
8.26
178
58.2
25.8
8.03
3.48
1.83
1.13
0.555
.372
.196
.146
.1o
.344
.183
.138
.106
.0940
.0840
.0769
.0669
.0919
.0828
.0762
.0664
.o599
.0595
3.81
2.99
0.032
.12
.35
0.0003
.0485
.0424
.0360
2.50
2.33
1.903
1.597
.56
.75
.91
1.17
.001
.003
.005
.o
.0330
.0313
.030%
.0295
1.326
o.981
.786
.572
1.39
1.81
2.10
2.52
.o16
.028
.038
.054
.0294
.0304
.154
•379
.067
.076
.085
.098
.0359
.M376
.0391.
.0412
.11
.0427
4.46
60
80
.326
.297
2.81
3.03
3.21
3.6
100
.213
3.64
50
With
coherent
3.64
1.9
1.20
0.595
.50
1.0
Electron
155
.40
.60
.80
Totalb
Pair production
Pwtolectrio
Scatterize
Roton
HOT
Iron
.0330
.0339
a Data in the first column is given by the sum of coherent scattering and of incoherent
scattering from the Klein-Nishina formula corrected for bindirn effects. In the second colum
incoherent scattering is given by the Klein-Niahina formula for free electrons.
=u 2 /g
b
Barns/aton x 0.0107-
#
1r9rgy region In Imbi.h dipole absorption attains a mxiw"i 'croc
section.
37
TAMW
27.
Copper
Photoelectric
KL and M shells
Photon
energy
WiIh
coherent
MW
Without
coherent
Barns/atom Barns/atom
0.01
Nucleu
arnh/atom
350
96
70
46
8.856
23600
182.4
17.92
17.31
8000
3580
1120
.04
.05
.06
.08
35
16.76
28
26.27
24
20.2
15.83
14.99
.10
17.9
14.5
12.8
10.7
14.29
12.86
11.79
10.25
.015
.02
.03
.15
.20
.30
.40
.50
.60
.80
1.0
1.5
2.0
3.0
4.0
5.0
6.0
8.0
Totalb
Fir production
Scatterine
Electron
Eazrn/atom Earns/atom
With
coherent
Without
coherent
cm2/g
225
22h
76.8
34.6
11.1
76.0
34.1
o0.8
474
242
113
6o.2
4.83
2.56
1.58
0.762
30.7
8.9
3.7
.461
.222
.156
.427
.206
.a
1.1
.132
.108
.0940
.0834
.0760
.0659
.0916
.0820
.0751
.0654
9.43
8.54
7.86
6.87
9.18
8.39
7.76
6.82
0.;48
.26
.16
.08
6.16
6.12
4.98
4.25
3.34
.05
.0589
2.45
1.51
0.713
0
0.0o41
.16
.43
0.0004
.0418
.0357
.0330
2.78
.70
.001
2.40
.93
.003
2.123
1.736
1.13
1.45
.006
.012
.0309
.0303
10
15
20
30
1.479
1.72
.018
.030o
1.0914
2.23
.031
.0328
40
50
.506
.422.364
.286
100
.238
0.877
.638
60
80
2.603
3.12
.0316
04.o3*
.060
.0362
3.48
.074
.0385
3.75
3.97
4.27
.085
.094
.11
.0404
.0420
.0A42
4.49
.12
.o016o
a Data in the first column is given by the sum of coherent scattering and of incoherent
scattering from the Klein-Nishina formula corrected for binding effects. In the second colum
incoherent scattering is given by the Knein-Nishina formula for free electrons.
88
4.65
b
Barns/atom x 0.009482 - em2 /g
*
Energy region in which dipole absorption attains a
imamm cross seotion.
TABLIE 2.
Scatteriga
Photon
energ
With
coherent
mv
0.01
.015
.0o20
.0200
.03
11400
3480
Without
coherent
cm 2/g
cm2/g
73.7
71.8
23.2
22.0
26.o
25.1
1.510
13000
4260
1o.5
82.6
27.4
9.64
81.8
26.9
24.3
1920
12.5
12.2
26.0
36
23.6
22.9
21.7
.10
30
23.2
19.8
16.1
20.7
18.63
17.08
14.85
14.0
23.30
12.15
n. 24
9.87
2.6
1.4
0.88
.45
8.96
8.87
7.25
7.21
6.15
4.83
.29
.14
.09
.05
0.095
.35
.93
4.03
3.48
3.08
.04
.03
.023
.002
.017
1.49
1.96
2.36
3.00
.02
.x349
.023
3.53
.03
.0359
5.32
.06
.0418
.40
.50
.60
.80
1.0
1.5
2.0
3.0
4.0
5.0
6.0
8.0
20
Barns/atom Barns/atom
With
coherent
71
.20
.30
15
160
160
96
26.9
26.4
Electron
.04
.05
.06
.08
.15
10
220
Nucleus
Barns/atom
Totaib
production
Pair
Photoelectric
KL and M shells
Without
coherent
Barns/atom Barns/atom
340
Molybdenum
56
46
12.6
11.5
10.0
2.52
2.14
1.585
1.270
1030
620
274
6.82
4.18
1.95
144
1.09
o.4.W
.242
.138
43.4
18.7
5.8
4.58
6.62
4.04
1.86
1.03
0.389
.225
.130
.104
.0879
.0rm7
.0656
.0998
.0851
.0761
.0648
.0581
.0575
.0470
.o467
0.0005
.005
.008
.04
.0414
.o365
.0349
.0344
.o344
.0390*
30
0.924
6.39
40
50
60
80
.733
.612
.527
.4is
7.121
7.65
8.08
8.69
.21
.12
.14
.16
.0499
.0526
100
.344
9.15
.18
.0607
.09
.0o65
.o549
.0582
a Data in the first column is given by the sum of coherent scattering and of incoherent
scattering from the Klein-Nishina formila corrected for binding effects. In the second column
incoherent scattering is given by the Klein-Nishina formula for free electrons.
b Barns/atom x 0.006279 - c22/g
o K edge; at this and lover energies data for the L and Mshells is given while at this and
higher energies data for the L1 M and K shells is given.
Snergy region in which dipole absorption attains a miusm crose section.
39
!ABLA 29.
Tin
Photoeleotric
K,L and M shells
GnrIIV
wioh
coher'ent
Nov
0.01
.015
.02
Totalb
Fair production
Scatteringa
photon
Barns/atom
Wrhohut
oherentcoherent
Barns/atom
Darns/atom
Nucleus
Elctron
With
Wi5thout
coherent
Barns/atom
Darns/aton
cm2 /g
oR2/g
510
340
32.0
31.14
71410
12t
39.3
240
30.9
3220
17.6
122
37.8
16.5
8580
8150
44.3
42.1
43.7
19.3
18.9
.0295e .50
.02925 35
.03
1W
30.0
30.0
29.8
w000
1050
6.09
5.48
41.5
.0o4
.05
.06
.08
10
79
28.9
28.0
3700
.10
.35
.20
.30
40
24.6
286
1.65
1.58
29.6
22.2
88.8
0.601
0.563
19.7
17.68
12.4
.163
.353
.40
17.0
.60
.80
13.8
12.0
35.84
14.46
.50
1.0
1.5
2.0
3.0
65
49
24.6
15.2
10.7
8.65
7.36
27.3
25.8
20.3
33.38
.1.75
lo.56
8.58
7.32
5.76
4.0
5.0
6.o
4.80
,4.14
3.66
8.0
2.99
10
35
2D
30
2.55
1.886
1.512
1.100
40
19900.5
1210
539
6.1
2.96
39.3
20.2
6.28
2.87
.324
.303
5.6
3.0
.-.5
1.9
.0797
.109
.0586
.0776
.06,7
.0576
.0462
.04110
.8
.0469
.0408
.0367
.0924
1.0
0.64
.32
.20
.12
.08
.0660
0.14
.51
1.35
0.0006
2.12
.002
.06
.05
.04
2.78
3.33
4.20
.03
.02
6.39
.015
.006
.01
.02
4.94
.03
7.40
.07
8.91
9.89
.0355
.0355
.0358
.0368
.05
.0383
.0o424
.10
.523
.04,57
50
60
8o
0.873
.728
.627
.4914
1o.6
11.2
12.1
.23
.35
.16
.19
.0553
.0583
.6O9
.0649
1M0
.1.10
12.7
.21
.O676
a Data in the first coluamn I given by the sum of coherent scattering and of incoherent
scattering from the flein-Nishina formula corrected for binding effects. In the second colum
incoherent scatering is given by the lesin-Nishinx formula for free electrons.
b
Barns/atox x 0.005076 - =, 2 /g
a K edge; at this and lower energies data for the L and Mshells 1i given while at this and
higher energies data for the L, X and K shells is given.
40
TIMD • 30.
Iodine
Fair production
Scatteringa
Photoelectrio
K,L and M shells
Photon
energy
with
coherent
Barns/atom Darns/atom
'V46
0.01
32.8
31.6
31.3
31.3
W3o
1260
933
7510
120
89
72
30.6
29.7
28.9
27.4
l449
44
26.1
23.5
21.5
18.874
033
010
-033230
150
.15
.20
.30
32
26.5
21.0
50
16.0
0.814
.41
9.10
7.76
6.10
4.39
3.88
3.17
20
30
40
o.92
50
60
80
1oo
2
44.6
19.8
6.23
4.58
36.4
35.8
21.9
12.1
7..6
3.4*7
21.a5
n.9
7.26
3.34
1.92
0.688
.363
.176
360
113
2.70
2.00
1.603
1.165
15
cm2/g
2470
.,00
6707
5.09
5.o
6.0
8.0
cm2 /g
Darns/atom Barns/atom
5.114
11.19
9.18
7.81
Without
coherent
4j6.2
20.9
6.74
U-.4
14.8
With
coherent
14o4
12.8
.60
.80
Electrn
9360
7.2
3.9
2.5
1.3
18.1
16.2
4.o
33.3
16.78
15.33
114.18
124.6
.50
1.5
2.0
3.0
10
514
.o40
1.0
Barns/atom
29"M
.02
.10
Nucleu
33.9
590
380
270
160
.015
.1
.
.06
.06
WithoutN
coherent
Totalb
.120
.0954
.0823
.0669
1.83
0.648
.339
.165
.114
.09M3
.0792
.0653
407
.O63
.0411
0.17
.26
.26
.59
1.53
0.0006
.11
.08
.07
.05
2.39
3.12
3.72
4.70
.003
.006
.01
.02
.0360
.0361
.0365
.0377
.04
5.52
.03
.03914
.03
.02
7.12
8.26
9.92
.06
.08
.n1
.0409
.3o70
.o1M
.,
.0532
11.0
31.9
12.5
.14,
.772
.665
.16
.17
.0573
.0609
.0633
.434
14.
.22
.0700
.524
23.5
.20
.0675
a Data in the first column is given by the sum of coherent scattering and of Incohere-t
scattering from the Klein-Nishina formula corrected for binding effects. In the second column
incoherent scattering is given by the flein-Nishina formla for free electrons.
b sarns/atom x o.0.74
- cm2 /g
SK edge; at this and lower energies data for the L and M shells is given while at thin and
higher energies data for the L, M and K shells in given.
Energy region in which dipole absorption attains a mximum cross section.
E
41
TADIZ 31.
Phetm scattering!
Tunmgteu
hotoelootrio
Totalb
Pair production
KL and M shells
oenerg
With
coherent
Without
coherent
Barns/atom Barns/atom
1%V
0.01
.010220
.0l12'd
.800
.02
.03
1300
1200
1000
O46.5
590
350
47.4
47.3
46.9
1.57
44.2
Barns/atom
Nucleus
Electron
With
coherent
Without
coherent
Barns/atom
Barna/atom
cm2 /g
42/g
17700
16800
647oo
36000
16000
5040
62.2
59.0
25
121
54.3
17.7
58.1
55.2
212
118
52.6
16.7
.04
.05
240
180
42.8
41.5
.06
.0696We
122
39.4
40.4
2220
1160
674
437
.06964
.08
122
104
39.4
3230
38.3
2250
7.71
7.49
.10
80
5I4
36.5
32.8
125C
408
4.36
1.51
4.21
.20
.30
42
31.5
30.1
26.2
186
63.1
0.74e7
.310
0.708
.293
.40
.5D
.60
.80
26.5
23.4
21.2
18.2
23.4
21.4
19.80
17.39
29.8
16.7
1.0
16.1
12.9
10.9
8.57
15.63
12.70
10.83
8.52
3.9
1.9
1.2
0.71
o.41
1.32
3.13
0.0009
7.10
6.13
5.42
4.43
.50
.38
.31
.23
4. 6 8
5.96
7.02
8.68
.004
.008
.01
.03
.A02
.0409
.0418
.0o38
20
30
3.77
2.79
2.24
1.627
.18
.11
.08
.06
10.2
23.1
15.2
18.3
.04
.08
.11
.0465
.0527
.0o78
.0660
40
1.292
.4
20.3
.19
50
1.077
21.8
.22
.o75
.0757
.15
1.0
.1.5
2.0
3.0
4.0
5.0
6.0
8.0
10
15
145
8.06
4.39
2.68
1.83
11.0
10.7
.174
.125
.0789
.0763
.0655
.0498
.0440
.0407
.15
60
80
0.928
.731
23.1
24.8
.24
.28
100
.606
26.1
.31
a Data in the first colum is given by the sum of coherent scattering and of incoherent
scattering from the Klein-Nishina formula corrected for binding effects. In the second colum
incoherent scattering is given by the Klein-Nishina formula for free electrons.
49
1.44
.a84
.131
.105
5.9
7.41
3.94
2.34
1.56
b
Barns/atom x 0.003276 - em/g
a
I
d
L1 edge; from this energy to the K edge energy data for the L and M shells is given.
e
K edge; at this and higher energies data for the L, M and K shells is given.
edge; at this and lover energies data for the M shell is given.
.101
.0640
.0492
.0437
.0405
.0795
.0845
.0885
TABLE 32.
Platinum
Scatteriega
Pair production
Photonu
energy
With
coherent
Nay
Without
coherent
K,L and M shells
Barns/atom Darns/atom
IWO
0.01
.0158.
*0o391a
.02
1200
1000
94o
670
.oA
280
.015
.03
400
.05
.06
Totapb
Photoelectrie
210
.07858s
.078580
.08
163
117
117
3.15
49.9
49.6
Nucleus
Barns/atom
l1.O00
53900
46.6
19700
62.0
45.1
2720
42.6
Barna/atom Darns/atom
22000
49.2
149.1
48.2
43.8
Electron
With
coherent
Without
coherent
cm2/g
cm2 /g
72.2
68.0
45.8
49.4
169
138
62.9
20.5
43800
166
135
60.9
19.is
9.26
1440
5.o9
836
3.08
40.6
40.6
W0.3
380
2860
275o
1500
498
226
77.3
8.53
4.58
2.71
1.53
9.19
8.84
1.30
8.95
8.61
h..90
1.72
0.836
.313
4.75
.10
.15
.20
.30
88
59
45
34
38.4
34.6
31.7
27.6
.40
.50
.60
.80
28.3
24.8
22.5
19.2
24.7
22.6
20.9
18.33
1.0
1.5
2.0
3.0
17.0
13.6
l1.6
9.04
16.4.7
13.38
11.42
8.98
4.9
2.4
1.5
0.90
0.4'7
1.51
3.52
0.001
4.0
5.0
6.0
8.0
7.52
7.18
6.46
5.71
4.6?
.63
.48
.39
.29
5.21
6.59
7.73
9.54
.004
.009
.02
.03
10
15
20
30
3.98
2.94
2.36
1.715
.22
.14
.10
.07
11.2
14.4
16.7
20.1
.05
.08
.31
.16
.01om
.0542
.0595
.0680
40
5o
60
8o
1.362
1.336
0.978
.770
.06
.04
22.3
24.0
25.4
27.3
.20
.Z3
.25
.29
.0738
.0784
.0822
.08.75
100
.639
28.6
.33
.0923
37.1
21.2
13.9
7.6
1.64
0.795
.324
.202
.142
.112
.0827
.191
.135
.107
.0800
.0676
.0508
.0451
.0415
.0659
.0501
.0445
.0414
.0412
.0411
.o18
.0A27
.A48
a Data in the first
colum is given by the sum of coherent scattering and of incoherent
scattering from the Kiein-Nishina formula corrected for binding effects.
In the second column
incoherent soattering is given by the Klein-Nishina formula for free electrons.
b Barns/atom x 0.003086 - em2 /g
a L3 edge; at this and lower energies data for the M shell is given.
d
L1 edge; from this energy to the K edge energy data for the L and M shells is
e
K edge; at this and higher energies data for the L, M and K shells is
given.
given.
43
TABLl 33.
Thallium
Fnotoeleotrio
K,L and M shells
lhoton
energl
Pmv
Totalb
Fair production
Scattering
With
coherent
Without
coherent
Barns/atom
Barns/atom
Barns/atom
Nucleus
Electron
Barns/atom
Barns/atom
With
coherent
Without
coherent-
cm2 /g
cm2/g
1500
51.8
26000
81.1
76.8
1200
990
51.3
50.7
13400
47200
43.0
142
39.7
139
.02
.03
730
430
50.1
48.4
22700
7220
69.1
22.6
67.1
21.4
.04k
.05
46.8
.06
.08
300
220
180
12
45.4
44.2
.085840
114
U.3
.0858V4
114
41.3
3200
1660
976
420
313.
2577
10.3
5.54
3.414
1.6o
1.34
7.93
9.57
5.o3
3.01
1.36
1.13
7.72
95
39.9
35.9
1710
576
5.32
1.88
5.16
1.80
261
88.9
0.911
.367
0.866
.346
0.01
.01268'
. 0 1 5 37 d
.10
.15
.20
.30
.80
1.0
2.0
63
'
141.9
48
32.9
28.6
29.6
26.0
23.4
20.0
25.6
z3.4
25.o
43.6
.216
.204
21.7
19.04
16.4
8.9
.117
.0852
.112
.0824
17.8
14.2
1.5
12.0
17.11
13.90
11.86
5.8
2.8
1.8
0.53
1.67
.0696
.0517
.0456
.x675
.05o8
.0452
35.5
.150
.143
3.0
9.40
9.32
1.1
3.83
0.001
.0422
.0420
4.0
5.0
6.0
8.0
7.81
7.77
6.71
5.93
4.85
0.72
.56
5.62
7.08
8.29
10.2
.004
.009
.02
.03
.0417
.o416
.0423
.0A33
4.13
.45
.32
.x454
.25
.oA84
3.06
2.451.781
.17
.12
.09
12.0
15.4
17.9
21.5
.o5
15
20
30
.09
.12
.17
.0552
.0607
.0694
40
50
.o7
.05
60
1.414
1.179
1.016
23.9
25.7
27.1
.21
.2l
.26
.a54
.0801
.o83
80
0.800
29.2
.31
.0894
100
.664
30.6
.34
.0932
10
a Data in the first column is given by the sum of coherent scattering sad of incoherent scattering from the Klein-Nishina formula corrected for binding effects. In the second colu"•1 incoherent
scattering is given by the --. ein-Nishina formula for free electrons.
44
b
Barns/atom x 0.002948 - cm2 /g
o
I. edge; at this and lower energies data for the M shell is given.
d
L1 edge; from this energy to the K edge energy data for the L and M shells is given.
*
K edge; at this and higher energies data for the L, M and K shells is given.
TABLZ 34.
energy
With
coherent
Photoeleotric
KL and M shells
Without
coherent
Earns/atom Barns/atom
MOv
0.01
.013070
01589d
.02
.03
.04
1600
1200
968
75o
450
310
.05
.06
230
180
Barns/atom
Electron
Barns/atom
Barns/atom
With
coherent
49.0
84.6
41.9
135
72.0
23.5
10.5
47.4
3310
46.o
44.8
174o
142.14
1040
144.4
.10
100
614
49
Without
coherent
cm2/g
27500
13200
45o400
240M0
7620
127
1.3
W41.6
1313
.20
.30
Nucleus
52.5
51.8
51.3
50.7
.08 0
.08823
088 23e
.15
Totalb
Pair production
Soatteringa
Photon
Lead
2
CM
/g
80.1
38.5
132
5.73
3.55
69.9
22.3
9.76
5.19
3.M5
1..6
334
2510
1.66
1.30
7.63
14
1.09
7.42
o40.14
36.,4
1780
596
5.47
1.92
5.29
275
0.942
0.896
1.814
36.2
33.3
29.0
.50
30.1
26.3
4.57
26.
.220
.152
.208
.145
.60
.80
26.0
23.7
23.8
20.3
21.9
19.27
17.3
.119
.0866
.114
.0836
18.0
17.32
6.2
.0704
.0684
.40
1.0
1.5
1.4
93.4
9.5
3.0
14•.07
2.0
3.0
12.2
9.51
12.00
9.44
2.0
1.1
4.0
7.91
7.87
6.79
0.80
.60
5.0
6.o
6.oo
4,.91
8.0
10
15
.377
.49
.35
0.55
1.72
3.93
0.001
5.76
7.25
.004
.009
8.47
.356
.o522
.0512
.0o63
.0423
.0457
.0421
.0421
.o420
.0426
.02
.0x36
10.5
.03
.0459
.o089
.0554*
14.18
.28
12.3
.05
20
30
3.09
2.48
1.803
.18
.13
.09
15.7
18.3
21.9
.09
.12
.17
140
50
1.432
1.194
.07
.05
24.4
26.2
.21
.214
.0759
60
80
1.028
0.810
27.7
29.8
.27
.31
.0843
.0899
100
.&72
31.3
.34
.M9
.06n
.0697
.0805
a Data in the first column is given by the sum of coherent scattering and of incoherent
scattering from the Klein-Nishina formula corrected for binding effects. 'n the second column
incoherent scattering is given by the Klein-Niahira formula for free electrons.
b Barns/atom x 0.002908
c
-
cm2 /g
L3 edge; at this and lower energies data for the M shell is given.
d L1 edgel from this energy to the K edge energy data for the L and M shells is given.
e K edge; at this and higher energies data for the L, Mand K shells is given.
,' Energy region in which dipole absorption attains a maximum cross section.
45
TABLE 35.
Uranium
enex1•
neith
MOT
0.01
0 17 20
coherent
Barns/atom
Barns/atom
1400
.o
,
.C2181-
Without
coherent
2100
.015.o3.m
1200
Totalb
Pair production
Scattering
Photon
58.9
Fotoelectric
,!. and K• shells
Barns/atom
Nucleus
Electron
Barns/atom
Barns/atom
44600
Without
coherent
cm2 /g
118
cm2 /g
440.2
28.3
113
36.8
25.5
12000
76.6
31.9
74.6
30.5
57.9
•4l
31o500
57.14
1000028355
29400
.03
880
590
.04
400
53.2
5250
14.3
.05
300
2780
.08
.10
230
163
123
51.6
50.2
47.6
1.3.4
7.17
4.28
1.93
1.06
1163e
.1163
.3.5
.06
56.5
With
coherent
54.9
45.3
716
374
7.79
4.73
2.22
1.26
103
103
43.8
43.8
239
1790
0.865
4.79
0.716
4.64
78
1640
59
40.8
37.4
916
.20
.30
425
2.52
1.22
2.42
1.17
42
32.5
146
0.476
0.452
.40
.50
.60
34.7
30.2
27.1
29.1
26.6
24.6
73.2
43.1
29.2
.273
.185
.142
.259
.176
.136
.80
23.0
21.6
16.0
.0987
.0952
1.0
1.5
2o.3
16.2
19.-43
15.79
10.5
5.1
.o779
.o559
.a757
.0548
2.0
13.7
13.47
3.3
0.77
2.35
1.9
5.09
.049o
.0484
0.001
.0448
.0445
8.83
7.62
6.74
1.3
1.0
0.81
.004
.01
.02
.ou
.59
7.26
9.00
10.4
12.8
.0440
.0446
.0455
10
4.69
.46
15.o
3.47
2.78
2.023
.30
.22
.15
19.3
22.4
26.8
.06
.10
.13
.19
.0511
15
20
30
50
60
80
40
1.606
1.34o
1.154
0.909
.1
.09
29.8
32.1
33.9
36.5
.24
.27
.30
.35
.0804
.0855
.0895
.o956
100
.754
38.3
.39
.0998
3.0
10.7
4.0
5.0
6.0
8.88
8.0
10.59
5.51
.04
.o479
.o586t
.0646
.0738
a Data in the first column is given by the sum of coherent scattering and of incoherent scattering from the Klein-Nishira formula corrected for binding effects. In the second column incoherent
scattering is given by the Klein-Nishina formula for free electrons.
b
Barns/atom x 0.002531 - cm2 /g
o
13 edge; at this and lower energies only M shell data is given.
d LI edge; from this to the K edge energy L~ata for the I,and N shells is given.
e K edge; at this and higher energies data for the L, I and K shells is given.
* Energy region in which dipole absorption attains a maximum cross section.
46
TABLE
36. Water
Scatteringa
Photon
energy
Mev
0.01
Pair production
Totalb
Photoelectric
K and L shells
With
coherent
Without
coherent
Nucleus
arns/mnlecule Barns/molecule Barns/molecule
12.8
Electron
Barns/molecule Barns/molecule
With
coherent
Without
coherent
cm2 /g
cm2/g
.02
.03
9.54
8.19
6.96
6.40
6.29
6.18
5.97
1J6
39.6
15.4
4.o9
5.31
1. 6 4
0.789
.370
5.1o
1.53
0.722
.336
.04
.05
.06
6.34
5.92
5.78
5.61
1.55
.264
.222
.245
.212
.08
5.33
5.17
.15
.2o4
.196
.183
.178
5.05
4.5o
4.93
.171
4.10
4.07
3.54
.071
.020
.010
.167
.149
.136
.118
.015
.10
.15
.20
.30
.40
.50
.60
.80
1.0
1.5
5.70
3.55
5. 4 6
4.4
0.73
.40
.151
.137
.119
3.17
2.89
2.68
2.35
2.11
1.716
.106
.0966
.0896
.0786
.0706
.o575
0.0029
.on
.033
0.0001
.0493
.0396
0.960
.828
.732
.599
.055
.072
.089
.116
.o0004
.001
.002
.003
.0339
.0301
.0275
.02h0
10
15
20
30
.510
.3,n
.302
.220
.138
.181
.213
.256
.006
.010
.014
.019
.0219
.0190
.0177
.0166
40
50
60
80
.174 6
.1456
.1254
.0988
.287
.310
.327
.355
.024
.026
.029
.034
.0162
.0161
.0161
.0163
100
.0820
.376
.038
.0166
2.0
3.0
4.0
5.o
6.0
8.0
M.64
1.151
a Data in the first column is given by the sum of coherent scattering and of incoherent scattering from the Klein-Nishina formula corrected for binding effects. In the second column incoherent
scattering is given by the Klein-Nishina formula for free electrons.
b Barns/mlecule x 0.0336 - em2/g
47
TABIZ 37.
Sodium Iodide
Scatterina
Pair production
Photom
energ
With
coherent
MOev
0.01
•03,5
.02
.0332a
.03323
.014
.05
40.3
39.6
38.2
37.8
37.8
130
37.0
.10
50
.20
.30
24.9
37
31
21.6
35.9
34.9
33.1
31.5
28.4
26.0
22.6
30.9
30.4
4500
18.6
10.3
18.2
10.1
2o70
1500
678
360
113
50.0
i6.0
0.84
9.42
.41
9.37
7.37
.26
.16
4.0
5.0
6.14
5.30
6.0
8.0
3.83
4.68
10
15
3.26
2.41
1.935
20
1.1407
30
40
i.n7
50
0.932
60
.8o3
80
.632
S.525
am2 /g
7520
13.52
10.98
cm2 /g
4.45
946
33.7
11.1
Barns/rolecule Barri/molecule
with without
coherent coherent
5.83
1.o04
17.7
Electron
125
39.9
18.0
15.4
.60
.80
Nucleus
30400
9530
4200
1280
7.2
3.9
2.5
1.3
19.4
2.0
3.0
Barns/molecule
20.3
18.51
17.12
.50
1.0
1.5
K,L and Mshells
1l.0
610
390
280
170
160
160
.06
.08
.40
Vithout
coherent
Barns/molecule Barns/molecule
96
79
60
.15
Totalb
Photoelectrio
0.18
.61
122
38.5
17.0
5.30
3.95
6.35
2.9?
6.17
2.86
1.65
0.603
.326
.16 4
1.97
0.568
.305
.155
.116
.0936
.0812
.671
.111
.0901
.0789
.069
.0584
.0470
.0414
.05n
.0465
.0412
1.59
0.0007
.0367
.11
.08
2.49
3.25
.004
.007
.035i
.0347
.07
.05
3.88
4.91
.01
.02
03
.0351
.04
5.77
.04
.0366
8.65
.07
.10
.o040
.0430
.03
.02
7.45
10.4
11.5
.23
.0480
.17
.0514
12.5
.19
13.1
.21
.0567
14.1
.24
.0602
14.8
.27
.0627
.O5W
a Data in the first column is given by the sum of coherent scattering and of incoherent scattering from the Klein-Nishima formula corrected for binding effects. In the second columa incoherent
scattering is given by the flein-Nishina formula for free electrons.
b
Barn./oleeule x 0.004019 - o=2 /g
a K-edge of Iodine; at this and lower energies data for the L and 4 shells is given while at
this and higher energies data for the L, M and I shells is given.
48
TABLE 38.
Calcium Fhoaphate
With
coherent
Nov
0.01
.015
.02
.03
Without
Photoelectric
K,L and 14 shells
Nucleus
Barns/molecule
144
.05
121
107
99.2
88.5
89.0
86.14
84.1
79.6
.10
.35
.20
.30
81.7
71.2
64.2
75.9
68.3
62.6
54.4
21.7
5.84
2.1a
0.72
.o40
.50
.60
.80
4.2
1.7
48.8
4.l.3
36.3
41.2
36.2
.32
.18
.11
.05
32.6
32.5
.03
.06
.08
1.0
1.5
2.0
3.0
55.2
44.5
Barnu/smlecule BarnB/molecule
24500
7580
3220
943
98.6
96.9
95.2
91.9
.04
Electron
coherent
Barns/molecule Barns/molecule
375
248
193
Total
NFar production
Scatteringa
Photom
energ
With
Without
coherent coherent
cm2 /g
15.2
6.63
2.11.
47.8
14.9
6.44
2.01
o.965
.567
.397
.254
0.903
.527
.367
.237
.201
.150
.129
.lo9
.144
.0962
.0872
.0804
.0706
.0954
.0868
.0802
.o7o4
.o634
.0632
48.3
376
185
105
42.2
cm2 /g
.190
.126
.107
26.4
22.5
17.73
0.10
.38
1.08
0.002
.0515
.0444
.03,46
14.78
1.79
.007
.0322
12.75
2.38
.02
.0294
6.0
8.0
11.27
9.22
2.91
3.75
.03
.06
.0276
.0253
10
15.81
20
30
7.85
4.50
5.86
6.86
8.23
.09
.17
.23
.32
.0242
.0230
.0228
.0232
40
60
2.69
2.24
1.931
9.22
9.92
10.5
.40
50
.50
.0239
.0245
.0251
80
1.522
11.4
.58
.0262
100
1.263
12.0
.65
.0270
4.0
5.0
4.66
3.39
.45
a Data in the first column is given by the sum of coherent scattering aril of incoherent scattering from the Klein-Nishina formuila corrected for binding effects. In the second colu.. incoherent
scattering is given by the Klein-Nishina formula for free electrons.
b
arns/mlecule x 0.001942 - oa 2 /g
49
TABIS 39.
Air
0.755 N, 0.232 0, 0.013 A by Weight
Mass Absorption Coefficient
Total
Total
"Fnoton
energy
'ith
coherent
Without
coherent
Photon
energy
With
coherent
Mev
cM2 /g
cM2 /g
Mey
c0 2 /g
Without
coherent
cm 2 /g
.0635
.0635
5.09
4.89
1.0
1.59
0.764
1.48
0.697
1.5
.0517
.349
.317
?.0
3.0
.0445
.0357
.245
.226
.194
5.0
4.0
. 06
.08
.186
.166
.178
.161
6 .o
8.0
X307
.0274
.0250
.0220
.10
.15
.151
.20
.30
.155
.136
.123
.107
.40
.50
.6c
.0954
.o868
.o084
.80
.0706
C.01
.X15
.e2
.03
.04
.204
.05
10
15
20
30
.0202
.0178
.0166
.0159
.0953
.0868
.0804
40
50
60
.0156
.0157
.0158
.0706
80
.0160
100
.0164
.134
.123
.106
Tatble 40.
Concrete
(0.56% H, 49.56% 0) 31.35% Si, 4.564 Al, 8.26% Ca, 1.22% Fe, 0.2h% g.,
1.71%Na, 1.92%K,0.12%S) (,0- 2.35 g/cv?)
Photon
Yass Absorption
Photon
Naas Absorption
Photon
Vass Absorption
energy
Coefficient
energy
Coefficient
energy
Coefficient
14.,
O.2/,
o
OU4.,
1ev
=1
0.01
24.6
.30
.107
6.o
.0268
8.o
.0243
.015
7.68
.40
.0954
.02
3.34
.50
.0870
10.0
.0229
.03
1.10
.6o
.0804
15
.0214
.80
.0706
20
.0209
1.0
.0635
30
.0209
1.5
o.o517
40
.0213
.197
2.0
.0445
50
.0217
.169
3.0
.0363
60
.0222
.139
4.0
.0317
80
.0230
a
5.0
.0287
100
.0237
.04
.542
.05
.350
.06
.08
.
Cohn.
50
, scattering is atot included in the calculations.
The data were not revised.
TABLz 41.
v.5qa
9
Thomas-
L.,enb
Incoherent scattering function, 8(v)
Koppe*
Fermi
UM
v.
qa
3hZ21
Thoma&
0.001
0.012
0.0068
0.00037
0.3
0.776
.005
.01
.051
.007
.074
.024
.4
.5
.839
.880
.160log
.-----
---------
.02
.03
.227
.217
.04
.0
.277
319
---------
.2
.674
.....
.1
.486
,6
.........
-------.196
.370
LenZb
Koppe.
0.890
--------
Fermi,
.095
0824
-. 6
7
.90A
.9
........
.......
8
.9
.944
.94-
-.......
-------
1.0
.962
........
1.0
........
.34
..688
a Values below P-0.05 are from Wheeler and LeAmb, and from v-0.05 to I from Bewilogua.
bValues ae caloulated for the Molire approximation to the Thomas-Ferml distribution.
@Values re calculated for analytcasl interpolation to give Correct values at low Pand Thomas-Ferml at bjgh P.
5. Appendix-Survey of Data on the Incoherent Scattering Function
Many effects of the interaction of radiations
with atoms depend on the so-called incoherent
scattering function S(q,Z). Among these are the
small-angle incoherent scattering of X-rays [73],
the small-angle inelastic scattering of charged
particles [74, 751, and the production of bremsstrahlung and of positron-electron pairs in the
field of electrons [761. Data on S(q,Z) are represented in the graphs of figures 6 and 7 and in
table 41.
The incoherent scattering function represents
the probability that an atom of a specified material be raised to any excited or ionized state as a
result of a sudden impulsive action which imparts
a recoil momentum ' to any of the atomic elec-
atoms with spherical symmetry or for an assembly
of atoms with random orientation.
In order to minimize the variation of the incoherent scattering function from one element to
another it is convenient to express the recoil
momentum q in terms of a suitable unit, namely,
to replace q by the variable
v=0.333 qa/1Z
2 13 ,
(16)
where a=0.53 X 10-8 cm is the Bohr radius.
The incoherent scattering function is then indicated as
" deIF.(j)l2
S(v)=S(O.333 qa/AZ1/3)=(1/Z)J(>1
trons.
The generalized form factor of an atom with
atomic number Z can be defined as a matrix
element
"."Tf
where the integral includes both a sum over the
discrete spectrum and an integral over the continuous spectrum. The function (17) still depends
on Z at constant v, but this dependence is not
(15)
explicitly.
indicated
This equation may be transformed by applica-
jth electron
of the
vectorand
i is theto position
where
with respect
the nucleus,
E indicates the
tion of a closure theorem (sum rule) so that it
defines S(v) in terms of properties of the ground
state only, specificall in terms of diagonal elements of matrices r taming to the ground state
F.(q)F.
Z
e -W10\5
0
energy of an excited (or ionized) stationary state,
as measured from the ground state. The expression (15) and ah of its applications in this appendix have been derived and should be considered
only in the frame of nonrelativistic quantum
mechanics.
The incoherent scattering function S(g,Z) is
the sum of the IF.(•) over-all excited states of
the atom, divided by the number of electrons, Z.
The sum is independent of the direction of ' for
(
*L.!i
S(v)=-(1IZ~j(0I.e
1110)-[ii.7)11]
(18)
i
where F~,Z) is the form factor that deterni..es
1_)e coherent scattering.
When the electron recoil momentum, q, is much
larger than the initial momentum of the electron
in its bound state, the electric forces that initial!
were binding the electron in the atom influence
the recoil only to a slight extent. The recoiling
electron is practically certain to leave the atom,
and the incoherent scattering function is very
nearly equal to 1. This feature is displayed by
every graph in figure 6. On the other hand, if
the recoil momentum is very small, the atom is
almost certain to absorb the recoil as though it
were a rigid body, that is, to remain in its ground
state. Accordingly, S(v) tends to vanish for small
values of v, as shown in figure 6.
Hydrogen atom. The incoherent scattering
function for the hydrogen atom can be calculated
analytically, because the H wave function is
known analytically, and has in fact a simple
algebraic form. The first term in the bracket of
eq (18) equals I for H and the second term equals
[I+qla 2'/411]-
4 =[l+9.04v2/41-1.
Therefore,
S(v)=1--[l+9.04d/4]- 4
(9.04v2/4)(2+9.04i2/4)(2+9.04v2/2+81.6v4 /16)
+ 46
(1+ 9.04v2/4)1
(19)
This expression is plotted in figure 6.
Thomas-Fermimodel. The incoherent scattering
function for an atom described by the ThomasFermi model has been calculated by Heisenberg
[77] and Bewilogua [78]. According to this model
the incoherent scattering function, S(v), is a
universal function independent of Z, i. e., valid for
all elements. It is plotted in figure 6 and tabulated
in table 41. It was stated by the authors that this
application of the Thomas-Fermi model should be
with screened
0 atoms
and basis
C the
for on
calculations
valid for Z>6,
of comparison
with
hydrogenic wave functions.
The Thomas-Fermi model yields an electron
distribution that is excessively smeared out at the
edge of the atom. This causes the incoherent
scattering function to be in error for small values
of v. The incorrect assumption that this part
of the electronic distribution is spread out with
low density, low binding energy, and low momenturn yields an erroneously large probability of incoherent scattering with low recoil momentum.
Therefore, the Thomas-Fermi S(v) tapers off much
too slowly for low v, that is, on the left side of
figure 6.
The Thomas-Fermi model also gives an incorrectly high density of electrons near the nucleus,
as though there were a portion of the electronic
charge with exce: ;ively high momentumn.
MTere
results an incorrectly large probability of coherent
scattering for comparatively large values of q and
v, and a corresponding incorrectly low pro'abilitv
of incoherent scattering. As a result the Thoias'Fermi S(v) approaches 1 in the region of r--=1 too
52
graduall
This is indicated by the comparison
of the Tomas-Fermi S(v) with the curves calculated from the Hartree model in figure 7.
Lenz [791 has suggested that simplified calculations be made utilizing the approximate formula
for the electron density of the Thomas-Fermi
atom introduced by Moliere [80]. In table 41 a
comparison is made of S(v) obtained by Bewilogua
for the Thomas-Fermi model and values from
the Moli~re type of approximation. The Molire
distribution of electrons drops off at the edge of
the atom faster, and therefore more realistically,
than the Thomas-.Fermi distribution. Accordingly the scattering function is more in line with
realistic expectation than is the original Bewilogua
curve.
Low-v approximation. Koppe [811 has suggested
that the incoherent scattering function be calculated, for low v, from an improved model. For
low v, that is for low q, the exponential in eq (15)
can be expanded into powers of q, disregarding
powers after the first. The first term of the expansion, namely -,jl=Z, contributes to FE(q) an
amount Z(E*1110), which vanishes owing to the
orthogonality of the eigenfunctions. The next
term
yields
-
F.(q),-•'(,0).
(20)
This expression vanishes for parity reasons when
E= 0. A closure theorem yields then
210)
S(v) =(1/Z)f!FE(q)i2de-•
-1
(0
1q'h , 1
O
(21)
2
h2(0jJ1•jj12j0),
where the last equality has been obtained by
averaging over-all directions of q and taking into
account the assumed spherical symmetry of the
atom.
Because the atomic electrons move very nearly
independently of one another, the square of Z'j
in eq (21) has an average value nearly equal to
that o2
vr- . This latter average can be obthat of
tained for various substances from experimental
values of the volume diamagnetic susceptibility xd,
according to the law that
-r
1 66 me'
e2
m.
i(0I•---jlr•20)
=
2
1.25X 106(-xd
1 ) Aa )
(22)
P
where N is the number of atoms per cubic centimeter, A is the atomic numper, p the density in
grams per cubic centimeter, and a is the Bohr
radius. Equation (22) differs from Koppe's eq
(14) by a factor of 2. This discrepancy is probably
due to an inconsistency between the normalizations involved in the various equations [81, p.
661].
A reasonable approach to obtain a complete
curve S(v) would be to draw S(v) for low v on the
basis of eq (21) and (22), for large v on the basis
of the Thomas-Fermi curve, and then join by
interpolation the parts of the curve thus obtained.
Koppe has suggested that this interpolation be
done simply by multiplying the Thomas-Fermi
S(v) by the factor vI(v+A), where the constant A
is adjusted to yield the correct behavior for low v.
However, this interpolation formula appears to
give values of S(v) that are too low for intermediate values of v (see table 41). Therefore, a more
realistic interpolation seems necessary.
Hartree model calculations. A more basic approach to the calculation of S(q, Z) utilizes electron atoms provided by the Hartree self-consistent
field method [82]. Data obtained by this method
are discussed in this section, but on the whole,
applications of the Hartree method to the incoherent scattering function appear much less advanced than one might believe.
The Hartree method starts from an independent
particle picture, which assumes that the excitation
or ionization involves one electron only, leaving
the other electrons undisturbed. From this standpoint the incoherent scatterinig function for a
material represents simply an average of the
incoherent scattering functions for its separate
electrons. One can then write
S(q,Z)=l1-(1/Z)'••,]"(q)j2,
(23)
where f (1(q) indicates the probability that the ith
electron gets neither excited nor detached, even
though it has received the recoil momentum q.
The quantity f (o'(q) is not quite the same as the
ordinary form factor f(O)(q), which represents the
contribution of the ith electron to coherent
scattering; the difference lies in the fact that the
excitation of an electron from one orbit to another
may be forbidden by the exclusion principle.
Data on the form factor f )(q) for electrons in a
few orbits and for a number of atoms have been
provided by James and Brindley [47] on the basis
of Hartree wave functions. Values of Elf(1)(q) 12
have been calculated from these data by Compton
and Allison [83]. However, it is not clear how
this data was obtained for the higher Z materials
because James and Brindley give practically no
data for shells higher than the M shell. The
combined difference between Ifs,(")I2 and If"1( )I2
for all electrons is treated by Waller and Hartree
[841 and indicated as a corrective term by Pirenne
[74]. The relative importance of this corrective
term decreases as the number of electrons in the
atom increases.
Calculations including the correction of WallerHartree have been made for neon and argon
f84,85]. In a limited region of the variable v the
values of S(v) thus obtained are in good agreement
with values from the Thomas-Fermi model (see
fig. 7).
Wentzel model. Lenz [74] suggested that one
assume a distribution of the electronic charge
within the atom according to a model introduced
by Wentzel. With this model a constant can be
adjusted so as to yield the experimental value 6f
the diamagnetic susceptibility, which implies a
correct behavior for S(v) at low v. This procedure
implies really that the atom behaves with respect
to incoherent scattering as though it contained
a single charged particle distributed in density as
described by the Wentzel formula [86]. This
density is
(24)
Z
"
(24)
P---4TR2
where
__r
1
0
(25)
6Z
then
S(v) =1-F
R2
L(1-I+q2R2/7j2)2J-
h2)2"q2R2/h2)
/) (22
(qIRW
(1 + q
R 2/ý)
2
r
J
1+9.04Ziv2(- 1.25 X 10o6,41:D
Z
(26)
Curves according to eq (26) for Pb and C (graphite) are plotted in figure 6.
It is difficult to assess the accuracy provided by
the Wentzel model. The density (24), being
singular at r=0, should yield an excessively slow
approach of S(v) to 1 as v increases. In practice
S(v) approaches 1 for lower values of v than in
other models but this is presumably due to more
serious inaccuracies of the model at medium
distances from the nucleus.
Conclusion. The preceding discussion indicates
that existing approximate models fail to yield
accurate data on the incoherent scattering function. Under the circumstances the values of
S(q,Z) derived from the Thomas-Fermi model
were used, because the final results lid not appear
to depend critically on the systematic errors of
these values for low and large q.
53
6. References
(3]
[41
(51
6
7]
[8]
(91
U Fano Nucleonics 11, 8 (1953); 11, 55 (1953).
the and J. Ashkin Experimental nuclear
H. A.
physics, I, part 1, E. Segrý, editor (John Wiley &
Sons Inc., New York, N. Y., 1953).
C. M. Daviason and R. D. Evans, Revs. Mod. Phys.
24, 102 (1952).
G. T. P. Tarrant, Proc. Cambridge Phil. Soc. 28, 475
(1932).
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WASHINGTON,
September 4, 1956.
54
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