I. INTRODUCTION
An elementary concept in radiation
dosimetry and allied fields is the radiation
exposure field resulting from a point source
in a vacuum. Such a concept is too elementa
ry to be adequate to most situations; and
thus a more complex concept, but one still
quite idealized, has been developed, studied,
and widely used  namely, the exposure field
from a point source imbedded in an infinite,
homogeneous, attenuating medium. For gamma
rays, calculations by Goldstein and Wilkins
are still considered the basis of most of our
information on such exposure fields.
In many practical cases, the situation
does not conform to the idealization of an
infinite, homogeneous, attenuating medium. In
particular, measurements of exposure dose are
often made under circumstances in which the
source and/or the detector are near an inter
face between two different media or between
two regions of radically different density
states of the same medium. Under such a cir
cumstance, even if the source and detector are
on the same side of the interface, the pres
ence of the interface may have a significant
effect on the exposure field in its vicinity.
This report concerns a study of the gamma
radiation exposure field under circumstances
in which the source and detector are on the
vacuum side of the interface between a vacuum
and a semiinfinite dense medium. The calcu
lations are specifically for concrete, but the
results are applicable also to material of
average atomic number near that of concrete,
such as earth. The study determines the ex
posure field on the vacuum side of the inter
face at all points, within certain limitations
as to distances involved. This represents an
idealization of a configuration, such as a
concreteair or groundair combination, in
which the sourcetodetector, sourceto
ground, or detectortoground distances are
small compared to a meanfreepath of the
radiation in air. This is thus a situation
in which the interface effect is the most
significant factor perturbing the direct
radiation. The air attenuating and scatter
ing effects under these circumstances are
negligible or can be separately accounted for
to the same degree of accuracy as the inter
face effect.
It is fairly obvious that the interface
effects may be quite small as compared to the
direct radiation field. (For example, as the
distance between source and detector approach
zero, both being a finite distance from the
interface, the interface "backscattering"
effect remains finite, while the direct radi
ation detected increases without limit.)
Nevertheless, there are many experiments,
calibration operations, and perhaps other
operational situations in which a knowledge
of the exposure field is needed to a degree
of accuracy such that interface effects cannot
be ignored. It is to such uses that this
study effort is pointed.
Most theoretical and experimental works
concerning the interface effect on radiation
from point sources have been devoted to the
situations in which the sourcedetector and
sourceinterface distances are on the order of
a meanfreepath or less in the ground, or in
which the distances involved are an apprecia
ble fraction of a meanfreepath of the radia
(28)
tion in air or greater. Very little work
has been published with regard to the scatter
ed radiation field when the source and detec
tor are near the interface but not within a
distance equal to a meanfreepath in the
ground, and under conditions in which the
source and detector are sufficiently close
together to permit ignoring the attenuating
and scattering effects of the air. The only
available works under these circumstances that
have been found are by Jones et al., by
Clarke and Batter, and by Henry and
Garrett, all experimental. No accurate
calculational results have been published, to
the knowledge of the writer.
A number of side benefits derive from
this study. For one thing, the approach to
the interface effect is through the Chilton
Huddleston(12) approximate formula for differ
ential gamma ray albedo (hereinafter called
the CH formula); and an examination of the
results of the present study and comparison
with experimental values give an understanding
of the degree of accuracy that might be expec
ted in its use. Other interesting and signif
icant aspects of the study will be brought out
in the body of the report.
*Appreciation is expressed to Dr. W. H. Henry
for sending me his data prior to its
appearance in the literature.
II. PRELIMINARY CONSIDERATIONS
As indicated above, the keystone to the
analysis in this report is the albedo concept,
through the use of the CH formula. The
albedo approach to calculation of radiation
reflected from a differentially small area
has been adequately covered by previous re
ports and articles in the literature.
We will start by pointing out that for the
situation illustrated in Figure la the differ
ential dose rate* in the detector contributed
by scattering from the differential area dA
is given by
D cose o dA
dD = 2 .
r
For the situation illustrated in Figure
lb:
D1 coseo d dA
dD= 12 2 d
2r r2
r r
This latter formula is somewhat approximate,
since in this case the radiation does not be
have as if reflected exactly from a surface
area element. If the distance of the source
from the differential area is appreciably
greater than a meanfreepath of the radiation
in the reflecting medium, the error involved
in the use of Equation 2 is quite small.
*The relationships calculated herein are, un
less noted otherwise, equally applicable to
dose rates or integrated doses. The term
"dose rate" will normally be used herein.
It is also to be noted that the calcula
tions will be in terms of "exposure dose
rate," although the concepts are more wide
ly applicable.
The differential dose albedo, ad' as
used in these formulas is approximated by the
CH formula:(12)
C10 *26.K (E ,e ) + C'
d cos@
1 + 
cose
Symbols for the above equations are defined
as follows:
is the dose rate measured in the
incident beam for the case in
Figure la;
is the direct dose rate measured
one unit distance from the source
for the case in the Figure lb;
is the photon energy for the
incident gamma rays;
is the polar angle of the incident
gamma ray direction with respect
to the normal (measured so that it
is < 900);
is the polar angle of the reflect
ed gamma ray direction with re
spect to the normal (measured so
that it is < 900);
is the scattering angle between
the incident and outgoing rays,
for single scattering, given by
the formula: cose = sin% sine
cos(  cosO cose ;
is the azimuthal angle of the re
flected ray, with respect to the
azimuthal component of the direct
ion of the incident ray;
K e(E , s) is the energy scattering Klein
Nishina crosssection per
electron, which is (E scat/E o)
times the photon scattering
crosssection per electron;(16)
C and C' are parameters for the formula,
empirical in nature and adjusted
to provide a good fit to specific
albedo data obtained otherwise.
In this report, the values of C and C'
will be those obtained for reflection by
Portland cement concrete of composition
assumed by Grodstein. However, on the
basis that all materials having approximately
the same average atomic number should have
approximately the same reflecting character
istics for gamma rays, the results derived
may be considered applicable to concrete of
any normal composition, as well as for most
earthy materials.*
Most of the calculations in this report
involve the use of the above formulas for the
contribution to detector response by reflec
tion from small, individual, incremental areas
in the interface plane, followed by summation
over a practically infinite portion of the
plane or, in some cases, over a prescribed
finite area. Thus, the formula for total re
flected dose becomes:
DI cos@ cose [C1026 *K(E ,0 )+C'I AA
2 2
total (cose + cose ) r r
area o o
(4)
An important consequence of this formula is
that the result is invariant under an exchange
of position of source and detector. Thus,
within the degree of approximation inherent in
*For this reason, the words "concrete", and
"earth", and "ground" are used indiscrimi
nately in this report.
the use of the CH formula for albedo, "recip
rocity" exists between source position and
detector position. This fact is of assistance
in mapping out the field with a minimum of cal
culation.
A socalled "similarity principle" given
by Fano(19) is also of assistance in minimizing
the work involved in establishing the exposure
dose rate field. This principle implies that
when considering the dose rate in air near the
interface, the ratio of reflected to direct
dose rate is independent of absolute geometric
scale of the configuration, but depends only
on the relative distances involved. This is
subject to the proviso that the distances in
volved are large compared with the meanfree
path length of the radiation in the ground but
small with respect to the meanfreepath length
of the radiation in the air. For gamma rays of
usual concern, this means that the distances of
primary concern (sourcetoground, detectorto
ground, and sourcetodetector) should be great
er than several inches but not more than a few
dozen feet.** Within these limitations, then,
it is feasible to normalize all calculations to
a source height of one foot, and obtain and
plot all data on this relative geometric basis.
**This statement is consistent with the crite
(19)
ria stated by Fano. Actually, there
appears to be no reason why the principle
should not be valid for a sourcedetector dis
tance less than a meanfreepath in the ground
as long as the other criteria are adhered to.
III. REVIEW OF EXISTING SETS OF PARAMETERS FOR ALBEDO FORMULA
At an early stage in the progress of this
work the question arose as to what values of
the parameters C and C' should be used in the
albedo formula. The work of Chilton and
Huddleston which first suggested the
formula specifies two possible sets of values,
which are slightly different and depend upon
the method used in obtaining a "best" fit to
Raso's Monte Carlo data. One set, called
the "nonweighted" set, is derived by the most
elementary least squares fitting process, with
out any weighting of the values of the squared
residuals which are summed and minimized; the
other set, called the "weighted" set, is de
rived by weighting the values of the squared
residuals by the reciprocal of the calculated
albedo values.
It was decided that the choice between
these two sets should be based upon which set
predicts most correct values of total albedo.
(This is on the quite reasonable grounds that,
after all, we are interested in computing total
scattered doses in this report.) In particular,
the set used should provide the closest possi
ble match to the values of total albedo obtain
able from Raso's Monte Carlo results.
It is desirable at this point to review
the total albedo concept and to derive a for
mula for total albedo equivalent to the CH
formula for differential albedo. It appears
quite natural that total albedo should be the
integral of the differential albedo, inte
grated over the hemisphere which represents the
outward direction from the interface. However,
since we have introduced differential albedo
in a rather indirect way through Equation 1
it would be desirable to demonstrate the consis
tency of Equation 1 with the statement in the
previous sentence.
Total albedo, which is a ratio of the
dose rate "current" reflected from the inter
face to the broad, parallel beam dose rate
"current" into the interface, is fundamentally
defined as follows:
A  1
d D coso
0 0
out of
J surf.
From Equation 1, we then get:
d cos6 D
o o
total
plane
cose dD . (5)
2
cose(D cosOo yd dA/r ) .
(6)
The terms are those as previously defined
and as indicated in Figure 2. From Figure 2 it
is readily seen that (dA cos@/r 2) equals d%,
and thus,
Ad d d
I ower
hemi.
The total albedo concept considered above
is a ratio of dose rate "currents," as previ
ously stated. This is somewhat artificial
since dose is actually defined in terms of
flux, and the concept of a dose "current" does
not appear to have operational significance.
It is possible, however, to define a different
type of albedo  one based on flux. Such a
definition is a ratio of dose rates  that
caused by reflected radiation to that caused
by incident radiation; and the concept thus
eliminates the cosine terms in the fundamental
formula, so as to give:
1
d D coso
out of
surf.
From Equation 1 and Figure 2, it
that:
A' =
lower
Shemi.
dD . (8)
can be se
cos9
cose 0
COQ .C "d
current could not be accurately obtained from
Raso's Monte Carlo results without further de
tailed processing of his data, but values of
total dose albedo based on flux could be quick
ly calculated. The latter values were there
fore used to determine the choice of the albedo
formula parameters. They were determined on
the basis that Raso's reflected dose rate re
sults are normalized to an incident number
current of 1 photon per square centimeter of
surface per second. The results are given in
Table 1. The reflected dose rates are tabu
lated in Raso's report, and are repeated herein
in Table 2. The ratios of the reflected to the
incident tabulated data for each energy and
angle of incidence give the total dose'albedo
based on flux, and are listed herein in Table
en The calculated value of total dose albedo,
which was compared to the information in Table
3, was obtained by substituting the CH formula
(9) (Equation 3)into Equation 9 . When this is
done the following formula is obtained:
Ad = C Fl(eo) + C' F2( o)
It is worth mentioning at this point that
not all workers are willing to call A' an
(20)
albedo. Raso, for example, calls this
concept a "dose rate ratio." Leimdorfer,(21)
on the other hand, considers A a "current
albedo" and A' a "flux albedo," which is
consistent with the point of view maintained
in this report. The literature is generally
confusing on this point, and still different
definitions for what is called "albedo" are
used. (Sometimes the specific numerical val
ues given are the only clues as to what is
meant by the terminology.) The subject is
further complicated by the fact that albedo
may refer to a ratio of numbers, energy, or
dose. In this report only dose albedo is
being considered.
Values of total dose albedo based on
where C and C' are the usual parameters for
the CH formula;
F ( ) = 10 26cose d(cose) K (9 )c;
So) = 10 o cos+cose e s)d;
(0 0
(11)
F2 (e ) = 2rr coso0 In (1 + seco )
2 o 0 0
(12)
and K (9 ) and s are as previously defined in
e s s
connection with Equation 3. The integral of
F cannot be readily obtained analytically,
but in particular cases it can be computed by
numerical techniques. This was readily accom
plished by digital computer. The techniques
for this calculation are so simple and
TABLE 1. CALCULATION OF INCIDENT DOSE VALUES
E (MeV) .2 .5 1.0 2.0 4.0 6.0 10.0
o
o (cm2/g)* .0268 .0297 .0280 .0234 .0186 .0163 .0144
oE E (keV/g) 5.36 14.85 28.0 46.8 74.4 97.8 144.
CosO Inc. dose rate (keV/gsec) = 0E /cose
1.0 5.36 14.85 28.0 46.8 74.7 97.8 144.
0.75 7.147 19.80 37.33 62.4 99.2 130.4 192.
0.50 10.72 29.70 56.0 93.6 148.8 195.6 288.
0.25 21.44 59.40 112.0 187.2 297.6 391.2 576.
0.10 53.60 148.5 280. 468. 744. 978. 1440.
* Values taken from Ref. 17.
TABLE 2. REFLECTED DOSE RATE
E (MeV) .2 .5 1.0 2.0 4.0 6.0 10.0
o
Cose
1.0 1.57 2.37 2.33 2.04 2.22 2.35 2.87
0.75 2.08 3.69 3.85 3.94 3.86 3.91 4.27
0.50 2.96 5.23 6.65 7.36 7.87 7.30 8.08
0.25 4.28 8.79 14.0 18.2 20.7 21.4 21.5
0.10 6.00 14.2 22.8 35.7 47.6 53.7 52.9
TABLE 3. TOTAL DOSE ALBEDO BASED ON FLUX
standardized as to require no detailed expla
nation.*
When these calculations were carried out
with the use of both the nonweighted and the
weighted sets of CH parameters, the results
shown in Table 4 were obtained. It is discon
certing to see that neither the results using
nonweighted parameters nor the results using
weighted parameters provide answers which are
consistently close to the data of Table 3, in
extreme cases giving results which deviate by
a factor of almost two. It is even more sur
prising to note that the variations are almost
entirely in one direction; that is, the data
obtainable directly from Raso's report are in
general appreciably lower than that obtained
*Machine computations indicated in this re
port were accomplished using the IBM7094
computer of the Digital Computer Laborato
ry of the University of Illinois (supported
in part by the National Science Foundation).
from the ChiltonHuddleston formulation. Fig
ures 3 and 4 illustrate the discrepancies for
two different values of the incident angle in
a very revealing fashion. The discrepancy is
not too bad at low energies, but, for the
energy values of 2.0 MeV and higher, the dis
crepancy is too substantial for comfort.
One can also see that there is no good
basis here for a choice between the two sets
of parameters  the weighted set is closer for
normal incidence, whereas the nonweighted set
is closer for slant incidence. Since the pa
rameters used in the CH formula are themselves
based on a best fit to Raso's differential
albedo data, it must be concluded that either
the ChiltonHuddleston formulation for differ
ential albedo is grossly inaccurate (especially
for higher gamma ray energies) or else the
methods used by Chilton and Huddleston for ob
taining a least squares estimate, whether non
E (MeV) .2 .5 1.0 2.0 4.0 6.0 10.0
Coso
1.0 .2929 .1596 .08321 .04359 .02984 .02403 .01993
0.75 .2911 .1864 .1031 .06314 .03891 .02998 .02224
0.50 .2761 .1761 .1188 .07863 .05289 .03732 .02806
0.25 .1996 .1480 .1250 .09722 .06956 .05470 .03733
0.10 .1119 .09562 .08143 .07628 .06398 .05491 .03674
TABLE 4. TOTAL DOSE ALBEDO BASED ON
FLUX, CALCULATED WITH CH FORMULA
E (MeV) .2 .5 1.0 2.0 4.0 6.0 10.0
Cose
Nonweighted Parameters
1.0 .2992 .1831 .1133 .0749 .0551 .0474 .0390
0.75 .2940 .1888 .1216 .0812 .0572 .0477 .0376
0.50 .2777 .1931 .1345 .0935 .0632 .0505 .0370
0.25 .2246 .1731 .1367 .1076 .0764 .0603 .0403
0.10 .1421 .1172 .1012 .0893 .0717 .0608 .0424
Weighted Parameters
1.0 .3007 .1832 .1069 .0665 .0477 .0411 .0341
0.75 .2951 .1885 .1174 .0755 .0519 .0429 .0338
0.50 .2783 .1925 .1337 .0925 .0617 .0486 .0350
0.25 .2248 .1722 .1398 .1135 .0820 .0645 .0428
0.10 .1421 .1169 .1053 .0975 .0812 .0694 .0491
weighted or weighted, are inadequate  at least
for estimating total albedo and any quantities
closely related to total albedo.
At this stage in the research, a review
was made of the existing ChiltonHuddleston
work. No error could be found in princi
ple nor in the computer program used at NCEL.
However, a spot check was made of the previous
results of parameter calculations by an indepen
dently formulated program and a calculation on
the IBM7094 of the University of Illinois.*
The results obtained by a separately written
computer program and sample calculation for
1MeV provided essentially the same result for
the nonweighted set of values of the empirical
constants as had been published by Chilton and
Huddleston.
In the hope that a different weighting
system might improve the agreement, a new set
of empirical coefficients were generated at
NCEL, with "double weighting," that is, with
the required residuals in the least squares
process being weighted by the square rather
than the first power of the computed differen
tial albedo values.** This tended to give bet
ter agreement in those cases for which the sin
gle weighted coefficients were better than the
nonweighted coefficients, but gave still poor
er agreement under circumstances in which the
nonweighted coefficients were better. It is
believed, therefore, that improvement in the
system of weighting is not capable of helping
the situation very much, if any.
Further considerations given to the gener
al subject has brought appreciation of the
*This work was undertaken by Mr. Riaz Khadem, a
Civil Engineering graduate student who was
employed for his familiarity with digital
computer programming and numerical mathema
tical techniques but who was not biased by
any previous familiarity with the subject of
gamma ray albedo.
**This was carried out at the suggestion of the
author by Dr. C. M. Huddleston and Mr.
W. L. Wilcoxson of the Naval Civil Engineer
following aspects. The original data of Raso
are compartmented in such a way that there are
many more values given for reflected photon
directions near the normal than for photon
directions substantially differing from the
normal. Furthermore, the photon directions
near the normal encompass smaller solid angles
than those emerging at substantial offnormal
angles. Fifty per cent of the data given, for
example, covers a region of the exit hemi
sphere which amounts to only 20% of the hemi
sphere. If, therefore, the CH formula pro
vides an imperfect fit to the differential
albedo function, the process of fitting to the
data of Raso is such as to force a better fit
for the narrow cone of directions around the
normal than for the remaining directions.
Since the contribution to the total albedo
generally includes a substantial amount coming
out at slant angles (a study of Raso's data
will confirm this), it should not be surprising
to find the fit for total albedo to be imper
fect. This is particularly true for total
albedo based on flux, since in such case, as
can be seen from Equation 9, the radiation
emergent is divided by the reciprocal of the
cosine of the polar angle of emergence  thus
giving strongest weight to the more extreme
slant directions.
ing Laboratory, who have cooperated willing
ly and generously in the effort to improve
the utilizability of the ChiltonHuddleston
formulation.
IV. PARAMETERS ADJUSTED TO TOTAL ALBEDO BASED ON FLUX
Because of the failure of any of the pre
viously available sets of CH parameters,
based on fitting differential albedo data, to
provide sufficiently close fit to values of
total albedo computed from Raso's tables, it
was decided that the proper course of action
was to derive a set of parameters which would
be obtained directly by a least squares fit
from the formula given by Equation 10 to the
data given in Table 3. This has been done in
manner described below.
The method of least squares curve fitting
is a rather standardized procedure and the
theory need not be reviewed in detail since it
(22)
is available in standard references. The
calculational theory and the digital computa
tion flow chart and program are given in Appen
dix A.
In carrying out this fitting process. the
question of proper weighting for the squared
residuals arises, just as it did for the pre
vious differential albedo fitting of Chilton
and Huddleston. The statistically proper way
to weight these squared residuals is by the
reciprocal of the variances (squares of the
standard deviations) of the individual values.
For reasons discussed in their original pa
(12)
per, Chilton and Huddleston felt unable to
obtain and use such weighting factors for
their calculations leading to the original set
of formula parameters. In the present case,
however, a somewhat better means of making
estimates of the proper weighting factors
appeared to be at hand.
(23)
The paper of Berger and Raso, which
presents results of a Monte Carlo calculation
of total albedo based on current, indicates
that a reasonable fractional error (standard
deviation divided by the given value) for to
tal dose albedo is estimated to be about twice
the fractional error for the equivalent number
albedo value. In the case of Raso's differen
tial albedo calculations, which were based on
5000 photon histories, the fractional number
albedo value would be:(23,24)
G .n I n A 1 A n .(13)
A / A (4999) ~ 70 A *
n n n
The fractional value for total dose rate re
flected (and therefore for total dose albedo)
should be approximately:
aCd 1 1  An
Ad 35 A
d n
Raso(25) has pointed out that the error
for total dose and total dose albedo based on
flux should be expected to be slightly higher
than that based on current, because of the
presence of the cosine term in the denominator
of Equation 9 and the slight spread of direc
tions within the angular "compartment" for
each single tabulation. However, he indicates
that in his opinion the value of 2, given
above for multiplying the error in number
albedo, is fairly conserxative; therefore the
expression given in Equation 14 is still con
TABLE 5. TOTAL NUMBER ALBEDO BASED ON CURRENT
E (MeV) .2 .5 1.0 2.0 4.0 6.0 10.0
Coseo
1.0 .285 .275 .207 .164 .152 .152 .181
0.75 .338 .338 .271 .228 .205 .210 .238
0.50 .419 .419 .365 .316 .293 .299 .340
0.25 .527 .541 .502 .462 .464 .471 .537
0.10 .615 .638 .620 .606 .627 .665 .755
TABLE 6. WEIGHTING FACTORS IN
LEAST SQUARES FITTING TO RASO'S DATA
E (MeV) .2 .5 1.0 2.0 4.0 6.0 10.0
Cose
0
1.0 .5693 .1824 .4618 .1265 .2466 .3803 .6816
x 104 x 105 x 105 x 106 x 106 x 106 x 106
0.75 .7388 .1801 .4282 .9075 .2087 .3622 .7735
x 104 x 105 x 105 x 105 x 106 x 106 x 106
0.50 .1159 .2849 .4993 .9154 .1816 .3751 .7324
x 105 x 105 x 105 x 105 x 106 x 106 x 106
0.25 .3428 .6594 .7903 .1113 .2192 .3645 .1020
x 105 x 105 x 105 x 106 x 106 x 106 x 10
0.10 .1562 .2361 .3014 .3238 .5030 .8064 .2798
x 106 x 106 x 106 x 106 x 106 x 106 x 107
sidered to be a fairly reasonable means of
estimating the proportionate error for the
total albedo based on flux used in this
study. Thus, .it can be concluded that:
A'  A
, d An
d 35 A n (15)
n
The number albedo for Raso's calculation
are directly available from his report.(20)
They are given herein as Table 5. The appli
cation of the formula in Equation 15 to the
data in Tables 3 and 5 gives information for
standard deviations of the Table 3 values,
which are then readily converted into recip
rocals of the variances. These are given in
Table 6. It might be noted in passing that
these weights represent values of standard
deviation which range from about 6% of the
albedo values at normal incidence (cos o=0.10),
to about 2% for coseo=0.10.
The results of the least squares fitting
are given in Table 7. It is seen that the
trends of the values as a function of energy
are similar to those previously published by
Chilton and Huddleston;(12,15) however, there
are definitely differences well beyond a rea
sonable range of error, based on values of
standard deviation given in Table 7. The
estimates of standard deviation of the values
of the parameters are greater for these calcu
lations than those quoted by Chilton and
Huddleston; however, since the weighting fac
tors for the ChiltonHuddleston results were
not directly related to individual errors of
the input data, the "standard deviations"
quoted in that paper are relative in character
and cannot have an absolute significance for
estimating error.
The values of the parameters in Table 7
were used to compute values of total dose
albedo based on flux for the standard values
of energy and cosine of incident angle. This
was accomplished in the same computer program
by which the "best fitted" parameter values
were obtained. The results are given in
Table 8.
Comparison between Tables 3 and 8 shows
that, in general, the computed values at normal
and highly grazing angles of incidence tend to
be somewhat high, whereas the values for inter
mediate angles of incidence tend to be a little
low. This trend is quite definite and cannot
be ascribed to statistical fluctuations. It
must therefore be recognized as an inability of
the basic CH formulation to give any appreci
ably better fit. This is also shown by an
examination of the values of q, the sum of the
weighted squared residuals, in Table 7. It can
be shown 22) that if the discrepancies between
input values and the least squared computed
values were due to random variations in the in
put data, the function q would be distributed
according to a Chisquared distribution with,
in this case, 3 degrees of freedom. For this
distribution, if variations are ascribable only
to statistical errors, there is only a 5%
chance that the value of q would be greater
than 7.81 in any single case, and only a 1%
chance that the value of q would be greater
than 11.34. The values of q, which range
from 8.2 to 55.0 in our case and generally run
around 15, are such as to indicate that random
variations have little likelihood of explaining
the total discrepancies between Table 3 and
Table 8. A rough estimate of the error inher
ent in the mismatch of the CH formulation and
input Monte Carlo data can be obtained by the
following relation, valid for this situation:
5 2
q = 3 + 12
i=l i
where R represents the difference between the
"best fit" value of the albedo and the "true"
value of albedo. (In this expression, the
number 3 represents the number of degrees of
freedom: five given points corresponding to
TABLE 7. VALUES OF CH PARAMETERS
BASED ON TOTAL DOSE ALBEDO (FLUX)
TABLE 8. TOTAL DOSE ALBEDO BASED ON PARAMETERS FROM TABLE 7
E (MeV) .2 .5 1.0 2.0 4.0 6.0 10.0
CosO
1.0 .3151 .1742 .0928 .0502 .0342 .0273 .0238
0.75 .2922 .1739 .1002 .0582 .0382 .0294 .0238
0.50 .2566 .1706 .1118 .0730 .0472 .0352 .0250
0.25 .1916 .1466 .1147 .0916 .0655 .0502 .0318
0.10 .1156 .0971 .0854 .0794 .0662 .0566 .0381
E
o C C' q
(MeV)
0.2 .0038 ± .0048 .0666 ± .0111 11.6
0.5 .0231 ± .0029 .0262 ± .0036 8.2
1.0 .0428 ± .0045 .0108 ± .0029 20.2
2.0 .0725 ± .0051 .0052 ± .0015 17.3
4.0 .1042 ± .0065 .0050 ± .0009 16.9
6.0 .1269 ± .0068 .0046 ± .0006 14.8
10.0 .1295 ± .0122 .0048 + .0008 55.0
five different angles of incidence for each
photon energy, minus two parameters computed).
Thus, for an average value of q around 15,
the average value of R/l turns out to be
about 14. A reasonable estimate of the varia
tion which might be expected between the "true"
value of the total albedo and that computed by
the CH formulation using the parameters given
in Table 7 would be roughly 14 times the stan
dard deviation of the input data. Since the
latter varies from about 2% to 6%, the error in
the CH formulation would be on the order of
some 3% to 10%. (The high value of q at 10
MeV cannot be included in this generalization.
For this case, the error of the semiempirical
curve would be on the order of some 7% to 20%.)
There are included in Figures 3 and 4
lines corresponding to the total dose albedo
(flux) results given in Table 8. The improve
ment over the results using the originally
published albedo formula parameters is readily
seen (see Tables 3 and 4).
Figure 5 is included to show the fit at
a certain value of energy on a plot of total
dose albedo (flux) versus the cosine of the
angle of incidence. The energy selected for
this comparison is 1.25 MeV. This permits com
parison with results given by Bulatov and
Leipunskii(26) and by Kirn, Kennedy, and
Wycoff(27) for cobalt60 experiments. (Results
in the literature are often given for energy
rather than dose albedo. The difference is
only a few per cent or less, and small correc
tion factors can be readily obtained from the
(28)
work of Berger and Doggett, who calculated
both types.) Monte Carlo results of Raso (from
(28)
Table 3), Berger and Doggett, and
(21)
Leimd~rfer, are included, obtained in part
by means of interpolation between given values
at other energies. In computing the values
from the CH formula, the values of the param
eters given in Table 9 were used, which were
obtained by interpolation between available
values obtained from Raso's data. (The details
of the interpolation method are discussed below
in connection with Table 10.)
Figure 6 is a similar display of total
dose albedo, this time based on current, for
1.25 MeV. The experimental value provided is
by Bulatov and Garusov for cobalt60 gamma
rays incident on aluminum, which should give
values very close to that from concrete. The
Monte Carlo work of Raso, of Berger and
Raso(23) and of Davisson and Beach(29) are
also included.
It is clear that at an energy of 1.25
MeV, the curves for total dose albedo based on
flux calculated using the original parameters
provided by Chilton and Huddleston are appre
ciably above the concensus of both experimental
and theoretical (Monte Carlo) values. The
curve based on the "total albedo" parameters
gives overall a much closer average fit, which
is consistent with previously expressed expec
tations. On the other hand, for total dose
albedo based on current, at 1.25 MeV, the
curves using original parameters seem to give
a better fit in the middle range of albedos,
whereas the curve calculated from the param
eters presented in this report appear to give
a better fit at the extremes, that is, for
either normal incidence or highly glancing
incidence. As an exception, one should note
that experimental values of the Russian experi
enters are consistently somewhat higher and
tend to fit the curves from the original
Chilton and Huddleston parameters a little
better. There are substantial experimental
difficulties connected with such determinations,
however; and existing experimental results are
not likely to be very accurate.
As a result of the above considerations,
the author's belief in the virtue of his choice
of formula parameters based on total dose
albedo (flux) has been reinforced. For the re
mainder of this report, all calculations in
TABLE 9. INTERPOLATED VALUES OF
PARAMETERS FOR 1.25 MeV GAMMA RAYS
volving the use of the CH formula uses this
set of parameters (Table 7).
Method of Obtaining Parameters C C'
"Nonweighted" fit to differential data .0576 .0123
"Weighted" fit to differential data .0645 .0090
Statistically weighted fit to total albedo (flux) .0503 .0081
V. CALCULATIONS OF REFLECTED DOSE FIELD
ABOVE INTERFACE
A program was written for the University
of Illinois IBM7094 digital computer which
permits calculations of dose reflection from
point source to point detector by the inter
face, with the use of Equation 4. The results
are best expressed as a ratio of the reflected
dose to the direct sourcetodetector dose.
In accordance with the "similarity princi
ple" of Fano,(19) it is adequate to make the
calculations on the basis of a detector height
of 1 unit distance; and the results are appli
cable, within the limitations of his principle
discussed in Section II, to all situations on
a direct geometric scaling basis. Furthermore,
because of the "reciprocity" relationship also
noted in Section II, it is necessary only to
calculate the field for a situation in which
the detector is at the same height as, or
higher than, the source. The situation in
which the detector is lower than the source
is examined by a simple exchange of the posi
tions of source and detector.
The program for the calculation is ex
plained in more detail in Appendix B. One
particular matter is worthy of special note.
In order to limit the amount of machine time
required for computation of the values for
each point in the field, two things must be
done in the calculation. The first is obvi
ously to limit the computation to a finite
portion of the interface, insuring in so doing
that the remaining portion of the infinite
region contributes to the results by less than
some predetermined value. For the purposes of
FOR MONOENERGETIC ISOTROPIC POINT SOURCES
this study, the infinite field contribution be
yond the calculated limits was generally held,
by a simple though somewhat approximate formula,
to a value less than 0.5% of the results. This
formula is derived in Appendix B. The "far
field" estimate can be conservatively estimated
and added to the finite field results; but for
most of the present results this was not done,
even though the computer program made the calcu
lations of the correction and printed it out.
The other step taken in the calculations
was to keep the size of the individual incre
ments of area down to that required for accura
cy (calculational accuracy on the order of 1%
was desired), but no smaller. Some trial and
error was necessary to bring this about. For
certain of the calculations, as indicated in
more detail in the appendices, the area incre
ments were all of the same size. However, for
the bulk of the calculations, the sizes of the
increments were varied in a fashion such that
those regions which were more likely to contrib
ute heavily to the total answer were divided in
to small increments, whereas those regions
which obviously should contribute less impor
tantly to the accumulated result were divided
into larger increments. The method of accom
plishing this is also described in Appendix B.
It was decided to limit the initial study
reported herein to gamma ray energies applicable
to monoenergetic radionuclide sources frequent
ly used in practice; that is, 1.25MeV and
0.662 MeV. The first is representative of co
balt60 and is a fairly close average between
the energies of the two photons emitted by co
balt60 nuclei, 1.172 MeV and 1.333 MeV. The
second represents cesium137. By choosing
these radionuclides for the basis of this
analysis, it was hoped that checks with experi
mental results would be available, or would be
most readily obtainable in the future. It was
necessary to obtain values of CH parameters
by interpolation between those previously cal
culated and appearing in Table 7. This inter
polation was a socalled "fifth order" inter
polation, obtained by fitting a fifth order
equation to the six closest known values. This
was accomplished by a library subroutine pro
vided by the Digital Computer Laboratory of
the University of Illinois. The method used
for this calculation is a standard mathematical
one and needs no particular explanation here.
(Reference to any good text on numerical methods
in mathematics is suggested for the interested
reader.) The results of the interpolation are
given in Table 10.
The calculations were divided into three
cases. The most general case is that in which
there is no restriction, conceptually, upon the
relative positions of the source and the detec
tor, with the exception that the program devis
ed does not lend itself to the situation in
which the source and the detector are in the
same vertical line ("vertical case"). The
second case is for that situation in which the
source and detector are at the same height
above the interface ("horizontal case"). The
third case is the "vertical case" previously
mentioned.
The horizontal and vertical cases, be
cause of their regularity, are capable of some
degree of simplicity in the method of calcula
tion, and the details of their computation on
the IBM7094 are discussed in more detail in
Appendices C and D. These cases could be han
dled more rapidly by the digital computer, with
greater accuracy, and with more leeway in trying
TABLE 10. VALUES OF CH
PARAMETERS FOR 0.662 AND 1.25 MeV
Gamma Ray
Energy C C'
0.662 .0307 .0186
1.25 .0503 .0081
out computational variations, such as in the
size of the incremental areas. Likewise, being
calculations by entirely separate computer
programs, their results could be used to check
the results of the general case. In case of a
vertical orientation, a precise check was not
possible; however, the general case could be
checked for trends as the horizontal distance
between source and detector approached zero, to
see if the vertical case results fit in with
such trends.
The results obtained for the general case
are given in Table 11; the results for the
horizontal case calculations are given in Table
12; the results for the vertical case calcula
tions are given in Table 13. The results for
all three cases fit together quite well. Fig
ures 7 and 8 give loglog plots of the dose
radiation fields for cesium137 and cobalt60,
respectively.
TABLE 11. POINT SOURCE, BACKSCATTERING FACTORS, GENERAL CASE
Hor. Dist., Source to Detector
Ht. of Detector Ht. of Detector
Ht. of Source .25 .5 1.0 2.0 4.0 8.0 16.0 32.0 64.0
Cesium137 (0.662 MeV)
1 .002 .007 .024 .059 .101 .125 .111 .079 .050
2 .016 .024 .046 .083 .118 .124 .097 .066 .041
4 .047 .055 .074 .104 .128 .118 .087 .057 .035
8 .078 .084 .098 .121 .132 .113 .080 .052 .032
16 .101 .104 .115 .131 .133 .109 .076 .049 .030
32 .115 .118 .125 .137 .133 .106 .074 .047 .029
64 .124 .126 .132 .140 .132 .104 .072 .046 .028
128 .129 .131 .136 .142 .132 .103 .071 .046 .028
Cobalt60 (1.25 MeV)
1 .001 .003 .012 .032 .062 .094 .098 .075 .049
2 .008 .012 .024 .047 .079 .101 .089 .063 .040
4 .023 .028 .040 .062 .092 .101 .081 .055 .034
8 .039 .043 .054 .075 .099 .098 .075 .050 .031
16 .052 .055 .064 .084 .103 .095 .071 .047 .029
32 .061 .063 .072 .090 .104 .092 .068 .045 .028
64 .066 .068 .076 .093 .104 .091 .066 .044 .027
128 .069 .072 .079 .095 .104 .090 .065 .043 .027
NOTE: The source and detector heights may be interchanged, without
affecting the tabulated data.
TABLE 12. POINT SOURCE, BACKSCATTERING FACTORS, HORIZONTAL CASE
Hor. Dist., Source to Detector
Ht. of Source and Detector
Source .125 .25 .50 1.0 2.0 4.0 8.0 16.0 32.0 64.0
Cesium137
(0.662 MeV) .0005 .0018 .0071 .024 .060 .102 .126 .112 .080 .051
Cobalt60
(1.25 MeV) .0002 .0009 .0035 .012 .032 .062 .095 .099 .075 .050
TABLE 13. POINT SOURCE, BACKSCATTERING FACTORS, VERTICAL CASE
Source
Ht. of Detector Cesium137 Cobalt60
Ht. of Source (0.662 MeV) (1.25 MeV)
1 .000 .000
2 .0135 .0066
4 .044 .022
8 .076 .039
16 .101 .052
32 .117 .061
64 .126 .066
128 .131 .070
256 .134 .072
512 .135 .073
1024 .136 .073
NOTE: The source and detector heights may be interchanged, without affecting
the tabulated data.
VI. CALCULATIONS FOR FINITE FIELD AND COMPARISON WITH INFINITE FIELD RESULTS
In many cases of practical interest, the
scattering plane is not sufficiently large to
be considered infinite, and in fact is small
enough to provide an amount of reflection sub
stantially less than an infinite plane would
provide. The summation of area contributions
by perimetric strips provides answers to the
contribution within each perimetric strip and
comparison of such contributions to the infi
nite plane result; the programs discussed in
the appendices are readily adjusted to provide
these data.
The horizontal and vertical cases are par
ticularly likely to be useful in this respect;
and since the perimetric strips for such cases
maintain a small constant width, good accuracy
in calculation is to be expected. The techni
ques of programming for these calculations are
quite straightforward. In fact, the program
described in Appendix C is already tailored to
provide the necessary finite field results for
the horizontal case. For the vertical case,
the program described in Appendix D is not
tailored to this purpose, but the slight
modification to the program described there is
so simple that a separate explanation for it is
not considered necessary.
Calculations carried out in the horizontal
case for cesium137 and cobalt60 give results
as indicated in Table 14. The data are given
as a percentage of the infinite field results,
and are listed as functions of the perimetric
distance. The perimetric distance for this
case is shown as "x" in Figure 9.
Similar calculations carried out in the
vertical case for cesium137 and cobalt60
give results as shown in Table 15. These also
are given as percentages of the infinite plane
cases.
These results may be graphed in a fashion
suitable for use in situations conforming
approximately to the basic idealizations. This
has been done, and the resulting graphs appear
herein as Figures 9 and 10. Since the results
are almost independent of energy when put on a
percentage basis, the graphs are an average of
the results for cesium137 and cobalt60 and
are considered valid within a few per cent for
any radioisotope gamma emitter.
Just as for previous calculations, these
results are unchanged if the positions of
source and detector are changed. This point is
trivial for the horizontal case; but for the
vertical case, it means that HI and H2 may re
place one another in the axis labels when the
source is above the detector.
TABLE 14. POINT SOURCE, BACKSCATTERING FACTORS,
FINITE HORIZONTAL CASE
Perimetric distance, x
Ht. of source
Srce.Det. Dist.
Ht. of Srce.
0.25 0.5 1 2 4 8 16 32 64
Cesium137 (0.662 MeV)
0.5 18.3 41.7 73.9 93.7 99.0
1 21.0 43.8 74.2 93.3 99.2
2 20.6 41.2 70.1 91.2 98.3
4 18.7 36.7 63.2 86.3 96.8 99.6
8 15.9 31.2 55.4 80.0 94.3 99.0 99.8
16 11.8 23.3 43.2 68.0 87.2 96.3 99.0 99.9
32 8.6 17.2 32.5 54.0 75.9 90.8 97.4 99.5 99.9
64 6.6 13.2 25.1 42.4 62.4 80.7 92.7 97.8 99.5
Cobalt60 (1.25 MeV)
0.5 18.4 42.0 74.0 93.5 99.0
1 21.3 44.1 74.4 93.7 98.7
2 21.8 43.1 71.6 91.8 98.6
4 20.9 40.3 67.1 88.7 97.4 99.5
8 19.1 36.5 62.1 84.6 95.8 99.3 99.9
16 14.4 28.0 50.3 75.6 91.6 97.8 99.6 99.9
32 10.4 20.4 37.8 61.1 82.6 94.4 98.6 99.7 99.9
64 15.5 28.8 47.9 69.0 86.5 95.8 99.0
TABLE 15. POINT SOURCE, BACKSCATTERING FACTORS,
FINITE VERTICAL CASE
Radius of Perimeter
Ht. of Source
Ht. of Det.
Ht. of Srce.
1 2 3 4 5 6 8 LO
Cesium137 (0.662 MeV)
2 48.3 81.2 92.3 96.2 97.9 98.7 99.5 99.7
4 35.6 86.6 83.9 91.0 94.6 96.5 98.4 99.1
8 28.9 58.5 74.6 83.6 88.9 92.2 95.8 97.5
16 25.5 52.3 67.8 77.0 83.0 87.0 91.9 94.7
32 23.6 48.7 63.5 72.5 78.5 82.7 88.1 91.3
64 22.6 46.7 61.0 69.9 75.8 79.9 85.4 88.0
Cobalt60 (1.25 MeV)
2 47.5 81.1 92.2 96.2 97.9 98.7 99.4 99.7
4 35.3 68.5 83.9 91.1 94.7 96.6 98.4 99.1
8 28.3 58.1 74.5 83.6 89.0 92.3 95.9 97.6
16 24.5 51.3 67.1 76.7 82.3 86.9 91.9 94.7
32 22.4 47.3 62.4 71.7 78.0 82.3 87.9 91.3
64 21.2 44.9 59.4 68.6 74.7 79.1 84.9 88.5
VII. LOCATION OF MOST IMPORTANT REFLECTION AREAS IN PLANE
The location of those regions of the plane
which contribute most heavily to the reflection
from a nearby point source is a matter of some
theoretical interest. Furthermore, this subject
deserves study from a practical point of view,
in that approximate analyses of interface
effects may be carried out by assuming only
certain regions of the interface to be of sig
nificance (e.g., for neutrons, see article by
Simon and Clifford and report by French.)
A study of this matter for the general
case of point source and detector near an inter
face probably would require a detailed calcula
tion for each individual case. For the special
case in which the source and detector are at
the same distance from the interface, called
the "horizontal case" in this report, it
appears possible to obtain an understanding of
the general situation without the necessity of
detailed individual calculations. This is
achieved by study of the basic formula for
the reflected contributions from a very small
area. If Equation 3 is combined with Equation
2 and dA is replaced by AA, the following
equation is obtained:
D [C.1026K (E ,s ) + C', AA
r1 2r2(e o s5
AD= 2 2
r r (secO + sece)
o o
(17)
It appears most reasonable first to inves
tigate the variation of dose rate with respect
to position of the incremental area along a
line in the plane which is an extended projec
tion of the line connecting the source and the
detector. Some things can be stated with re
spect to a plot of dose rate as a function of
position along this line, as being fairly obvi
ous without the necessity of detailed analysis.
For one thing, the fact that Equation 4 is
invariant under an exchange of position of
source and detector makes it clear that the
function AD is symmetric with respect to a
point on the line halfway between source and
detector, at least within the degree of approxi
mation inherent in the use of the CH albedo
formula. Also it is quite clear that, for
points on the line which are outside the inter
cept between source and detector projection
points and which are far away from source and
detector, the dose rate contributions are small
and approach zero as the distance from the
sourcedetector configuration increases.
For points within or near the intercept of
the line between source and detector projection
points, positive statements without careful
proof are a little dangerous. In fact, the
situation is governed rather strongly by the
nature of the differential albedo function; and
it is possible to conceive of hypothetical
functions which would lead to quite vacillatory
sorts of reflection dose rate functions. For
a smoothly varying function, such as that which
applies to gamma ray reflection, one would not
expect the resulting scattering contribution
curve to be highly vacillatory, but would ex
pect it to contain perhaps no more than one or
two maximum points. One can readily imagine
that there might be two maxima in the curve,
one near the source projection point and a
corresponding one near the detector projection
point. On the other hand one could quite
possibly expect a single maximum in the curve
at the point halfway between source and detec
tor projection points. Figure 11 indicates
these two possible cases.
It could also be imagined that the next
case in the hierarchy of complexity might be
possible, that of a maximum in three places 
one at the point of symmetry and one each near
source and detector projection points. For the
isotopes studied herein, this is not considered
likely; and, for a number of varied geometric
situations within the "horizontal" category
which were studied individually on the basis
of a pointbypoint calculation, no results
have been found which did not correspond to
either the onemaximum or the twomaxima case.
The important problem which one needs to
study rigorously is to determine the circum
stances under which each of the two cases is
valid. This is most easily approached by
studying Equation 17. A distinguishing feature
which characterizes this equation as being of
the onemaximum type or the twomaxima type is
whether the function AD reaches a maximum or a
minimum when AA is at the midpoint of the pro
jection of the sourcedetector line on the
interface. This can be readily seen in Figure
11. Thus, the sign of the value of the second
derivative at this midpoint establishes which
of these cases is valid for given values of
the pertinent geometric distances and albedo
parameters.
It is shown in detail in Appendix E that
the second derivative of AD with respect to
position of the incremental reflecting area
along the interface projection of the source
detector line, evaluated at the midpoint of
this line, is proportional to a function H(p),
where p = Yd2/4H2 . In this expression Yd
is the distance between source and detector
and H is the height above the interface.
Thus, assuming that values of the albedo
parameters C and C' are fixed, the exis
tence of the onemaximum or twomaxima case is
established by the value of the ratio of the
sourcedetector separation distance to their
height above the interface. The function H(p)
is as follows:
H(p) = (4p5)
2
3.97023 C (4
(p+1)
3.97023 C P 2 p 4  3 p 3 (p  + C'
I m m m m p+1
p)
.51079 2P3 + 4P5  3P4 + 3Pm4 (
.51079 m m m m P+1
+ 2P 3 (P l
m p+l1
where P is the ratio of singly scattered pho
m
ton energy to Eo, the incident photon energy,
for a single scattering occurring at the center
point of the interface projection of the source
detector line. When H(p) is negative, the one
maximum case is valid; when positive, the two
maxima case. The critical value of p which
separates the two cases occurs when H(p) is
zero.
If C is not equal to zero, one can use a
2
slightly simpler function, G(p), where G(p) is H(p) divided by 3.97023 GP . Then,
m
pm m pl )2
a m L \P+1/ J
+ C P
3.97023 C P 2
m
SEo0 P [2 + 4P 2  3P + 3P (1\2  2P (
1+p .51079 m m mp+ mp+
Just as for the function H(p), a value of p
such that G(p) is negative insures that the one
maximum case is valid; a value such that G(p)
is positive insures that the twomaxima case is
valid; and the value of p which makes G(p)
equal to zero is the critical value which sep
arates the two cases.
Figure 12 shows a plot of G(p) versus p.
It is seen that the critical values of p are
5.78 for cesium137 and 10.47 for cobalt60.
This corresponds to values of Y d/H of 4.8 for
cesium137 and 6.5 for cobalt60.
A question arises as to whether points
which are not on the line connecting the source
and detector projections points can become posi
tions of localized maxima. This can easily be
settled, by comparing the terms in Equation 17
for any assumed point off the line with those
for the point on the line nearest to the assumed
point. It is easily seen that for a point off
the line each term in the numerator of Equation
17 is smaller and each term in the denominator
is larger than for the nearest point on the
line.* Thus one need not consider points off
the line connecting source and detector projec
tions points as locations of possible maxima.
Although it is not directly related to the
purposes of the present report, an extension of
*This statement is true for situations in which
one can be sure that K (6 ) is always decreas
ing with increasing e. This is certainly true
for photon energies greater than 0.4 MeV, and
is probably true in practical cases for any
energy within the usual gamma ray region.
this important reflection area problem to neu
trons is so straightforward as to permit its
inclusion without difficulty. Song(32) has
shown that differential neutron dose albedo
can be approximated reasonably well by a formu
la of the following type:**
(20)
C"
°d c ose
1 + 
cose
This has the same form as the formula for gamma
rays (Equation 3), if C in that formula is
taken to be zero. Thus, the above consider
ations are still valid, except we must use
H(p) rather than G(0). It is easily seen that
in this case H(p) reduces to:
(21)
H(p) = 4p  C"
2
It is readily seen that this critical value is
1.25, which means that for values of Yd greater
than r/5 H, the halfway point between source
and detector projection positions is no longer
a position of maximum neutron dose rate reflec
tion. This provides additional support for the
usual practice in duct or plane interface prob
lems of assuming that the important reflection
regions for neutrons are near the source and
the detector. (30,31)
**Song has demonstrated the approximate validi
ty of this formula for incident neutron energies
from 0.1 MeV to 14 MeV. However, since the for
mula is based primarily on an assumption of iso
tropic scattering, it should also be valid for
energies down to thermal.
G(p) = 45
2
(19)
VIII. COMPARISONS WITH EXPERIMENT AND DISCUSSION
Experimental data adequate for comparison
with the results of this paper are difficult to
find in the literature. Comparisons are possi
ble only under certain circumstances. Prima
rily, the restrictions previously mentioned
with respect to the upper and lower limits of
the significant distances should be observed.
An equally important point involves the matter
of how one relates experiments done with air as
the overlying medium to theoretical results ob
tained under an assumption of a vacuum over the
dense medium. This aspect requires some prelim
inary discussion.
For experiments done in air, there are
several possible variations as to what is being
reported and what is taken as the reference
"direct" dose rate. Experimental data reported
may be:
(a) the socalled "scattered" dose rate,
including air scatter as well as the
interface reflection; or
(b) only the interface reflection.
Likewise, the "direct" dose rate, used as the
basis of the comparison with the "scattered"
or "reflected" dose rate, may indicate:
(a) estimated measurements in a vacuum;
(b) measurements in an infinite air medi
um;
(c) measurements in the air medium in
cluding the effects of the air above
the interface but not including the
interface reflection effects; or
(d) measurements in air excluding all
effects except those due to the un
collided radiation.
If the distances involved are only on the
order of one or a few feet, the air effect can
be considered small in comparison with the
interface reflection, and the air can be con
sidered as essentially a vacuum. However, for
distances even on the order of five or ten feet,
air interactions begin to be significant in com
parison with the interface reflection effect.
The best way to provide a comparison is to do
it in such a way as either to include the air
effects in the calculation or exclude them
from the experimental data. Since the compu
tations in this report are based upon the
assumption of a vacuum, the latter approach is
necessary. Clarke and Batter (10) have summa
rized this point in their article:
"... [One should] calculate the increase
in radiation intensity caused by the inter
face relative to the intensity expected in
an infinite homogeneous air medium. This
[has] the virtue of reducing to the scat
tered to direct radiation ratio for the
case where air is absent."
It might be added that the "intensity expected
in an infinite homogeneous air medium" is
practically the same as the intensity obtained
in the actual airconcrete case with the inter
face effect subtracted out, because most of the
air scattering is small angle scattering and
the region of space filled by the concrete would
have very little additional influence on the air
scattered component if it were filled with air
instead. The only exception to this is for
cases in which the line from source to detector
is almost parallel to the interface and is
appreciably longer than its distance above the
interface.
The foregoing has been a necessary preamble
to discussing the comparison of this report's
results with those few applicable experimental
reports which are available in the literature.
Further examination of this matter will be
deferred until the discussion of how far the
theoretical results can be applied in practice.
(9)
Jones et al. have published measurements
on dose rates from a cobalt60 source in air
under circumstances in which the source and
detector were the same height above the inter
face. This height varied from 9 to 57 feet and
the sourcedetector separation distance was from
7.5 to 70 feet. Their reported data was of
"scattered gamma dose rate," which included both
air scattering and ground reflection. They are
thus not suitable for precise comparison with
the results of the present report, although
Clarke and Batter, by making a plausible
assumption as to the approximate amount of the
air scattering contribution, found the Jones et
al. data to be approximately consistent with
their experimental results for comparable
sourcedetector heights.
Clarke and Batter's (10) experiments were
with cobalt60 and iridium192, source and de
tector always being at the same height above
the concrete surface. The sourcedetector sep
aration distance was varied from 1 foot to 6
feet; the height above the interface, from li
inches to 75 inches. Their results for iridium
192 were analyzed and presented in a fashion
consistent with their statement quoted above,
which would provide a good comparison with cal
culations for the vacuumconcrete case. Unfor
tunately, they did not extend this procedure to
the cobalt60 case. For the latter isotope,
their results were presented in a fashion which
included air scattering along with interface
reflection in the "scattered dose rate," and
they considered as the "direct dose rate" only
that due to the uncollided photon flux. This
tended to give a higher value of the ratio of
"scattered" dose rate to "direct" dose rate for
the airconcrete case than would be present in
the vacuumconcrete case.
The original data for Clarke and Batter's
cobalt60 results have been reanalyzed accord
ing to the same procedure they used for iridium
192; and the results are presented in Figure 13,
along with the curve based on calculations in
this report. It is seen that the experimental
results are somewhat scattered but provide in
general a fairly good fit to the theoretical
curve, the principle exception being of those
data for which the sourcedetector heights
above the interface were of the same order of
magnitude or less than a meanfreepath of the
source radiation in concrete. This discrepancy
is to be expected, since such a situation does
not conform to the assumptions inherent in this
report's theoretical development, as previously
indicated.* (The reason that Clarke and Batters
data for iridium192 are more selfconsistent is
that the average meanfreepath for photons
from the latter isotope is much smaller than for
cobalt60).
Henry and Garrett, by a series of ex
*It should be mentioned that an early report by
(33)
this author, as well as the paper by Clarke
(10)
and Batter, showed excellent comparison be
tween their experimental results and this au
thor's preliminary calculations using albedo
parameters based on differential rather than
total albedo information. For reasons as ex
plained in the present report, the early set of
parameters is not the most suitable to use for
present purposes, nor are the experimental data
as presented by Clarke and Batter in a fashion
suitable for the best comparison with our theo
ry. Thus the good fit noted in the aforemen
tioned works must be considered somewhat fortu
itous and must be considered as having only an
approximate significance.
Detector
, :  . "."^ ' , C o n c r e t e  S lo b I. l . .. 'p..* . ",; , ' ,
(a) Broad, parallel beam source
FIGURE 1.
Sourcez
S oo Detector
SdA Concrete Slab''." ' ' ..
(b) Point source
CONFIGURATIONS FOR REFLECTION CONTRIBUTIONS TO DOSE RATE
cos8 = r2 dO
FIGURE 2. GEOMETRIC RELATIONSHIPS FOR DIFFERENTIAL AREA CONTRIBUTION
Result Using "nonweighted" Parameters
 Result Using "singly weighted" Parameters
    Result Using "doubly weighted" Parameters
a Data from Raso ( See Table 3)
   Result Using Parameters Based on Total Albedo
(Flux) (See Table 8)
Cosine of Incident Angle = 1.0
Energy (MeV)
FIGURE 3. TOTAL ALBEDO BASED ON FLUX, NORMAL INCIDENCE
K
a
Z
0
0
0
0
I
I I I I 11111 I ii
Result Using "nonweighted" Parameters
Result Using "singly weighted" Parameters
   Result Using "doubly weighted" Parameters
Sao Data from Roso ( See Table 3)
S ^   Result Using Parameters Based on Total Albedo
tFlux)(See Table 8)
Cosine of Incident Angle 0.25
0 N.
I I I I I I I
I I I I I I I I
Energy (MeV)
FIGURE 4. TOTAL ALBEDO BASED ON FLUX, SLANT INCIDENCE
24
16
12
.08
.04
U
N


I I I I I I I I
I I I I I I I I
J ] I f T
/
/
S
Colculated Curves Using:
SNonWeighted" parameters
    "Weighted" parometers
   " Total Dose Albedo"
fitted parameters
I
Roso, Monte Corlo
INCIDENT BEAM ENERGY 1.25 MoV
1=] ^ ^ ^ S
CI Berger a Doggett, Monte Carlo
Leimdfrfer, Monte Corlo, (Interpol.)
Kirn, Kennedy B Wycoff, Exp.
Bulatov a Leipunski, Exp.
0.4
Cosine Angle of Incidence
FIGURE 5. TOTAL ALBEDO BASED ON FLUX, 1.25 MeV PHOTONS
MI
.12
.08
0
.04
.02
I
.02
/
I
//
II
/
0.4
 = i
I I
I  I
I
I
'F
*L
JI
Calculated Curves Using:
"Non Weighted" parameters
    "Weighted" parameters
   "Total Dose Albedo
fitted parameters
INCIDENT BEAM ENERGY 1.25 MeV
O Rosa, Monte Carlo
0 Berger & Roso, Monte Carlo
X Dovisson & Beach, Monte Carlo (interpol.)
+ Bulatov and Gorusov, Eap.
Cosine Angle of Incidence
FIGURE 6. TOTAL ALBEDO BASED ON CURRENT, L.25 MeV PHOTONS
£
0
o
o
0
o
a
I
Horizontal Projection Source  Detector Distance / Source Height
FIGURE 7. GAMMA BACKSCATTERING FIELD, CESIUM137 SOURCE
t
S
S
O
O
L.
Q
0
U_

9
a
S
I
0
0
a,
'a
Cesium137
Point
Isotropic Source
)0
4
£
0'
0
I
0
(/)
L_
0
4e
Oh
0
'5
L.
0i
o
U£
0
Horizontal Projection Source  Detector Distance / Source Height
FIGURE 8. GAMMA BACKSCATTERING FIELD, COBALT60 SOURCE
H1177
0.3 1.0 10 100
Source To Detector Distance (YD) / Height Above Slob (H)
FIGURE 9. FINITE RECTANGULAR AREA CONTRIBUTIONS, HORIZONTAL CASE
i
.0
0
.0
C
0
E
0
I
0
U)
U
O
0
**
4
0
QT
3
'*
Ig
2 4 6 8 10 12
Radius of Circular Slab (R)/ Height of Source ( H 1)
FIGURE 10. FINITE RECTANGULAR AREA CONTRIBUTIONS, VERTICAL CASE
11. POSSIBLE SHAPES FOR
NCREMENTAL AREA REFLECTION
FIGURE 12. PLOT OF G(p) VERSUS p
0
0
0a
X0 :2ft
o 0 :3 f~
+  0:4 ft
Aa :6ft
H/D
FIGURE 13. COMPARISONS OF RESULTS WITH CLARKE AND BATTER'S DATA
FIGURE 14. COMPARISONS OF RESULTS WITH HENRY AND GARRETT'S HORIZONTAL CASE
I 5
FIGURE 15. COMPARISONS OF RESULTS WITH HENRY AND GARRETT'S VERTICAL CASE
w
w
Idl
0
0
I
z
w
a
IJJ
I.
hi
U)
0
&LI
(0
0
I
U
0
,J
.I
ut.
w 0.2
UJ
w
U)
2
tli
w
0
w
C
'i
I
llJ
LL
.
IL
UiJ
(0
0
Qr
0.5 I 10
(H + 1.7cm.)
YD
FIGURE 16. COMPARISON WITH HENRY AND GARRETT, ADJUSTED, HORIZONTAL CASE
w
I)
0
0
a
U
U,
0
0
w
0
a
Q
UJ
w
1
w 0.2 I
(H2 + 1.7cm.)
(HI + 1.7cm.)
FIGURE 17. COMPARISON WITH HENRY AND GARRETT, ADJUSTED, VERTICAL CASE
5
( Read in
data
Initialize SWXX,
S'*, SWZZ at zero
DOloop to computt
functions h and g.
Initialize XHAY at
zero
DOloop to perform inte
gration (summation) with
respect to cosO
I Initialize QAY at zero
H
Complete integral
over cosO
Compute X(N) and Y(N)
Accumulate SWXX. **
SWZZ, to give elements
of h and g
I
Compute C, C', Q. and
std. dev. for C and C'
< DOloop to compute adjusted
values of total albedo
(l Write out answers
Go back to beginning for
data for new problem
e
FIGURE 18. LOGIC FLOW DIAGRAM FOR LEAST SQUARES ESTIMATE
jx
I
_01_ý
A
i I

=
SDOloop to perform inte
Sgration (summation) with
respect to
[ ~Complete integral
over
I
R
q B
>40
__b<
a I
E
D
r.
B
Detector
projection A
I A
A
A
Sour
pr
I
D
E
FIGURE 19. INCREMENTAL AREA PLAN, EQUAL SIZED AREAS
E
D
B
B
B
B
B
C
D
E
E
D
C
C
C
C
C
C
0D
E
D
D
D
D
D
D
D
D
E
E
E
E
E
E
E
E
E
E
I
£7
D
D
D
D
D
D
D
D
I
E
E
D
C
0
C
C
C
AB
AB
/
/
H H
/ /
/^
G1,G
FG H
F
F G
G 7G
H H
FIGURE 20. INCREMENTAL AREA PLAN, EXPANDING INCREMENTS
I 
£7
(a) Initial grouping of
areas.
(b) Symmetric plan with respect to source position.
(c) Transition to plan in (d).
I __________  I
(d) Symmetric plan with respect to point
halfway between source and detector.
(e) Transition from plan in (d) to plan in (b).
FIGURE 21. INCREMENTAL AREA PLAN, EXPANDING INCREMENTS, REVISED
F
m
I I
Detector
5K\
Source
hl
FIGURE 22. GEOMETRIC RELATIONSHIPS IN CALCULATION OF TRUNCATION ERROR
Sourco
hI
FIGURE 23. LOGIC FLOW DIAGRAM FOR GENERAL CASE COMPUTATION
S 
 I
FIGURE 24.
AREA PLAN,
DETECTOR AT
INCREMENTAL
SOURCE AND
SAME HEIGHT
Read in
data
SInitialize YD at 0.5
DOloop to carry out cal
^ ~ culation for each value 
of YD
Initialize all D(N) to
zero, X at onehalf of
DEL, DIST at slightly
more than YD
DOloop to accumulate
ont ibio"rns of the
Nperimetric strips (PER)
I
Accumulate Incremental area
contributions, DI(X,Y) for
each perimetric strip.
rCheck
S criteria for
No runcation of accu Yes
Nmulation pro
cess
I
I Write out results
SMultiply YD by 2
Read in data for
\. new problem
FIGURE 25. LOGIC FLOW DIAGRAM,
SOURCE AND DETECTOR AT SAME HEIGHT
FIGURE 26.
OF CASE, SOU
DETECTOR IN VERTICAL LINE
L
FIGURE 27. LOGIC FLOW DIAGRAM,
SOURCE AND DETECTOR IN VERTICAL LINE
(GD
2m
FIGURE 28. GEOMETRIC
RELATIONSHIPS FOR STUDYING
INCREMENTAL AREA CONTRIBUT
2m
04
FIGURE 28. GEOMETRIC
RELATIONSHIPS FOR STUDYING
INCREMENTAL AREA CONTRIBUT
ceedingly careful experiments with cobalt60,
have measured dose rates for situations in
which the sourcetodetector line was either
parallel or perpendicular to a concrete inter
face. In the parallel case the sourcedetector
distances were from 7 to 57 cm. and the heights
above the interface from 3 cm. to somewhat over
200 cm. In the perpendicular case the source
detector distance was either 7 or 20 cm., and
the heights above the interface ranged from
about a centimeter to about 25 cm. Their re
sults were normalized to the dose rate corres
ponding to complete absence of radiation scat
tered by the interface, and the fractional in
crease in dose rate due to the introduction of
the barrier was the basis for their plotted
data. Thus, their data were suitable for com
parison with the vacuumconcrete results of this
report in accordance with the discussion at the
beginning of this section. Comparisons of the
HenryGarrett data with a curve representing the
results of this paper's calculations are given
in Figures 14 and 15 for the horizontal and
vertical cases, respectively.
It is to be noted that the data in Figure
14 correspond to those for the right "tail" of
Figure 13, which are presented to an enlarged
scale in the upper right hand corner of the
latter figure. In this region the data of
Henry and Garrett appear to be less scattered
and probably more accurate.
Just as for Clarke and Batter's work, some
of the experimental situations studied by Henry
and Garrett involve distances of source and/or
detector a few centimeters above the interface.
These data thus cannot be expected to correspond
accurately to the results calculated in the
present report. For the experiments in which
the sourcedetector distance was 57 centimeters,
which in turn generally implies distances to the
interface greater than a meanfreepath in con
crete, the data ol Henry and Garrett for the
horizontal case are in excellent agreement with
the results of this paper.
The data given in Figure 15 for the per
pendicular case do not appear to agree as well
as might be desired; however, these data are
for experiments in which the sourcedetector
distances are only 7 to 20 centimeters, which
implies that either source or detector is quite
close to the interface. This is particularly
true for the "wings" of the plot. Thus these
data also do not, in general, conform to the
criteria necessary for the adequacy of the cal
culations of this paper.
Henry and Garrett have found that their
own data fall together much more consistently
if one assumes that the radiation, because it
actually penetrates and is reemitted from with
in the concrete slab, is reflected on the aver
age by a plane 1.7 centimeters below the actual
interface. Thus, by adding 1.7 centimeters to
the experimental detector and source heights,
they shift the location of the experimental
data horizontally on the graphs. Figures 16
and 17 are revisions of Figures 14 and 15 after
making this adjustment. It is seen that the
data tend to match this paper's calculated re
sults to a much better extent. In fact, except
for the left "wing" of the plot in Figure 17,
the theoretical and experimental results are
quite consistent. (It must be noted that in
plotting this paper's calculated results no
height correction is applied, since it is in
herent in the theoretical approach that heights
are measured above the reflecting plane, wherev
er that might be. Thus the theoretical curves
in Figures 16 and 17 are in the same place on
their respective graphs as the theoretical
curves in Figures 14 and 15.)
The excellent agreement on the right "wing"
of Figure 17 is especially interesting, since
this involves situations in which the source was
very close to the interface  at distances from
it on the order of a centimeter in some cases.
This would seem to show that the rule forbidding
the source height above the interface to be
less than a meanfreepath of the radiation in
the denser medium may be evaded by the height
correction mentioned above.
Unfortunately, it appears that one cannot
say the same about the case in which the detec
tor is very near the ground, as indicated by
the fact that there is still not very close
agreement on the left "wing" of Figure 17. This
lack of consistency between right and left wing
is only partially understood. It is known that
there are special sources of experimental error
under conditions in which the detector is near
the interface and the source is above the detec
tor: uncertainty in the precise location above
the interface of the point corresponding to the
35cm3 detector; the lack of complete isotropi
city of the detector. Errors from these diffi
culties are estimated (34)to be separately as
much as 4 or 5 per cent in the worst cases,
both in the negative direction. These errors
are significant when the detector is close to
the interface and when it is between the inter
face and the source. Therefore an appreciable
portion of the discrepancy on the left wing may
be ascribed to them.
There is also a possible theoretical basis
for some discrepancy. The results of the pres
ent paper provide a symmetrical curve in Fig
ures 15 and 17 because, within the degree of
approximation inherent im the use of the CH
formula for albedo, Equation 4 brings about
such symmetry. This is a result of the invari
ance of the equation with respect to an exchange
of positions of source and detector previously
mentioned. Such an invariance is clearly true
for the horizontal case because of the geomet
ric similarities of the configuration when
source and detector exchange position. However,
for the vertical case no such geometric similar
ity exists; and there is reason to believe that
such invariance may not be exactly true. For a
single scattered photon, for example, the path
length of the incident photon through the dense
medium before collision is not in general the
same as the path length of the scattered photon.
Since the attenuation coefficients of the inci
dent and reflected photons are different, the
probability of a photon proceeding from the
source along a certain path, undergoing a
scattering collision, and then proceeding to
the detector is not generally the same as the
probability of a photon surviving along the
same path in the reverse direction. The invar
iance predicted by the present theory is partly
(15)
due to the fact that Chilton and Huddleston,
in the development of their albedo formula,
assumed that the attenuation coefficients of
the incident and scattered photons are the
same. The precise nature of the discrepancy
between theory and experiment for the left wing
of the plotted results in the vertical case
must be considered an open question at present.
Good experimental work with which to com
pare results of the calculations for finite
planes has not been found.
It is very desirable to be able to use
this paper's results in order to solve practi
cal problems involving an airconcrete or air
earth interface. A question arises as to how
great the sourcedetector, source interface,
and/or the detectorinterface distances in such
situations can be without substantial loss of
accuracy in using the present approach, which
assumes a vacuum above the interface. Detailed
comparisons have been possible only with experi
ments in which these critical distances are on
the order of a halfdozen feet or less. For
distances beyond this, one can at present only
use theoretical reasoning on a somewhat quali
tative basis to answer the question.
For gamma rays penetrating an infinite
medium of low atomic number, such as air, the
removal of radiation from the sourcetodetec
tor path is roughly compensated by inscatter
ing, which directs radiation into the detector.
A more mathematical way of putting this fact is
that for distances up to an appreciable fraction
of a meanfreepath in light media, use of the
dose rate buildup factor approximately compen
sates for the exponential attenuation of the
uncollided photons. Since photon scattering
is largely small angle scattering, these same
considerations should apply to a great extent
even in the presence of a nearby interface, ex
cept under the condition previously noted in
which source and detector are at heights above
the interface much less than the distance be
tween them and the line joining them is almost
parallel to the interface.
Also, in an experimental situation, the
perturbing effect of the air is similar in sign,
though not in exact amount, for both the direct
and the reflected radiation; the ratio of the
two, which is what is calculated in this report,
is less sensitive to the perturbations caused
by the air than is each component separately.
Such considerations lead the author to be
lieve that the results of the present report
may be used for practical situations out to
several dozen feet in air, which include prob
lems involving enclosed rooms, air ducts pene
trating a shield, and shelter entranceways. At
close distances the errors involved, except in
special circumstances previously noted, appear
to be quite small. This conclusion, it is em
phasized, is subject to review in the future
after critical comparison with further very
precise experimental investigations.
IX. SUMMARY
This report explains the development of
a machine computational method for calculating
the backscattering by a concrete (or earth)
surface of gamma radiation from a point source.
The source and detector are assumed to be at a
distance from the interface somewhat greater
than a meanfreepath in the concrete, but the
sourcetodetector distance and their heights
above the interface are assumed to be appreci
ably less than a meanfreepath in air. The
primary results obtained give the reflected
dose rate field, as a fraction of the dose rate
provided directly by a cesium137 or cobalt60
source, for a surface of infinite extent. Ex
cellent agreement is found with experiments for
closein distances, especially if the experi
mental data are referred to an average reflect
ing plane slightly below the actual interface.
With this correction, good agreement is obtain
ed for some circumstances in which the source
is closer to the interface than a meanfree
path in the concrete. Accurate experimental
data for a complete check over the whole field
are lacking, however.
In simple cases, such as when the source
and detector are on a line either parallel or
perpendicular to the reflecting surface, special
methods are provided for getting finite plane
backscattering also. Results for cesium137 and
cobalt60 are provided for these cases, present
ed in the form of percentages of the results for
corresponding infinite field cases.
The resulting methods developed in this re
port are applicable to any radioisotopic source
of gamma rays, provided the gamma photon emis
sion energies and relative numbers are known.
It is also necessary to know values of the CH
formula parameters, which restricts the tech
niques to gamma ray photons of energy between
0.2 MeV and 10 MeV.
These results are of significance in radi
ation protection technology in the following
respects:
(a) They provide a basis for checking on
the adequacy of the ChiltonHuddleston
formula (12) for gamma ray differential
albedo by concrete or other materials
of approximately the same average
atomic number. This formula can now
be used with some confidence as a
simple, basic tool in studying the
effect of airground or airconcrete
interface effects on radiation levels
in the vicinity of a source, in deter
mining the streaming of gamma radia
tion through ducts, and in the analy
sis of internal gamma ray reflections
within a compartmented structure.
(b) They provide information useful in
the calibration of gamma radiation
measuring instruments, since such
instruments are usually calibrated in
the vicinity of one or more surfaces.
These surfaces may be finite or effec
tively infinite, depending upon
whether the calibration is performed
in a small room or in a large room
(or out of doors).
(c) They supply the means of making fair
ly precise estimates of the radiation
hazard from any point source radioiso
tope, under many practical conditions.
X. REFERENCES
1. H. Goldstein and J. E. Wilkins, Jr., "Cal
culations of the Penetration of Gamma
Rays," NYO3075. Nuclear Development
Associates, Incorporated, (1954).
2. M. A. Van Dilla and G. J. Hine, "Gamma Ray
Diffusion Experiments in Water,"
Nucleonics, Vol. 10, No. 7, (1952), p. 54.
3. M. J. Berger, "Effects of Boundaries and
Inhomogeneities on the Penetration of
Gamma Radiation," National Bureau of
Standards Report No. 4942, (1956).
4. M. J. Berger, "Calculation of Energy Dissi
pation by Gamma Radiation near the Inter
face between Two Media," Journal of Ap
plied Physics, Vol. 28, No. 12, (1957),
p. 1502.
5. F. Titus, "Measurement of the GammaRay
Dose near the Interface between Two Media,,
Nuclear Science and Engineering, Vol. 3,
(1958), pp. 609619.
6. B. P. Bulatov, "The Albedos of Various Sub
stances for Gamma Rays from Isotropic
Co60, Cs137, and Cr51 Sources,"
Atomnaya Energiya, Vol. 7, (1959), p. 369,
and Reactor Science, Vol. 13, (1960),
p. 82.
7. R. E. Rexroad and M. A. Schmoke, "Scattered
Radiation and Free Field Dose Rates from
Distributed Cobalt"0 and Cesiuma37
Sources," NDLTR2. Nuclear Defense Lab
oratory (1960).
8. J. Batter, "Cobalt and Iridium Buildup
Factors Near the Ground/Air Interface,"
Transactions of the American Nuclear
Society, Vol. 6, No. 1, (1963), p. 198.
9. B. L. Jones, J. W. Harris, and W. P. Kunkel,
"Air and Ground Scattering of Cobalt60
Gamma Radiation," Consolidated Vultee Air
craft Corporation Report No. 170T, (1955).
10. E. T. Clarke and J. F. Batter, "GammaRay
Scattering by Concrete Surfaces," Nuclear
Science and Engineering, Vol. 17, (1963),
p. 125.
11. W. H. Henry and C. Garrett, "Scattering of
Uncollimated Cobalt60 Gamma Radiation by
Concrete and Lead Barriers," to be
published in Acta Radiologica.
12. A. B. Chilton and C. M. Huddleston, "A
Semiempirical Formula for Differential
Dose Albedo for Gamma Rays on Concrete,"
Nuclear Science and Engineering, Vol.
17, (1963), p. 419.
13. T. Rockwell III, (ed.), Reactor Shielding
Design Manual, 1st ed., Princeton, New
Jersey: D. Van Nostrand Company, 1956,
p. 335.
14. J. C. LeDoux and A. B. Chilton, "Gamma Ray
Streaming Through TwoLegged Rectangular
Ducts," Nuclear Science and Engineering,
Vol. 11, (1959), p. 362.
15. A. B. Chilton and C. M. Huddleston, "A
Semiempirical Formula for Differential
Dose Albedo of Gamma Rays on Concrete,"
R228. Naval Civil Engineering Labora
tory (1962).
16. W. Heitler, The Quantum Theory of Radia
tion, 2nd ed., London: Oxford University
Press, 1944, p. 154.
17. G. W. Grodstein, "Xray Attenuation Coef
ficients from 10 key to 100 Mev,"
National Bureau of Standards Circular
No. 583, (1957).
18. G. W. Grodstein, "Xray Attenuation Coef
ficients from 10 key to 100 Mev,"
Supplement to National Bureau of Stand
ards Circular No. 583, (1959).
19. U. Fano, "A Similarity Principle in the
Backscattering of yRays," Radiation
Research, Vol. 1, No. 5, (1954), p. 495.
20. D. J. Raso, "Monte Carlo Calculations on
the Reflection and Transmission of
Scattered Gamma Radiations," TOB 6139.
Technical Operations, Incorporated,
(1962).
21. M. Leimdtrfer, "The Backscattering of
Gamma Radiation from Plane Concrete
Walls," Nuclear Science and Engineering,
Vol. 17, (1963), p. 345.
22. P. Cziffra and M. J. Moravscik, "A Practi
cal Guide to the Method of Least
Squares," University of California
Radiation Laboratory Report No. 8523,
(1958).
23. M. J. Berger and D. J. Raso, "Monte Carlo
Calculations of GammaRay Backscattering,
Radiation Research, Vol. 12, (1960), p.
20.
24. H. A. Meyer, (ed.), Symposium on Monte
Carlo Methods, New York and London: John
Wiley and Sons, Incorporated, 1956,
p. 148.
25. D. J. Raso, personal communication (April,
1963).
26. B. P. Bulatov and 0. I. Leipunskii, "The
Albedo of Gamma Rays and the Reflection
Buildup Factor," Atomnaya Energiya,
Vol. 7, No. 6, (1959), p. 551, translated
in Soviet Journal of Atomic Energy, Vol.
7, (1961), p. 1015.
27. F. S. Kirn, R. J. Kennedy, and H. 0. Wyckoff,
private communication referred to in (28).
28. M. J. Berger and J. Doggett, "Reflection
and Transmission of Gamma Radiation by
Barriers: Semianalytic Monte Carlo Cal
culation," Journal of Research of the
National Bureau of Standards, Vol. 56,
No. 2, (1956), p. 89.
29. C. M. Davisson and L. A. Beach, "A Monte
Carlo Study of BackScattered Gamma
Radiation," Transactions of the American
Nuclear Society, Vol. 5, No. 2, (1962),
p. 391.
30. A. Simon and C. E. Clifford, "The Attenua
tion of Neutrons by Air Ducts in Shields,"
Nuclear Science and Engineering, Vol. 1,
(1956), p. 156.
31. R. L. French, "A FirstLast Collision Model
of the Air/Ground Interface Effects on
FastNeutron Distributions," Nuclear
Science and Engineering, Vol. 19, (1964),
p. 151.
32. Y. T. Song, "A Semiempirical Formula for
Differential Dose Albedo for Neutrons on
Concrete," TN589. Naval Civil Engineer
ing Laboratory (1964).
33. A. B. Chilton, "Backscattering by an In
finite Concrete Plane of Gamma Radiation
from a Point Isotropic Source," Transac
tions of the American Nuclear Society,
Vol. 6, No. 1, (1963).
34. W. H. Henry, personal communications (July,
1964).
XI. APPENDIX A. CALCULATION OF PARAMETERS FOR CH FORMULA, BASED ON BEST FIT TO
TOTAL ALBEDO BASED ON FLUX
DETAILS OF THEORY.
The ChiltonHuddleston expression for
differential albedo can be written as
d = C f 1(0o;,9) + C' f2( 0o;) , (A1)
where
K (e )1026
S= cos (A2)
2 cose9
1 + 
cose
Also, it is well known that K (@ ), the Klein
e s
Nishina energy scattering crosssection per
electron is given by:
K (O9) = r 2 p2 [ + P2  P( cos20e)]
(A4)
where
2 26
(r ) = 3.97023 x 10
p = , (A5)
1+ (1  coses)
.51079 s
E = incident photon energy (MeV).
o
Likewise, from spherical trigonometry, one finds
cose = sine sine cosc,  cos9 cos0 .
(A6)
Substitution of the expression for d
(Equation A1) into the formula for A' (Equa
tion 9) gives:
Ad = C F(8o) + C' F2 () , (A7)
where
F I o(e
(A8)
F (9) = 2 cosO ln(l + )
2 0 cos
o
(A9)
For a given initial energy, there are
given five "experimental" (Monte Carlo) values
of A', one for each value of coseo used by Raso.
The data are given in Table 3. The problem in
volves obtaining the values of the coefficients
C and C' which gives a best fit (in the
least squares sense) of the formula (Equation
A5) for total albedo to the five values of A'
d
The points are to be weighted in proper statis
tical fashion by the reciprocals of the vari
ances as given by Table 6.
The following formulas apply:
Let xi = (cos@o) = cosei, i = 1, *", 5 ;
(A10)
F.(x.) = F.(e.), j  1,2
(A11)
SA = A'(ei) , experimental values.
(A12)
We let w. be the weighting factor for the ex
1
perimental value A. . Then, let
1
h = w Fk(x.) Fm(xi)
km i k 1 m i
; (A13)
(A 14)
g. = w. A. F.(x.)
J 1 1 3 1
Matrices and determinants may be defi
follows:
h h1 h 12
I h= h11 h22  (h12 )2
1 h22 h 12
9 22
[h h h h
Then,o
Thus,
C = (h22 g1  hl2 g2)
C' = (h21 gl + hll g2)
The statistic q is also neede<
the formula
5 5
q = wiA2CZwlFl(x )AiC
i=l i=l
The variances for C
following formulas:
5
wiF
i=l
and C' are g:
(AC)2 = q (h1 11 /3 = q h22/(3
(AC')2= q (h1 22/3 = q h11/(3
h )
h )
In these expressions, the value 3 re
number of degrees of freedom (5  2)
values of standard deviation on C
are of course the square roots of th
The computed values of total al
ined as  five velues of incident angle are computed by
using the formula in Equation A7 with the de
rived values of C and C'.
(A15) EXPLANATION OF COMPUTER PROGRAM
The logic flow diagram for this program is
given in Figure 18. The symbols used in the
; (A16) program are defined in Table A1. The program
itself is displayed below. It is tailored for
use on the IBM7094 of the University of
; (A17) Illinois and is adapted from one developed by
the Naval Civil Engineering Laboratory.
The program is set up to take any number
of input experimental values up to five. It
(A18) could readily be altered to include more, if
desired. The input data required is as follows,
presented card by card:
(a) The number of input values.
(b) The number of equal finite increments
(A19) into which the ranges of the two vari
ables, B and cose, are divided.
(Experience indicates that 48 and 40
; (A20) respectively provide an adequate num
ber of increments for reasonable
(A21) accuracy.)
(c) The values of cosOo, which is the in
d, given by dependent variable for the Monte
Carlo "experiment." The appropriate
gamma ray incident energy in MeV,
2(x )Ai * followed by the values of the total
albedo based on flux, which is the
(A22) dependent variable resulting from the
iven by the Monte Carlo "experiment." The order
of listing of these data is that
appropriate to the order of listing
; (A23) of the values of coseo .
(d) The same value of gamma ray incident
(A24) energy in MeV, followed by the stat
istical weights for the input albedo
presents the values. The order of listing of
The these data is that appropriate to the
and C' order of listing of the albedo data.
e variances. The output of the program, besides repeat
bedo at the ing the input data, provides the following
information: The incident energy in MeV; the
value of C; the standard deviation in the
value of C; the value of C'; the standard
deviation in the value of C'; the value of
q; and the computed values of total albedo
based on flux, obtained by use of the derived
values of C and C', listed in the same order
as that of the input experimental albedo data
which it is supposed to closely fit.
TABLE Al. DEFINITIONS OF SYMBOLS USED IN COMPUTER PROGRAM
Definition
C
K (e )1026
e s
Cose
Coso
s
C'
C
CAY
COTH
COTHS
CP
CPHI
CTHZ (N)
DET
D(N)
EO, EN
JPH, GPH
KTH, QTH
P
QAY, XHAY
QUE
SIGC
SIGCP
SWXX
SWXY
SWXZ
SWYY
SWYZ
Symbol
Input values of cosOo.
Ihi
Computed values of total albedo, using best fit
values of C and C'
Energy of incident photons in MeV.
Number of increments in 0.
Number of increments in cose.
P
Values of successive integrations (summations)
in the computation of Fl.
q
Standard deviation in C
Standard deviation in C'
h 11
h12 = h21
COMPUTER PROGRAM.
C GAMMA BArKcSCATTFRING DPORLFMCOMP. OF CH PARAMFTERS (TOTAL FLUV)
WRITE OUTDUT TAPE 6.5
50FORMATt47HI BEST FIT VALUFS OF ALPEDO FORMULA PARAMETFRS/)
ITMFNSTON CTHZt(5) .9 (5).w(f ).n(,). YXqlY(Oi)
READ INPUT TAPE 7.69M
6 FORMAT(I6)
RFAD INPUT TAPE 7.A8JPH.KTH
A FORMAT(216)
PFAO INPUT TAPF 7.Q,((TH7(N).N=I.M)
9 FOPMAT(5F6.3)
OTH=KTH
nPH=JPH
WRITF OUTPUT TAPE 6,IOJPH.KTH
100FORMAT(6H JPH=.T656H KTH=t,6//)
11 RFAD INPUT TAPE 7.1?.FO.(7(N)*N=1.M)
12 FOPMAT(FAE3.9F1?.4)
11 WPTTP OUTPUT TAPE 6.14.O*(7(N),N=1.M)
14 FOPQAT(FA.3,iEle4)
14 DFAD INPUT TAPF 7,16* N.(W(N).N=1.M)
16 FOQMAT(F8*.3,*9E 4)
17 WRTTF OUTPUT TAPF 6.1q.FN.(W(N).N=1.M)
1A rOPMAT(FP*3.3P1 ?.4//)
TFfFNFO)?P.O??0
20 WRITF OUTPUT TAP 6,?21
1OFOPRMAT(PH INrON'IST$TNT nATA 'f0C/f/)
22 GO TO 110
25 SWXW=O.
SWXVY=0
VWW7=0.
SWYY7O.
tWY7=M.
SW77=0.
Symbol Definition
5
SWZZ A2
1=1
VARC Variance in C
VARCP Variance in C'
W(N) Input statistical weighting factors for total
albedo data.
X(N), Y(N) F1, F2
Z(N) Input "experimental" total albedo data.
00 00 N=1.M
STHZ=SORT(l.CTH7(N)*CTH7(N))
XHAY=0.
00 AS K=I,KTH
COTH=(O.5)/OTH
STTH=SORT(I.COTH*COTH)
OAY=0.
00 70 J=IJPH
GwJ
PHt=.?AR31 B3*(G.5)/GoH
CPHT=COS(PHT)
COTHS=STHZ7*STH*CPHICTH7(N)*rOTH
Pat./( .+EN( 1 eCOTHS)/.'107q9)
CAY3.97023*P*P*( 1 .+P*PP*( I .COTHS*COTHS))
70 OAYvOAY+CAY
80 XHAY=OAY/(COTH+CTHZ(N))+XHAY
X(N)=CTHZ(N)*XHAY*6.2R31 A3/OTH/GPH
Y(N)=6.*831 "3*CTHZ(N)*FLOG(I.+1./CTH7(N))
SWXX=SWXY+WfN)*X(N)*X(N)
SWXY=SWXY+W(N)*X(N)*Y(N)
SWX'Z=SWX7+W(N) *XN)*7(N)
SWYYV=SWYY+W(N)*Y(N)*Y(N)
SWY7»SWYZ+W(N)*Y(N)*Z(N)
90 SW77wSWZZ+W(N)*7tN)*Z(N)
DET=SWXX*SWYYSWXY*SWXY
Cu(SWYY*SVXZ7SWXY*SWY7)/DFT
CP=(SWXX*SWYZSWXY*SWY7)/DET
OUF SW77ZZC*SWXZCP*SWY7Z
VARPC= SWYY*OUF/ET/3.
VARCP=SWXX*OUE/DET/3.
STGC=SORT(VAPC)
SIGCP=SORT(VARCP)
DO 95 N=t.M
0<5 0(N)=CCX(N)+CP*Y(N)
WiTTE OUTPUT TAPE 6,I00l,(TH7CN).N=IM)
IO0FORMAT(63H ENERGY C ST.DEV.C Co CT.DFV.CP 0
I CTH7=o6.o44H =«F6.4.4H =rFA.4.4H =.FA.4.4H =,oA.4//)
WRITF OUTPUT TAPE 6105.EO*C*STGC*CP*STGCP*OUF. (rN) N=1 M)
i105 FOWATCE6.2.4FIO.FI.4.4X.4F10.4///)
GO TO 11
110 CONTINUE
END
XII. APPENDIX B. DETAILS OF CALCULATION OF REFLECTED DOSE FIELD  GENERAL CASE
DETAILS OF THEORY.
The use of Equation 4 for the total Scat
tered dose involves the summation over small,
individual areas of the interface which may con
tribute appreciably to the total result. There
are many ways in which the scattering plane may
be divided into the incremental areas, for the
purposes of the computer calculation. Three
methods are described below.
In order to make these schemes as simple
and effective as possible, it is desirable to
make the distances involved simple multiple or
ratios of one another. The solutions are nor
malized to a source height of unity, and are
otherwise simplified by making detector height
some power of two, such as 1, 2, 4, 8, etc.
Likewise, the horizontal projection of the
sourcetodetector distance is made a simple
multiple of the detector height which is some
power of two; i.e., L, i, 1, 2, 4, etc. After
results are calculated for these distances, in
terpolation may be used for intermediate dis
tances.
The most elementary method of dividing the
plane is into small, square areas of equal size.
It appears desirable, in summing the contribu
tion of the area increments in such case, to do
this in a fashion which will tend to include the
more significantly contributing increments at
the earliest part of the compilation, and grad
ually work out toward the less significant por
tion of the plane. A simple way of doing this
is indicated in Figure 19, which is a view of
the plane cut into increments. The order of
summing the contributions is first to include
all increments designated with the letter A,
include all marked with a B, etc. Each letter
designates a "perimetric strip"; and it is
fairly obvious without detailed proof that
those perimetric strips of higher order con
tribute less to the total reflected dose at the
detector point. In fact, the summing of the
perimetric strip contributions involves a
converging series, and one can make calcula
tions as to the error involved in cutting off
the calculation at a finite number of such
perimetric strips. (This is discussed a little
later in this appendix.) The incremental size
should be small with respect to the height of
source and of detector, and it is convenient to
make it an integral divisor of the horizontal
sourcetodetector distance.
The advantage of this scheme is its sim
plicity and its usefulness in studying the rel
ative contribution per unit area of various
spots on the plane. The disadvantage is in the
number of individual computations required,
giving just as much time per unit area to cal
culations far from source and detector as to
nearby points.
The second scheme is such as to overcome
the disadvantage of the more simple approach
given above. It follows some such plan as in
dicated in Figure 20, with successive peri
metric strips being composed of larger incre
mental areas. In this plan, each perimetric
strip, after a somewhat complex system of sum
ming the areas closer to the source and detector,
becomes a simple square shape of double the
width of the previous strip. The system of
dividing the plane into increments is such
that, under the source, the increment dimen
sions are not greater than some predetermined
fraction of the source height, and the same
rule applies to the area underneath the detec
tor. Since for these calculations the detector
height is always equal to or greater than the
source height, the scheme indicated in Figure
20 is quite adequate.
This scheme is much more rapid than the
original, simple one in the machine computation;
and since the contributions per unit area from
the distant regions fall off with distance much
more rapidly than the incremental areas increase
in size, the convergence is reasonably fast.
The system is, however, more complicated in its
logic, in order to insure that areas are not
skipped or duplicated and are given roughly the
significance they deserve. (Greater potential
contribution requires smaller increments.)
It was established, in spite of its prom
ise, that the above scheme was not as accurate
as desired, because the increments tend to in
crease in size so rapidly that the albedo and
inverse square relationship, calculated at the
middle of each increment, is not always a good
average for the whole increment. The scheme
finally used was a slightly more complicated
variant of the second scheme which would double
the width of every other perimetric strip rather
than each perimetric strip. (The discussions
above therefore have no direct bearing upon the
results reported in this case, but help to ex
plain and justify the scheme finally employed
for the computer program used.) This technique
for dividing the plane into incremental areas
is indicated in Figures 21a through e, which
indicate steps in the process. These steps are
as follows:
(a) First a central area beneath the
source is accumulated, with the in
cremental size being a simple binary
fraction (1/8 has been found suffi
ciently small) of the source height.
(b) Perimetric strips about this central
area are added, in a square symmetric
arrangement, until the size of the
incremental area becomes the same
fractional value (1/8) of the detec
tor height.
(c) At this juncture, a scheme is employ
ed which adds incremental areas of
this size, so that the accumulated
area is symmetric in a rectangular
(nonsquare) fashion with respect to
a point on the plane halfway between
the source and detector projection
points.
(d) Then the process of adding on peri
metric strips and doubling the width
of every second strip is resumed as
long as it is possible.
(e) At a certain juncture (i.e., when the
increment size becomes equal to the
distance between the planar project
ion points of the source and detec
tor), the previous process runs into
difficulties; and a transition scheme
is employed to bring the system back
to a square type of symmetry about
the source projection point.
The other important aspect of this problem
is that of determining how far to continue
accumulating perimetric strip contributions be
fore terminating the process with a known (at
least approximately) truncation error. The
following development is based on the geometric
situation given in Figure 22, and we calculate
dose contributions from Equation 4.
The differential area is taken to be a
perimetric strip of circular character center
ed at a point between the projections ol the
source and detector positions on the plane. It
is assumed that the radius of the circular
strip is substantially larger than the spatial
distance between source and detector and also
substantially larger than both source and detec
tor heights (in practice a radial distance
greater than twice the sourcetodetector dis
tance, the source height, and the detector
height was maintained when using the truncation
criterion). At this distance it would be ex
pected that the contribution to the scattered
dose from each part of the circular strip would
be approximately of equal importance. Under the
given circumstances, the following approximate
relationships are also valid:
(a) r1 =r r r x ; (B1)
(b) AA = 2rrx (Ax)
(c) cosEo = h /rl h /x
(d) cose = h2/r2 h2/x
(e) es is sufficiently close to
that K (e s) is approximately
(B2)
(B3)
(B4)
h1
(f) from c, d, and e, ad = G/(1 +  ),
2
where G is a constant. (B5
From these relationships, substitution in Eq
uation 2 gives:
D1 2nx (Ax) h G
D = h (B6
5 h1
x (1 + 2 )
= MkHx)/x , (B7)
where M is the collection of all factors
which are considered to be constant for a speci
fic problem.
We would also like to know the value of
the sum of the increments beyond x. Taking
Equation B7 in its differential form, one
readily finds that this value is given by
D(x to o) = M dx' = M/3x3
x,4
= AD)x , from Equation B7. (B9)
3(Ax)
As seen in Figure 21b, x/Ax is always equal
to 2. Thus,
D(x to om) = AD/I.5. (B10)
If it is desired to obtain a certain degree of
accuracy, one may continue the summing of the
perimetric strip contributions until the esti
mated truncation error is less than a certain
proportion of the cumulated area contributions
up to the distance x. In other words, we
select some value of truncation error  for
this work, always taken to be 0.5%  and re
quire that
(AD/1.5)/(Accumulated dose) < .005 .
(B11)
An explanatory comment is needed here. In
the first place since Ax is onehalf of x,
it can hardly be considered as a small incre
ment, and the approximations inherent in the
above development may become rather appreciable.
However, it is clear that the error is in the
direction of giving too large a value for
D(x to ), which means that the error is on the
safe side. Likewise, it is appropriate to point
out that the perimetric strip has a square
rather than circular shape; however, the varia
tion due to this fact is not considered highly
significant as long as the perimetric strips
are far enough away from source and detector to
insure that dose contributions per unit area do
not vary rapidly with distance. Thirdly, it
must be noted that the center for the circular
strip development is between source and detec
tor projections, whereas for the square case,
the source projection is used as center of sym
metry. This, too, is not considered highly
significant as long as the perimeter is far
from both source and detector. As a matter of
(B8)
linE= i
8 m
fact, as another criterion, the truncation pro
cess is only allowed to proceed if the value of
x is larger than twice the spatial distance
from the source to the detector.
To summarize the above points: simple,
though approximate, criteria are at hand for
keeping the truncation error small; and, as
long as the percentage error allowed is an or
der of magnitude smaller than the likely errors
inherent in the basic formula itself, the ap
proximate nature of the criteria is not severe
enough to be of concern.
EXPLANATION OF COMPUTER PROGRAM.
The logic flow diagram for this program is
given in Figure 23 . The symbols used in the
program are defined in Table B1. The program
itself is displayed below. It is tailored for
use on the IBM7094 of the University of
Illinois.
The program is designed for making compu
tations under a wide variation of the parameters
of detector height and horizontal projection of
sourcetodetector distance. The value of the
source height is kept as one unit of distance,
since the results as obtained are the same for
scaled up or scaled down geometric relation
ships, within the limitations discussed in the
main body of the report. The program first
computes the result for a detector height of 1,
with a horizontal projection distance of 0.25
times the detector height. The result is ex
pressed as a ratio of the reflected dose rate
divided by the direct dose rate, and is thus
independent of source strength. The horizontal
projection distance is then doubled and the cal
culation repeated. This process is continued
until the horizontal projection distance is 64
times the detector height. The detector height
is then doubled and the above process repeated.
The detector height doublings are repeated until
a height of 128 is obtained and calculations ob
tained therefor.
The initial value of the incremental square
block widths is taken as 0.125 for each compu
tation. This has been found to be sufficiently
small for the sake of accuracy without requir
ing an unduly long computation.
Only one input card is required. It spec
ifies the value of the source photon energy and
the values of C and C' used in the differ
ential albedo formula (Equation 3).
The output of the program presents the re
sults for each computation in a tabular form,
showing the ratio of reflected to direct dose
rate at each value of horizontal projection
distance and detector height. It also presents,
under each result, the computed value of the
truncation error (by use of Equation B10) as
a proportion of the primary result obtained.
TABLE Bi. DEFINITIONS OF SYMBOLS USED IN COMPUTER PROGRAM
Symbol
Definition
Differential dose albedo.
K(s )1026
The cosine of the angle of scatter, os, of a photon
from the source singly scattered at the center of
the square incremental area toward the detector.
The cosine of the angle with respect to the normal
of the line from source to center of a square
incremental area.
The cosine of the angle with respect to the normal
of the line from detector to center of a square
incremental area.
A reference value used in determining how far to
extend the calculation before truncation. It is
always somewhat greater than the square of the
sourcetodetector distance.
C in CH albedo formula.
C' in CH albedo formula.
Width of incremental square areas.
The reflected dose rate contribution from an
individual square area increment, as a function
of the coordinates of the center of the incremental
area.
ALB
CAY
COSTHS
0OSTH1
COSTH2
CRIT
Cl
C2
DEL
DI(X,Y), DI
Symbol
DOSE
EN
ERR(N)
H2
K
P
PER
REDO
RESID
R1
RlSQ
R2
R2SQ
X
Definition
The accumulated value of the reflected dose rate
calculations of the perimetric strips.
Source photon energy.
The final stored and read out value of RESID for
each computation.
Height of detector.
YD/H2
The ratio of final to initial photon energy after
a single Compton scattering through angle s.
The accumulated value of the reflected dose rate
calculations from individual square area increments,
making up the contribution from a perimetric strip.
The reciprocal of the direct dose rate, on the basis
of a unit dose rate at one unit distance. It there
fore equals the square of the sourcetodetector distance
The computed value of the truncation error in reflected
dose rate, as a proportion of the total accumulated
dose rate.
The distance from the source to the center of a
square incremental area.
The square of R1.
The distance from the detector to the center of a
square incremental area.
The square of R2.
The coordinate distance on the plane of the center
Symbol Definition
X (con't)
Y
YD
of an incremental square area, in a direction
. perpendicular to the horizontal projection of the
sourcetodetector line, and measured with respect
to an origin at the projection of the source.
The coordinate distance on the plane of the center
of an incremental square area, in a direction
parallel to the horizontal projection of the source
todetector line, and measured with respect to
an origin at the projection of the source.
Horizontal projection of sourcetodetector distance.
COMPUTER PROGRAM.
C EAMMA PAY BACKSCATTFPTNG PROPLFM  4ENFDAL CASF
WPRIT OUTPUT TAPE 6.5
ROFORMAT(43H1 GAMMA BACKSCATTERTNG  ASYMMFTRICAL CASP///)
COMMON H?.DFL,EN.C1,C?P.Y
YTMENSION D(9)9 FP(P)
7 PFAM TNPUT TAPF 7*8AFN,C1,C?
8 FOPMAT(F6.2.2F6.4)
WRITF OUTPUT TArF 6.10
IOFORMAT(4cH HI MIN.nFL FN C1 CI YV=K(HP)
WRITF OUTPUT TAPF 6.1P.FNC1 .P
1?OFOPMAT(12H 1.00 0.12~.*F.?*PF6.4//)
WRITF OUTPUT TAPF 6.1
150FOPMAT(OSH HP? =0O.? =0. =1.0 =?.0
14.0 =8.0 =16.0 =32.0 =64.0//)
H2=1.
no00 o J=1.8
YD=0. 2?;*
00 70 N=1,
CPIT= .+H2*H2+Yn*YD
P0nO=(H?1 )**P+Yn*Yn
nOSF=o.
DFL=0 l?P
X=0 .5*DFL
DO 30 L=1.?
Y=I .*rFL
DO ?P K=1.4
rOSF=OOSF+DI (X.Y)
20 Y=Y+0FL
I0 Y=Y+0FL
00 60 T=1.20
T F (YOPR. ) 1 7n.lP? I 1
Symbol
Definition
T131 TF(nDFLH2/P. 1) l',n ?IO1 n
C nOLOOP FOR SOUAPE SYMMFTDICAL AOANrFEMN=T wITM rF.rT TO C(UDor
170 DO 1S M=I.4
YV=3. I*DFL
DPER=PER+DlfX Y)
Y=Y+DEL
DFR=pFp+rt (wy.Y)
IF(2.*DELX)176.172.174
17? WROTF OUTPUT TAPP 6.173
173 FOPMATi(1H GOOF NO.6)
GO TO 100
174 Y=2.*DEL
PFR=FRP+01(XY)
Y=Y+DfFL
PFP=PFp4+nT (yv)
GO TO 180
176 Y=Y+÷EL
PEP=PFR+ODI(XY)
F(Yv3.*FL)176.t?77,IM
177 WPTTF OUTPUT TAPF 6.17P
I 7M AURa I DIIIH JUI NO.7)
GO TO 100
180 X=X+TtL
DOSE=DOSr+PFp
GO TO 5n
C nOLOOP FOP TRANSITTON FOOM SOUAOF TO DFCT, SYMMFToy
230 no ?0 0 L=1,4
Y=3.5*DEL
PFR=PER+t(Xe.Y)
Y=Y+DEL
PER=PFR+DI (XY)
fF( 2.*DFLX) ?4?. ?3.9 47
?15 WPRITTF OUTPUT TAPF 6.P36
236 FOPMATtO1H GOOF NO*.)
GO TO 100
e37 Y=Z.5*UEL
PFR=PER+DI(X.Y)
PIS TF(YYD3.*DFL)?42?93P,'
23q WRITF OUTPUT TAPF 6.240
240 FORMAT(IOH GOOF NO.0)
GO TO 100
242 Y=Y+DEL
PF=PFPR+OT(0XY)
GO TO 238
250 X=X+÷FL
DOSE=DOSF+PFP
GO TO 60
C DOLOOP FOR RECT, SYM* ARRPNGT. PSPT. PT. RTWN. rPCF. AND OFT.
130 DO 39 M=1.4
v=1. £#FL
PFP=OEP+nO(XY)
Y=Y+nEL
PER=PEP+DT(X*Y)
TF(2g.*DLX)« ,31 9 .
31 WRITE OUTPUT TADF 6.3P
3? FOPMAT(10OH nGOOF NO.?)
GO TO 100
33 Y=YD+2.**DFL
pr=OFR+nT(x.Y)
Y=Y÷nL
PFP=DFR+DnfXY)
(Irn T r :i
35 Y=Y+÷FL
PFO=MFP+lT (X.Y)
TF(YYn3.OFL )136.37,10
A3 GO TO I
37 WRITT OUTPUT TAPE 6.3*
3S FORM!AT(IOH GOOF N10.3)
GO TO onn
3q Y=X+nFL
DOSSF=DOSF+PER
GO TO 60
C 00LOOP FOP TRANSITION FROM RFCT. TO SOUAPF SYMMETRY
13I UU 1e)" M=Io"
Y=3. 5*DrFL
PFR=PER+DT (X*Y)
Y=Y+OEL
PER=PER+T1 (XVY)
TF(2.*oELX)140.135.137
I13 WPr1E OUrPUIT TAPt &.136
136 FOPMAT(IOH GOOF NO.4)
(O TO In0
137 Y=YD+2.5'DEL
' H =rW+UJ I I A 9 T I
GO TO 160
14U T=T+fIEL
PFR=PERP+DI(XY)
11 I YTYL '*UlrL !4ti In"* I
150 WRITE OUTPUT TAPE 6.151
1 1 I1M lD I , JIMH M Fr N1.906 )
GO TO 100
160 X=X4DEL
DOSE=DOSE+PE
GO TO 60
50 ESID=PEP/I .5/DOSE
tF(PRESID.005)52?52.60
P5? tF(CPITtnFL*rFL)6?.6?qP6
60 rFL=*?.*FL
6p "<(N)=?.*PFDO*nOSF
FRR (N) =PFS ID
70 YD=2.YD
WRITF OUTPUT TAPE 6*79*H2((0(N)IN=1,0)
75 FOQMAT(F68.2g.2XOF 10*3)
WRITE OUTPUT TAPE 6.77.(FPP(N).N=tP9)
77 rOMAT(lIX,OF10s.'/)
80 H2=2.*H2
WRITF OUTbUT TAPF 699.
AB5 FORMAT(lH?)
9O TO 7
100 CONTINUF
END
rUNLI ruN UI *Y)
COMMON H.PDFL*FN*Cl.CP*YO
PISO=1.+X*(+Y*Y
PPSO=H?*HP+X*X+(YDY)**P
PI=SORT(PfSOI
COSTH1=1./RI
COSTH?=H?/0?
COSTHS=f(YYDVYYX*XH2)/01/0?
P= 1 ./( 1 .+FN* (1 .OSTHS)/or.'1 1 )
CAy= o 1,O P»( i ,+O»D ( i { *C STHS*Ca TH<))
ALP =(Ci*CAY+C?)/(tl.+rOTHI/COTH2)
OT=ALR*COSTHID*FLUflL/DI O/oD?P
DFTURN
F Nfl  
XIII. APPENDIX C. DETAILS OF CALCULATION OF THE REFLECTED DOSE RATE FIELD 
SOURCE AND DETECTOR AT SAME HEIGHT ABOVE INTERFACE
DETAILS OF THEORY.
The basic principles behind this calcula
tion are the same as for the general case, ex
plained in Appendix B, but certain minor varia
tions are primarily occasioned by the require
ment to determine the contribution from a finite
field as well as from an infinite field. It
was also desired to have a method which uti
lizes the same small sized incremental areas
throughout, in order to provide a comparison
with some of the results obtained by the general
method. Since the general method uses increas
ing sized incremental areas, it is subject to
suspicion of introducing an inaccuracy thereby;
and an independent check of this sort is con
sidered very worthwhile.
The scheme for dividing the plane into in
cremental areas and perimetric strips is essen
tially the simple method illustrated in Figure
19. Even further simplification than this is
permitted. Because of the "reciprocity" re
lationship between position of source and detec
tor, and because of the geometric symmetry of
this case with respect to a position halfway
between source and detector, only a quarter of
the infinite plane field needs to be studied
rather than a half, as shown in Figure 24.
The method of expressing results and the
method of keeping the truncation error below
some specified minimum percentage is essentially
the same as indicated in Appendix B. However,
since we no longer can specify that x = 2(Ax),
we must use Equation B9 rather than Equation
B10 (see Figure 24).
EXPLANATION OF THE COMPUTER PROGRAM.
The logic flow diagram for this program is
given in Figure 25. The symbols used in the
program are defined in Table C1. The program
itself is displayed below. It is tailored for
use on the IBM7094 of the University of
Illinois.
The program assumes that the source and
detector heights are both one unit of distance.
Geometric scaling by the same factor is possible
for all dimensions, so that the results are
readily applicable to other heights and other
sourcetodetector distances, with equal results
for the same ratio of separation distance to
height above the interface plane.
Two input cards are required. The first
gives the value of the incremental area size.
From a practical point of view, it should be
sufficiently small that the reflected dose rate
density over a single incremental area does not
vary substantially; and at the same time it
should be no smaller than necessary for accura
cy, to minimize the amount of computer time re
quired. From the standpoint of its fitting in
to the logical scheme of the program, it must
be equal to the sourcetodetector distance
divided by some even integer. Sourcetodetec
tor distances considered are 0.5, 1.0, and so
on up to 64, by powers of 2. It has been found
that the linear size of the incremental area
may be as large as .25 without undue loss of
accuracy.
The second input card gives the data need
ed for the albedo computation  the value of
the photon energy and the constants for use in
the ChiltonHuddleston formula.
The output, aside from a readout of the
input data, is in the form of a table in which
the reflected dose rate, relative to the direct
dose rate, is presented as a function of the
sourceto detector distance. The data is given
not only for the infinite plane case, but also
for finite plane cases; that is, the accumulated
relative reflected dose rate is presented for
values of the perimetric distance (see Figure
24) equal to 1, 2, 4, etc., up to 256 times
the size of the incremental area. However, the
accumulation of relative reflected dose rate
may approach the infinite limit with sufficient
closeness before the ultimate perimetric dis
tance is reached, so that zeros will appear in
the output table for values of the perimetric
distance beyond this point of sufficient close
ness. The table also provides the estimate of
truncation error, as a proportion to the accumu
lated dose rate, and the number of the perimet
ric strip which was the last included prior to
the truncation process.
TABLE Cl. DEFINITIONS OF SYMBOLS USED IN COMPUTER PROGRAM
Symbol
ALB
CAY
COSTHS
COSTHI
COSTH2
Cl
C2
DEL
DI (X,Y)
DI
DIST
Definition
Differential dose albedo.
K(e ) * 1026.
from the source singly scattered at the center of
the square incremental area toward the detector.
The cosine of the angle with respect to the normal
of the line from source to center of a square
incremental area.
The cosine of the angle with respect to the normal
of the line from detector to center of a square
incremental area.
C in CH albedo formula.
C' in CH albedo formula.
Width of incremental square areas.
The reflected dose rate contribution from an individual
square area increment, as a function of the coordinates
of the center of the incremental area.
A reference value used in determining how far to
extend the calculation before truncation. It is
always somewhat greater than the sourcetodetector
distance.
D(N)
EN
H1
H2
LIMIT
P
PER
RESID
Rl
R1SQ
R2
R2SQ
The accumulated value of the reflected dose rate
contributions from the perimetric strips numbering
from 1 through N.
Source photon energy.
Height of source (1.0 in this problem).
Height of detector (1.0 in this problem).
Ratio of distance from source to perimeter ("x" in
Figure 24) to incremental square width (DEL, or
"A x" in Figure 24).
The ratio of final to initial photon energy after
a single Compton scattering through angle es.
The accumulated value of the reflected dose rate
calculations from individual square area increments,
making up the contribution from a perimetric strip.
The computed value of the truncation error in
reflected dose rate, as a proportion of the total
accumulated dose rate:.
The distance from source to the center of a square
incremental area.
The square of Rl.
The distance from the detector to the center of a
square incremental area.
The square of R2.
Symbol
Definition
COMPUTER PROGRAM.
C GAMMA RAY BACKSCATTERING PROBLEM  SOURCE AND DETECTOR SAME HEIGHT
WRITE OUTPUT TAPE 6.100
O100FORMAT(/43H1 GAMMA BACK5CATTERING FROM CONCRETE SLABS//)
A COMMON DEL.EN.CIC2*YD
DIMENSION D(560)
READ INPUT TAPE 7.1.DEL
1 FORMAT(F6.2)
2 READ INPUT TAPE 7.3.EN.C1.C2
J3 FORMAIIF6.2,F6e.4)
WRITE OUTPUT TAPE 6.4
40FORMAT(//36H HI H2 EN CI C2 DEL)
WRITE OUTPUT TAPE 6,5.EN.CI.C2.DEL
50FORMAT(12H 1.00 1.00,F6.2.2F6.4.F6.2/)
WRITE OUTPUT TAPE 6*101
101 FORMAT(/50H DIST. FROM SOURCE TO PERIMETER = LIMIT*DEL/)
WRITE OUTPUT TAPE 6.7
70FORMAT(/120H YD LIMIT=1 =2 =4 =8 =1
16 =32 =64 =128 =256 =INF RESIDUAL NO.
2/)
YD=e5
DO 45 J=1.8
DO 102 K=.1500
102 D(K)=0.
X=DEL/2.
DISb1=ORIUWd.+YU*YU
DO 32 N=1.500
Y=tYU+ULL)/Z.
PER=0O
Symbol Definition
X The coordinate distance on the plane of the center
of an incremental square area, in a direction
perpendicular to the horizontal projection of the
sourcetodetector line, and measured with respect
to an origin at the projection of the source.
Y The coordinate distance on the plane of the center
of an incremental square area, in a direction
parallel to the horizontal projection of the
sourcetodetector line, and measured with respect
to an origin at the projection of the source.
YD Sourcetodetector distance.
8 IF(YYDX)16.10,15
10 PER=PER+D01(XY)
Y=Y+DEL
GO TO 8
15 Y=YDEL
X=XDEL
17 IF(X)30.25.20
20 PER=PER+DI(X*Y)
X=XDEL
GO TO 17
25 WRITE OUTPUT TAPE 6.26
26 FORMAT(IIH YOU GOOFED)
27 GO TO 50
30 D(N+1)=PER+0(N)
G=N
X=X+(G+I.)*DEL
RESID=fXDEL/Z.)*PER/3./`DEL/D(N+I)
IF(RESID.005)31.31.32
31 IF(2.*DISTX)35.35.32
32 CONTINUE
35 WRITE OUTPUT TAPE 6*36.YDD(2),D(3),D(5)*D(9).D(17),D(33).D(65),
ID 129) .D(257) D(N+1) RESID.N
36 FORMAT(F6.2.1IE1O.3.14)
45 YDzYD*2.
GO TO 2
50 CONTINUE
END
FUNCTION UI(X.Y)
COMMON DEL*EN.Cl.C2*YD
RISO=Ie+X*X+Y*Y
R2SQ=I.+X*X+(YDY)**2
Rl=SORT(RI50)
R2=SORT(R2SO)
COSTH2=1./R2
COSTHS=(Y*YDY*YX*X1o)/RI/R2
Pale/fl.+EN*(e.COSTHS)/*Sll>
CAY=3.97*P*P*(e +P*PP*( IeCOSTHS*COSTHS))
ALBS(CI*CAY+C2)/(I.+COSTHI/COSTH2)
Die*1 YD*YDeALB*COSTHI*DEL*DEL/RISQ/R2SO
RETURN
END
XIV. APPENDIX D. DETAILS OF CALCULATION OF THE DOSE RATE FIELD  SOURCE AND
DETECTOR ON SAME VERTICAL LINE
DETAILS OF THEORY.
This case is in principle similar to the
general case, in that Equation 4 is the basic
formula for determining and summing the con
tribution of individual incremental areas to
the reflected dose rate. However, a radial
symmetry exists with respect to the vertical
line containing both the source and detector;
and therefore the perimetric strips may be cir
cular in shape about the foot of this vertical
line (see Figure 26). Furthermore, since the
contribution from each portion of the perimet
ric strip is the same per unit circumferential
length, there is no need to break down the peri
metric strips into smaller increments.
Since the computation is rather simple and
involves few steps, it is not considered neces
sary to increase the width of the perimetric
strips as one proceeds, although this could
easily be done. Perimeter width has therefore
been kept the same and made sufficiently small
to insure accuracy.
The same criteria for truncation of the
accumulation of perimetric strip contributions
are used as described for the previous cases,
described in Appendices B and C.
EXPLANATION OF COMPUTER PROGRAM.
The logic flow diagram for this program is
given in Figure 27. The symbols used in the
program are defined in Table D1. The program
itself is displayed below. It is tailored for
use on the IBM7094 of the University of
Illinois.
As for other cases, the source and detec
tor positions may be reversed, with similar re
sults, because of the "reciprocity" relationship
discussed in the main body of this report.
In the computation, the height of the
source is normalized at one unit, which is
quite general because of the similarity princi
ple of Fano, discussed in the text. The source
height is varied from 1 to 64, by a factor of 2
increase after each computation.
The input consists of two cards. The
first gives the value of the width of each per
imetric strip. It has been found that a width
of 0.20 is adequate for accuracy. The second
card gives the value of the source photon ener
gy and the values of C and C' for use in the
CH albedo formula.
The output, in addition to a readout of
the input data, gives the values of three vari
ables, as a function of detector height. The
first of these is the dose rate at the detector
position, assuming a source strength sufficient
to give one unit of dose rate directly at a dis
tance of one unit from the source. The second
variable is the estimate of truncation error,
as a proportion of the total accumulated value
of reflected dose rate from the perimetric
strips out to the truncation point. The third
variable is the ratio of reflected dose rate to
direct dose rate. The latter value is zero for
a detector height of 1, because for that case
the source and detector are superimposed, giv
ing a theoretical direct dose rate value of
infinity. This is the reason for the separate
listing of the absolute reflected dose rate:
an interpolation for the absolute dose rate is
much easier for a sourcetodetector distance
between one and two feet, and this interpolated
value can then be readily converted to a value
relative to the direct dose rate.
TABLE Dl. DEFINITIONS OF SYMBOLS USED IN COMPUTER PROGRAM
Symbol
ALB
BACK
CAY
COSTHS
COSTH1
COSTH2
Cl
C2
D
DEL
EN
HI
H2
P
PER
Definition
Differential dose albedo.
The ratio between the total reflected dose rate
and the direct dose rate.
26
K(e ) * 1026
Cosine of the angle of scatter, )s, of a photon
from the source singly scattered at a point on
the midcircumference of a perimetric strip.
The cosine of the angle with respect to the normal
of the line from source to the midcircumference
of a perimetric strip.
The cosine of the angle with respect to the normal
of the line from detector to the midcircumference
of a perimetric strip.
C in the CH albedo formula.
C' in the CH albedo formula.
The accumulated reflected dose rate.
Width of circular perimetric strip.
Source photon energy.
Height of source.
Height of detector.
The ratio of final to initial photon energy after
a single Compton scattering through angle e .
Contribution to the reflected dose rate from a
perimetric strip.
COMPUTER PROGRAM.
rC GAIMA RAY HALK5LAIIfHING PRUOLLL  Vt RI ALIbNIT UP 5RCE. ANU DEIt.
WRITE OUTPUT TAPE 6.100
100 FORMAT (/66H1 GAMMA BACKSCATTERING FROM CONCRETE SLABS  VERITCAL
I ORIENTATION//)
READ INPUT TAPE 7.1eDEL
I FORMAT(F6.2)
2 READ INPUT TAPE 7».3EN*CI*C2
3 FORMAT(F6.2.2F6*4)
WRITE OUTPUT TAPE 6.4
4 FORMAT (//30H HI EN C1 C2 DEL)
WRITE OUTPUT TAPE 6,5.HI.EN.CI.C2.DEL
5 FORMAT(2F6.2*2F6.4.F6.2/)
WRIIE OUIPUI IAPE 6.7
7 FORMAT (/40H H2 DOSE RESIDUAL BACKSCATTER/)
HI=1.
H2al.
UU 4b J=9f/
D=O.
RHO DEL/Ze
DO 32 N=1.500
RI Q=HI HI+RHO*RHO
R2SO=H2*H2+RHO*RHO
R2=SQRT(R2SO)
COSTHI=HI/RI
COSTH2=H2/R2
COS IHS=RHO*RHOH1*H2)/RWI/R
P=1./(1.+EN*(1.COSTHS)/.5I1)
CAY=3.97*P*P(I.l+P*PP*C( .COSTHS*CUSIHS))
ALB=(CI*CAY+C2)/(1.+COSTHI/COSTH2)
Symbol Definition
RESID The computed value of the truncation error in
reflected dose rate, as a proportion of the total
accumulated dose rate.
RHO Distance from projection of source on the plane
to the midcircumference of a perimetric strip.
R1l Distance from the source to the midcircumference
of a perimetric strip.
R1SQ Square of Rl.
R2 Distance from the detector to the midcircumference
of a perimetric strip.
R2SQ Square of R2.
PER=6.283185*RHODEL*ALB*C I HI/WlbUQ/R'SU
D=D+PER
RHO=RHU+ULL
RESID=(RHODEL/2. ) *PER/3/DEL/D
IF(RE1SI0.00)31 e3J .3
31 IF (2.*H2RHO) 35.35932
32 CONIINUE
35 BACK=(H2HI)**2*D
WRITE OUTPUT TAPE 6.36.H2D.*RESID.BACK
36 FORMAT (F6.2*3E1I4)
45 H2=H2*2.
GO TO 2
ENUI
XV. APPENDIX E. DERIVATION OF CRITERION FOR DETERMINING IMPORTANCE FOR CONTRIBU
TION TO REFLECTED DOSE RATE OF THE REGION HALFWAY BETWEEN PROJECTIONS OF
SOURCE AND DETECTOR ON INTERFACE
As indicated in Section VII, the criterion
for determining the importance for contribution
to the reflected dose rate of the region half
way between the projections of source and detec
tor on the interface is the sign of the second
derivative to the dose rate function given in
Equation 17 with respect to a change in positon
of the reflecting area increment along the line
connecting the source and detector projection
points.
This statement can be clarified if refer
ence is made to Figures 11 and 28. We note
that the first derivative of reflected dose
rate from an incremental area is zero at the
point x=m, and that this point is important when
the dose rate curve is a maximum; that is, the
second derivative of the curve is negative. Let
us now obtain this second derivative.
From Equations 2 and 3, it is noted that:
AD = (D AA)  r co cose]
and m, since ro, r, 0, 9s are all functions
of x and the parameters.
Let us define Q as AD/k, and find the
second derivative of Q with respect to x,
evaluated at the point x=m. Because of the
fact that at this point the first derivative of
both S and T are zero, it is apparent that
dx I dx dx
x=m x=m
One can readily find that:
S(x) =
h
(h2+x2) h2+(2mx)2 [ 2+x /+ h2+(2mx)2
(E4)
It is then readily determined that:
and
c. 1026.K(9 ) + C'
(E1)
= k S(x) T(x) . (E2)
In this latter expression for AD, k is the
collection of constants in parenthesis in Eq
uation El; and the functions S and T are
the two factors in brackets, in corresponding
order. These functions are dependent upon the
variable x, as well as on the parameters h
S(m) = 2h 25/2
2(h + m2)52
d2S(m) h(4m2  5 h2)
dx2 2(h2 2+ m9/2
dx 2(h + m )
(E5)
(E6)
In order to find the expression for T
and its second derivative, at x=m, one must note
that (see Ref. 12):
K(@s) = 3.970231026 (p2p4_p3+p3 cos2s) ,
(E7)
where
1
p = E
1 + 0 (1coss)
.51079 s
It is therefore obvious that:
T
Thus, one obtains:
= C*3.97023. (P2 + p4  P3 cos2 ) + C'
[ / ~2 2\~
T(m) = C*3.97023* P 2 + P 4 _ p 3 + p3 / m  h 2 + C'
mm m l 2 2
I + h
where
m E 2h2
1 +
.51079 m2 + h
E = incident photon energy (MeV).
It can also be determined that:
d2T(m)  26 dK
dx2 d(cosG
x=B
d2(coso )
dx2
(E12)
ixm
S3.97023 .51079 2P3 + 4P 5  3P 4 + 3P4 ( 2 h
m m m + h2
+ 2P3 (2  h
m m2 + h 2
 82 h2
L m __
2 + 2 3
m + h
If a new parameter p is defined as
2
m
p h2
h
then one can combine Equations E3, E5, E6, E10, E13, and E14, to obtain:
d2Q(m) H(p)
dt2 h6 (l+p)9/2
where
) (4p5) [,2 3 +2
H(p) (45) 3.97023 C P 2 + 4 3 + 3 +1 +
2 m m m mn +1
C
3.97023 C (4p)
(p+l)
S2P 3 + 4P 5  3P 4 + 3P 4 P71 + 2P3 3
S.51079 m m m p+1 m p+l
It is possible to simplify this slightly, if C W 0, by defining a new function
(E8)
(E9)
(110)
(E11)
(E13)
(E14)
(E15)
. (E16)
G(p)  H(p) . (E17)
3.97023 CP
m
G(p) is given in detail by Equation 19 of the main body of the text.
d2(AD)
It is to be noted that if d2( is zero, positive, or negative, the values of H(p)
dx x=m
and G(p) are correspondingly zero, positive, or negative.