AN ANALYSlS OF CERTAN MWEMATICAL ASSUMPTHDNS UNDERLYENG THE DESIGN AND OPERATEON OF GAMMAuRAY SURFACE DENSUY GAGES Thesis for the Door» 6% M. S. MECHIGAN STATE QMS‘E’ERSEW Bryant“ Walker Pocock 1956 AN ANALYSIS OF CERTAIN MATIIE MATICAL ASSUMPTIONS UNDERLYING THE DESIGN AND OPERATION OF GAMMA-RAY SURFACE DENSITY GAGES by Bryant Walker ggcock AN ABSTRACT Submitted to Michigan State University of Agriculture and Applied Science in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE Department of Agricultural Engineering Year 19 56 Approved by WM/fjw ii Since the pioneer work of Belcher, Cuykendall, and Sack at Cornell Univer- sity in 1950 (1), a number of papers have appeared (2-11) describing various instruments for use in measuring densities of soil masses by means of their interactions with gamma radiation. These instruments have been either of the surface type (2) or of the depth probe type (3-11). A few such instruments have found their way into commercial availability and have met with varying degrees of success. Without exception, all the papers describing these so-called gamma-ray density gages have indicated a need for further insight into those fundamental relationships which exist among the various parameters concerned with effective design and use of the equipment being reported. It was with the hope of filling a portion of this need that the investigation outlined in this report was conducted. Certain mathematical relationships were assumed on theoretical grounds to ezdst among the several independent variables involved in the design and opera- tion of a satisfactory gamma-ray surface density gage. These relationships were explored analytically and a number of tentative conclusions were made. On the basis of these conclusions, a gamma-ray surface density gage was designed and constructed. The performance of the gage was compared with its predicted pre- dicted performance, and the degree of similarity between actual and predicted calibration curves was taken as justification for the original assumptions. AN ANALYSIS OF CERTAIN MATHEMATICAL ASSUMPTIONS UNDERLYING THE DESIGN AND OPERATION OF GAMMA-RAY SURFACE DENSITY GAGES by Bryant Walker Pocock A THESIS Submitted to Michigan State University of Agriculture and Applied Science in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE Department of Agricultural Engineering 19 56 (thaws ACKNOWLEDGMENTS The investigations reported in this thesis comprised a phase of one project in the program of research adapted by the isotopes section of the Michigan State Highway Department research laboratory on the campus of MichiganState University, in coopera- tion with the University of Michigan-Phoenix Memorial Project staff in Ann Arbor. The isotopes section was established in 1952 by executive order of Commissioner Charles M. Ziegler, following far-sighted recommendations by Messrs. W. W. McLaughlin and E.A. Finney, respectively testing and research engineer and director of the research laboratory of the MichiganState Highway Department, andby Dr. Henry J. Gomberg, assistant director of the Phoenix Memorial Project. The author gratefully acknowledges his indebtedness to the following individuals for their generous assistance in the preparation of the thesis: Professor Ernest H. Kidder, for his guidance and encouragement; Professor George B. Beard, for checking the physical principles involved; Professor James S. Frame and Mr. Walter E. Sowerman, for checking the the mathematics; Professor Lloyd M. Turk and Mr- E.A. Finney, for their critical reviews of the thesis; Messrs. Ruell Ormsby and Thomas Holmes, for preparation of the artwork; and Mrs. Rosales Burr, for typing the manuscript. TABLE OF CONTENTS TABLE OF CONTENTS INTRODUCTION................... THEORETICAL DEVELOPMENT ...... . . . . . . Preliminary Considerations ........................ Attainment of Maximum Efficiency ....... Relationship of the Substrate Attenuation Coefficients . . .......... . . . . . . Use of the count rate meter . . . ..... . . The "Count-in—Soil to Count-inStandard" Ratio The Effect of Lead Thickness . . . . ...... Equation of curve® . ..... . ...... Equationofcurve® Equation of curve 6) ................. . ........ The magnitude of the g functions . . . . . . Relations between curves® , ® , and® . . Determination of Source Strength ........ OOOOOOOOOOOOO EXPE RIMENTA L DEVE LOPMEN T ..................... Equipment and Procedure ......................... Analysis of Results ........... . ............ Equationofcurve®............. The magnitude of the f functions . ..... A predicted calibration curve . . . . ..... PA GE 14. 16 19 22 24 25 27 32 32 33 35 36 37 DEVELOPMENT OF A GAMMA-RAY SURFACE DENSITY GAGE BASED ON THE FOREGOING DATA ...... . ..... . . ...... Construction of the Gage ............... . . . . . . Laboratory and Field Applications . . . . . . . . . . . . . . . . SUMMARY...... ........ CONCLUSIONS............ BIBLIOGRAPHY . . . . ...... APPENDIX A: GRAPHS FOR DETERMINATION OF f1 AND f2 FUNCTIONS FOR CONCRETE, SAND, CLAY, AND WOOD SUBSTRA TES 000000000000 O O O 0 O O O O O O O O O O 0 APPENDIX B: DATA FOR DEVELOPMENT OF EMPIRICAL CALIBRATIONCURVE. ....... .. ........ ..... APPENDIX C: CHARACTERISTICS OF SOIIS USED IN DEVELOPMENT OF EXPERIMENTAL CALIBRATION PA GE 40 40 46 49 51 53 56 61 63 LIST OF TABLES TABLE PAGE I. Substrate Materials Used in Experiment 1 . . . . . . . . . . . . . . 33 II. Values of f1 and f2 for Substrate Materials ListedinTableI. ....... ....... . 36 LIST OF FIGURES Effect of Lead Thickness on Activity Through the Substrate Relation Between Total Count Rate and Rate Through Lead Laboratory Count Rate on Concrete Substrate E Thickness Diagram of Experimental Gamma-Ray Surface Density Gage--Sheet 1 .............. . ....... . . . . . Diagram of Experimental Gamma—Ray Surface Density ........ Gage- -She et 2 ............................. Operator Placing Gamma-Ray Surface Density Gage for Field Determination of Soil Density .............. . . . Determination of Soil Density ................. . . Operator Extracting Field Soil Sample for Analysis in Laboratory ..... . .................. . . . . . Gamma—Ray Surface Density Gage in Use for Determination of Count Rate onSteel (Point on Experimental Calibration Curve) . . . . FIGURE 1. 20 at Zero Thickness of Lead ..... . . . . 30 0f Lead 00000 0 O O O O O O O 4. f Functions E". Substrate Density . . . . . . . . . 5. Theoretical Calibration Curve 6. 7. 8. 9. Gamma-Ray Surface Density Gage in Use for Field 10. 11. Operator Removing Soil Sample from Ground 12. 13. Experimental Calibration Curve PAGE 31 34 38 39 41 42 43 43 44 44 45 48 INTRODUCTION During the past six or seven years, several papers have appeared (1-11) describing the deve10pment of equipment for use in measuring densities of soils by means of certain interactions of gamma radiation with matter. The instru- ments reportedin these papers have been of two types: (1) the surface gage, and (2) the depth probe. Although the surface gage is intended for use in assessing densities near the surfaces of land masses, whereas the depth probe is designed for employment at various elevations beneath the surface, both instruments are based upon the same theory. This theory states in its simplest form that the manner in which gamma radiationinteracts with matteris related to the density of the matter, that this relationship can be discovered, and that it can be used for the purpose of estimating densities with sufficient precision to be of value. That these instruments possess both advantages and shortcomings is a fact openly conceded in all the above-mentioned reports. For example, in §_i_tg measurements are possible, yet the equipment may be expensive or difficult to procure. Determinations can be made rapidly, yet difficulties have arisen which are traceable to a lack of complete understanding of certain principles underlying the method. These principles are fundamental, and are extremely complex. There is disagreement among the amhors who have described gamma-ray density gages. Where one writer reports that they are too sensitive, another states they are not sensitive enough. Where one says the method is simple, another says it is difficult. Four report the method to be independent of soil type, yet two others point out that the presence of rocks affects the results. Six had no comment on such an effect. Nine authors used cobalt 60 as a source of gamma radiation, and two used radium. One used X-rays. Source strengths varied from one to seventy milli- curies. Detectors employed included Geiger-Muller tubes, scintillation counters, boron 10 trifluoride tubes, X-ray film, and pocket-type ionization chambers. Two authors divided all their count rates on soil by count rates on standard concrete blocks, to eliminate the effects of changes in source strength with time. Four used fifty-five-gallon soil samples for calibration, three used samples less than fifty- five gallons, one used various quantities, and two did not report sample sizes. Accuracies of density measurements claimed in the papers varied fromplus or minus 0. 5 lb/cu ft to plus or minus 5. 5 lb/cu ft. Improvements suggested by the authors included use of boron-type counters and/or scintillation counters; of compact, battery~operated scalers; of smaller, or more rugged equipment; of count rate meters; and of improved designs. Suggested fields for further study embraced means for increasing sensitivity, for decreasing sensitivity, for elimi- nating the effect of rocks, and for limiting gamma energies; and studies of the size and shape of the field of influence and of the effect of soil moisture on density read- ings. In spite of the imcertain status of the gamma-ray density gage, as is evident from the above comments, a number of these have become available commercially 3 -—especially of the surface type--and certain supply and equipment firms are work— ing on improved models at this date. The commercial models are expensive and have met with varying degrees of success. A survey of the literature discloses that much remains to be done. Many refinements are needed before it can be said thata truly adequate gamma-ray surface density gage is available. Perhaps one way to approach this problem is to under- take a critical evaluation of certain basic assumptions which must be appraised before a satisfactory design can be realized. THE ORETICA L DE VE LOPME NT Preliminary Considerations It has been shown (12, 13) that the intensity, AX, of gamma radiation pene- trating an absorber varies closely in accordance with the relation, — x AX = Aoe P , (1) where A0 is the intensity at zero thickness of absorber, [.118 the total attenuation coefficient of absorber, and x is the thickness of absorber. The degree of precision with which a Geiger-Mllller tube can be used to measure the term, A, in equation (1) will depend to some extent upon the constants or para- meters associated with the experiment in question. Conventional gamma-ray surface density g‘ages which have been described (2) make use of a lead absorber, which separates the source from the counter tube. This assembly is placed in contact with some material such as soil, the density of which is to be measured. The substrate material actually comprises an additional absorber in parallel with, and beneath the lead. The total radiation received by the counter tube is that which is reflected by and transmitted through both the lead and the substrate. It obviously becomes of importance to create a design capable of distinguishing minute differences in substrate density most efficiently, yet with minimum weight. Usually the density range of interest is not large. 5 Since in general the activity detected through the substrate only, for a given thickness of lead between source and counter tube, is a function of the density of the substrate and of the strength and energy of the source, this activity may be given closely by the relation, f1(D, E) f2(D, E)x AB = g (D,E) A e , (2) where E is source energy. For a given source energy, equation (2) becomes f1 (D) f2 (D) x A, = g(D) A e . (a) where As is the activity thraigh the substrate only, of density, D, at thickness of lead absorber, x; g(D) is the coefficient of reflection of the substrate, always positive; f1(D) is the log scattering ratio of the substrate, always positive; f2(D) is the negative total attenuation coefficient of the substrate; and A is the activity through the lead only at zero thickness of lead (a function of source strength). I The total intensity, At , received by the counter tube then will be the sum of that through the substrate and that through the lead absorber; or, f1(D) f2(D)x - px At = g(D)A e + A e (4) 6 It is desirable to compare the activities scattered by and transmitted through substrates of two different densities, D = D1 and D = D2, where D1 < D2. Experi- mental evidence shows that when D1 < 132, Asl"__'>;-Asz within the limits of this discussion. Exceptions to the general statement will appear subsequently (Figure 13). For D = D1, f1 1 f2 1x A31 = glA e 3 (5) where g1 is g(Dl, E), f1 1 is f1(D1,E), and f2 1 is f2(D1, E). For D = D2 , f12 fzzx A82 = ng e ! . (6) where g2 is g(D2, E), £12 18 f1(D2, E), and Attainment of Maximum Efficiency In order thata surface density gage operate with maximum efficiency Within the range of densities between D1 and D2, inclusive, it is necessary to maximize the extent of A A8 within this density range. A suitable parameter for this purpose con- sists of the independent variable, x. f11 f21" :12 t22x A“s"‘s1’As2'fl g1 ° “2 e ' Letting , ‘11 I. 3 81A . ‘21! u a e , £12 I o - .73.:— > 1 (tobeshown). then AA,‘ au - bucl , (AAIK - a - bou° ' 1, and A“ $18.)“ :- -bo(o - 1)u° ' 2 < 0. Therefore (AA.)u-a-bou°" 1«o (7) is a maximum. From this, uo-1__B%_. 712-: we) F f1:1 _ T'Z'T 1 f12 'f11 I -f f -f 21 22 21 22 f g = In (22 2) A f2131 f12 ' f11 _ 1n (f22g2/ f2 15-351)A _ (3) f21 ' f22 Equation (8) gives the value, x, of that thickness of lead which will provide maximum A A 8 within any given range of substrate densities. It is a most useful relation. It will be referred to henceforth as x(opt)' Relationship of the Substrate Attenuation Coefficients It is also instructive to develop the relation which exists between the activi- ties, Asl and A 82, and the total linear attenuation coefficients of the two substrate materials concerned at the value of x, x = x(opt)° A82 ‘ E 1 c - 1 At u = CH?) , A -—1 81 : 1(L) ‘-'- C = -———f22 o (9) A82 b be f21 Equation (9) states that when the thickness, ‘ x, of absorber separating the source from the counter is so chosen that the difference between the activity through a substrate only, of density, D = D1, and that through a substrate only, of density, D : D2, is at a maximum (x = X(opt))’ the ratio of the activities through the sub- strates is equaltothe reciprocal of the ratio of the total linear attenuation coefficients ofthe two substrate materials. It is significant that this relationholds true only for x = "(optr It can be seen from equation (9) that the ratio of activities through any two substrates of different densities is a constant with respect to source strength at x = x(opt)’ although the value of x will vary according to equation (8). Equa- (Opt) tion(8) contains anA term, thus causing x(opt) tobe partially dependent upon source strength. Use of the count rate meter. It is of interest at this point that such a ratio of two different activities through substrate materials of different densities can be measured most efficiently with a count rate meter possessing a linear scale only when the latter is adjustedto read the greater activity at the upper limit of its scale. Comparisons between activities at other scale settings diminish in precision as the highest reading falls below the upper scale limit. These considerations, in addition to the generally low order of precision of rate meters and the difficulty of applying statistical treatments to data derived with their use, quite definitely indicate the smeriority of scaling equipment where small differences in density must be deter- mined. The "Count-in—Soil to Comt-in—Standard" Ratio The U. S. Army Corps of Engineers, in their Field Tests o_fNuclear Instru- ments £21.: .fll‘i Measurement 9_f_ Soil and Density (6), report (page 10) that a "count- 10 in-soil to count-in-standard ratio is used to eliminate variations in count that may be caused by changes in . . . source strength." The Cornell group (1) and*(2) also reported results based on the ratio of activity through an unknown substrate to activity through a standard substrate. Both groups used cobalt 60 as a source of gamma radiation. It is important to examine the validity of the use of a "count-in-soilto count- in-standard" ratio. Changes in source strength are due to the effect of half-life only, although changes in observed count rate may be a result also of background changes. If the effects of all these changes could be eliminated by the use of .a stan— dard, as suggested, this might well avoid the necessity for making periodic calibra- tions of equipment. ' Equations (5) and (6) may be employed to yield activities through an unknown A substrate, A81, and through a standard substrate, A82. Use of the ratio, 81 , A82 leads to equation (9), as shown, provided x = x(opt)‘ At other values of the term, x, use of the ratio results in the following equation: 31 _ g1 A(f11‘f12) (le'fzz)x A __ _ e . A32 32 (10) Since the value of A will change with time as a function of the half-life of the source used, At At2 = Atle' ,. (11) where A is the decay constant of the source (12, 13). 11 Upon substitution in equation (10) of the expression for At2 obtained from equation (11), the ratio becomes, (f 412) "If —f12)(- /\t) + (f 422)): A81 = E An 11 eL11 21 :l . (12) A32 g2 It is possible to consider the significance of the ratio expressed in equation (12) under three sets of conditions, as follows. Condition (1) . Dsl = Ds2 . Under this condition, the ratio A31 - 1 . As2 Therefore, where the density of the sample is equal to that of the standard, the value of the term, t, in equation(12) is without effect, and the consequence of half-life on the calibration curve is eliminated. This condition, however, is not attainable in practice. Condition (2). D81 > D32 Under this condition, f1 1 > f12 0 Therefore, (£11 -f12) <- it) < o, and the effect of the value of the term, t, in equation (12) must be to reduce the value of the ratio as the magnitude of t increases. 12 Condition (3). D81 < D82 . Under this condition, f1 1 < f1. 2 ° Therefore, (f11 ‘ f12)(-/\t) >0. and the effect of the value of the term, t, in equation (12) must be to increase the value of the ratio as the magnitude of t increases. As a result of these considerations, it becomes apparent that use of the "count-in-soil to count-inasmndard" ratio will not eliminate the effect on the cali— bration curve of half-life in reducing source strength in practice. Yet, although use of the ratio will not eliminate the effect of half-ulife, it is possible that its use may reduce this effect. It is of value to examine this possibility more closely. The relative effect of time on the calibration curve can be ascertained by examining the relative magnitudes of the coefficients of the term, t, in equation (12), and in equation (5) modified by substitution of the expression for At2 from equation (11). It can be shovm for allthree possible density relations listed above under conditions (1) through (3) that ~l 0 . , Therefore, g1 cancels, and the sense of the inequality remains the same. As will be shown in inequality (24) , £12 > f1 1 > o . Therefore, ———flf <0. 11 12 this term cancels, and the sense of the inequality is reversed: $2 ‘ f2 282 21 f 2 181 By definition, g2 > 0 . Therefore, f _ 22 12-1— > 1 . (18) By definition, £21 < 0. Therefore, £22 < £21, and (19) 22 f11 f11 f11' f12 . -- f11' f12 f g ' g g1(_£§_§_ must < g1 (.32.) f2 131 1 Equation for curve® . The equation for curve @may be developed as follows. Substitution inequation (6) of the expression for A derived from equation (16) shows that f12 f11'f12 £224.13” f22‘1'521 x ‘ f f (Opt) f g 222 '- sz 2( f21g1) 6" -( ) Equation (20) is the equation of curve@ . The term, f12 -f f11 12 g f2282 , 2 I2 181 . gives the ordinate of curve®at x = x0, or its intersection with the ln A,3 -axis. The slope of curve®is given by the term, f H f22"f21 Since the intersection of curve @with the lnAl5 -axis is less in magnitude than the intersection of curve 61) , f12 ) ( f11 f11"f12 ‘f11'f12 f f g 2 232 2 2 2 f2 121) 2 1g1 23 It will be instructive to show analytically that this is true. As will be shown in inequality (24), “11' £12) < 0' Therefore , f12 f11 f g f g2 g2 22 2 > g1 f22 ’ and f2 1g1 2 181 (£12 " f1 1) .52.(__fZ_2_g.2_) >1. 81 f21g1 Recalling from inequality (18) that f2 2 1'21 1, and anticipating, as will be shown in inequality (24), that f12 > f11' and recalling that by definition f12 > o and f11 > 0 and 12’ 11) > 0, it follows that f (72—2 > 1 . <21) 24 At this point, it is convenient to let (£12 ‘ f1 1) f22 _ z I21 ’ ' As shown, z > 1 Therefore, «12- o [rm-f > + a g2 22.53 = 2 £2. 11 > 1 g1 f21g1 g1 Ef12’f11)+ 1] 8 -1 (A) > Z , and g1 ( -1 g2 f12 ‘ f11+1 ‘gT Z >°' (22’ Therefore, _ f12 f11 f g 111- 112 f f11- £12 22 2 2232 s —— g 7—— 2 f2191 < 1( 2131 ) The magnitude of the g functions. It may be shown that inequality (22) will hold true if it be assumed that g2 =1 81 ’ as follows. 25 As shown, 1 < Z . Therefore, (f12 - f11+ 1) 1:1 >2"1 >0,and -1 f -f +1) 1 >z(12 11 >0. Therefore, inequality (22) holds true if .:_i. = Recalling that by definition, g1 > O , and 1:2 > 0 . g Omay < imay < 1, g 1 but this would not satisfy all possible values of Z. Therefore, where the values of g1 and g2 are not known, although there is nothing to indicate that g _g_2— cannot > 1 , 1 inequality (22) is always satisfied by the assumption that g1 = g2 = 1. (23) Relations between curves @ , ® , and C) . Since the intersectionof curve ® with the In A 8 -axis is less than that of curve 63) , 26 f s 22 2 2 g2 . < 31(1- (£2131) 1 Therefore, f12 f12 f11 f11' f12 f11" f12 f11 " f12 g 2 32 an (E?) < (31') ' d A Fort-h NN HM v H {2.2. > 1, f21 f 12 f f < o 11 12 Since, by definition, f11 > 0 , and f12 > 0 , (£11 ‘ £12) < 0 Therefore , f1 1 < 112 . (24) 27 Comparisons of equations (14), (15), (17), and (18) show that the slopes of curves® , G) , and @are equal; therefore, the curves are parallel on semi- logarithmic coordinate paper. Determination of Source Strelfl In order to obtain the maximum counting rate with available equipment, it is necessary to consider the rate through the lead absorber and the rate through the substrate material. The rate through the lead is established by the term, Ae—rx, where [a is the total attenuation coefficient of lead, and x is the thickness of lead. The rate through a substrate material of density, D = D(min) in the range of interest (so that A s is maximum within this range), is given by the term, 81A e . in which x is that thickness of lead absorber, x = x which will provide (090’ maximum AAs within the density range. The total count rate, At’ is given by the relation, 28 maul Ham Hm mHa-NHcV 2 «NH I HNH HAM I NHM J< Nfiufifl Hum muc-ama J NWN NH \ m HMH Nu A: «$-Hfl Nam: Ham Ham: Nam Nam 1 Hum NN cuauc H HN J .NMILI 5 04 HwHN N m 53x1 04 II 29 E #3on u x 0323 NN IHNH AHHH I NHHV HNM xm< + “Pat H rm 5: H 3 4 N «NH Hm + la< Nwlgml fl «4 3 «mm -32 m a «we - an in «3.23 w a + a illnma 1:2- 2:1 1 - . anemone. 30 It is seen that equation (25) is the equation of a curve, @ , which is the sum of two separate curves, @and. . These curves can be plotted onlogarith- mic coordinate paper, as shownin Figure 2. In the case of curves®and , the coefficient gives the intersection with the In At —axis at A = 1, and the exponent gives the slope. Once a value of At has been selected as convenient for the equipment to be used, the value of A for this value of At at x = x(opt) can be read from the graph. To determine the indicated source strength to be used, this value of A should be divided by the efficiency of the detection unit in the geometry under consideration. It should be emphasized that the value of r1 in equation (25) has been modified by varying the source-counter distance, and must be determined by experiment. _h_ r In A reaching counter tube, on In desired A, l I l 1 2 In indicated A In A Figure 2. Relation between total comt rate and rate through lead at zero thickness of lead. 31 EXPERIMENTAL DEVELOPMENT Equipment and Procedure A series of laboratory experiments was designed in order to evaluate the optimum thickness of lead absorber, X(opt)3 the magnitude of the functions, f1 1 , f2 1, f1 2, and f2 2; and the amount of the activity A. A nominally five-millicurie sealed cesium 137 source was purchased, to be used in conjunction with aTGC 2 GM counter tube. It was decided to evaluate x(opt) within the range of substrate densi- ties between 60 and 180 lb/cu ft, inclusive. A number of three-inch by nine-inch by one -eighth—inch lead plates were prepared, for use as separators between the source and the counter tube. The assembly was mmmted on a forty-inch by fourteen—inch by three-quarter-inch ply- wood base in such a manner that the number of lead plates, standing upright on their long edges between the source and the counter, could be varied. Rather than keep the source-to-counter distance constant for all numbers of lead absorber plates, this distance was allowed to change with the total lead thick- ness. In this way, for each thickness of lead, the typical geometry ofa gamma-ray surface density gage containing that thickness of lead absorber was approximated. In all cases, provision was made for packing the sandwich of lead plates together as tightly as possible without introducing extraneous reflecting materials. The thick- ness of the plywood base, constant throughout the experiment, was neglected. 33 Samples of substrate materials of different bulk densities were provided, contained in wood boxes possessing more than infinite volume for the field of influ- ence employed, upon which the assembly could be placed for obtaining count rates at various thicknesses of lead absorber. Background activities were substrefted from gross activities in all cases. Total counts ranged from N = 16, 000 to N = 256, 000, depending upon the count rate. Substrate materials used included those shown in Table I. TABLE I SUBSTRATE MATERIALS USED IN EXPERIMENT 1 Material Bulk density, lb/cu ft Concrete 150. 2 Sand 93. 8 Clay 63. 09 Wood 29. 2 Analysis of Results For each determination, net activity in counts per second was plotted on semilogarithmic coordinate paper against total thickness of lead absorber in inches, as shown in Figure 3 for a concrete substrate. That portion of the curve (@) which is a straight line (above approximately two and one=quarter inches of lead) was interpreted as the curve of A s including negligible quantities of Ax. The differ» ence between its extrapolation and the balance of the curve below approximately two and one-quarterinches of lead (curve .) was interpreted to be the curve of Ax only. Activity, ca O I 2 3 4 5 O 7 O 0 l0 ll Thickness of Lead, Inches Figure 3. Laboratory count rate on concrete substrate v_s thickness of lead. 34 35 The intersection of curve with the ln A-axis gives the magnitude of A for the substrate used. This value was found to be 79, 000 counts per second in the case of all substrate materials listed in Table I. The intersection of curve@ extrapolated to the ln A-axis gives the magni- tude of A8 at x = 0 (activity through the substrate only at zero thickness of lead) for the substrate used. The value of A s at x = 0 varied with the substrate material, being 790 counts per secondin the case of concrete, 568 in the case of sand, 410 in the case of clay, and 198 in the case of wood. The fact that curve®has a straight-line portion above x equals approxi- mately two and one-quarter inches of lead when plotted on semilogarithmic coordi- nate paper effectively precludes the possibility that equation (2) have an x-term in its first term, either as a coefficient of or as an exponent of A . Equation of curve@ . For the purpose of developingthe equation ofcurve @ , equation (3) can be written, __ 1 A s — A e , (26) which is the equation of curve@ . When x = 0, the intersection of curve®with the ln A—axis may be deter- mined by the relation, f]. In AS = in A . (27) The slope of curve®is given by the term, f2 . 36 The magnitude of the f functions. It can be seen from equation (27) that at x = 0, 1n AS = _____ 28 f1 1nA ( ) Therefore, in the case of concrete substrate, £1 = 1“ 79° = 0.5916, (29) 1n 79, 000 and the value of f2 may be determined between any two points, as, for example, x = 6andx = 8.7. Then, 2. 30259 - 3. 66356 = . 4O 30 2.7 .050 ( ) f2: Similarly, curves corresponding to curve® for a concrete substrate maybe plotted from data derived from use of the other substrate materials listedin Table I. The values of the functions, f1 and f2, calculated from these data in a manner simi- lar to that outlined in equations (29) and (30), are given in Table H. TABLE II VALUES OF f1 AND f2 FOR SUBSTRATE MATERIALS LISTED IN TABLE I Material f1 f2 Concrete 0. 59 16 -0. 5040 Sand 0. 5623 -0. 4115 Clay 0. 5334 -0. 3442 Wood 0. 4689 -0. 2290 37 A predicted calibration curve. The values of the fimctions, f1 and f2, shown in Table II were plotted on logarithmic coordinate paper against the bulk densities of the substrate materials from which they were derived, as shown in Figure 4. It willbe noted that the values of f1 and f2 for wood fall slightly below their correspond- ing curves, a fact which may be attributed to the probability that the volume of the wood sample used was insufficiently infinite for the geometry employed. The curves for f1 and f2 in Figure 4 were used to obtain values for these flmctions at other densities, in order to establish a quasi-theoretical calibration curve for an instrument designed on the basis of the above data. This predicted calibration curve is shown in Figure 5. Data for deriving the curve are available in the Appendix. 38 0:3. .3350 22.35 58:3 3453—; m 23383 H .2. 0.3mm. 0. rfis . ~P32uo musics.» 85.14513 .52 J_(.ruo ¢w>OU “t. .1....“ . 1... L (on .=(huo «Bus...» 2U:- .cl Im u I». Tl canIIL «u: .50.. 4.2 cu V3 L .3 80.... no; r max...” uHua - H - -1 \lml II :II/ fl‘MII] c .m. .. .7 .7 a fl\ .1 W a] r .aaauao a m------huauul.Huuuquuuuan S; . - IITI. 5. .Il.' I lllll #1 ----i n J - — 42 0 $3882.. smxoom 85 .N. 28E .62: u “I” one. a ozon— .N d .3 in 22284 >Jm§UWm< Fuzwmfl Nab... '— (‘0. ’\ l o o . Figure 8. Operator placing gamma-ray surface density gage for field determination of soil density. 1 N‘s'.‘ -. 3“} ',_:»:::‘ {bfi . ,- .1. N‘x-Co’lz .."-t Figure 9. Gamma-ray surface density gage in use for field determination of soil density. 43 Figure 10. Operator extracting field soil sample for analysis in laboratory. ' ' $3,...“I'QH'5: ‘ - ‘ ‘. 0.0.“. ' O i q o - s Q a‘ n .. ‘ -~“'¢§~.o— ".;*'~r 0:. " ‘ l I. ~ . B can .'~ -- . IlJ-~-—_A..--A ‘ . , . . - . ~ . . . - . . s‘ -- s “ ‘..-O-'.‘.u O‘.‘.‘..go . . . - ‘ ' . o ' ‘ .. s . ‘ ‘ ‘ , - Figure 11. Operator removing soil sample from ground. 44 45 . .. .. ¢.. . .. ~..tenv...-oc.c .‘...oQUo50ss u~sv_.a.a‘. .oo......saooao \ .uoo.pc.....us4.atax--.-.ul- . . I . ~ 00s.... \ .5 o .... . .. .. p . TVs-“Wane... ; a a I _ _ .‘. ussrlc. a?“ O .o .2. .<. . ._..._.m.... _ .. . . . . _.. . Tit? . .. . .. ... . ._,. .~~. , ...-\s.-.. . .. I. . . .. . .... .Q. 1 u.... . .. v— .1 a . s . .0... .‘.”vfi4 .... .ao .‘.. ..’>. . . I. . on. . ,. . .I\~.O§tv¢¢ ._. «H. v . ._ . . . . . . . .. _se _ .. . . _ . ... .. ._. ... . is.“ o 2...... ... n. . _. . . O... ._ .. ._ . .‘. o ’0: .l .. . , I... ........._.. .. : ..... L. . .... ... . .. . . . ~ . J‘. ...... ¢.o.. n....¢.. c . ’54.... ._ . a~.. _.. . _... .‘.‘.<-‘.’.-. a... .—-.a.o-ocflv._%‘s.coo¢oo¢ai. .¢........ . .2. . . p.... Gamma-ray surface density gage in use for deter- mination of count rate on steel (point on experimental calibration curve). Figure 12. 46 field use. A cord length of ten feet was used between the TGC 2 counter tube and the Nuclear model 183 sealer. Laboratory and Field Applications Count rates obtained with the instrument were plotted against bulk densities for a number of materials. Substrates employed included air, wood shavings, wood boards, paper (stacks of identical journals), concrete, steel, and lead; and various soils under several degrees of compaction. The soils included laboratory and field samples. In the laboratory, they were contained in wood boxes measuring twenty by twenty-four inches. These were filled to a depth of approximately six inches and struck off flush with the tops. Control densities were determined by volume and weight measurements on a wet basis. Field gage measurements were made with the soil in situ. After gage measure- ments had been completed, samples were taken at the same sites for conventional density determinations and for laboratory analyses. Occasionally, the soil was tamped, and both types of density determination were repeated to evaluate the effect of tamping on soil compaction. It was found that tamping increased the density of soil No. 21 by approximately sixteen pounds per cubic foot. Conventional density determinations made at the site were based on volume- weight relationships taken immediately, by use of a portable balance and the so- called "can" technique. Both tops and bottoms were removed from twelve-ounce number one cans, and one end of each can was ground sharp. It was found possible 47 to insert the can into the soil with minimum disturbance to a depth of four inches (the depth of the field of influence of the density gage in soil) and to extract asample of sufficient size for independent determinations. By striking the soil off at the bottom of the can with a sharp knife, a right cylinder could be produced, whose height could easily be determined. The volumes of less cohesive soils were measured by filling the cavity with 1miformly graded Ottawa sand. Figure 13 shows the calibration curve for the density gage based upon experi- mental determinations in the laboratory and in the field. It can be seen that this curve agrees fairly well with the theoretical calibration curve shown in Figure 5. The degree of similarity between the two curves is an indication of the degree of validity of the basic assumptions contained in equation (2) with respect to the develop- ments here reported. Experimental evidence indicated, as noted above, that the depth of the field of influence in soils was about four inches. In concrete, the depth was three and one-half inches, and in air it was about twenty inches. A procedure adepted for plotting points on the experimental calibration curve consisted of referring all count rates to a selected count rate with the gage suspended in air. The selected rate in air included the background rate at the time it was determined. As subsequent determinations in air fell above or below the selected rate, due to variations in back~ gromid activity, all other rates on substrates determined at. approximately the same time were raised or lowered accordingly. 48 13.5.2.8 gum 2750 noflmunfiao FEoEEonm .2 8&3 8a 8a. 8m 8.». 3. l3... £7 / :2. l/ . ducal”. / 35523 IA T. o _ laoz_><:n on coo; O O 4. 356m .1 I cook 36» L m... 0 Kb an _ m cunt ~\ 0 II E rIIT o cow 8.. 8: 3o coo. 8m 8.. W H. M. I. coo .A n 000 m Sam 8.... 8.. can 8o 82.. OON SUMMARY A survey of the literature on the development of the gamma-ray density gage disclosed a variety of conflicting reports. Investigators disagreed as to the best design, the best type and strength of source, the extent of the field of influence, the effect of soil type, the ease of making determinations, the sensitivity of the method, and the use of reference standards. As a result of the survey, it was felt that a fundamental approach to the entire subject would have to be considered. Certain basic assumptions were made on theoretical grounds only, related to known interactions of gamma radiation with matter. An equation was developed from these assumptions and subjected to rigorous treatment. Expressions were obtained stating the best thickness of lead absorber, the relation between density and total linear attenuation coefficient, certain limitations on the use of standards, and the effect of source strength. The magnitudes of certain functions contained in the equation were measured experimentally, and a predicted calibration curve was estab- lished for agamma-ray surface density gage designed on the basis of the equation. A gage was constructed, based upon the fundamental assumptions, designed to operate most efficiently within a range of densities between 60 and 180 pounds per cubic foot. A nominally five-millicurie sealed source of cesium 137 was used in the gage, separated from a TGC 2 counter tube by 6. 0 inches of lead. Weight of the gage was ten pounds. 50 The gage was used to measure densities of various materials which ranged in bulk density from that of air to that of lead. The experimental calibration curve for the instrument was compared with the predicted calibration curve, and the extent of similarity between the two curves was taken as indicative of the degree of validity of the assumptions underlying the fundamental equation. CONCLUSIONS It would appear on the basis of experimental evidence cited that the following conclusions are justified; 1. Information of assistance in the designing of a satisfactory gamma-ray surface density gage can be obtained by the assumption that f1(D, E) f2(D, E) x AS = g(D, E) A e . (2) 2. The best thickness of lead absorber separating the source and Geiger- Mliller counter tube of a gamma-ray surface density gage is given by the relation, [ f12 ‘ fli’ 1“ (f22g2/f21g1)A . (8) x :— (opt) - f2 1 f2 2 For a gage employing five millicuries of cesium 137 and a TGC 2 Geiger-Muller counter tube, the value of x(opt) is 6. 348 inches when the gage is intended to operate at maximum efficiency withina range of densities between 60 and 180 pounds per cubic foot. 3. The ratio of activities through any two substrates of different densities, when optimum thickness of lead absorber is used, is equal to the reciprocal of the ratio of the total linear attenuation coefficients of the two substrate materials. This relation holds true at no other thickness of lead. 52 4. Use of a "count-in-soil to count-in-standard" ratio will not eliminate the necessity of subtracting background count rates, nor the need for periodic recalibra- tions of a gamma-ray surface density gage; it will merely serve to reduce the requir- ed frequency of recalibration. Use of such a ratio is not justified with long-lived sources of gamma radiation. Its use with short-lived and medium-lived isotopes is of questionable value. 5. The indicated source strengthto be employed can be determined by divid- ing the value of A by the efficiency of the detection unit in the geometry to be used. The required value of A can be ready directly from a plot of At vs A (Figure 2), where the value of A(t0t) represents the desired maximum total count rate for the instrument. 6. Rate meters employing linear scales can be used with maximum efficiency in conjunction with gamma-ray surface density gages only if adjusted so that, when comparing the densities of two materials, the greater activity is read at the upper scale limit. The superiority of scaling equipment over count rate meters for use with density gages appears to be indicated. 7. The need for a commercially available, portable, battery-operatedscaler of modest cost and of rugged, though light, design is definitely indicated. BIBLIOGRAPHY 1. Belcher, D.J.; Cuykendall, T. R.; and Sack, H. S. ; The Measurement of Soil Moisture and Density by Neutron and Gamma Ray Scattering. Technical Development Report No. 127, Civil Aeronautics Administration Technical Develop- ment and Evaluation Center, Indianapolis, Indiana, October, 1950. 2. Belcher, D. J .; Cuykendall, T. R.; and Sack, H. 8.; Nuclear Meters for Measuring Soil Density and Moisture in Thin Surface Layers. Technical Develop-_ ment Report No. 161, Civil Aeronautics Administration Technical Development and Evaluation Center, Indianapolis, Indiana, February, 1952. 3. Carlton, Paul F.; Belcher,. D. J. ; Cuykendall, T. R.; and Sack, H. 8.; Modifications and Tests of Radioactive Probes for Measuring Soil Moisture and Density. Technical Development Report No. 194, Civil Aeronautics Administration Technical Development and Evaluation Center, Indianapolis, Indiana, March, 1953. 4. Vomocil, J. A. . "In Situ Measurement of Soil Bulk Density, "Agricultural Engineering, Vol. 35, No. 9, September, 1954, pp. 651-4. 5. Berdan, D.; and Bernhard, R. K.; PilotStudies of Soil Density Measure- ments by Means of X~Rays. Presented at the Fifty-third Annual Meeting of the American Society for Testing Materials, June 26-30, 1950. 6. U. S. Army Corps of Engineers, Field Tests of Nuclear Instruments for the Measurement of Soil Moisture and Density. Miscellaneous Paper No. 4—117, Vicksburg Infiltration Project, Forest Service, U.S. Department of Agriculture, for Waterways Experiment Station, Vicksburg, Mississippi, March, 1955. 7. Hosticka, Harold E.; "Radioisotopes and Nuclear Reactions Applied to Soil Mechanics Problems, " Symposium on the Use of Radioisotopes in Soil Mech- anics, ASTM Special Technical Publication No. 134, American Society for Testing Materials. Presented ata meeting of Committee D- 18 on Soils for Engineering Pur- poses, Cleveland, Ohio, March 5, 1952. 8. Bernhard, R. K. ; and Chasek, M. ; Soil Density Determination by Direct Transmission of Gamma Rays. Presented at the FiftyeeighthAnnual Meeting of the American Society for Testing Materials, June 26~July 1, 1955; 18 pp. 9. Miles, M. E. ; Energy Distribution of Gamma Rays Scattered Around a Soil Density Probe. Master's Thesis, Cornell University, June, 1952. 54 10. Goldberg, Irving; Trescony, Louis J.; Campbell, James 8., Jr.; and Whyte, Gordon J.; Measurement of Moisture Content and Density of Soil Masses Using Radioactivity Methods. Paper prepared for presentation at the 1954 Pacific Coast Regional Conference on Clays and Clay Technology, June 25-6, 1954, Univer- sity of California, Berkeley. 11. Horonjeff, Robert; Goldberg, Irving; and Trescony, Louis J.; The Use of Radioactive Material for the Measurement of Water Content and Density of Soil. Paper prepared for presentation at the Sixth Annual Street and Highway Conference, February 3-5, 1954, University of California, Los Angeles. , 12. Evans, Robley D. The Atomic Nucleus. (p. 711) New York; McGraw- Hill Book Company, Inc. , 1955. 972 pp. 13. Friedlander, Gerhart, and Joseph W. Kennedy. Introduction to Radio- chemistry. (p. 170) New York: John Wiley 8: Sons, Inc. , 1949. 412 pp. APPENDIXES APPENDIX A GRA PHS FOR DETERMINATION OF f1 AND f2 FUNCTIONS FOR CONCRETE, SAND, CLAY, AND WOOD SUBSTRATES Activity 013 I000 900 000 700 5W?" 400*- 0 30m- C8137on concrete (.900 January 30, 1956 200'— Density = 150.21bs/ft3 A0 = 79,000 c/s lnA lnA 1 = % =0. 5916389 = fl Pb(x=0) . 76 100— 9°— T—‘Afil = 0.01 got. 70— Pb