SEPTEMBER 1979
'q0 )
BULLETIN 807 (TECHNICAL)
Determining Water Content
and
Bulk Density of Soil by
Gamma Ray
Attenuation Methods
E. S. B. Ferraz and R. S. Mansell
\~t ~ `f
AGRICULTURAL EXPERIMENT STATIONS
INSTITUTE OF FOOD AND AGRICULTURAL SCIENCES
UNIVERSITY OF FLORIDA, GAINESVILLE
F. A. WOOD, DEAN FOR RESEARCH
/ 0
e63"
Determining Water Content and Bulk
Density of Soil By Gamma Ray
Attenuation Methods
Determining Water Content and Bulk
Density of Soil by Gamma Ray
Attenuation Methods
By
E. S. B. Ferraz and R. S. Mansell
Dr. Ferraz is a professor, Centro De Energia Nuclear Na Agricultura,
University of Sio Paulo, Piracicaba, Brazil, and previously visiting professor
in the Soil Science Department, University of Florida.
Dr. Mansell is a professor in the Soil Science Department, University of
Florida.
IFA
This public document was promulgated at an annual cost of
$2761.90 or a cost of $2.76 per copy to report an extensive re
view of gamma radiation methods used in determinations of soil
water content and bulk density.
CONTENTS
I. Introduction ........ ............................. 1
II. Historical Review of the Development and Use of
Gamma Radiation Methods ......................... 2
S III. Theory of Radiation Attenuation Methods in Soils ....... 3
A. Interaction of Gamma Rays with Matter ........... 3
B. Attenuation of Monoenergetic Gamma Radiation ..... 5
M: C. DualEnergy Gamma Attenuation Method .......... 6
D. Experimental Factors Associated with Gamma
.a Attenuation M ethods ........................... 7
E. Optimum Thickness of the Soil Column ............ 8
IV. System Parameters ................................ 10
A. Gamma Radiation Sources ....................... 10
B. System Geometry and Collimation of the Radiation .... 14
C. Detection of Gamma Radiation Intensity ............ 16
V. Mass Attenuation Coefficients for Gamma Photons in
W ater and Soils ................................. 18
A. Theoretical Values ............................. 18
0 B. Experimental Values ........................... 21
VI. SingleEnergy Method for Separate Determinations of
Water Content and Bulk Density of Soil ............... 23
SA. General Considerations ........................ 23
SB. Example of Water Content Determination ........... 24
S; C. Examples of Density Determination ................ 26
VII. DualEnergy Method for Simultaneous Determinations of
Soil Water Content and Bulk Density .................. 27
A. General Considerations .......................... 27
B. Accuracy and Precision ......................... 28
1. Sensitivity of the Method for Determinations for
Soil W ater Content ........................... 29
2. Sensitivity of the Method for Determination of
Soil Bulk Density ............................ 31
C. Examples of Simultaneous Determinations of 0 and p .. 32
VIII. An Uncollimated Radiation Attenuation Method for In Situ
Determination of Soil Water Content .................. 34
A. Description of the Method ..................... 34
B. Example of an In Situ Method ................... 35
IX. Other Gamma Radiation Methods for Determining Soil
Water Content and Bulk Density ..................... 36
X. Summary and Conclusions .................. ....... 37
XI. References Cited ................................. 39
XII. Appendices ...................................... 45
Appendix A. Mathematical Equations for Soil Water
Content (0) and Bulk Density (p) Determinations
Using the Monoenergetic GammaRay
Attenuation Method ........................... 46
Appendix B. Mathematical equations for Soil Water
Content (0) and Bulk Density (p) Determinations
Using the DualEnergy Gamma Attenuation Method .. 47
Appendix C. Mathematical Description of Experimental
Errors Associated with Determinations of Soil Water
Content (0) and Bulk Density (p) by GammaRay
Attenuation Methods ........................... 47
Appendix D. Theoretical Consideration of the Accuracy and
Precision of Water Content and Density
Determinations of Soil by the DualEnergy
Gamma Attenuation Method ............ ....... 50
I. Introduction
Measurements of soil water content, 0, and bulk density, p, with time
and depth in the soil profile are valuable inputs to water management of
agricultural soils for the production of crops. From water content dis
tribution data, soil water flow, water uptake by plant roots (transpira
tion), and soil water retention can be estimated for periods of redis
tribution following a given rainfall or irrigation event. Direct methods
(31) such as gravimetric determination, thermoelectric, neutron scatter
ing, gamma ray attenuation, and electrical resistance as well as indirect
methods such as tensiometers have been used to measure soil water con
tent. The gravimetric or weighing procedure is usually considered as
the standard for comparison with other methods for water content mea
surements. Although the neutron scattering technique is widely used to
provide in situ determinations of soil water content, this method has
certain inherent disadvantages which can be overcome by the gamma at
tenuation method. For example, a severe restriction of the neutron
method is that the volume of soil which is measured for water content
increases as the soil becomes drier; whereas the gamma attenuation
method measures the water content of a soil volume which is inde
pendent of the soil water content.
The gammaray attenuation technique is one of the most important
of the direct methods for the determination of water content of soils.
Attenuation of gamma radiation during transmission through soil has
been used for nearly three decades as a method for determining volu
metric water content and dry bulk density. Although this method is best
suited for laboratory determinations in soil columns, it has also been
used for in situ measurements under field conditions. The gamma at
tenuation method offers many advantages for determining soil water con
tent since rapid, nondestructive determinations are provided for small
volumes of soil.
For stable soils which do not swell upon wetting with water or shrink
upon drying, bulk density may be assumed to remain constant during
water flow through the soil, and thus changes in measured intensity of
transmitted radiation may be attributed to change in water content.
However, for unstable soils, the bulk deniit\ is .ubject to changes dur
ing the time that water flows through the soil and cannot be assumed to
be a constant. Either a single beam of dJual.enri\ gamma photons or
two separate beams with greatly different energies can be used to simul
taneously determine water content and bulk density in these unstable
soils.
A general review of gammaray attenuation methods used for de
termining water content and bulk density of soil is reported in this bul
letin. These methods are most applicable to measurements under labora
tory conditions, but can also be used for measurements in soil profiles
under field conditions. Examples are also presented for determinations
of density of materials other than soil, such as wood from a pine tree
and geological material such as marble. Theoretical equations which de
scribe the attenuation of beam intensity as gamma radiation is trans
mitted through soil as well as other materials are presented for both
single and dualenergy beams of gamma photons. Sensitivity, precision,
accuracy and experimental errors for the method are evaluated and dis
cussed with respect to the theory.
II. Historical Review of the Development and Use of
Gamma Radiation Methods
One of the first successful experiments based upon radiation inter
action with matter phenomena for the purpose of measuring bulk density
was performed by Belcher et al. (2) in 1950. They developed a method
for measuring water content in soil using neutron moderation and bulk
density by gammaray scattering. Later gammaray attenuation methods
were used for a wide range of purposes: to determine fluid level in
tanks (80), uniformity of several different materials (3), the density of
concrete (37), the concentration of solids in fluids (1), the concentra
tion of heavy metals in aqueous solution (75 and 85), density and water
content of pieces of wood (55), and the concentration of tungsten sus
pensions (91). These early studies provided important contributions to
the development and dissemination of radiation methodology in soil
science.
Gammaray attenuation was first used by Vomocil (90) in 1954 and
later by Bernhard and Chasek (4) to determine bulk density in field
soils. These workers used uncollimated radiation from a 60Co source and
used a Geiger Muller detector to measure radiation intensity. The radia
tion source was placed inside a vertical tube in the soil and the radiation
detector was placed in a nearby second tube. In situ determinations of
soil bulk density were determined for the soil between the two tubes. In
1957, Van Bavel, Underwood and Ragar (89) used a13Cs as a gamma
ray source and a sodium iodide crystal scintillator detector to determine
soil bulk density under field conditions. Theoretical aspects of the use
of gamma attenuation to determine soil bulk density were investigated
by Korobochkin (52) in 1959 and by Van Bavel (87 and 88) in 1959
and 1960.
A narrow collimated beam of gamma radiation was first used in lab
oratory investigations by Gurr (42, 43) and Ferguson and Gardner (25)
during the period 19601962. These workers determined bulk density
and water content in soil columns under conditions of steadystate water
flow. Davidson, Biggar and Nielsen (17) used a 200 mCi 137Cs source
to determine soil water content under conditions of transient water
flow. In each of these investigations, '37Cs was selected as the radiation
source because of the relatively high 660 KeV primary energy of the
photons. Lead collimators were used to provide a beam of monoener
getic photons.
The radioisotope 241Am was proposed by King (50) in 1962 as a
gamma radiation source for use in determining bulk density of soil.
Earlier in 1955, Miller (62) proposed the use of 241Am for measuring
concentrations of uranium and plutonium salts in aqueous solutions. The
low 60 keV primary energy for radiation from the 241Am radioisotope
and the long halflife of 460 years are advantageous for measurements
of water content and density of soils. Probably because of its limited
availability at the time, 241Am did not receive wide attention until about
1966. Gardner and collaborators (32, 33), Soane (77), and Groene
velt, DeSwart, and Cisler (40), Vauchaud et al. (86), Stroosnijder (82),
and Cisler et al. (8) used 241Am as a radiation source. Today, the 241Am
gamma ray is commonly used by many researchers to determine water
content and density of soils and of low density materials such as wood
(27).
For stable soils, bulk density may be assumed to remain constant dur
ing water flow, and thus changes in radiation intensity may be attributed
to changes in volumetric water content. However, for soils which swell
upon wetting and shrink upon drying or moist soils undergoing cycles of
freezing and thawing, changes in intensity of a single beam of mono
energetic radiation may result from changes in both water content and
density. Because of this limitation, either a single beam of dualenergy
photons or two monoenergetic beams must be used to provide simul
taneous measurement of both bulk density and water content. The pos
sibility for using radiation from two gamma ray sources each with dif
ferent primary energy spectra was first considered theoretically by
Durante et al. (22) in 1957 and later by Gardner and Calissendorf (34)
in 1967. Soane (77, 78) in 1967, Nofziger and Swartzendruber (64) in
1976 and Gardner and Fischer (32) in 1966 successfully tested this
method to simultaneously determine water content and bulk density of
soil columns.
III. Theory of Radiation Attenuation Methods in Soils
A. Interaction of Gamma Rays with Matter
When monoenergetic gamma radiation passes through a homogeneous
material gamma photons interact with electrons, nuclei and electrical
3
fields to become absorbed and scattered. These interactions result in an
attenuation of the intensity, I, of the incident beam of photons. The
fraction (dl/I) of gamma rays attenuated is directly proportional to
the thickness of the material, x, where the proportionality constant, k,
is referred to as the linear attenuation coefficient (6)
dl/ = k dx. [1]
Many interaction processes (19, 48, 65) are known to occur between
gamma photons and matter, but only a few of these are important to the
measurement of volumetric water content and bulk density of soil.
Compton effect, which includes scalteillng and absorption, is the main
attenuation process, but photoelectric, p:lirproduc(ioii and Rayleigh
effects also may become important, depending upon the energy level.
The linear attenuation coefficient termed k (units of 1/cm) for a
given radiation energy and for a specific homogeneous (example: liquid
water) material is the sum of several individual attenuation coefficients,
which represent all of the individually occurring attenuation processes:
k = ke + k + kp +k ..... [2]
where the subscripts C, E, P, and R represent the Compton, photo
electric, pairproduction and Rayleigh effects, respectively. Each of these
coefficients taken separately represents the statistical prob:ibilit\ for oc
currence of each of the individual interactions. The sum of all thc
efficients represents the total probability\ for attenuation of gamma rays
with a specific energy during transmission though a specific material
with a given chemical composition.
Gammaray attenuation coefficients may also be expressed as a mass
attenuation coefficient, A, which is simply the linear coefficient (k)
divided by the densitl, (p). The coefficient termed A has units of cm2/g
and for a given radiation energy, depends upon the chemical properties
of the absorber material. The mass attenuation coefficient is generally
independent of the physical state of the material. For most purposes, the
mass attenuation coefficient, u, can be used more conveniently than the
linear coefficient (k).
For a given absorber material, the fractional contributions of each of
the various absorption and scattering mechanisms to the total magnitude
of the mass attenuation coefficient (t) is greatly influenced by the pri
mary vnerg\ level of the gamma photons. For example, during inter
actions between 60 KeV photons and water the Compton Effect (ko/p)
coniributc' about 89% (81% of Compton scattering and 8% of Comp
ton absorption) of total attenuation (L) and photoelectric (kW/p) and
Rayleigh (kg/p) effects with only 7% and 4%, respectively. For the
medium energy) region, such as 662 KeV of 137Cs, the Compton effect
(ko/p) accounts for about 100% of total attenuation (62% due to
4
Compton scattering and 38% due to Compton absorption), because the
occurrence of other processes are negligible. For gamma photons with
energy levels greater than 1 MeV, the pairproduction (kp/p) effect can
occur but only becomes important for energy levels greater than 3 or
4 MeV.
Knowledge of these interaction processes which result in absorption
and scattering of photons during movement through specific absorber
material provides the basis for using gammaray attenuation as an ex
cellent method for studying selected physical properties of matter.
B. Attenuation of Monoenergetic Gamma Radiation
After a collimated beam of singleenergy gamma photons has passed
through a selected absorber material, measurements of the attenuated
radiation permit calculation of the density of that material. As a colli
mated beam of monoenergetic photons with intensity Io (number of
photons per cm2 per sec) is transmitted through a homogeneous sub
stance having density p (g/cm3) and thickness x (cm), the radiation
becomes attenuated in accordance with the wellknown LambertBeer
equation used in physical chemistry:
I = I. exp[/zpx] [3]
where I is the resulting intensity (number of photons per cm2 per sec)
and A is the mass attenuation coefficient (cm2/g). Equation [3] is the
integral form of equation [1].
When gamma radiation is transmitted through a column of hetero
geneous material such as soil, gamma ray attenuation occurs not only
by the soil' but also by other absorbers along the path of the photons.
In a sample of moist soil, the intensity of a collimated beam undergoes
attenuation by soil, water, and air components as well as the walls of
the container and by the air separating the soil column from both the
radiation source and the detector (see Figure 1). Since attenuation of
gamma radiation by air is very, very small compared to that by the soil
and soil water components, the influence of the air within the soil and
the air surrounding the soil column is usually ignored in determinations
of soil water content and bulk density. Equuaions [Aiv] and [Av] in
Appendix A can be used to calculate the water content (0) of the soil,
provided the soil density is known and remains constant with time,
0 = 1 I(n + xp, [4]
or to calculate the soil bulk density (p), provided the water content is
known and remains constant with time,
5
P = In [(l + x,]. [5]1
Thus for a collimated beam of gamma photons with a specific primary
energy transmitted through an undisturbed core or packed column of a
given soil, the value of 0 or p can be calculated from the ratio of mea
sured intensities of radiation that has passed through the container filled
with soil and through the empty container. Prior to measurements of
the required radiation intensities, the thickness of the bulk soil (x), the
mass attenuation coefficient for water (t,) and the specific photon en
ergy, and the mass attenuation coefficient for water ( ,,) and the spe
cific photon energy must be known as input parameters to either equa
tion [4] or [5]. Once the magnitude of these parameters is known, the
magnitude of the water content (0) or bulk density (p) will vary accord
ing to changes in intensities (1) of the attenuated radiation. For example,
if the water content (0) is being measured at given times under transient
water flow conditions in a soil column (example: redistribution of soil
water following infiltration) measurements of the radiation intensity (I)
must be made at the required times.
I I
// 1 . .
0 *10 *
S* ..*1... .. Xc
1. I Y Y .
Figure 1. Schematic photon path in a sample of soil. The thickness of the
soil sample is represented as x. The equivalent thicknesses of soil, air, and
water in the sample are represented as x,, x,, and x,c. The radiation source is
designated as S and the detector as D. The wall thickness of the container
is given as xc. The distances between the soil column and the source and
between the column and the detector are given as x,,1 and x,2, respectively.
C. DualEnergy Gamma Attenuation Method
If gamma radiation is transmitted through a soil column either as a
single collimated beam of dualenergy photons or as two separate beams
of monoenergetic photons with different energies, mathematical equa
6
tions for determinations of soil water content (0) and bulk density (p)
develop which are slightly more complicated than those for the single
energy method. These equations are presented as equations [Biii] and
[Biv] in Appendix B. For the dualenergy method these two equations
can be used to simultaneously determine both soil water content (0) and
bulk density (p) from measurements of radiation intensity. This method
is particularly useful for determining water content (0) changes in the
surface layers of a soil which swells upon wetting or shrinks upon dry
ing. For such a soil both the water content and the density undergo
change during cycles of wetting or drying. Use of the dualenergy gamma
attenuation method to simultaneously determine 0 and p is also advan
tageous for soils that undergo compaction in the surface horizons due to
S the movement of heavy agricultural equipment.
Americium241 which emits 60 KeV photons and Cesium137 which
emits 662 KeV photons are commonly (35, 58, 26) used as radiation
sources for gamma ray attenuation methods of determining 0 and p.
D. Experimental Errors Associated With Gamma
Attenuation Methods
Determinations of soil water content (0) and bulk density (p) have
associated experimental errors. Mathematical equations presented in Ap
pendix C relate 0 and p determinations to these errors. For the single
energy gamma attenuation method, a value determined for water con
tent (0) is primarily attributable to a measured value of the radiation
intensity ratio, I/Io, but improper values used for soil thickness (x),
soil density (p), mass attenuation coefficient for water (g), and mass
attenuation coefficient for the soil (us) will contribute to errors in the
determination of 0. Precision in the values of x, p, g and L have been
shown (36) to be particularly important to determinations of water
content.
The minimum resolvable change in soil water content, ao, is given
in Appendix C as equation [Cviii]
"1 e [i I
a \Ioexp 2 (Ap 0) [6]
An important observation from this equation is that the minimum re
solvable change in water content becomes smaller as the unattenuated
radiation intensity (lo) becomes larger. Thus relatively large radiation
S sources are generally used to provide values of Io in the range of 104 to
106 cpm. The value of ac also is influenced by the thickness of the soil
column, the mass attenuation coefficients for soil and water, the density
of the soil, and the soil water content. For most soils the magnitude of
the minimum resolvable change in water content ranges from 0.002 to
0.005 cm3/cm3.
The minimum resolvable change in soil bulk density, ap, is also given
in Appendix C as equation [Cxv]
a / exp (,p + 1,0) [7]
which differs from equation [6] only in that g, rather than ~t, appears
in the denominator. The magnitude of the minimum resolvable change
in bulk density ranges from 0.10 to 0.20 g/cm3 for most soils.
For dualenergy gamma ray attenuation methods, errors in concur
rent determinations of soil water content 0 and bulk density p have been
shown (36) to result from the random emission of gamma rays from the
sources, random errors in mass attenuation coefficients, and resolution
time of the gamma photon counting system. For counts greater than 106
cpm measured in air from 241Am and 137Cs sources, Gardner et al. (36)
stated that the standard deviation in both water content and bulk den
sity determinations becomes limited by the precision of measurement of
soil and water attenuation coefficients and of soil column thickness.
An important observation is that the errors in bulk density or in water
content for measurements with the dualenergy method are nearly always
greater (see Appendix C) than the errors when the singleenergy method
is used. In the singleenergy method, the error in determination of p
does not influence the determination of 0, since p is taken to be constant.
However, for the dualenergy method, errors in the determination of p
increase errors in 0 determinations (see equations [Cxviii] and [Cxix]
in Appendix C). Therefore the dualenergy method should only be used
under conditions where the soil bulk density changes with the water
content, e.g. soils with montmorillonitic clay minerals which undergo
swelling upon wetting and shrinking upon drying or where the bulk
density of the surface soil changes with time due to compaction by agri
cultural equipment. Under more frigid winter climates than normally
occur in Florida, the bulk density of the surface soil may also change
due to freezing and thawing of soil water. Thus, the dualenergy gamma
attenuation method could also be used under conditions of freezing and
thawing of the soil.
E. Optimum Thickness of the Soil Column
The best results for determining soil water content using equation
[Aiv] in Appendix A, will be obtained when the value of the minimum
resolvable change in water content (ae) takes the smallest possible value.
A convenient means to minimize the value of as is to optimize the
thickness of the soil column. Equation [Cviii] in Appendix C expresses
a minimum value of as when the exponent equals unity:
('sP + ,eO) 1[8]
which can be written in a simpler form:
x* [9]
SpP + ^9O
where x* represents the optimum thickness of the soil sample. Thus not
only does the optimum thickness of the soil sample depend upon the
variables of bulk density (p) and water content (0) but also upon the
mass attenuation coefficients for water and the soil. The mass attenua
tion coefficients .o and /k, are greatly dependent upon the primary
energy of the gamma photons and upon the chemical composition of the
absorber material.
For a given soil the optimum thickness of the column or core tends
S to be greater for high energy photons such as 662 KeV than for lower
energy photons such as 60 KeV. With increasing energy of the photons,
the magnitude of t, and g, have been observed to decrease (Fig. 2).
For example, the value of A, is much greater for 60 KeV gamma pho
tons than for 662 KeV. Therefore the optimum thickness of the soil
sample is influenced by the energy of the gamma photons and the chem
ical composition of the soil and the water in the soil. Using equation
[Cviii] in Appendix C, error in the water content determination was
plotted in Figure 3 as a function of the soil sample thickness, x, for sev
eral different soils using 60 and 662 KeV photons. For low energy pho
Stons such as the 60 KeV, the chemical composition of the soil is ob
served to strongly affect the magnitude of the mass attenuation coefficient,
A. In Figure 3 curves relating ca with x are presented for 60 KeV pho
tons with a mineral soil high in iron content (A), with the same soil
when compacted to a higher bulk density (B), and with an organic soil
(C). A fourth curve (D) is given for any soil using 662 KeV photons.
The minima values for the curves represent the optimum thickness, x*,
for the soil samples, and these values were approximately 1.5, 1.8, 3.0,
and a range from 6 to 13 cm for curves A, B, C, and D. Thus the opti
mum thickness of the soil sample is an important consideration when low
ene .i ga:nmn. photons are used in the gamma attenuation method for
determining the soil water content. For higher energy photons such as
662 KeV, variations in the sample thickness may not cause appreciable
error in the determination of 0. Thus the optimum thickness of the soil
sample will vary with the primary energy of the gamma radiation used
in the method. Generally for soil columns less than 10 cm thick, 60 KeV
photons from 241Am provide best determinations for either 0 or p, but
for columns 1025 cm thick, 662 KeV photons from 137Cs provide best
results.
0 10
o
5
O
0 .5 \
(0
S.1 ^^ WATER
.05F SOILL '
.05
U)
O 60 662
10 100 1000
Primary Gamma Photon Energy
E(KeV)
Figure 2. Mass attenuation coefficients for water and a representative soil as
functions of the primary energy of gamma photons (Ferraz, 26).
IV. System Parameters
A. Gamma Radiation Sources
Most gammaray attenuation determinations of 0 by soil scientists
have been made with 137Cs (662 KeV) sources; however, 241Am (60
KeV) is frequently used as a gamma ray source. Preference for these
radioisotopes may be attributed to several inherent advantages over other
sources.
In the selection of a radiation source, several factors should be con
sidered. First, the radiation spectrum must show a well distinguished
primary energy peak in a region free of interfering radiations, because
the commonly used solid NaI(Tl) scintillator detector has high effi
ciency but limited resolution. Secondly, the radioisotope halflife must
be of the same or greater order than the duration of the programmed
experiments to minimize or eliminate corrections for decay. Also, the
cost for the system assemblage must be considered.
E
U
R A
.002
C0C
2 .001
E D
5 10
Soil Sample Thickness, x(cm)
Figure 3. Error in water content determination as a function of soil column
thickness. Curve A corresponds to a mineral soil having large values for the
mass attenuation coefficient (,= 0.4260 cm2/g for 60 KeV photons, p= 1.2
g/cm3, and 0=0.2 cm3/cm3) due to a high content of iron. Curve B is for
the same soil but with a higher bulk density (p= 1.4 g/cm3). Curve C is for
an organic soil (,,=0.2552 cm2/g for 60 KeV photons, p=1.2 g/cm3, and
0=0.20 cm3/cm3). Curve D represents any soil for high energy gamma
radiation (662 KeV photons). This data was obtained by Ferraz (26).
A third consideration is that both the total and specific activities of
the radiation source are important considerations. Because of errors due
to the random nature of radioactive disintegration, a large number of
photons must reach the detector. The actual number of photons counted
is a function of source activity and specific activity, geometry, collima
tion, and electronic discrimination. Usually a source with 100 to 200
mCi is used for a collimated beam in laboratories and 3 to 7 mCi for in
situ 0 determinations in the field using uncollimated gamma radiation.
Certain investigators have used stronger sources in the laboratory, such
as Finnemore and Schaaf (28) with 250 mCi, Mansell et al. (58) and
Gardner and Calissendorff (34) 300 mCi and Nofziger and Swartzen
druber (63) 280 and 389 mCi; Gardner, Campbell and Calissendorf
(36) 500 mCi. Herkelrath and Miller (47) used a 1300 mCi 138Cs
source and recommended 2000 mCi for use with plastic scintillator
detectors having low deadtime.
The thickness of the soil sample, its density and the gamma radiation
energy determines the optimum experimental conditions. The product
of density and thickness determines the best gamma energy to be used.
Thus the primary energy of the gamma radiation is a fourth considera
tion when selecting a radiation source. For most soils with columns 4
to 8 cm thick, the 60 KeV 241Am is recommended. For columns with
thickness between 10 to 25 cm or for dense materials, the 662 KeV
137Cs is normally used and sometimes in special cases, 1170 and 1310
KeV of 6oCo or 1250 KeV of 22Na. For very thick samples of lowdensity
materials, Xray sources can also be used.
Detailed discussion about sources of gamma transmission methods
has been given by Christensen (7). King (50) reported the advantages
for 241Am source in comparison with others and especially with 137Cs.
Gardner and Calissendorf (34) and Ferraz (26) discussed the possi
bility for using sources other than 241Am and 837Cs in the dualenergy
gamma beam method for simultaneous measurements of bulk density
and water content. Careful examination of a radioisotope table verifies
that only a few radioisotopes can be used (Table 1).
For the dualenergy gamma energy method, a collimated beam with
two very well distinguished energies is required. These energies must
provide adequate mass attenuation coefficients for soil and water, in
order to give acceptable resolution. Figure 2 shows variations in the
mass attenuation coefficients, for water and for a typical soil, as a func
tion of photon energy. Note that these two mass attenuation coefficients
become greatly different only for low energies, that is below 70 KeV.
However, 203Hg, 170Tm and 109Cd, with energies 78, 84, and 88 KeV,
respectively, are just above this point. The 210Pb isotope with a halflife
of 20 years, has a peak at 47 KeV that could be an ideal source, but its
spectrum is complex and this peak represents only 4% of the entire
12
Table 1. A list of radioisotopes used as sources of gamma radiation for
gammaray attenuation methods.
HalfLife MainPeaks
Radioisotopes (years) (%) (KeV)
Americium 241Am 458 86 60
Cadmium 109Cd 1.24 100 88
Cerium 144Ce 0.78 11 134
Cesium 134Cs 2.50 23 570
98 605
99 796
Cesium 137Cs 30 85 662
Lead 210Pb 20 4 47
Cobalt 60Co 5.30 100 1173
100 1332
Iridium 192Ir 0.20 29 296
20 308
81 317
49 468
Mercury 203Hg 0.13 38 78
77 279
Sodium 22Na 2.60 180 511
100 1275
Thulium 170Tm 0.37 3 84
spectrum. Among all of the low energy sources, only the 241Am has the
desirable characteristics of a long halflife, clean spectrum, low cost and
relatively large values of mass attenuation coefficients which are signifi
cantly different between values for soil and for water. Since 241Am is
probably the best radioisotope to be used as a low energy source, the
obvious question is what is the best second source with higher energy
for the dual energy method? Calculations using the sensitivity equations
show that 110 KeV energy source would be ideal, but unfortunately
such an ideal radioisotope with this energy and practical conditions is
not available. Among the available gamma radiation sources (Table 1),
theoretically the best would be 144Ce, which has a peak at 134 KeV, but
the peak represents only a portion of the spectrum and has a short half
life. The next best choice is '37Cs which is suited for practical conditions.
Today, most of the investigations with the dualenergy method for simul
taneous measurements of bulk density and water content in soils are
made with paired 24'Am and 137Cs gammaray sources.
B. System Geometry and Collimation of the Radiation
Collimation is especially important in the singleenergy gamma ray
attenuation method for determining water content or bulk density in
soil columns and particularly in simultaneous measurements with dual
energy photons. The collimator must provide either a single monoener
getic or two parallel monoenergetic beams or a single dualenergy beam,
as thin as possible, but with maximum intensity. This ideal collimator is
impossible to obtain due to the practical geometry limitations but many
authors have obtained good collimation with materials such as lead.
For water content or bulk density measurements by a monoenergetic
gamma beam, configurations for the geometry and collimation of the
beam have been discussed by Gurr (43) and Davidson et al. (17). Fer
guson (25) used radiation collimation only on the detector side. In order
to meet specific objectives, many researchers have also optimized colli
mation by adjustment of the length, position, size, and shape of colli
mators.
For the simultaneous measurements of water content and bulk density
with dualenergy photons, several important configurations have been
used. Gardner and Calissendorf (34) and other investigators, separately
passed two gamma beams having different energies through the same
collimator and used a single detector. Ferraz (26) used two parallel
beams which penetrated the sample in two locations separated by dis
tances of 10 mm and detected by one or two scintillator detectors.
Stroosnijder and DeSwart (83) used two perpendicular beams which
penetrated the sample on different sides and was detected by two separate
detectors. Bridge and CollisGeorge (5) used a fixed geometry with only
one collimator in conjunction with a mechanical mechanism to alter
nately change the sources.
Under field conditions, the relative position of the source and detector
was investigated by Van Bavel et al. (87 and 89), Reginato et al. (70
and 71), and Ryhiner and Pankow (74). De Vries (21) used collima
tors in the field and Reginato (69) used simultaneous measurements
with two sources in the field.
Collimators are generally constructed of lead, because of its high
density, and sometimes of tungsten as well as other heavy metals. Sev
eral collimator geometries have been used. The collimator cross section
and its length is determined by resolution and beam intensity. Some sci
entists have used a circular cross section with different sizes. Two such
cases are Mansell et al. (58) and Ferraz (26), who used 78.5 and 4.45
mm2 cross sections, respectively. Others such as Gardner et al. (35) and
Nofziger and Swartzendruber (63) used rectangular slits with 6.75 and
18 mm2 cross sections, respectively.
For best results, collimation of the radiation is required on both source
and detector sides. Collimating slits or holes should be parallel, of the
Figure 4. Energy spectrum taken with a Nal detector and a multichannel
analyzer for a dualenergy beam of gamma radiation from 241Am and 13aCs.
The peak on the left represents the 60 KeV primary energy for 241Am and
the peak on the right is the 662 KeV energy for 137 Cs.
same size and exactly in line. On the source side, good collimation is
necessary to provide less divergence for the beam of gamma rays leaving
the source. On the detector side, good collimation minimizes the effect
of gamma scattering by providing a smaller selected area through which
gamma rays can reach the detector without interactions.
The ratio between the maximum size of the opening and the length of
the collimator is very important because it will alter the possibility for
gamma rays to reach the detector after one or more Compton or Raleigh
scattering events. The probability for a Compton or Raleigh scattering
event to occur is a function of the photon energy and occurs for angles
close to or less than 180 (24). A good procedure for testing collima
tion involves a comparison between theoretical and experimental mass
attenuation coefficients for a known material, such as water. Experi
mental values of u will usually be smaller than the theoretical value, and
a high experimental value close to the theoretical one will indicate good
collimation. Groenevelt et al. (40) performed an extensive study on
collimation of 60 KeV gamma rays.
Figure 4 shows the energy spectrum of a collimated beam of gamma
photons obtained in the apparatus described by Mansell et al. (58) with
24'Am and ''Cs. This photograph illustrates the effectiveness of lead
collimators (on both the radiation source and detector) for gamma ra
diation.
C. Detection of Gamma Radiation Intensity
In the early work of Vomocil (90) a 60Co source and a Geiger Muller
detector were used for field measurements of soil bulk density. Van
Bavel et al. (89) later introduced the use of a sodium iodide (Nal)
scintillator detector, optically coupled to a photomultiplier tube and
connected to a single channel pulse analyzer. This kind of equipment is
currently used by soil physicists. The single channel gamma ray spec
trometer with a sodium iodide detector has been described by Crouthanel
(15) or Davisson and Evans (18).
The NaI(TI) detectors used for measuring intensity of narrow colli
mated beams of medium energy such as 662 KeV (137Cs) require the
use of a thick Nal crystal (5.08 or 7.62 cm) to provide proper energy
discrimination. Ferraz (26) used a 7.62cm thick scintillator with a
diameter of 7.62 cm for simultaneous detection of 60 KeV photons from
241Am and 662 KeV photons from 137Cs. Resolutions of 16% and 10%,
respectively, were obtained for 60 and 662 KeV photons. Mansell et al.
(58) used a 5.08cm thick crystal with a diameter of 5.08 cm for the
same objective, and the resolution of this detector may be seen in Fig
ure 4.
Although used extensively in gamma spectroscopy analysis, the high
resolution (less than 1%) germanium Ge(Li) detector has not been
used in the majority of soil science laboratories, primarily because of its
low efficiency (0.2% for 662 KeV gamma photons), which is about 100
times less than for the Nal(TI). The Ge(Li) detector also needs per
manent cooling with liquid nitrogen which results in many operational
difficulties in the laboratory and it is much more expensive than NaI(TI)
detectors. Corey et al. (13) examined advantages and disadvantages for
the Ge(Li) and NaI(TI) detectors with single channel and multichannel
analyzers of 400 and 4096 channels. For simultaneous measurements of
1'8Cs and 241Am peaks, they recommended the NaI(TI) detector with
a 400 channel pulse analyzer. Herkelrath and Miller (47) reported that
plastic scintillator detectors have the advantage of a low dead time (about
30 nanoseconds) but the disadvantage of very low efficiency. Plastic
scintillators operate best for strong (2 Ci) radioisotope sources. Kirkham
and Corey (51) have reviewed the types of radiation detectors used by
soil scientists.
Measurements of intensity of a beam of medium energy gamma radia
tion such as 137Cs is not difficult; however, for low energy radiations or
the use of two monoenergetic beams of radiation, simultaneous deter
minations of bulk density and water content require special techniques.
16
For the detection of 241Am at 60 KeV for example, the pulse shape,
electronic dead time, discriminator window, and Compton effect in the
sample are very important (14 and 81).
The intensities of gamma rays and Xrays in 241Am alpha decay has
been reported by Magnusson (57) detected with a 0.3 cm thick NaI(T1)
scintillator. Presently more accurate decay schemes are available such as
in Lederer et al. (53). A detailed spectrum of 241Am radiation shows
many other peaks beyond the main peak energy of 59.6 KeV. The most
important of these are the 17.5 and 103 KeV energy peaks. When the
spectrum was taken with a NaI(Tl) detector, sometimes a 32 KeV
escape peak and a region of summation were observed. Because of these
unwanted radiation peaks, the window discriminator for the gamma
spectrophotometer must be correctly adjusted. A good rule is to adjust
the energy window with the same thickness as the crystal resolution for
that ..neig\. A thin window decreases the photon counts due to the
scattering provided by the Compton effect in the sample. This is par
ticularly important if the collimator solid angle is large.
The Compton effect in the sample is important for low energy pho
tons, i.e. when E/Eo < 1 where E is the incident photon energy, Eo is
the electron rest energy moC2, and C is the speed of light. For low
energy photons, the Compton effect equation which shows the photon
wave length variation as a fraction of the scattering angle, a, may be
presented as:
AE E
(1 cosa). [10]
E Eo
This equation states that loss of energy is very small for large values of
the angle a and the scattered photon energy, E', is approximately the
same as the incident photon energy E. The angular distribution of the
Compton effect is not isotropic and has been described with the Klein
Nishina equation by Evans (24) and by Heitler (46).
Ignoring the small influence of multiple scattering, the contribution
due to the Compton effect is less than 0.5% for collimators with solid
angles less than 6 on the detector side. Cisler et al. (8) reported the
influence of the Compton effect upon the counts and Groenevelt et al,
(40) reported the influence of the Compton effect upon mass attenua
tion coefficients measured with IJtlerent collimator geometry. After ex
tending the work of Conner et al. (9), Gopal and Sanjeevaiah (38) con
cluded that the effect of multiple scattering upon determinations of mass
attenuation coefficients could be greatly minimized by always maintain
ing the product tkx (mass attenuation coefficient and sample thickness)
as less than one mean free path.
A finite resolving time in measurements of radiation intensity for the
garnma ray spectrometer contributes to a coincidence loss or counting
17
error. The observed counting rate, R, in the detection system can be
corrected for this coincidence loss with the quadratic equation given by
Chase and Rabinowitz (6) for the true count rate I
1= f V 1 4TR [11]
27 L I
where r = pulse resolving time (microseconds/photon). A commonly
accepted (29) approximation of this equation
I = R/(1 rR) [12]
is valid only when used with low count rates. A plot of R versus I re
veals that at high count rates equation [12] overcompensates for coinci
dence counting loss. For example if R = 0.25/r where 7 is the pulse
resolving time (with r = 2 microseconds per pulse, R = 7.5 x 106
cpm), equation [12] yields only I = 0.33/r whereas equation [11] yields
I = 0.50/r.
V. Mass Attenuation Coefficients for Gamma Photons
in Water and Soils
A. Theoretical Values
The mass attenuation coefficient, x, may be defined as the probability
per unit volume of a given absorber for collision with a photon of a
specific energy. This probability is controlled by the nature of the ab
sorber and by the energy of the photon. For a given photon energy, the
mass attenuation coefficient for a heterogeneous material such as soil is
directly related to the chemical composition of that material. Because of
this, a theoretical mass attenuation coefficient for a given soil can be
calculated by summing the products of the mass attenuation coefficients
and the contents for the respective chemical elements by the equation
S= E (MiAi) = M + 2+ Mn [13]
where n is the number of different elements in the absorber, Mj is the
percentage of the ith chemical element in the soil and p is the mass
attenuation coefficient for the ith element.
In order to illustrate the validity of equation [13], /. values for several
geologic materials, soils and other gamma absorbers were determined
experimentally by Ferraz (26) using a gamma ray attenuation apparatus
with a 60 KeV 241Am source. The absorber materials were then an
alyzed for chemical composition. Using the chemical analyses and mass
attenuation coefficients for the elements present (Table 2), theoretical
18
Table 2. Mass attenuation coefficients of 60 and 662 KeV gamma photons
for those chemical elements frequently found in soils (Hubbell,
49).
Mass Attenuation Coefficient, /j
(cm2/g)
Element 60 KeV 662 KeV
H 0.326 0.1538
B 0.158 0.0716
C 0.179 0.0774
N 0.181 0.0774
0 0.190 0.0775
F 0.191 0.0735
Na 0.227 0.0741
Mg 0.257 0.0765
Al 0.277 0.0748
Si 0.219 0.0772
P 0.347 0.0750
S 0.404 0.0775
Cl 0.434 0.0745
K 0.557 0.0756
Ca 0.646 0.0778
Ti 0.752 0.0716
V 0.829 0.0706
Cr 0.971 0.0723
Mn 1.07 0.0714
Fe 1.20 0.0732
Co 1.33 0.0722
Ni 1.52 0.0754
values of riass attenuation coefficients for the absorber materials were
calculated using equation [13]. The results of that comparison are pre
sented in Table 3, and a plot of values of theoretical (g,) versus ex
perimental (1kE) mass attenuation coefficients for the soils and other
absorber materials is presented in Figure 5. Theoretical and experimen
tal values were closely correlated and were easily fitted by linear regres
sion to the straight line
E = 0.0119 + 0.9573 I'T. [14]
Note that most of the experimental values are lower than the theoretical
ones, and this is due to experimental difficulties which will be discussed
in another section. The correlation coefficient for the LeastSquares fit
of equation [14] to the data was r = 0.994. Thus Ferraz (26) con
cluded that equation [13] is a valid means for estimating the mass at
tenuation coefficient for water and soils of the type used to generate the
data in Table 3.
Table 3. Theoretical and experimental data of mass attenuation coefficients
of 60 KeV gamma photons for water, soil and geologic materials
(Ferraz, 26).
Number of Absorber Mass Attenuation Coefficient, f
Determinations Material Theoretical Experimental
(cm2/g) (cm2/g)
14 water 0.20522 0.2001
15 sand 0.25030 0.25008
16 horizon A2, soil 0,26498 0.26361
17 horizon B1, soil 0,26628 0.27392
18 horizon C2, soil 0.26607 0.28005
19 horizon C6, soil 0.28814 0.27718
20 fresh rock 0.26692 0.26659
23 calcium carbonate 0.37068 0.35342
1 red soil 0.42450 0.41771
0)
E
.1
Theoretical Attenuation
PT (cm2/g)
Coefficient
Figure 5. A plot of theoretical and experimental mass attenuation coefficient
values for soils and other absorber materials shown in Table 3.
20
The influence of chemical composition of a soil upon the mass attenua
tion coefficient was also reported by Reginato and Van Bavel (72). They
calculated theoretical values of g for 662 KeV photons for nine repre
sentative soils of the United States. Reginato (69), calculated mass at
tenuation coefficients of 60 KeV photons for the same soils. Coppola
and Reiniger (10) extensively discussed the dependence of k upon
chemical composition of soils and calculated attenuation coefficients for
four soils over a range from 10 to 3000 KeV.
B. Experimental Values
Experimental determinations of mass attenuation coefficients for soil
have provided values with magnitudes similar to theoretical values.
Proper determination of mass attenuation coefficients requires a small
standard deviation, good collimation of the radiation beam, a small dis
criminator window for the spectrometer, high intensity of the incident
gamma beam, correction of counts for dead time, and use of optimum
thickness of the soil sample. These factors have been discussed in other
sections.
In subsection B of section II the theoretical optimum thickness of
the soil sample was discussed. However, if very great accuracy and pre
cision is needed in the measurement of mass attenuation coefficients,
especially when the dualenergy gamma beam method is used, addi
tional considerations of the Compton effect is necessary. For example, if
the sample thickness is large and the photon energy low, the Compton
scattering influence increases and may become important to the ratio of
radiation intensities (I/Io).
Gopal and Sanjeevaiah (38) reported that an increase in the experi
mental value of g is due to the multiple scattering of gamma photons
within the soil sample, which increases the counts. They studied the
sample thickness influence upon f measurements and their investigations
indicate that multiple scattering of photons could be minimized if the
product of the mass attenuation coefficient times the sample thickness is
less than one mean free path ([x
In addition to consideration of the cited factors, the accuracy of a
measurement for A depends upon the nature of the absorber. Determina
tions of A for homogeneous and pure materials such as water is con
siderably simpler than for heterogeneous materials such as soil. Conner
et al. (9) determined mass attenuation coefficients for several pure
metals using equipment similar to that used by soil physicists, and they
obtained a very low standard deviation. For example, they found for
aluminum the value for the mass attenuation coefficient plus or minus
the value of the standard deviation to be 0.1827 0.0008 cm2/g at
88.09 KeV energy. For soils, a standard deviation value of 0.4% of the
21
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attenuation coefficient is difficult to attain and normally an error of
1.6 to 2.0% is considered experimentally acceptable. The soil hetero
geneity, the size of its particles and the sample compaction within the
container, are some of the possible sources of errors. For some soils it
is necessary to obtain a large number of determinations with samples of
differing bulk density. A common procedure is to prepare 4 to 8 sam
ples of different bulk densities and make 6 to 10 determinations for u at
different points along each sample. Using 30 to 100 data points, the
means and the standard deviations can be calculated for t.
In summary, low experimental values for the mass attenuation co
efficient, /, indicate poor collimation and detection, and large magnitudes
of the standard deviation may indicate nonuniform bulk density of the
soil sample, chemical heterogeneity of the soil material, insufficient num
ber of determinations or low intensity for the incident beam of the
gamma radiation.
Ferraz (26, 27) determined experimental values of mass attenuation
(Table 4) coefficients of water, wood, geologic materials, and 13 typical
soils of Sao Paulo State, Brazil. The high values for the first five soils
were due to the high percentage of iron and clay minerals, and the high
value for soil sample number 12 was due to high contents of both cal
cium and iron. Soils numbered 6 to 10 are sandy in texture with low
concentrations of Fe, and consequently values of A were low, similar to
those for the pure sand. The gt value for a soil high in organic matter
(sample No. 13) was low, similar to that for wood. It is important to
observe that experimental values for mass attenuation coefficients for
60 KeV gamma photons differed greatly between soils with different
chemical compositions relative to the a. values for 662 KeV radiation.
The high value of standard deviation for soil sample 12 was due to prob
lems of nonuniform compaction and heterogeneity, even though the
number of determinations was twice that for the others.
VI. SingleEnergy Method For Separate Determinations of
Water Content and Bulk Density of Soil
A. General Considerations
When correctly used, the singleenergy gamma attenuation method
provides rapid, easy, practical, and precise determinations of soil water
content or bulk density. This method does not need empirical calibra
tion and offers the advantage of being nondestructive (73). The pre
cision and accuracy of determinations of either water content or bulk
density is effected by the intensity of the incident radiation, time of
23
counting, precision of mass attenuation coefficients used for soil and
water, heterogeneity of the soil sample, and proper instrumentation for
measurements of radiation intensity.
The theory presented in Appendices A, B and C shows that high
count rates are needed for measurement of either 0 or p, but a high in
tensity beam causes Compton scattering and dead time problems. For
determinations of water content in soil undergoing no or slow flow,
count rates of approximately 200,000 cpm for lo and I maintained be
tween 80,000 and 20,000 cpm are usually sufficient. For rapid soil water
movement however, a higher intensity beam is needed in order to de
crease the actual time of counting. Measurements of intensity may be
performed by preset time or preset count modes and the statistical treat
ment differs for each one as indicated by Shackel (76).
The precision of 0 or p measurements depends upon the mass attenua
tion coefficient (x) measurement precision. As reported in section V,
many factors affect /measurements and measurement precision of I is
very important when a 60 KeV energy is used. In this case, mass at
tenuation coefficient values should be checked for each sample collected.
Heterogeneity of soil samples tends to decrease the accuracy for mea
surements of p or 0. Size and shape of soil particles, water retention
properties, chemical composition, porosity, compacted layers, and other
factors affect measurement errors. Effects of heterogeneity upon deter
minations of p or 0 have been minimized by using special geometries
such as a collimated beam with a large cross section (58); or collima
tion in a slit (63); or a cylindrical column with rotating movement
(84) of the soil column.
Under optimum conditions, experimental errors in the determination
of soil water content can be close to the minimal theoretical error at
tributable only to the statistical fluctuation of radioactive disintegrations.
Errors as small as 0.005 cm3/cm3 have been obtained for water con
tent measurements, but errors 10 times greater than this value also have
been reported.
B. Example of Water Content Determination
Elzeftawy, Mansell and Selim (23) used the singleenergy gamma at
tenuation method (g,=0.0772 cm2/g and L,,=0.0837 cm2/g for 662
KeV photons from 137Cs) to determine distributions of soil water content
in 100 cm long columns of Lakeland sand during the initial phase of
water infiltration. Resulting distributions of water content are shown in
Figure 6 for a steady water application rate (2 cm/hr) into soil with ini
tial water contents of 0.20, 0.11 and 0.01 cm'/cm3. Experimentally de
termined values of 0 were similar to those calculated from a numerical
solution of the continuity equation for water. For a given period of in
24
Cd
C
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'a
Cr
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fn
73
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co
C
C)
~
it
CB tl
M^
m
E
E
u
0
U
L.)
6
uI
in
filtration, the wetting front moved deepest into the soil with the higher
initial water content. During the very early stages (0.5 hr) of infiltration,
the water content at the soil surface was highest in the soil with the
highest initial water content; however, with increasing time, the surface
water content approached the same ultimate value in all three soil col
umns.
C. Examples of Density Determination
The singleenergy gamma attenuation method has also been used to
measure the bulk density of several materials other than soils. Distribu
tions of bulk density of wood measured along the diameter of the cross
section of a 9yearold tree and across a sample of marble are shown in
Figures 7 and 8, respectively, as examples of the utility of the gamma
ray attenuation technique (Ferraz, 26). In both cases a narrow gamma
beam of 60 KeV (241Am) with 1.6mm diameter cross section was used.
E
U
0)
4,
co
Cr.
S '1 2 3 4
Radius of Trere CrossSection, r(cm)
Figure 7. Distribution of wood density measured with distance across a disc
of Pinus elliottii (3.1 cm thick sample from a 9year old tree) showing
growth rings (Ferraz, 26).
ut 3.2
0)
C
S0 5 10
Thickness of Marble Sample, y(cm)
Figure 8. Distribution of density measured with distance across a piece of
white marble 2.8 cm thick.
In Figure 7, nine annual growth rings are clearly distinguishable from
the distributions of bulk density of the wood along the diameter of the
cross section. Depressions on the curve of bulk density correspond to
the rapid growth of the wet summers (normally with high rainfall) of
Brazil and peaks in the curve correspond to slower growth rates during
the relatively dry winter months. The bulk density of the wood has an
average value of approximately 0.5 g/cm3. Such distributions of the
density of wood in trees can be compared with long term weather data
for evaluations of the influence of climate upon the growth of trees in
commercial forests.
VII. DualEnergy Method for Simultaneous Determinations
of Soil Water Content and Bulk Density
A. General Considerations
The simultaneous measurement of water content and bulk density
using the dualenergy gamma attenuation method requires good resolu
tion, precise values for water and soil mass attenuationcoefficients for
both energies, precise measurement of the thickness of the soil sample,
and accurate measurements of the beam intensities for both energies.
For the dualenergy gamma attenuation method, there are at least two
commonly used ways to measure the radiation intensities for the two
photon energies termed Ia and I,, i.e., simnultaneousl. and alternately. For
simultaneous measurements of I, and Ic, the complex energy spectrum
with two prominent peaks (see Fig. 4) can be electronically separated
with a twochannel Lanjl',,er and two scalers to record the countrates.
For this kind of measurement, special attention must be given to mini
mizing and correcting the high energy peak interference in the low
energy peak counts. These interference have been documented by sev
eral inmestig.atos. Gardner et al. (36), Corey et al. (13), Mansell et al.
(58) and Nofziger and Swartzendruber (63),
Another way to obtain m tenItti.es 1 and 1, is to obtain the attenuated
beam intensity in the 662 KeV channel (I,) alternately with that in the
60 KeV channel (I ), in order to prevent Cesium interference in the
Americium counts. Ferraz (2') used a geometry with two parallel beams,
where the Cesium radiation is shielded when the Americium peak is
being counted. Bridge and CollisGeorge (5) used a mechanical device
in order to change the position of the sources in the collimator. Stroos
nijder and De Swart (83) used a crossbeam device with two inde
pendent counting systems to obtain the same objectii eV. When the cross
beam device is used, the soil column must be rotated during measure
ment of radiation intensity. If the incident intensities for both photon
energie, are high, sufficiently high counts in the attenuated beam may
be obtained in only 10 to 20 seconds to permit measurements of 0 in
soil during transient water flow.
B. Accuracy and Precision
Error in water content determinations by the dualenergy method
appear to be restricted to standard Je\iations of appro\imntel. 0.010
cm '/cm8 (58) due to practical limitations upon measurement precision
of attenuation coefficients for soil and water. Calculated values for bulk
den'irt (p) and water content (0) should have an accuracy or standard
deviation approaching the limit imposed by the random nature of the
sources (36). In order to obtain p and 0 within a useful degree of ac
curac\ (< 1%) in simultaneous measurements (see Fig. 9 and Ap
pendix D), the mass absorption coefficients for the soil and water at
both gamma energies must be known with a standard deviation of better
than 104 cm2/g, if the total counts for low and high energy) photons
through air are about 4 X 106. The counting time must be kept short
in order to measure rapidly changing water contents and/or soil bulk
densities. Thus the incident gamma intensities must be high and the
processing of high counting rates becomes more complicated.
Another important consideration concerns the water mass attenutation
coefficient value for low energy In most soils, the water which moves
in the column carries electrol\i e in solution, and the mass attenuation
28
coefficient for the solution may be different from that of pure water. For
example, iron in solution can increase significantly the mass attenua
tion coefficient for water. A 0.5 M solution (0.076 g/cm3) of FeSO4
increases the ft water value by 12%, and this increase of / is propor
tional to the concentration.
In order to test the dualenergy gamma method for determining water
content and bulk density, two experimental procedures were performed.
These are described in the following sections.
1. Sensitivity of the Method for Determinations for
Soil Water Content
Ferraz (26) used two widely different soils to obtain the sensitivity
of the method for determination of 0: a Red soil (sample No. 2 in Table
n 10o: 10 bf
a I I'' [ ,
0 10 10 l
L L
> >
Po
4 ' 3 ~
L L
10 6
10 10S 10 10 u'
Precision of Attenuation Coefficients (cm2/g)
Figure 9. Dependence of standard deviations in measurements of soil bulk
density and water content upon precision of measurement of 60 and 662
KeV mass attenuation coefficients for soil and water.
29
4) with a high A value due to high contents of iron and clay and a
Latosol (sample No. 7) sand with a very low value for A. Samples of
the soils were prepared with several different water contents: 13 samples
of the Red soil (a range of water contents from 0.0 to 0.39 cm3/cm3
water content with 1.0 g/cm3 bulk density) and 6 samples of the
Latosol (a range of water contents from 0.002 to 0.24 cm3/cm3 water
content with 1.30 g/cm3 density). Actual water contents, OA, were
measured by weighing and the experimental water content, OE, was ob
tained by the dualenergy gamma beam method, with eight determina
tions per sample.
Actual water contents were plotted against values obtained by the
dualenergy gamma method (Fig. 10). A linear regression equation of
the form
OA = 0.0059 + 1.0095 OE
E
E 4
CD
4. .3
C
0
v.2
L
4,
C<
5 K1
jo
Experimental Water Content, eE(cm3/cm3)
Figure 10. Sensitivity of soil water content determinations. Actual water
content, 0A, were determined by weighing and experimental values, OE, were
determined by the dualenergy attenuation method (Ferraz, 26). Data are
presented for a Red soil and a Latosol, each for a range of water contents.
The straight line was determined by linear regression (r=0.995).
30
was observed to fit this data well with a correlation coefficient of 0.9950.
Thus, Ferraz (26) concluded that the dualenergy gamma attenuation
method is a sensitive means to determine soil water content.
2. Sensitivity of the Method for Determination of
Soil Bulk Density
Ferraz (26) used 41 samples of 11 different soils with bulk densities
ranging between 0.989 and 1.733 g/cm3 to determine the sensitivity of
the dualenergy gamma attenuation method for determining soil bulk
density. The soils were first dried at 1050 C for 3 days, then the sam
ples were prepared with homogeneous compaction and the actual density
E7
u 1.6
C3 "
~ *0
n 
S1.0
1O. .*
0
1.0 1.2 1.4 1.6
Experimental Bulk Density, p (g/cm3)
Figure 11. Sensitivity of soil bulk density determinations for 41 samples
from eleven soils. The actual density, d, was determined by weighing and
the experimental bulk density, p, was determined by dualenergy gamma
attenuation (Ferraz, 26). The straight line was determined by linear re
gression (r=0.989).
(d) was calculated by the gravimetric method. The same sample was
placed in the gamma equipment and the experimental bulk density (p)
was determined. The data for actual bulk density (d) and experimental
bulk density (p) were fitted using linear regression to obtain the straight
line
d = 0.0252 + 1.0234 p.
[16]
The linear regression had a correlation coefficient of 0.9890. A plot of
actual soil bulk density versus that determined by the dual energy method
is presented in Figure 11. Ferraz concluded that the dual energy gamma
method is a sensitive means to determine soil bulk density.
C. Examples of Simultaneous Determinations of 0 and p
The main utility of the dualenergy gamma method is for determina
tions of 0 and p in shrinkingswelling soils, such as reported by Corey et
al. (13), Groenevelt (39), Olesen (66), Stroosnijder and de Swart
(83), and Guerrini (41). Ferraz (26) used this method to simulta
neously determine 0 and p at a depth of 1 cm from the surface of a
shrinkingswelling soil column during horizontal infiltration. As 0 in
o
01.
r0
.30 .35 .40 .45 .50
Soil Water Content, E (cm3/cm3)
Figure 12. Experimental data (Ferraz, 26) showing the dependence of the
relative bulk density, p/po (po is the dry bulk density which was 1.3 g/cm3)
upon water content, 0, during water infiltration into a soil which swells upon
wetting.
Soil Water Content, (cm3/cm3)
.2 .4
i I < *
Wn. I I I I I1 0 1
.8 1.0 1.2 1.4
Soil Bulk Density, p (g/cm3)
Figure 13. Distributions of 0 and p
tural equipment (Ferraz, 26).
in a soil compacted by heavy agricul
creased from airdryness to 50%, the bulk density of the soil decreased
from 1.30 to 1.17 g/cm8 (fI 12). Thus, as the soil became wet by
the infiltration, the soil was observed to swell,
Another kind of study, possible only with the dualenerg. gamma
method is that of determining 0 and p in soils that have undergone com
paction under field conditions due to heavy equipment such as harvest
machines and agricultural mechanization. Ferraz (26) took samples of
undisturbed Red soil (soil No. 2 in Table 4) in a sugarcane field, using
C
O r
20.
40.
60.
80.
100.
120
)
a special Ushaped aluminum sampler. The aluminum sampler was 60
cm long with an inside width of 6 cm. Two samplers were used to
sample the soil profile from 0 to 100 cm depths. These samples were
carried to a laboratory where a dualenergy gamma system was used to
determine water content and bulk density simultaneously.
Distributions of 0 and p with soil depth are shown in Figure 13. Bulk
densities as high as 1.4 g/cm" were observed in the top 30 cm of soil
as compared to densities of approximately 1.0 g/cm3 between soil depths
of 60100 cm. Assuming a soil particle density of 2.65 g/cm3, the bulk
densities of 1.4 and 1.0 g/cm3 represent soil porosities of 47 and 62%,
respectively. The higher densities and lower porosities in the surface soil
were due primarily to the compacting effect of heavy agricultural equip
ment.
VII. An Uncollimated Radiation Attenuation Method for In
Situ Determination of Soil Water Content
A. Description of the Method
Uncollimated gamma radiation has not been used for in situ deter
mination of absolute water content but it has been used to determine
relative differences in soil water content (54, 79). The uncollimated
radiation method indicates relative changes in water rather precisely, but
measurements of the absolute water content may be subject to large
errors (72). When the gamma attenuation method is to be used for
measurements in situ, many difficulties may be encountered, such as
temperature effects in the detector and in the instruments, lack of colli
mation, low activity sources, standardization, alignment of access tubes
for source and detector, and accurate values for mass attenuation coeffi
cients. Despite these problems, this method has been useful for field
studies of drainage and evapotranspiration.
Basically, this method consists of placing two access tubes into the
soil at 20 to 30 cm distance apart, one for the gamma radiation source
and another for the detector. When the source and detector are placed
at the same soil depth, the attenuated fraction of the gamma beam is
measured, and, by the LambertBeer law, changes in water content can
be determined.
If two measurements of radiation intensity (1) of the attenuated
beam II and 12 are taken at two different times tl and t2, the difference
between 11 and 12, can be attributed to water content variation at this
soil depth. The change in water content from 01 to 02 can be expressed
according to the equation
02 01 In3 [17]
34
B. Example of an In Situ Method
Several limitations of the use of the in situ method in the field require
that the method be performed with care. Reginato and Jackson (70)
reported changes in gain of the photomultipliertube and resolution of
the scintillator crystal due to temperature effects in the detector. Prob
lems of geometry, nonparallelism of the access tubes and errors were
reported by Ryhiner and Pankow (74). De Vries (21) attempted to
use partial collimation. Standardization problems have been studied by
Reginato and Van Bavel (72) and general description of this method
may be found in papers of the latter author (88 and 89). McHenry and
Gill (60) reported the use of portable equipment for soil water content
changes in a 100cm profile as a function of rainfall, soil cover, and
profile characteristics. Rawls and Brooks (68) reported the use of this
method for measuring bulk density, and McHenry and Dendy (59) re
/
o ;E 1:1 0'
c 0 E
O( .04 
tJ' 
S0 05
0 .02 .04 .06 .08
Experimental Water Content
Change, 6e (cm3/cm3)
Figure 14. Actual difference in soil water content, AO,, versus water con
tent difference determined by uncollimated radiation. The measurement error
was 0.005 cm3/cm3.
was 0.005 cm3! cm3.
ported an adaptation for inmasiuring sediment densities. When correctly
used, the in situ method can detect changes in water content as small as
1% (0.01 cm3/cm3) and with a theoretical error of about 0.5%.
Figure 14 shows the results of a laboratory experiment to observe the
s.ensii\ii\ of this method for determining a change in water content.
A sandy soil was placed in a large container and its water content was
changed by addition of known volumes of water, homogeneously dis
tributed in the soil. The change in water content was measured by the
g.lmiinm attenuation method (A\) and these results were compared with
the actual water content change (AB,) determined gravimetrically. The
error in this determination was 0.005 cm3/cm3. These results show that
the in situ method can be used to determine changes in soil water con
tent under field conditions.
IX. Other Gamma Radiation Methods for Determining Soil
Water Content and Bulk Density
Some other gamma radiation methods for determining soil water con
tent are Jeserving of mention, despite the fact that they are not attenua
tion methods. The combined method for determining water content by
neutron moderation and density by gamma cateilinL (2, 30, 44) is
frequently" uved in field research in\ o1\ ing soil moistureplantatmosphere
rel:tiornhilp such as those by Denmead and Shaw (20) and Maertens
et al. (56). Recently, two other methods for deiermnining water content
by gamma radiation were introduced, the neutron .critai.ion, and the
gamma photoneutron methods. The neutron activation method is based
upon bombardment of hydrogen ('H) atoms with neutrons (n) to pro
duce deuterium (2H) and 2.225 MeV gamma rays (y) according to the
equation
iH + n 2H + y [18]
By measuring this 2.225 MeV peak of the gamma radiation, the amount
of hydrogen, and thus the water content, can be determined dirc.tl\.
Parker et al. (67) and Metzger et al. (61) reported studies of this
neutron activation methodology for determining water content in soils
of the Moon and Mars.
The gamma phlooneutron method (12), is based upon the bombard
ment of deuterium (QH) nuclei in water with high energy (greater than
2.225 MeV) gamma rays (y) to produce li\drogen atoms ('H) and
neutrons (n) according to the equation
H + y ) IH + n [19]
The production of photoneutrons is proportional to the deuterium con
tent in water. The natural abundance of deuterium is 0.015% but, in
36
soil water, this value is lower and can be determined for each soil. This
method needs a high energy and high activity gamma source due to the
low crosssection of the reaction and the low deuterium content in nat
ural waters. This method was first used by Haskell and Hawkins (45)
in a study of deuterium as a tracer, with a 24Na (2.75 MeV) source.
Corey et al. (11, 12) used a 1.4 Curies 208T1 source in secular equilib
rium (4.3 Curies of 22sTh).
X. Summary and Conclusions
During the transmission of collimated monoenergetic gamma radia
tion through a homogeneous material such as water the gamma photons
undergo scattering and absorption (primarily due to the Compton ef
fect). Thus the intensity of the radiation decreases as the path length
for the transmission increases. The LambertBeer equation states that
the fraction of gamma rays attenuated is directly proportional to the
thickness of the material, x. The proportionality constant is the linear
attenuation coefficient. Dividing this value by the density provides the
wellknown mass attenuation coefficient which is controlled by the
chemical composition of the material and by the primary energy of the
gamma photons. For a given material, an effective mass attenuation co
efficient can be calculated by summing the individual products of at
tenuation coefficients and contents for each chemical element. High
contents of elements like iron tend to give higher values for the effective
attenuation coefficient for the material. Also, for a given material, the
mass attenuation coefficient tends to be highest for low energy gamma
photons. For example, mass attenuation coefficients for water tend to be
higher for 60 KeV photons than for 662 KeV photons.
For a heterogeneous material, such as soil, which is comprised of air,
water, and solid components the LambertBeer equation can be used to
obtain an attenuation equation which attributes changes in radiation in
tensity to each of the air, water, and solids components. The attenuating
influence of soil air is infinitesimally small compared to water and solids
and can thus be ignored. Thus the attenuation equation can be rear
ranged to provide an expression for water content if the density is known,
or for bulk density if the water content is known.
Attenuation of the incident intensity of a collimated beam of mono
energetic gamma rays transmitted through a nonswelling soil provides
a convenient method to determine the dry bulk density and water con
tent of the soil. This technique is well developed for use under tem
peraturecontrolled laboratory conditions where precise, rapid, and non
destructive determinations are needed for packed columns or undisturbed
cores of soil. Attenuation of the intensity of gamma radiation is not
specific with respect to water or soil solids since an increase in the overall
wet soil bulk density (mass of solids plus water per unit bulk volume of
soil) results in a decrease in radiation intensity. For stable (nonswell
ing) soils, dry bulk density may be assumed to remain constant during
water flow, and thus changes in wet bulk density are attributed to changes
in volumetric water content. For soils which swell upon wetting and
shrink upon drying, a beam of dualenergy gamma radiation or two
parallel beams with different radiation energies should be used to simul
taneously determine soil dry bulk density and water content. The dual
energy method can be used to determine soil water content and bulk
density in surface soils compacted by heavy agricultural tractors and
machinery. Commercial equipment is also available for using the gamma
attenuation method to determine water content in situ under field con
ditions.
The precise and accurate determinations of soil water content or bulk
density by the singleenergy gamma attenuation method require precise
measurements of the soil sample thickness, accurate measurements of
radiation intensity, and precise determinations of mass attenuation co
efficients for soil and water. For measurements of water content under
conditions of transient soil water flow, the incident gamma intensity
must be high in order to permit measurements of a large number of
photon counts in a short time. For determinations of water content, the
minimum resolvable change in soil water content ranges from 0.002 to
0.005 cm3/cm3 for most soils. For determinations of bulk density, the
minimum resolvable change in soil bulk density ranges from 0.10 to
0.20 g/cm3. For best accuracy, the thickness of the soil column or sam
ple must be optimized for the primary energy of the radiation. For ex
ample the optimum thickness of soil samples for 60 KeV photons from
241Am range from 510 cm; whereas for 662 KeV photons from 137Cs
the optimum thickness ranges from 10 to 25 cm.
Errors associated with determinations of soil water content or bulk
density by the dualenergy method are normally greater than when the
singleenergy method is used. The precision of determinations of water
content and bulk density by the dualenergy method is strongly de
pendent upon the accuracy with which the mass attenuation coefficients
are determined for each photon energy. Thus the dualenergy method
should only be used to determine water content under conditions where
the bulk density changes with the soil water content such as occurs dur
ing water infiltration into initially dry soils with high contents of mont
morillonitic clay minerals.
Although the in situ gamma attenuation method for determining
changes in water content does not utilize collimation, changes in water
content can be detected as small as 0.01 cm"/cm' with an error of about
0.005 cm3/cm3 when correctly used.
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74. Ryhiner, A. H., and J. Pankow. 1969. Soil moisture measurement by
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XII. Appendices
Appendix A. Mathematical equations for soil water content
(0) and bulk density (p) determinations using
the monoenergetic gammaray attenuation method
For the monoenergetic gammaray attenuation method, equation [4]
can be rewritten to describe attenuation of collimated monoenergetic
gamma radiation in soil as
I = I, exp { f/ ps X Iw pw X,, /La pa Xa 2c,, pc xn
a, pa (x,, + Xa2)} I[Ai]
where the subscripts s, w, a, and c represent soil, water, air, and the wall
of the container, respectively. The distances from the radiation source
to the soil column and from the soil column to the radiation detector are
represented as x,L and xa2. The third and fifth terms in the exponent of
equation [Ai] are insignificantly small compared to the remaining terms
and can thus be excluded. During routine measurements of the intensity
of the gamma radiation, the incident radiation, Io, is commonly ob
tained after the radiation beam has passed through the empty soil con
tainer. Thus the fourth term of the exponent of equation [Ai] which
represents the attenuating influence of the container can be eliminated
by rewriting the equation in a simpler form as
I =Io exp { js ps Xs w ptw xw} [Aii]
where x, is the equivalent thickness of the soil and x,, is the equivalent
thickness of water in the bulk soil. The thickness of the bulk soil, x, is
thus the sum of x,, x,,, and x, as shown in Fig. 1, or expressed in other
words, the sum of the volumetric fractions of soil, 0,, water, 0, and air,
0G, for the bulk soil is unity. Since the product of the particle density
(ps) for soil and the equivalent soil thickness (x,) is the same as the
product of the soil bulk density (p) and the thickness of the bulk soil
sample (x), and since the product of the water density (pw) and the
equivalent water thickness (x,,,) equals the product of 0 and x, equation
[Aii] can be further simplified as
I = Io exp { x(,sp + p,, 9)) [Aiii]
Thus attenuation of a collimated beam of monoenergetic gamma rays
during transmission through a soil column can be attributed to the com
bined effects of the bulk density, p and the volumetric water content, 0,
of the soil.
Assuming p to be invariant with time and using carefully measured
values for the parameters x, /, and f,, equation [Aiii] can be rearranged
to obtain the water content of the soil
0 = [1 In ( +x) p [Aiv]
46
Or if 0 is constant, then the soil bulk density can be obtained using
p =  [In ( x 09 [Av]
Appendix B. Mathematical equations for soil water content
(0) and bulk density (p) determinations using
the dualenergy gamma attenuation method
When gamma photons are transmitted through a soil core or column
either as a single collimated beam of dualenergy photons or as two
separate beams of monoenergetic photons with different energies, two
equations similar to equation [Aiii] in Appendix A result:
for gamma photons with primary energy a
"= exp { x (/.p + /wa 0)} [Bi]
and for gamma photons with energy c
I
 = exp { x (mop + pc 0) [Bii]
1oc
These two equations can be solved simultaneously to obtain the two un
known variables 0 and p, as reported by Soane (81)
14a I y t sc 8C In
0 = in i[Biii]
X (GLsa ILwc 11sc Ilwa
In In In()
P 0 [Biv]
X (ftsa itwe Ilse flva
Appendix C. Mathematical description of experimental errors
associated with determinations of soil water content (0) and
bulk density (p) by gammaray attenuation methods
Every determination of soil water content, 0, by the gamma ray at
tenuation method has associated experimental errors. An infinitesimally
small water content determination dO, can be attributed to several
sources:
dO= dA (+ ) ( dp) + ( d, + d,,) [C
47
I
where A = In T Under ideal or perfect conditions the parameters
x, p, t,, and A, may be considered to be constant such that the second,
third, fourth and fifth terms on the right hand side of equation [Ci] should
be negligible. For such an ideal case a measured change in water con
tent, dO, would be attributable completely to a change in the logarithm
of the ratio of attenuated and unattenuated radiation intensities dA. In
other words, an increase in the measured 0 would be attributed to a de
I
crease in the relative radiation intensity, / Under nonideal or actual
laboratory conditions, a measured change in soil water content will still
be primarily attributable to a measured change in A, but changes in x,
p, t,w and p,s will contribute errors to the measured change dO. Gardner
and Calissendorf (36) have examined several components of equation
[Ci] which contribute errors to measurements of soil water content by
the singleenergy gamma radiation attenuation method. They concluded
that precision in the measurements of x, pw, 1, and p are particularly
important to determinations of water content. Spatial rates of change
of 0 with respect to A, x, p/,, p, and p may be obtained by taking the
respective partial differentials for 0 in equation [Aiv] in Appendix A:
30 1
[Cii]
aA Xyp,,
X I X2 [I x [Ciii]
30 1 (1)+ a
x. + xP [Cv]
19 1 r, a xI,
1\ 31 x [Cvi]
dp x/w I p gw J
Thus one may easily perform a sensitivity analysis for 0 determinations
by solving equations [Ci] through [Cvi] for given values of 1, 1o, x, IAw,
A,, and p and then using these solutions to solve equation [Ci] for dO
Such analyses show the importance for precise measurements of 1, x, t,,
.s, and p.
Experimental measurements of I and I, are subject to errors due to
the statistical fluctuations in the radioactive disintegration events. Dur
ing the measurement of gamma radiation intensity I, the uncertainty, at
48
the 68% of probability level, is equal to the square root of I. For small
increases of I, it is possible to make the following approximation;
dl 1
i [Cvii]
A fractional change in intensity measurement, dl/I, causes variations in
the determinations of density, dp, or thickness dx, or water content, dO.
By combining equations [Aiii] in Appendix A and [Cvii] as proposed by
Watt and Lawter (91) and with the statistical consideration of equation
[Cii] the error in water content measurements can be determined using
a, = x, exp (pP + iPw0) [Cviii]
xl,,i. \ / L 2
where ao is the minimum resolvable change of water content.
Similar to equation [Ci], the change in soil bulk density p, can be
analyzed using the following relationship
dp ( dA) +( d) + de)+( ,) d,)
[Cix]
As expected, equation [Cix] reveals that the sources of experimental
errors associated with determinations of p are also the same ones that
influence determinations of 0. Spatial rates of change of p with respect to
A, x, tp., t., and p may be obtained by taking the respective partial dif
ferentials for p in equation [Bi] in Appendix B:
ap 1 [Cx]
aA xAjS
p X~1 x I
Tx + x0 [Cxi]
Dp Ix, I ]xi
iW, I _I +i 1 J
a= I + xe x [Cxii]
YO 1 1 I 'a
= + xLT ,, [Cxiv]
Thus the minimum resolvable change, o,, of soil bulk density can be
calculated from the equation
a =;xt, L 0 exp (ap + G 4 wO) [Cxv]
which differs only from equation [Cviii] in that p/ rather than /, appears
in the denominator.
Relationships similar to equations [Ci] to [Cviii] and [Cix] to [Cxv]
for the singleenergy gammaray attenuation method can also be ob
tained for the dualenergy method for simultaneous determinations of
soil p and 0. If the attenuated beam intensities for both sources (Ia and
I,) occur with random fluctuations (dla and dIl), then fluctuations will
also occur in determinations of water content (dO) and bulk density
(dp). By differencing equations [Biii] and [Biv] in Appendix B and
adopting criteria similar to that used for Cvii:
Jsc bLsa
dO = /I v/I [Cxvi]
X (psa iPwc Psc fPwa)
/Awc Izwo.
and dp = /I, v/Ic [Cxvii]
x (tsga Pwc lsc Vtwa)
Using standard error propagation theory, equations [Cxvi] and [Cxvii]
become:
I1/2
=x c Ja [Cxviii]
X (ps) P2wc 2j 12a)
[(fw2 (i2)2] ^ 2
and a, = c la [Cxix]
X (l/sa w w 1sc /twa)
where a, and a, represent the minimum resolvable changes in 0 and p
respectively. The relative sensitivities for water content and bulk density
may be calculated, by dividing equation [Cxviii] by [Aiv] in Appendix A
and [Cxix] by [Av] in Appendix A.
Appendix D. Theoretical consideration of the accuracy and
precision of water content and density determinations of
soil by the dualenergy gamma attenuation method
Theoretical considerations of errors in 0 and p determinations due to
random fluctuations, are given in section III. Correct measurements of
mass attenuation coefficients are essential to the dualenergy gamma at
tenuation method. Rewriting equations [Biii] and [Biv] of Appendix B
in matrix form, provides the following mathematical form:
50
ln( lTa Al =x [Di]
\n (IT)_ sc /wcL 0
This form of the equations clearly reveals that the precision of water
content (0) and bulk density (p) determinations depend strongly upon
the precision of measured values of soil and water mass attenuation co
efficients.
Using equations [Cxviii] and [Cxix] in Appendix C, Mansell et al.
(58) calculated contributions to standard deviations in determinations
of 0 and p due to random error in measuring attenuation coefficients
(Fig. 9) for the special case of a soil column with thickness of 10 cm
known to within 0.01 cm, bulk density of 1.50 g/cm3, attenuation co
efficients of 0.2500 () and 0.0780 ( C,) cm2/g for 60 and 662 KeV
gamma rays, and with nonswelling properties. Attenuation coefficients
of 0.2040 (p,) and 0.0857 (p,) cm2/g for water were used for 60
and 662 KeV gamma rays. Count rates through the empty soil container
were 4.6 x 106 cpm for 241Am and 3.2 x 106 cpm for '37Cs. For
equal precision of measurement in the four attenuation coefficients for
soil and water, the standard deviation for soil water content was essen
tially constant at about 0.8% water content for precisions greater than
2 x 105 cm2/g in the attenuation coefficients, but for precisions less
than 1 x 104 cm2/g, the standard deviation for 0 increases exponen
tially. For precisions of the attenuation coefficients within the range from
2 X 106 to 1 X 104 cm2/g, values for the standard deviation for 0
were observed to occur within the interval from about 0.8 to 1.1%
water content. Values of the standard deviation for p over this practical
range of precision occur within the interval from approximately 8 to 9
mg/cm3. The intervals indicated for each point on the standard devia
tion curves for 0 and p show the small but significant influence of in
creasing 0 from 3 to 40% water content by volume with p remaining
constant. Figure 9 implies that the precisions of measurement in t,,s for
60 KeV photons and soil, E,, for 662 KeV and soil, ju, for 60 KeV
and water, and .,we for 662 KeV and water, have equal effects upon the
standard deviations for 0 and p. Gardner et al. (36), however, showed
that the precision of I A, has a much greater influence upon the standard
deviations for 0 and p than either of the precisions of measurement for
ASsa, Lwa, or M.... For example, where the precision of ,sa, s, !1a, and
,, are each 103 cm2/g the standard deviation for 0 has a value of
7.27%, but by simply changing the precision for p,. to 104 cm2/g, the
value of the standard deviation decreases to 2.65%.
