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- https://ufdc.ufl.edu/UF00027323/00001
## Material Information- Title:
- Determining water content and bulk density of soil by gamma ray attenuation methods
- Series Title:
- Bulletin University of Florida. Agricultural Experiment Station
- Added title page title:
- Gamma ray attenuation methods
- Creator:
- Ferraz, E. S. B
Mansell, R. S University of Florida -- Institute of Food and Agricultural Sciences - Place of Publication:
- Gainsville Fla
- Publisher:
- Agricultural Experiment Stations, Institute of Food and Agricultural Sciences, University of Florida
- Publication Date:
- 1979
- Language:
- English
- Physical Description:
- 51 p. : ill. ; 23 cm.
## Subjects- Subjects / Keywords:
- Soil moisture -- Measurement ( lcsh )
Soils -- Density ( lcsh ) - Genre:
- bibliography ( marcgt )
non-fiction ( marcgt )
## Notes- Bibliography:
- Bibliography: p. 39-43.
- General Note:
- Cover title.
- General Note:
- "September 1979."
- Funding:
- Bulletin (University of Florida. Agricultural Experiment Station)
- Statement of Responsibility:
- E.S.B. Ferraz and R.S. Mansell.
## Record Information- Source Institution:
- University of Florida
- Holding Location:
- University of Florida
- Rights Management:
- All applicable rights reserved by the source institution and holding location.
- Resource Identifier:
- 023757238 ( ALEPH )
08368793 ( OCLC ) AHM1170 ( NOTIS )
## UFDC Membership |

Full Text |

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. Dual-Energy 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. Single-Energy 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. Dual-Energy 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 Gamma-Ray Attenuation Method ........................... 46 Appendix B. Mathematical equations for Soil Water Content (0) and Bulk Density (p) Determinations Using the Dual-Energy Gamma Attenuation Method .. 47 Appendix C. Mathematical Description of Experimental Errors Associated with Determinations of Soil Water Content (0) and Bulk Density (p) by Gamma-Ray Attenuation Methods ........................... 47 Appendix D. Theoretical Consideration of the Accuracy and Precision of Water Content and Density Determinations of Soil by the Dual-Energy 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 gamma-ray 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-.en-ri\ 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 gamma-ray 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 dual-energy 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 gamma-ray scattering. Later gamma-ray 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. Gamma-ray 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 1960-1962. These workers determined bulk density and water content in soil columns under conditions of steady-state 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 half-life 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 dual-energy photons or two mono-energetic 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 photo-electric, p:lir-produc(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, pair-production 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 with a specific energy during transmission though a specific material with a given chemical composition. Gamma-ray 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 photo-electric (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 pair-production (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 gamma-ray attenuation as an ex- cellent method for studying selected physical properties of matter. B. Attenuation of Monoenergetic Gamma Radiation After a collimated beam of single-energy 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 mono-energetic 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 well-known Lambert-Beer 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. Dual-Energy Gamma Attenuation Method If gamma radiation is transmitted through a soil column either as a single collimated beam of dual-energy 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 dual-energy 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 dual-energy 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. Americium-241 which emits 60 KeV photons and Cesium-137 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 dual-energy 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 dual-energy method are nearly always greater (see Appendix C) than the errors when the single-energy method is used. In the single-energy method, the error in determination of p does not influence the determination of 0, since p is taken to be constant. However, for the dual-energy method, errors in the determination of p increase errors in 0 determinations (see equations [Cxviii] and [Cxix] in Appendix C). Therefore the dual-energy 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 dual-energy 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 10-25 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 gamma-ray 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 half-life 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 low-density materials, X-ray 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 dual-energy 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 dual-energy 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 half-life 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 gamma-ray attenuation methods. Half-Life Main-Peaks 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 half-life, 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 dual-energy method for simul- taneous measurements of bulk density and water content in soils are made with paired 24'Am and 137Cs gamma-ray sources. B. System Geometry and Collimation of the Radiation Collimation is especially important in the single-energy 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 dual-energy 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 dual-energy 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 Collis-George (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 dual-energy 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 7.62-cm 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 5.08-cm 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 nano-seconds) 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 X-rays 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 Least-Squares 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 dual-energy 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 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 I-I 0 m m m m m00 m m m m w04 o M.0/ m wr D V m -t 4 M cr 4N N NA c 3 Im I o-0 C) N 00 CiO i' m CiO N O.- m mOOO OmOItO I IcI D r00 00000 06c6-6066x06w 0) 000000000000000000000000 - CO dddddddddddddddddodd ,Z I 0 W C N) In N V 000) 0) m^ 'C N 00 N ') i m r/ Q \C oooooooo6ooo6 ooooocococq vM ovc dddddddddd cm- 'i3* o 0 t- 't- cr I'D C 0 CoN C CNre m 130 N 00000000000 00 SN \00v ooooooooooooooooooocN 00 ( 0000i-00)x0000 x N000 SN N m c *" 0 a 0c c Cc COCOc -oo 0 > 7; . U C ga 0, )m -, - ca 22 0) 1) 1) 0 g 0 7 O0 0) z cl m m CCCi0 0 1 0 < 4" ^5^000m 00 U0 S- - -3 3i N N N EY yi 0 c3 c3 -.b ta c38 ? a ca ca er c3ac 33a 3 %>^ -i .S ObUUUUUU OUUU UC 'a E ,,-og S ^^r^-i-^^-
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. Single-Energy Method For Separate Determinations of Water Content and Bulk Density of Soil A. General Considerations When correctly used, the single-energy 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 non-destructive (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/cm-3 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 single-energy 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 -< -. 'a Cr 0 fn 73 0 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 single-energy 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 9-year-old 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 1.6-mm diameter cross section was used. E U 0) 4, co Cr. S '1 2 3 4 Radius of Trere Cross-Section, r(cm) Figure 7. Distribution of wood density measured with distance across a disc of Pinus elliottii (3.1 cm thick sample from a 9-year 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. Dual-Energy 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 dual-energy gamma attenuation method requires good resolu- tion, precise values for water and soil mass attenuation-coefficients for both energies, precise measurement of the thickness of the soil sample, and accurate measurements of the beam intensities for both energies. For the dual-energy 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 two-channel Lanjl',,er and two scalers to record the count-rates. 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 Collis-George (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 cross-beam 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 dual-energy 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 10-4 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 dual-energy 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 dual-energy gamma beam method, with eight determina- tions per sample. Actual water contents were plotted against values obtained by the dual-energy 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 dual-energy 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 dual-energy 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 dual-energy 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 dual-energy 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 dual-energy gamma method is for determina- tions of 0 and p in shrinking-swelling 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 shrinking-swelling 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 air-dryness 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 dual-energ. 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 sugar-cane field, using C O r 20. 40. 60. 80. 100. 120 ) a special U-shaped 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 dual-energy 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 60-100 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 Lambert-Beer 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 photomultiplier-tube and resolution of the scintillator crystal due to temperature effects in the detector. Prob- lems of geometry, non-parallelism 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 100-cm 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 moisture-plant-atmosphere 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 cross-section 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 Lambert-Beer 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 well-known 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 Lambert-Beer 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 non-swelling 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- perature-controlled 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 (non-swell- 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 dual-energy 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 single-energy 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 5-10 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 dual-energy method are normally greater than when the single-energy method is used. The precision of determinations of water content and bulk density by the dual-energy method is strongly de- pendent upon the accuracy with which the mass attenuation coefficients are determined for each photon energy. Thus the dual-energy 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. XI. References Cited 1. Bartolomew, R. N., and R. M. Casagrande. 1957. Measuring solids con- centration in fluid systems by gamma-ray absorption. Ind. Eng. Chem. 49 (30):428-431. 2. Belcher, D. J., T. R. Cuykendall, and H. S. Sack. 1950. 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Measurement of concentration of tungsten suspensions and density of liquid sodium by gamma ray absorption. In Symposium on the peaceful uses of atomic energy in Australia, Canberra, pp. 610-614. (n L i : I r XII. Appendices Appendix A. Mathematical equations for soil water content (0) and bulk density (p) determinations using the monoenergetic gamma-ray attenuation method For the monoenergetic gamma-ray attenuation method, equation [4] can be rewritten to describe attenuation of collimated monoenergetic gamma radiation in soil as I = I, exp { f-/ ps X I-w 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 dual-energy gamma attenuation method When gamma photons are transmitted through a soil core or column either as a single collimated beam of dual-energy 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 gamma-ray 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 single-energy 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 single-energy gamma-ray attenuation method can also be ob- tained for the dual-energy 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 l-sc 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 dual-energy 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 dual-energy 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 10-5 cm2/g in the attenuation coefficients, but for precisions less than 1 x 10-4 cm2/g, the standard deviation for 0 increases exponen- tially. For precisions of the attenuation coefficients within the range from 2 X 10-6 to 1 X 10-4 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 10-3 cm2/g the standard deviation for 0 has a value of 7.27%, but by simply changing the precision for p,. to 10-4 cm2/g, the value of the standard deviation decreases to 2.65%. |