MONTE CARLO SIMULATION OF COMPTON BACKSCATTERING AT THE COMET NUCLEUS SURFACE

Transcription

1 Republic of Iraq Ministry of Higher Education and Scientific Research University of Baghdad College of Science Department of Astronomy and Space Science MONTE CARLO SIMULATION OF COMPTON BACKSCATTERING AT THE COMET NUCLEUS SURFACE A THESIS SUBMITTED TO THE DEPARTMENT OF ASTRONOMY AND SPACE SCIENCE COLLEGE OF SCIENCE, UNIVERSITY OF BAGHDAD IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE IN ASTRONOMY BY Sarah Salah AL- Hussany B.Sc. 28 Supervised by: Ass. Prof. Dr. Alaa B. Kadhim 211 AD 1432 HD

2 هللا الرحمن الرحيم (و ق ل ر ب ا د خ ل ن ي م د خ ل ص د ق و ا خ ر ج ن ي م خ ر ج ص د ق و اج ع ل ل ي م ن ل د ن ك س ل ط انا ن ص يرا ) هللا العلي العظيم (سورة الا سراء: ۸۰)

3 Supervisor Certification I certify that this thesis is prepared by Sarah Salah Al Hussany under my supervision at the Department of Astronomy, College of Science, University of Baghdad as a partial fulfillment of the requirements needed to award the degree of Master of Science in Astronomy. Signature : Name : Dr. Alaa.B. Kadhim Title : Ass. Professor Address : Department of Astronomy, College of Science, University of Baghdad Date : / / 211 Certification of the Head of the Department In view of the available recommendation, I forward this thesis for debate by the examining committee. Signature : Name : Dr. Kamal M.Abod Title : Ass. Professor Address : Head of Astronomy Department, College of Science, University of Baghdad Date : / / 211

4 Examining Committee We, members of the Examining Committee, certify that after reading this thesis and examining the student (Sarah Salah Al Hussany) in its contents and, in our opinion that it is adequate for the award of the degree of Master of Science in Astronomy. Signature: Signature: Name: Dr. Mazin M. Elias Name: Faisel Ghazee Mohammed Title: Professor. Title: Ass. Professor. Date: / /211 Date: / /211 (Chairman) (Member) Signature: Signature: Name: Salman Sidan Khalaf Name: Dr. Alaa B. Kadhim Title: Ass. Professor. Title: Ass. Professor. Date: / /211 Date: / /211 (Member) (Supervisor) Approved by the University Committee of Graduate studies Signature: Name: Prof. Dr. Dean of College of Science Address: University of Baghdad, College of Science Date: / /211

5 Indeed, all praise is due to Allah for His countless blessings, we praise Him,seek His aid and asked His forgiveness,we thank Him for granting me success and facilitate my mission to complete this project. I feel deeply indebted to my Supervisor, Dr. Alaa B. Kadhim for proposing this project, and for the support and patience continuously exhibited since we have first moment met, my thanks are for his guidance. I also would like to thank everyone that help me in this research in the departments of Astronomy, Physics and Chemistry. Also thanks are due to the Dean of the College of Science and the student affairs of the College. Sincere appreciations are due to :my family,the people who have kept supporting me unconditionally. I owe it to them. I shall always be grateful to the University of Baghdad for granting me this chance to upgrade my scientific certificate. Sarah

6 Abstract Monte Carlo simulation of scattering and absorption in bulk materials has been used to investigate the design parameters specially the variation of backscattered count rate with density and composition. Gamma backscatter densitometer use the Compton scattering of gamma ray photons in bulk material. A source of gamma photons 662 kev from a collimated 137 Cs and 6 kev from a collimated 241 Am source are placed at the surface of the bulk sample to inject gamma photons into material. A detector is placed a short distance along the surface from the source to count photons scattered out of the material shielding the source to prevent photons from reaching the detector directly. A FORTRAN program has been written to model the interaction of photons in a bulk material. The elements and compositions of a comet material have been used such as (H2O, C, SiO 2, Mg 2 SiO 4, and Fe 2 SiO 4 ). The results of calculation which have been obtained are compared with the published results. It appears in a good agreement with behavior. The backscatter and attenuation techniques have been merged in one geometry arrangement, consider that the new and not work previous and appeared the results of tests very good. i

7 List of Symbols SYMBOL Rv Ee Eb hv Random Variables Energy of photoelectron Binding energy of photoelectron Energy of incident photon µ Linear attenuation coefficient ρ µ/ρ τ σ κ Wi I I E Ē Density of medium Mass attenuation coefficient Probability of photoelectric effect Probability of Compton scattering Probability of pair production Weight fraction of element Number of transmitted photons Number of incident photon Energy of incident photon Energy of scattered photon DEFINITION µ1,µ2 Mass attenuation coefficient of photon in the material before and after scattering m oc 2 α,β φ r1,r 2 d R D Electron mass energy =511 kev Angles of primary and scattered photon paths to the surface The angle of scattering,equal to α+β Length of primary and scattered photon paths Source detector separation (sonde length) Radius of detector ii

8 CONTENTS SUBJECT Abstract List of symbols Contents PAGES I Ii v-vi Chapter One: General Introduction 1 Introduction The Comet Structure of Comets The Nucleus The Coma The Tail The Origin of Comets Previous work Aim of the work Thesis layout 8 Chapter Two: Principle of the Model 1 Introduction 2-1 Monte Carlo Method Probability Random Variables Sampling Technique Interaction of Gamma rays Interaction Mechanisms 14 v

9 2-2-2 Gamma ray Attenuation Select Geometry for the System Compton Backscatter Technique Attenuation Technique Gamma Backscatter Density Gauge Single Scattering Model(SSM) Parametric Equation 24 Chapter Three :Monte Carlo Simulation 25 Introduction FORTRAN PROGRAM 32 Chapter Four: Results and Discussion Results of calculation Composition Effect Comparison 61 Chapter Five: Conclusions and Future Work Conclusions Future Work 71 References 72 Appendices 76 vi

10 Introduction Chapter One General Introduction A Compton backscatter densitometer has been proposed for the Roland probe in order to measure the bulk density of material near the surface of comet nucleus. Roland is to be employed from a spacecraft in orbit around the target comet, as part of international Rosetta mission. The Roland design aims to employ a lower energy radioisotope source so that a light weight detection system can be used. Monte Carlo simulation of scattering and absorption in semi infinite bulk material has been used to investigate the design parameter, specifically the variation of backscattered count rate with density and composition comet nuclei are thought to contain primordial material which has remained relativity unchanged since the formation of solar system. The method of Compton backscattering densitometry has been chosen in preference to other methods of measuring density because the surface gravity of target nucleus is both uncertain and very low, making it very difficult to weigh a sample of known volume reliably to obtain the local density of a surface material. Because the nature of comet nucleus material is brittle, sampling a known volume of material particularly difficult. On the earth gamma ray density gauges have been in use since early 195,in fields such as geophysics, soil science and manufacturing and constructing industries [1-5]. 1

11 1.1 The Comets Comets are primitive conglomerates of solar system containing mostly a mixture of ice water, snow and dust [6]. When a comet is far from the sun, it is inert ice and snow. As it approaches the sun, heat causes ices in the nucleus to sublimate, creating a cloud of gas and dust known as the coma. Sunlight and solar wind will push the coma gas and dust away from the sun, resulting is the typical long tailed shape of the comet that is always pointing away from the sun [7]. 1.2 Structure of Comets Comets consist of three main parts: The Nucleus Comet nuclei are primitive bodies or in the solar system containing a mixture of frozen gases and particles of refractory silicates (Mg 2SiO 4, Fe 2 SiO 4 ) and complex organic molecules [8]. Table (1-1) shows the organic compound in the comet and relative abundance for approximate nucleus diameter 1 kilometers with masses in the range ( kg). Density estimates range between 2-15 kg.m -3 that reasonable densities was taken as arrange over which a nucleus surface densitometer must operate [9]. The nucleus of the comet has been described as a giant "iceberg" or "dirty snowball and contains most of the comet's mass [1] The Coma A spherical cloud of gas and dust surround the nucleus is called coma [11].Figure (1-1) show the nucleus of comet is a few kilometrs across,while the coma is about 1,-1, km wide[12]. 2

12 Figure (1-1): Structure of the comet [12]. Much of the science interest in comets comes from their significant role in explaining the processes responsible for the formation and evolution of the solar system. Bulk density is one of the important physical properties of cometary nucleus. The measurements of bulk density increase our knowledge about the formation and evolution of the solar system[13]. 3

13 Table (1-1): Organic compounds in comets and molecule relative abundance [14]. Compounds H 2 O CO CO 2 CH 4 C 2 H 2 C 2 H 6 CH 3 OH H 2 CO HCOOH CH 3 CHO NH 3 HCN HNC HNCO CH 3 CN HC 3 N H2S CS 2 CS SO 2 SO OCS H 2 CS Relative Abundance %

14 1.2.3 The Tail A comet tail is its most distinctive feature. As it approaches the sun it develops an enormous tail of luminous material that extends for millions of kilometers away from the sun. The intense solar wind from the sun (stream of charged particle mostly electron and protons generated by the sun) interact strongly with gases that exists in comet head, producing ionized gases and form the ions tail. It also called plasma tail because it is made of ions and electrons. It appears blue because its light is dominated by emission from carbon monoxide ions. The color of plasma tail is blue is produced by ionized carbon monoxide, which emits strongly in blue part of spectrum. The ion tail pushed away of the sun by the solar wind and extends of millions of kilometers. While the dust particle that exists in the nucleus is pushed outward by the force of solar wind and that reason of dust tail being pointing away from the sun. It appears yellowish because the dust particles are being scattered by the dust tail has a length less plasma tail because the velocities of the particles of the dust tail are slower than the velocities in the ions tail, the dust tail is more curved than the ion tail [6,7]. 1.3 The Origin of Comets Dutch astronomer Jan Hendrik Oort noted in 195 the source of most comets come from the Oort cloud[7] which he proposed to be a spherical region that completely surrounds the solar system and extends to perhaps as much as 1, AU from the sun. Astronomers thought the comet in the Oort cloud is swarm of trillions of icy bodies and lies far beyond the orbit of Pluto and the origin of long period comet is Oort cloud. The long period comets have cycles much longer than 2 years. These comets enter the solar system by passing stars that disturb the orbits of some of the comets from the Oort cloud and create new comets. Some come from 5

15 a disk like swarm of icy objects that lies just beyond the orbit of Neptune about the sun and extend to 1 AU from the sun, a region is called Kuiper Belt, which is a reservoir for short-period comets (those making complete orbits around the sun in less than 2 years) such as Halley which has a period of 76 years. It has visited the solar system in 1986 [1]. 1.4 Previous Work The ratio of mass to volume yields the average density of cometary nuclei. In practice, the density of comet nucleus is very difficult to determine because both comet's mass and volume are so uncertain, attempts to measure the mass of cometary nuclei. Whipple in 195[15] was the first to suggest that tail produced by a comet activity could be used to estimate its mass. Rickman in1986 used this method to estimate the bulk density of comet Halley about (1-2 Kg.m -3 ) [16]. Taylor and Kansara 1967 suggested a method to measure the density of soils and found that the calibration curve connecting density and instrument response varies with soil type and certain other factor[17]. The characteristics of gamma ray backscatter density have been encountered in a new study in 1969 by K. Lin et al. and found that there is a good correlation between the count rate and density [18]. Devlin and Taylor 197 investigate the characteristic curves for backscatter gauges of varies geometries[19]. Gardner et al.1971 put the major sources of error of gauges in gamma backscatter density as due to sensitivity to soil composition, surface roughness and counting rate measurement errors[2]. 6

16 Rickman 1989 revised his earlier value for Comet Halley nucleus density and re-estimated the density to kg.m -3 [21]. Festou and et al. 1992, from observation of p/temepl 2 from 1899 to 1988 cross pounding to 13 apparitions they were able to estimate the density and found it to be 3 kg.m 3 [22]. Sphaug and Benz 1996, showed that the tidal breaks up and reassembly of comet Shoemaker-Levy 9 into~21 major fragments in 1992 providing another means for indirectly estimating the bulk density of the nucleus, yielding values about 6 kg.m -3 [23]. Benkhoff and Huebner (1995) showed that bulk density can not only decrease due to the loss of volatiles but can also increase just below the sublimation fronts of volatile components, due to the inward transport and condensation of volatiles[24]. Ball et al 1996 Monte Carlo simulation of scattering and absorption in semi infinite bulk materials they used to investigate the design parameter specifically the variation of count rate with density [5]. Skorov and Rickman1999 found that a correction factor for the local momentum transfer correction is approximately 1.8 relative to the published paper of Rickman. The density range was about 5 12 kg.m -3 [25]. Harmon et al 1999 could use radar albedo of small number of comet nuclei to derive surface bulk material density in the range 5 to 1 kg.m -3 [26]. Ball et al 21 they report the measurement of the bulk density instrument and the results of related Monte Carlo simulations, laboratory tests and calculations of the instrument s performance [13]. 7

17 Spohn et al 27 showed that MUPUS, the multi Purpose sensor package onboard the Rosetta lander PHILAE, will measure the energy balance and the physical parameters in the near-surface layers up to about 3 cm depth- of the nucleus of Rosetta s target comet Churyumov- Gerasimenko. Moreover it will monitor changes in these parameters over time as the comet approaches the sun. Among the parameters studied are the density, the porosity, and temperature[27] 1.5 Aim of the Work The aim of the recent research work is to investigate the design parameter, more specifically the variation of count rate with density and composition. There are some studied parameters that are very important to give a suitable geometry such as the following : 1. Energy spectrum of photons in the primary beam. 2. Geometrical arrangement of the radiation source, attenuating material and detector. 3. Material layer composition. 4. Type of calculation response. In the present research work calculation of the variation count rate with density was taken in two techniques namely, Compton backscatter densitometer and attenuation densitometer. 1.6 Thesis Layout In this research five chapters are involved. In chapter one after the introduction of research, explanation of the idea of technique was given, 8

18 then introduction of comets and then aim of the work after we review of previous researches about the subject of research. Chapter Two contains the most important theoretical and physical basis that have been adopted in the program. Chapter Three involves the general and complete details of the program used in this research and the flowchart for all the steps in the program. In Chapter Four the results of the program are discussed (tables and charts) of different geometric sites of the source and detector to ratio of distance and depth between them and test the accuracy of technical calculations and results. Finally, a verification of the program was completely made to confirm the results. Chapter Five contains the conclusions of this research and future work. 9

19 Chapter two The Theoretical Principles Introduction The main function of the program has been written to model the interaction of photon in bulk material. Several issues required investigation, including the Monte Carlo method to describe the interaction of photons with matter and geometrical arrangement considered in the calculation dimension of the detector and density of elemental composition. The main way of the work program for this research is to follow the path of the photon until detected or absorbed or escaped from the material. 2.1 Monte Carlo Method The term "Monte Carlo" was introduced by the American mathematicians Neumann and Ulam after gambling casinos at the city of Monte Carlo in Monaco in 1949 [28]. In the former Soviet Union, the first articles on the Monte Carlo method were published in The Monte Carlo method is a method of approximately solving mathematical and physical problems by the same mathematical and physical variables as some random numbers. The theoretical basis of the method has long been known. In the nineteenth and early twentieth century, statistical problems were sometimes solved with the help of random selections, that is, in fact, by the Monte Carlo method. Thus called sometime the method of statistical trials. Prior to the appearance of electronic computers, this method was not widely applicable since the simulation of random quantities by hand is a very laborious process. Thus the beginning of the 1

20 Monte Carlo method as a highly universal numerical technique became possible only with the appearance of computers. [29] The Monte Carlo method can be used not only for solution of statistical problems, but also solution of deterministic problem. A deterministic problem can be solved by the Monte Carlo method if it has the same formal expression as same statistical process [3]. Monte Carlo simulation can be employed to model an inserted attenuation densitometer. While the attenuation of primary photons is easily calculated analytically, The Monte Carlo method can also model the scattered radiation.the detection and counting system of the attenuation densitometer should aim to avoid counting too high a proportion of scattered photon, so it is useful to characterize the problem they pose. by Monte Carlo method, one can write a program to follow the scattering and absorption histories of a large number of photons (emitted by a point source) in a semi infinite bulk material of uniform bulk density. The probability and random variable, which are bases of mathematical model of this research Probability Suppose that an experiment was repeated N of times and M represent the number of times the success of this experiment. Thus the probability of success of this experiment when N is approaching to infinity is [31]: The probability of failure is: P= M.. (2-1) N Q= (N M) N =1- M N (2-2) 11

21 On the condition that (P+Q=1) the modern treatment of the theory of probability begins from the group axioms to build a mathematical model. If one assumes that S represents the sample space, p(e) represents the probability of occurrence E then the conditions that verify the hypotheses will be as following[32]: 1) p (E) 1 2) P(S) =1 3) Let E1 and E2 be mutually exclusive. The probability that either E1 or E2 occurs must then be equal to the sum of their respective probabilities: P [E1 + E2] =P [E1] +P [E2] There is a possible probability called conditional probability based on the selection of a random number. The probability can be calculated in accordance with the terms of certain, and then try to test whether these conditions are achieved? If yes, the experiment has succeeded and continues accounts. If not achieved, the experiment should be repeated by choosing a new random number and retest the previous Random Variables The random variable refers to the outcome of any process which proceeds without any discernible aim or direction [28]. There are two types of random variables [33]: 1) Discrete Random Variables The random variable (RV) is called discrete if it can assume any of a discrete set of values (rv1, rv2,.rvn) where (rv1, rv2,.rvn) are the possible values of the variable (RV) and (p1, p2,.pn) are the 12

22 probabilities corresponding to them. The numbers (rv1, rv2,.rvn) are arbitrary. However, the probabilities (p1, p2,.pn) must satisfy two conditions: a) All (pi) are non negative: p i, i = 1,2,.. n b) The sum of all the (p i ) equals 1: P 1 +P 2 +.P n =1 The expected value of the random variable (RV). E(RV)= n i (rv) i p i n i p i (2-3) 2) Continuous Random Variables The random variable is continuous if any value in some continuous interval. The first type of these variables has been achieved in the mathematical model presented where it was processed according to the condition and assumption of mathematics Sampling Technique Based on the technology preview to simulate the statistical operations on the random number (R) which are distributed regularly <R<1 by the random numbers it is possible to find a random variable X of the probability distribution f(x) by the relationship [34]: R = X f(x)dx f(x)dx (2-4) at X= R=1, at X=- R= 13

23 2.2 Interaction of Gamma Rays with Matter Although a large number of possible interaction mechanisms are known for gamma rays in matter, only three major types play an important role in radiation measurements: photoelectric absorption, Compton scattering, and pair production. All of these processes lead to the partial or complete transfer of the gamma ray photon energy to electron energy. They result are sudden and abrupt changes in the gamma ray photon history, so that the photon either disappears entirely or it is scattered through a large average angle Interaction Mechanisms 1. The Photoelectric Absorption: In the photoelectric absorption process, an incoming gamma ray photon undergoes an interaction with an absorber atom in which the photon completely disappears. In place, an energetic photoelectron is ejected by the atom from one of its bound shells as shown in figure (2-1). The interaction cannot take place with a free electron but with the atom as a whole. For gamma rays of sufficient energy, the most probable origin of photoelectron is the most tightly bound or K shell of the atom. The photoelectron appears with an energy given by [35] : Ee = h u -Eb.(2-5) Where Eb represents the binding energy of the photoelectron in its original shell. For gamma ray energies of more than a few hundred kev the photoelectron carries off the majority of the original photon energy. 14

24 De Excitation X rays Incident Photon Photoelectron from K shell Figure (2-1): Photoelectric Effect [35]. 2. Compton Scattering The interaction process of Compton scattering takes place between the incident gamma ray photon and an electron in the absorbing material. It is most often the predominant interaction mechanism for gamma ray energies typical of radioisotope source. In Compton scattering the incoming gamma ray photon is deflected through an angle θ with respect to its original direction. The photon transfers a portion of its energy to the electron (assumed to be initially at rest), which is then known as a recoil electron as shown in figure(2-2). Because all angles of scattering are possible, the energy transferred to the electron can vary from zero to a large fraction of the gamma ray energy. The expression which relates the energy transfer and the scattering angle for any given interaction can be simply derived by writing simultaneous equations for the conservation of energy and momentum.using the symbols defined in the sketch below [35]: 15

25 Figure (2-2) : Compton scattering. It can be shown that where: E g hv = hv = 1+ g ( 1- cosq ) g = hv 2 m c... (2-6) and 2 m c is the rest mass energy of the electron (.511 MeV). For small scattering angles θ, very little energy is transferred. Some of the original energy is always retained by the incident photon, even in the extreme of θ=π. 16

26 3. Pair Production If the gamma ray energy exceeds twice the rest mass energy of an electron (1.2 MeV), the process of pair production is energetically possible. As a practical matter, the probability of this interaction remains very low until the gamma ray energy approaches twice this value,and therefore pair production is predominantly confined to high energy gamma rays. In the interaction (which must take place in the coulomb field of the nucleus), the gamma ray photon disappears and is replaced by an electron positron pair. All the excess energy carried by the photon above the 1.2 MeV required to create the pair goes into kinetic energy shared by the positron and electron. Because the positron will subsequently annihilate after slowing down in the absorbing medium, two annihilation photons are normally produced as secondary products of the interaction. The subsequent fate of this annihilation radiation has an important effect on the response of gamma ray detectors as shown in figure (2-3) [35]. γ > 1.22 MeV Figure(2-3): Pair Production [36]. 17

27 Figure (2-4): The relative importance of the three major of gamma ray interactions. The lines show the values of Z and h v for which the two neighboring effects are just equal [36] Gamma ray attenuation Attenuation Coefficients When monoenergetic gamma rays are collimated1 into a narrow beam and allowed to strike a detector after passing through an absorber of variable thickness, the result should be simple exponential attenuation of the gamma rays. Each of the interaction process removes the gamma ray photon from the beam either by absorption or by scattering away from the detector direction, and can be characterized by a fixed probability of occurrence per unit path length in the absorber. The sum of these probabilities is simply the probability per unit path length that the gamma ray photon is removed from the beam [35]: μ = τ(photoelectric) + σ(compton) + k(pair) (2 8) 18

28 where µ is called "the linear attenuation coefficient". The number of transmitted photons I is then given in terms of the number without an absorber (I o ) as : where x is thickness of absorber I Io = e μx 1..(2-9) The gamma ray photons can also be characterized by their mean free path λ, defined as the average distance traveled in the absorber before an interaction take place. Its value can be obtained from λ = xe μx dx e μx dx = 1 µ..(2-1) and is simply the reciprocal of the linear attenuation coefficient. Typical values of λ range from a few mm to tens of cm in solids for common gamma ray energies. The use of the linear attenuation coefficient is limited by the fact that it varies with the density of the absorber, even though the absorber material is the same. Therefore the mass attenuation coefficient is much more widely used and is defined as : mass attenuation coefficient = μ.. ( 2-11) ρ where ρ represents the density of the medium. For a given gamma ray energy the mass attenuation coefficient does not change with the physical state of a given absorber. For example it is the same for water whether present in liquid or vapor form. The mass attenuation coefficient of a compound or mixture of elements can be calculated from: μ compound = w ρ i( µ i ) ρ i..(2-12) 19

29 Where the w i factors represent the weight fraction of element i in the compound or mixture. 2.3 Select Geometry for the System It is necessary to take into consideration that the variation of backscattered count rate depends on: a) Energy of photons. b) Geometrical arrangement of the radiation source, attenuation material and detector. c) The density and composition of Materials. d) Type of calculation response. There are three kinds of geometries as a follow: 1) Compton backscattered densitometer. 2) Gamma backscatter density gauge. 3) Attenuation densitometry Compton Backscattering Technique: The Compton backscattering technique relies on the detection and analysis of Compton scattered photons at the surface of bulk material that is being irradiated by a source placed at the surface of the bulk sample to inject gamma photons into material. A detector is placed a short distance along the surface from the source to count photons scattered out of the material. Shielding of the source prevents photons reaching the detector directly. 2

30 Sonde length Source shield Detector R1 R2 Incident photon Scattered photon φ Bulk material p Recoil electron Figure (2-5) Basic geometry for a Compton backscattering densitometer. The shield prevents photons reaching the detector directly. Sl=sonde length (distance between source and detector) The backscattering technique is useful for semi infinite bulk material ( such as soil and concrete surface)or boreholes where linear geometry of Source, sample and detector is not achievable it may be difficult to dig out a simple or insert a probe without changing the bulk density [5]. 21

31 2.3.2 Attenuation Technique The attenuation technique relies on the measurement of the change in detected count rate from a source when a material (of a known thickness) is introduced in between. This technique requires no moving parts other than deployment to the surface of the material. Figure (2-6): Basic geometry of attenuation densitometry. The measurement relies on detection of unscattered photons (1), through singly (2) and multiply (3) scattered photons (of lower energy) also enter the detector as contamination hence the need for energy discrimination. Most of the emitted photons are of course lost by scattering and absorption (4) [13] Gamma backscattering density gauge In this geometry a detector is placed a short distance along the surface from the source to count photons scattered out of the material. Shielding of the source prevents photons reaching the detector directly as in figure (2-7). 22

32 Figure 2.7. Diagram showing the basic geometry of gamma backscattering density gauges. Emitted photons are either 1) detected having undergone the surface by single a) or multiple b)scattering in the material,(2) lost through the surface by single or multiple scattering in the material 3) lost by scattering and absorption in the material, or 4) stopped by the source shielding [37 ]. 2.4 Single Scattering Model (SSM) The SSM has been used for many years to explain the basic behavior of backscatter densitometers. Monte Carlo methods are preferred for modeling real devices but the SSM can nevertheless to examine basic features of the measurement technique. The SSM assumes that photons reaching the detector have been scattered only once in the material. this assumption is not valid for real instruments which have large source detector separations or operate on particularly high densities (the upper limit scale length for single scattering being the attenuation length of source photons in the material). However, the competition between scattering and absorption must still exist outside the dominate of the SSM. Hence one may expect the SSM retain some qualitative importance even in the multiple scattering regimes [37]. 23

33 2.6 Parametric Equations The parametric equation of a straight line is : x = x o + at r = ro + At or y = y o + bt (2 19) z = z o + ct The parametric equation has a particularly useful interpretation when the parameter (t) means time. Consider a particle m(electron or billiard ball) moving along the straight line L in figure (2-8), position yourself at the origin and watch m move from po to p along L. your line of sight is the vector r; it swings from ro at t = to r = ro + At at time t.the velocity of m is dr = A; A is a vector along the line of motion[38]. dt m Po ro p L z r y x Figure (2-8): A particle m moving from p O to p along the line L. 24

34 Chapter Three Gamma Backscattering Simulation Introduction The ideas mentioned in the previous chapters led us to carry out the computer program backscatter gamma rays designated for the calculation in the geometrical arrangement of Compton backscattered densitometer as shown in Figure 3.1. A source of gamma photons (usually 662 kev from a collimated 137 Cs source ) is placed at the surface of the bulk sample to inject gamma photons into the material. A photon detector with radius (RD) is placed a short distance (d) along the surface from the source to count photons scattered out of the material. Shielding of the source prevents photons reaching the detector directly. Sonde length (d) x source detector shield z α Incident photon µ 1,r 1 R3 β Φ µ2,r2 Scattered photon p Recoil electron Figure (3.1.A): Basic geometry for a Compton backscattering densitometer. 25

35 Sonde length (d) source S x d-x D z α Incident photon r 3=y 2 + z 2 β Φ µ2,r2 Scattered photon µ 1,r 1 Recoil electron Figure (3.1.B): showing the photon SPD through the material from the source S to the detector D. A general path for singly scattered photons is shown in figure 3.1.B, the direction of the emitted photon being at an angle α to the baseline SD. Compton scattering is assumed to occur at a point p in the material, though some proportion of the photons may not reach p, having undergone absorption or scattering somewhere along the path SP of length r 1. Those photons scattered at P towards the detector make an angle β with the baseline,and may of course be lost along the path PD (of length r 2 ). The material under investigation in assumed to be of a uniform density ρ. The mass attenuation coefficient μ for photons is a function of their energy. Since one assumes a mono-energetic source (such as the most commonly used radioisotope 137 Cs which emits at 662 kev), The attenuation coefficient μ1 for primary photons was fixed at about m 2 Kg 1 of SiO 2 for example. However the mass attenuation coefficient for scattered 26

36 photons μ2 varies with scattering angle φ, since the scattered photon energy Ē is related to φ by the Compton formula(equation 2-6). For computing μ2 an energy dependence approximated by a cubic function fitted to tabulate Compton cross-section data were used for the appropriate (H 2 O, C, SiO 2, Mg 2 SiO 4, Fe 2 SiO 4 ) [35, 4, 41]. The fit was done in log-log space so the function μ(e) was of the following form: log 1 μ(e) = a1 + a2 log 1 E + a3 (log 1 E) 2 + a4((log 1 E) 3 (3 1) Where μ(e)is the appropriate mass attenuation coefficient (in m 2 Kg 1 ),E is reduced energy (The reduced energy is the photon energy divided by the rest mass energy (mo c 2 ) of an electron which equals.511 KeV ) the range of energy between 1 kev and 1 MeV and a1,2,3,4 are fitting coefficients (a1 = , a2 = ,a3 = ,a4 =.89562). The fitting was done using the general linear least squares method. Using simple trigonometry one can obtain the basic relations (3-2 to 3-6) between the angular and linear parameter in the diagram. These are useful when transforming between angular and Cartesian co-ordinate system and when writing computer program. r1 2 = x 2 + y 2 + z 2, r2 2 = (x d) 2 +y 2 + z 2.(3-2) d sin β r1= sin φ d sin α, r2= sin φ.... (3-3) r3= y 2 + z (3-4) φ = α + β....(3-5) α = tan 1 y2 +z 2, β = tan 1 y2 +z 2.. (3-6) x d x 27

37 The Compton backscatter program is based on the Monte Carlo method. The flowchart is shown in figure (3-2). This chart contains all auxiliary operations beginning from the data reading and ending by the printing of the results. It is divided into a main program and ten subprograms for the data preparation and interpolation. The appendices (A1-A5) shown the attenuation coefficient for various elements (H 2 O, C, SiO 2, Mg 2 SiO 4, and Fe 2 SiO 4 ) with respect to gamma ray energy was used as the input data of the program. The program calculated single scatterings of each photon and gains the values of the Compton backscatter simultaneously for various values of five elements, detector positions and sizes. Seven sizes and five detector positions horizontally and vertically have been used simultaneously the information concerning the energy of the reaching photon. The program was designed in FORTRAN language (77-9) for PC computer. 28

38 Definitions: d =source detector separation. Start X=distance between source and R3. Xd=distance between R3 and detector. RD =radius of detector. Xb and Zb= dimension of the block. No.of evt=no.of event. Eg=energy of gamma ray. R3=distance between p and start of Xd. SL1,Xd1=are the first sonde length and Xd 1 distance for the first position. Ro: density of materials. µu:mass attenuation coefficient of materials. x,y,z=coordinates of the emitted photon in x,y,z dimension. Eğ= energy of the scattered photon. µ2=the attenuation of the scattered photon. PL2=is the path length of the scattered photon. Input data SL,SX, Xd,RD, Xb,Zb,no of evt,eg,r3 Call Material calculate µu H 2 O,µu C,µu SiO 2,µu Mg 2 SiO 4,µu Fe 2 SiO 4 Loop 1 SL1,2,3,4,5 Xd1,2,3,4,5 Loop 2 Ro 1,2,3,4, 5 µu 1,2,3,4,5 1 Calculate the mass attenuation coefficient of materials Figure (3-2): The flowchart of program. 29

39 1 Loop 3 Start with no. of photon 1 6 Call Random number Calculate (R) Generate random number between -1 Call emtang calculate ө em,φ em Calculate emitting angles of photon from the source (polar and azimuthal) Call unvect calculate n x,n y,n z Calculate units vector ( n x, n y,n z ) of photon path from the source to the detector Call Traj calculate Trajectory Calculate the distance of path length of photon from the source to the point p 2 Figure (3.2): Continues. 3

40 2 Call coord calculate x,y,z Calculate emitting direction of photon x,y,z Calculate path calculate PL1 Calculate the path length of photon from the source to the point p Call scat calculate α,β,r1,r2,r3 Calculate these by trigonometry relations Calculate Escat calculate Eğ Calculate the energy of scattered photon by Kahn method Call Material calculate µ2 Calculate the mass attenuation of scattered photon Call path calculate PL2 Calculate the path length of scattered photon 3 Figure (3.2): Continues. 31

41 Loop 3 3 Loop 2 Loop 1 Call check No If the no.of photons are complete Yes If the no. of elements are complete No Yes If these the no.of position are compete No Print the mass attenuation coefficient of scattered photon of materials and the count rate of photon that reached the detector Output Data Yes Stop End 32

42 3-1 FORTRAN PROGRAM To model the densitometer design as shown in figure 3.1, Monte Carlo program has been written to describe the absorption and scattering of photons in bulk material. Simulation of a single scattering photon history requires to classify the program into following steps: a. In order to run the program it is necessary to input a certain data such as dimension and density the bulk material, radius of the detector used and number of photons incident. b. Sampling for emission (polar and azimuthally angles θe and e ) θe = cos 1 (2R 1).(3-7) Where R is a random number ( R 1) and sampling for azimuthal angle of scattering (uniform distribution from to 2π, so e = 2 πr) This is done by subroutine EMTANG. c. Computing the mass attenuation coefficient of Compton interaction with the materials (H 2 O, C, SiO 2, Mg 2 SiO 4, and Fe 2 SiO 4 ), by mass attenuation that put in the program of Compton interaction by function SCNDR between(.1-2) MeV (see Appendices A1-A5). d. Sampling for source energy. This would only be used for monochromatic sources, e.g. 137 Cs which has one emission peak. e. Determination of the attenuation length l=μ(e) 1, where μ(e) is the mass attenuation coefficient multiplied by the density and E the photon energy. f. Sampling for the actual path length pl of the photon in the material, using 33

43 pl = l ln(1 R)..(3-8) The calculation of pl was done using the subroutine path. g. Calculating the coordinate of the emitted photon by using the parametric equation. r=ro+t n x=xo+t nx y=yo+t ny (3-9) z=zo+t nz Where (xo, yo, zo)are coordinates of the emitting point in the source and t is the trajectory of the emitted photon from the source to the point p. There coordinates have been calculated by subroutine COORD. h. Sampling for the new photon energy after Compton scattering. The most appropriate sampling method for this application is the Khan method as shown in figure (3-3) and works for any incident photon energy, where v 1 are random numbers uniformly distributed in the range v i < 1, E and Ē are the initial and final photon energies, respectively (in the unit of the electron rest mass energy) and R is the ratio of E/ Ē. The method requires the generate and analysis at least one set of three random numbers (v1, v2, v3) in the range vi 1. The procedure for a single Compton event is shown in figure (3-3). i. Calculation of the polar angle of scattering θ using the Compton formula cos θ = 1 ( 1 Ē 1 E )moc2 (3-1) 34

44 Where E is the initial energy of photon and Ē is the new photon energy and moc 2 is the electron rest mass energy (subroutine ANGLES SCAT). Figure (3-3): The Method used for random sampling of the Klein Nishina distribution by the Kahn method. 35

45 Photons are discarded if: 1.They exceed some limiting radial distance from the source. 2. If they fall below some limiting energy. 3. If the photon escapes up through the surface. 4. After a large number of photon histories have been tracked, this data can be analyzed to characterize the backscatter response of the system. Figure (3-4): The photons scattered at p towards the detector and incident on it, RM must be less or equal to RD. 36

46 Chapter Four Results and Discussion Introduction As explained before, the technique of Compton backscatter densitometer is used to write a program that calculates bulk density for selected materials that may be found in the composition of the cometary nucleus. The results of calculations comprise the study five elements, which are the components of nucleus of comet such as (H 2O, C, SiO 2, Mg 2 SiO 4, and Fe 2 SiO 4 ). Monte Carlo simulation of scattering and absorption in bulk material has been used to investigate the design parameters, specifically the variation of backscattered count rate with density and composition. Also the system was examined according to the sonde length (distance separation between source and detector) and depth, in addition to the response to gamma backscatter with different values detector radius (RD). 4.1 The Results of Calculation The results have been obtained to examine the effect of the variation of the source- detector separation (horizontally and vertically). This is shown in figures (4-1 to 4-5) and tables (4-1 to 4-5) for each of the five sonde lengths. For the distance 2.5 cm is higher than the other distances and started to decrease gradually until it reaches the value of distance 25cm. This is because the count rate is a function of distance and density. Figures (4-6 to 4-1) and tables (4-6 to 4-1) show the effect of count rate with depth of the source from the detector. Also the 37

47 response was examined for the system with different values of the detector radius as shown in figure (4-11). Table (4.1): Count rate with sonde length for H 2 O and RD=12.5cm. Sonde length (cm) (d) Sx(cm) (d-sx)cm No.of count rate H 2 O ρ=1. g/cm 3 Count rate Sonde length (cm) Figure (4.1): Variation of count rate as a function of sonde length for H 2 O. 38

48 Table (4.2): Count rate with sonde length for C and RD=12.5 cm. Sonde length (cm) (d) Sx(cm) (d-sx)cm No.of count rate Count rate C ρ =1.7 g/cm Sonde length (cm) Figure (4.2): Variation of count rate as a function of sonde length for C. 39

49 Table (4.3): Count rate with sonde length for SiO 2 and RD=12.5 cm. Sonde length (cm) (d) Sx(cm) (d-sx)cm No.of count rate Count rate SiO 2 ρ =2.4 g/cm Sonde length (cm) Figure (4.3): Variation of count rate as a function of sonde length for SiO 2. 4

50 Table (4.4): Count rate with sonde length for Mg 2SiO 4 and RD=12.5cm. Sonde length (cm) (d) Sx(cm) (d-sx)cm No.of count rate Count rate Mg 2 SiO 4 ρ =3.21 g/cm Sonde length (cm) Figure (4.4): Variation of count rate as a function of sonde length for Mg 2SiO 4. 41

51 Table (4.5): Count rate with sonde length for Fe 2SiO 4 and RD=12.5cm. Sonde length (cm) (d) Sx(cm) (d-sx)cm No.of count rate Fe 2 SiO 4 ρ =4.34 g/cm 3 Count rate Sonde length(cm) Figure (4.5): Variation of count rate as a function of sonde length for Fe 2SiO 4. 42

52 Table (4.6): Count rate with depth for H 2O and RD=3.81cm. D(cm) No. of count rate H 2 O ρ=1. g/cm 3 Count rate Depth (cm) Figure (4.6): Variation of count rate as a function of depth for H 2O. 43

53 Table (4.7): Count rate with depth for C and RD=12.5 cm. D(cm) No. of count rate C ρ =1.7 g/cm 3 Count rate Depth (cm) Figure (4.7): Variation of count rate as a function of depth for C. 44

54 Table (4.8): Count rate with depth for SiO 2 and RD=12.5 cm. D(cm) No. of count rate Count rate SiO 2 ρ =2.4 g/cm Depth (cm) Figure (4.8): Variation of count rate as a function of depth for SiO 2. 45

55 Table (4.9): Count rate with depth for Mg2SiO4 and RD=12.5 cm. D(cm) No. of count rate Mg 2 SiO 4 2 ρ =3.21 g/cm 3 Count rate Depth (cm) Figure (4.9): Variation of count rate as a function of depth for Mg 2SiO 4. 46

56 Table (4.1): Count rate with depth for Fe2SiO4 and RD=12.5 cm. D(cm) No. of count rate Count rate Fe 2 SiO 4 ρ =4.34 g/cm Depth (cm) Figure (4.1): Variation of count rate as a function of depth for Fe 2SiO 4. 47

57 Table (4.11): Count rate with radius of the detector. RD H 2 O C SiO 2 Mg 2 sio 4 Fe 2 SiO Count rate H2OH 2 O C SiO2 SiO 2 Mg2SiO4 2 4 Fe2SiO4 2 SiO Radius(cm) Figure (4.11): Shows the count rate as a function of the radius of the detector. 48

58 To examine the effect of the density of material and composition on backscatter gamma rays, the backscatter count rate was found as a function of the bulk density. An approximate functional form for this calibration curve has been used in this research [37]: I(ρ) = K1ρe c1ρ k2ρ 2 e c2ρ (4 1) Where I(ρ) is the total count rate detected, ρ is the material density, and k1, 2 and c1, 2 are constants. This function is plotted in figure 4.12 according to the values that have been gathered from [39] which are C1 = 3.45kg 1 m 3, C2 = 2.6 kg 1 m 3, K1 =.475 s 1 kg 1 m 3, K2 = s 1 kg 2 m 6 that have been substituted in equation (4-1). The count rate reaches a maximum at some critical density, dependent on the sonde length and the source energy.below this density the count rate falls due to the reduced concentration of electrons to scatter photons into the detector, while above this density the count rate falls due to the increased attenuation of the photon. The two competing effects-scattering and attenuation hence in general particular count rate can correspond to two alternative densities depending on whether scattering or attenuation dominates. This can cause confusion if both densities are within the range expected for the material in question. The solution to this ambiguity is to use detectors at more than one sonde length. In this way the true density is identified by the coincidence of values from each of the detectors. Figure 4.13 shows the variation of count rate with density for the three detectors. Backscatter densitometers usually operate on materials with a density above the critical value,i.e. where an increase in count rate implies a decrease in density [39]. 49

59 5 Table (4.12): Count rate(n) with density of H 2 O (ρ). I(ρ) Normlize (ρ) ρ Normalize I(ρ)

60 Normlized count rate I(ρ) H 2 O Density ρ(kg.m -3 ) Figure (4-12): The curve showing the relation between normalize count rate as a function of bulk density of material [39]. 51

61 Table (4.13): Count rate with three sonde length and relative function. RELITIVE COUNT RATE COUNT RATE COUNT RATE ρ(g/cm 3 ) function (25 cm) 1-3 (2 cm) 1-3 (15 cm)

62 count rate(n) density(g/cm 3 ) I (SL=15 CM) II(SL=2 CM) III(SL=25 CM) IV(II/III) Figure(4-13) :The curve shows the relation between count rate as a function of density at three sonde length(15,2,25)cm. the ratio of count rate I and III was found to be convenient function of density (IV). 53

63 4.2 Composition Effect The effect of composition on the backscatter counts has been examined, As shown in figure Photoelectric absorption by high Z elements result in to reduce the count density quite dramatically.the extreme curves 1 to 5 correspond to carbon and Fe2SiO4 respectively. Actual cometary material is likely to produce a profile somewhere in between, whether it is predominantly low Z volatiles compound or refractor minerals such as magnesium silicates. Table (4-14):Count rate vs.sonde length for different materials, density=1. g/cm 3, E=.6 MeV. SL H 2 O C SiO 2 Mg 2 SiO Fe 2 SiO H2OH 2 O C log Count rate SiO2 SiO 2 Mg2SiO 2 SiO 4 4 Fe2SiO Sonde length (cm) Figure (4-14): Count rate as a function of sonde length for different materials. 54

64 Table (4-15): Count rate vs.sonde length for three densities of (C,Mg2SiO 4). C(1. ) C(1.5 ) Mg Sl C(.5 ) g/cm 3 g/cm 3 g/cm 3 2SiO 4 (.5 ) g/cm 3 Mg 2SiO 4 (1.) g/cm Mg 2SiO4 (1.5 ) g/cm 3 C(.5 g/cm 3 ) C(1. g/cm 3 ) C(1.5 g/cm 3 ) Mg 2SiO 4(.5 g/cm 3 ) Mg 2SiO 4(1. g/cm 3 ) 15 Sonde length(cm) Mg 2SiO 4(1.5 g/cm 3 ) Figure (4-15) :Count rate vs.sonde length for 1)carbon and 2) Mg 2 SiO 4 densities (.5,1.,1.5 g/cm 3 )respectively,e=.6 MeV. 55

65 Table (4-16): Count rate vs. sonde length for three densities for (H 2 O, Mg2SiO4). Sl H 2 O (.5 ) g/cm 3 H 2 O (1. ) g/cm 3 H 2 O (1.5 ) g/cm 3 Mg 2 SiO 4 (.5 ) g/cm 3 Mg 2 SiO 4 (1.) g/cm Mg 2 SiO ) g/cm H2O(.5 H 2 g/cm 3 ). gm/cm^3) H2O(1 H 2 O (1. gm/cm^3) g/cm 3 ) log Count rate H2O(1.5 H 2 g/cm 3 ) gm/cm^3) Mg2SiO4( g/cm 3 ) gm/cm^3) Mg2SiO4( g/cm 3 ) gm/cm^3) Mg2SiO4( g/cm 3 ) gm/cm^3) Sonde length (cm) Figure (4-16): Count rate vs.sonde length for 1) H2O and 2) Mg 2SiO 4 densities (.5,1.,1.5 g/cm 3 )respectively,e=.6 MeV. 56

66 Table (4.17): Count rate with density for (H2O), RD=12.5 cm, energy of source=.6 MeV. ρ COUNT COUNT COUNT COUNT COUNT RATE RATE RATE RATE RATE (SL=5 cm) (SL=1 cm) (SL=15 cm) (SL=2 cm) (SL=25 cm) Count rate H2O Density(g/cm 3 ) sl=5 cm sl=1 cm sl=15 cm sl=15 cm sl=2 cm Figure (4-17): Number of counts passing through each of five simulated detectors, plotted as a function of a range of densities of H 2 O. Energy of source=.6 MeV. 57

67 Table (4.18): Count rate with density for (C),RD=12.5 cm, energy of source=.6 MeV. ρ COUNT COUNT COUNT COUNT COUNT RATE RATE RATE RATE RATE (SL=5 cm) (SL=1 cm) (SL=15 cm) (SL=2 cm) (SL=25 cm) Count rate C Density(g/cm 3 ) SL=5 CM SL=1 CM SL=15 CM SL=2 CM SL=25 CM Figure (4-18): Number of counts passing through each of five simulated detectors, for a range of densities of C, energy of source=.6 MeV. 58

68 Table (4.19): Count rate with density for (SiO 2 ), RD=12.5 cm, energy of source=.6 MeV. COUNT COUNT COUNT COUNT COUNT ρ RATE RATE RATE RATE RATE (SL=5 cm) (SL=1 cm) (SL=15 cm) (SL=2 cm) (SL=25 cm) Count Rate SiO Density (g/cm 3 ) sl=5 cm sl=1 cm sl=15 cm sl=2 cm sl=25 cm Figure (4-19): Number of counts passing through each of five simulated detectors, plotted as a function of a range of densities of H2O, energy of source=.6 MeV. 59

69 Table 4.2 Count rate with density for (Mg2SiO4), RD=12.5 cm, energy of source=.6 MeV. ρ COUNT COUNT COUNT COUNT COUNT RATE RATE RATE RATE RATE (SL=5 CM) (SL=1 CM) (SL=15 CM) (SL=2 CM) (SL=25 CM) Mg 2 SiO 4 SL=5 CM 12 SL=1CM count rate SL=15CM SL=2CM SL=25 CM density(g/cm 3 ) Figure (4-2) Number of counts passing through each of five simulated detector, plotted as a function of a range of densities of Mg2SiO4, energy of source=.6 MeV. Table 4.21 Count rate with density for (Fe 2 SiO 4 ), RD=12.5 cm, energy of source=.6 MeV. 6

70 COUNT COUNT COUNT COUNT COUNT ρ RATE RATE RATE RATE RATE (SL=5 cm) (SL=1 cm) (SL=15 cm) (SL=2 cm) (SL=25 cm) Count rate Fe 2 SiO 4 SL=5 CM SL=1 CM SL=15 CM SL=2 CM SL=25 CM Density(Kg/m 3 x 1 3 ) Figure (4-21): Number of counts passing through each of five simulated detectors, plotted as a function of a range of densities of (Fe 2 SiO 4 ), energy of source=.6 MeV. 61

71 Figure (4-18) shows the variation of number of counts with density for each of the five detectors. using carbon as the sample material. Figure (4-21) shows the results of Fe 2 SiO 4 to illustrate the effect of a more strongly absorbing material. For carbon the peak count rate for each detector lies at a higher density than for Fe 2 SiO 4, where the peaks for all detectors with density.2 g/cm 3.Without prior knowledge of the elemental composition the turning point in a detectors calibration curve cannot be predicted. However, the turning point does move to higher densities for detectors closer to the source. In order to be as sensitive as possible, the densitometer needs to measure count rate at a distance where the fractional change in count rate per unit change in density is large for the material under examination.detectors at a range of C will thus ensure better coverage of the possible materials. For no material will all the detectors be near their turning points and thus insensitive to change in density. Figure (4-21) shows that while a material with high photoelectric absorption produces too low a count rate in the farthest detector, a sufficient count rate may still be registered by detectors closer to the source. This is another advantage of the multiple detector arrangement. 4.3 Comparison From comparing the Monte Carlo (4-13,14,15,18,21) results with the experimental and theoretical results [5], it is shown that there is a good agreement with the shape and behavior. This means that the new geometry design is in a good condition. With respect to the figure (4-26) shows example calibration curves from densitometer. In general a particular count rate can correspond to two alternative densities, depending on whether scattering or attenuation dominates. This can cause confusion if both densities are within the range expected for the material 62

72 in question. The solution to this ambiguity is to use detectors at more than one sonde length. In this way the true density is identified by the coincidence of values from each of the detectors. The three curve, reaching a maximum at some critical value (.53) g/cm 3 of part (A).Figure (4-27) show that the count rate reaches a maximum at some critical density, dependent on the sonde length and the source energy. Below this count rate falls due to the reduced concentration of electrons to scatter photons into the detector, while above this density the count rate falls due to the increased attenuation of the photons. Figure (4-22) shows the composition between the experimental results and our work. Also the figure explains the effect of composition on the backscatter profile. Photoelectric absorption by high Z elements can reduce the count quite dramatically. The extreme curves for C to Mg 2 SiO 4.Figure(4-23) shows count density profile for C and Mg 2 SiO 4 at three different densities. It is clear that the effect due to density variation. While significant is smaller than that due to differences in composition. 63

73 count rate(n) A Density (Kg /m 3 x 1 3 ) I (SL=15 CM) II(SL=2 CM) III(SL=25 CM) IV(II/III) Figure (4-26): Variation of count rate with density for the three detectors of the sonde length are (15,2,25)cm for A and (1,18,25) cm for B. A) Present work,b)experimental published result [5]. 64

74 Normlize count rate I(ρ) B SL=15 cm density(kg/m 3 x 1 3 ) Figure (4-27) Calibration curve for a gamma backscatter densitometer,showing detected count rate vs. density,normalized to the maximum. B)Present work,a) Theoretical result[37]. 65

75 (m 1 2 ) Figure (4-22): Count rate vs. sonde length for the materials (H 2 O, C, SiO 2, Mg 2 SiO 4, Fe 2 SiO 4 ). Density =1. g/cm 3,E=6 kev. A) Present work B) Experimental published result [5]. 66