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Monte-Carlo Simulation for reproducing image of an object using Compton back scattering Effect Le Hong Khiem, Nguyen Tuan Khai Institute of Physics P.O.Box 429 BoHo, Hanoi 10000, Vietnam Contact

A Monte-Carlo simulation has been developed to describe the interaction processes of gamma rays when coming in a medium with infinite thickness. The photons emitted from a radiation source or x-ray tube are followed until they are either lost by the photoelectric absorption in the medium or registered by the back scattering detector array. The registered back scattered events are used to determine the projection, i.e. reproduce the shape, of the defects existing in the testing material. Our simulation results are useful not only to evaluate the optimal configuration of a radiographic system using Compton back scattered radiation, but also to access the feasibility for setting up experimentally this system.

Introduction

The gamma ray transmission method has been widely used in radiographic technique to determine the defects existing in the testing specimen as well as to measure density or thickness of the specimen. The method is based on the differences between the gamma attenuation coefficients or the absorption thickness at the various positions in the specimen. In practice, these are existing the objects located at the positions where the transmission method could not be used. In these cases the radiography using Compton back scattering method becomes more favorable. In principle, this method is also based on the difference between the Compton scattering cross-sections and mass densities of the medium and object. However, due to the fact that for a given absorber the intensity of Compton back scattered radiation which depend on the energy of the incident gamma rays and the scattering angle, is rather low (some percents at 10 MeV to maximum 50 % of the intensity of scattered beam at 10 KeV [1], especially when the beam collimator is required. For this reason, the use of Compton back scattering measurements has been limited in the practical applications.

In practice, the use of ¹³⁷Cs source emitting the mono energetic gamma rays of 661 KeV or x-ray tube having the high voltage 400 kV may produce the Compton back scattering intensity about 10-30 % of the scattered beam [1], i.e. sufficiently high. If the activity of the irradiation source is high and the irradiating-measuring conditions are optimally designed, the use of the Compton back-scattered radiation becomes possible in radiography.

In this work, the configuration proposed for the simulation is showed in figure 1 including a collimated 10 mCi ¹³⁷Cs source, a collimated back scattering detector array (sensitive film) and a steel layer with the infinite thickness containing a defect hole (object) inside. The distances between the source - the steel layer, the detector - the steel layer and the source - the detector can be changed. Based on the proposed configuration, we have developed Monte-Carlo simulation to describe the interaction processes occurred when the photon beam coming in the medium containing the testing objects, the results of which are that only the photons that are back scattered within a solid angle covered by the detector collimator are to be registered by the corresponding detector in the array. The registered back scattered events allowed then us to reproduce the two dimension image of the object, to determine saturation thickness of material for a given energy of gamma rays and to plot energy spectrum of the back scattered photons. In general our simulation program can be used for the material with multi elemental constituents.

Procedure for simulation

As showed in figure 1, when the collimated mono energetic gamma rays of 661 KeV emitted from ¹³⁷Cs source enter into a steel layer with infinite thickness, the interaction processes of the gamma rays occurred in this medium will include only photoelectric absorption and scattering. For an isotopic material of independent atoms, the total scattering coefficient, s _s, is as follow:


s s = s c + s I	(1)
where s s and s I are coherent or Rayleigh scattering and incoherent or Compton scattering cross-sections, respectively. Then, the total interaction cross-section is given by:
s t = s s + s a	(2)
where sa is photoelectric absorption cross-section.

Fig 1: Geometrical Simulation 1. Source collimator,2. Detector collimator, 3. Object ,4. Detector in array

Because in a solid angle shielded by a detector of the array the back scattered radiation is distributed randomly, in order to have an image with good definition and contras, some important suppositions included in our simulation are as follow:

1. the activity of ¹³⁷Cs source should be chosen at least 10 mCi
2. the deepness of the testing object must be less than the saturation thickness of the gamma rays in the material
3. the differences between Compton scattering cross-sections or mass densities of the material and object should be sufficiently great and
4. the diameter of the testing object should be greater than 3 times that of the source and detector collimators.

The point source of ¹³⁷ Cs which is placed at the origin of coordinates (X_o,Y_o,Z_o) emits photons uniformly distributed in space 4p , only the incident photons that are emitted within the solid angle W ₁ shielded by the source collimator are considered. The intensity of this primary photon beam can be determined as:

I = 3.7*107IoW 1/(4p )	(3)
Where Io is the activity of the source in mCi

The procedure for simulating the life history of a large number of photons emitted by the source according to random walks can be summarized by a following scheme:

COMPUTATIONAL SIMULATION SCHEME


In this program, the free path of the photon in a given medium, L, has been simulated as:

L = -LN(q)/S t

(4)

where q is a random number uniformly distributed between 0 and 1

S _t(cm^-1) is linear attenuation coefficient for gamma rays,

S t = r (g/cm3)* s t(cm2/g)

(5)

where r is mass density of the medium and

s t = s s + s a

(6)

s _a and s _s are photoelectric absorption and Compton scattering cross-sections, respectively. Their values are determined in appendix 1.

If at an interaction point in the medium Compton scattering occurred, then cosine of the polar angle of the scattering is given by [3]:

COS(q _S) = 1 + ( 1/E - 1/E' )m_oc² (7),

Where m_oc² is rest mass energy of electron

E is energy of incident photon

E' is energy of Compton scattered photon determined in appendix 2.

Results and discussion

In order to determine the position of defect in the material and reproduce its image, we can imagine that the testing material is divided into many prism pixels through which the complex of ¹³⁷ Cs source, detector array and corresponding collimators will be scanned on the whole plane (X,Y) of the material. At each step of scanning, the photon history from its emission is simulated by the above mentioned computational scheme. It is obvious that the intensity of back scattered radiation will be varied suddenly when the complex scans through pixel having the defect. Thus, when the whole plane (X,Y) of the material is scanned, the projection (two dimension image) of the defect can be reproduced on the detector array.

Fig 2: Two dimension image of object green points are back scattered events from steel red points back scattered events from carbon hole

Figure 2 showed the two dimension image of cylindrical defect hole (defect) containing carbon in steel layer with infinite thickness. On this picture, blue points are denoted back scattering from steel and red points are from defect.

From the picture we can see that the better the image contras is, the greater difference between intensities of back scattered radiation from the material and the object is, i.e. the greater differences between the scattering cross-sections or mass densities are . For our case, ratio s _S(steel) / s _S(carbon) is about 4.5 for photon energy range from 0.6 down to 0.1 MeV and ratio r (steel) / r (carbon) is equal to 3.5

As mentioned above, the image is processed by scanning method on whole plane of testing material, therefore, in order to have better definition (sharpness) of the image the diameters of source and detector collimators should be as small as possible compared with the diameter of object. However, in order to compromise with limited intensity of radiation source and CPU time, ratio of object diameter / collimator diameters should be chosen from 2.5 to 3.5

Fig 3: Monte - Carlo simulation for determination of saturation thickness of steel for 661 keV for gamma rays

Figure 3 showed a curve for determination of saturation thickness of steel for 661 KeV gamma rays by the back scattering method. The curve give a valid description for variation of intensity of back scattered radiation with steel thickness. From the figure we can see that this saturation thickness of steel is about 0.9 cm. Our simulation program can be used to select a photon source with appropriate energy and intensity for measuring thickness of material layers.

Appendix 1 :

The data for photon cross-sections in a given material in the wide range of energies from some KeV to 2 MeV are taken from [2]. In order to be convenient in our calculation, these data of the cross-sections are parameterized by a cubic function of the photon energy E:

Ln( s _a) = a_o + a₁Ln(E) + a₂Ln²(E) + a₃Ln³(E)

Ln( s _s) = b_o + b₁Ln(E) + b₂Ln²(E) + b₃Ln³(E)

Where (a_o, a₁, a₂, a₃) and (b_o, b₁, b₂, b₃) are fitting coefficients for photoelectric absorption and scattering cross-sections, respectively.

s _t = s _a + s _s

These values of cross-section are given in barn/atom unit. To convert into cm²/g unit they must be multiplied with corresponding conversion coefficients. For Fe and C the conversion coefficients are 0.010780 and 0.050140 respectively.

Appendix 2:

The energy of Compton back scattered photon can be determined by Monte-Carlo simulation for transportation of gamma radiation through matter [3] as follow:

a '=a /[1+Sq+(2a -S)q3] + (a -4)(1-q)2q2

(2-1),

where S = a /(1 + 0.5625a ),
a = E/m_oc² and a ' = E'/m_oc²
E and E' are energies of incident and Compton scattered photons, respectively.
m_oc² is the rest-mass energy of the electron.
The second term in formula (2-1) will be included if E > 4m_oc².

Reference

1. Glen F. Knoll, Radiation Detection and Measurement, Second edition, John Wiley & Sons, Inc 1989
2. E. Storm and H. Israel, Photon Cross-sections From 1 keV to 100 MeV for Elements Z =1 to Z=100, Los Alamos Scientific Laboratory, University of California, Los Alamos, New Mexico 87544
3. N. G. Guxev, V. A. Klimanov, V. P. Maskovich and A. P. Xuvorov, Physical Basis of Radiation Protection

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