# Patent

## Title
Method for direct dosimetry of radioactive nuclei in different types of samples using solid-state detectors of nuclear traces

## Abstract

The invention concerns a novel method adapted to experimental conditions, for determining the function defining the actual detecting sensitivity of a solid-state detector of nuclear traces and which represents the detecting performances for each type of nucleus emitting energy alpha particles Ei, located in each position inside a sample to be analyzed, and hence precision dosimetry of the radioactive nuclei in a sample using nuclear trace solid-state detectors. The method provides an improvement of the quality of sample analyses, a significant reduction of analysis costs and an important widening of the scope of application of this type of detectors. Thus, the method will bring about the development of a new generation of detectors having a better price-quality ratio which can be found in the field of radioactive nuclei analyses in different types of samples.

## Background
RELATED U.S. APPLICATIONS
Not applicable.
STATEMENT REGARDING FEDERALLY SPONSORED RESEARCH OR DEVELOPMENT
Not applicable.
REFERENCE TO MICROFICHE APPENDIX
Not applicable.
FIELD OF THE INVENTION
The present invention relates to a method for direct dosimetry of radioactive nuclei in different types of samples using solid-state detectors of nuclear traces.
BACKGROUND OF THE INVENTION
The history of solid state nuclear track detectors began in 1958 after initial observations by D. A. Young of fission fragment tracks of lithium fluoride (LiF) irradiated by thermal neutrons. In 1959, Silk and Barnes made these same observations on mica using an electron microscope. Numerous observations have been made since then on a large number of materials. The detectors have the ability to record the passage of particles such as alpha particles, protons, and fission fragments.
In the years that have passed, work by the international scientific community on solid state nuclear track detectors made it possible to develop several dosimetry methods for radioactive nuclei with the use of these detectors. Though these methods use basic knowledge of calculation of the probability that an alpha particle emitted from a radioactive nucleus of a sample can reach and be recorded on a solid state nuclear track detector, they are based on estimation of the global critical angle depending on the sample, the characteristics of the particles to be detected, and the detector. The estimation thus induces an error in the calculation and makes the solid state nuclear track detectors (SSNTDs) imprecise.
Other methods of detection that do not use solid state nuclear track detectors already exist. They involve, for example, geochemical analyses and instrumental techniques such as analysis by neutron activation, analysis by mass spectometry or analysis by gamma spectrometry and liquid scintillation analysis. All of these techniques, often destructive to the samples analyzed, use equipment that is heavy, sophisticated, and especially costly.
BRIEF SUMMARY OF THE INVENTION
In order to correct these disadvantages, a new method has been developed. It is based on the implementation of a new technique specified for calculating the sensitivity of a solid state nuclear track detector for each type of nucleus emitting alpha particles having energy E1 located in each position in the sample to be analyzed. This method does not allow the estimation but instead the calculation without errors of the critical angle of incidence depending on the sample, the characteristics of the alpha particles to be detected, and the detector. Thus, this method is based on the actual function of the critical angle of incidence which makes it possible to solve the problem of the sample-detector geometry, and as a result, determination of the function that defines the real sensitivity of detection of a solid state nuclear track detector (SSTND) and which represents the efficiency of detection for each type of nucleus emitting alpha particles having energy E1 located in each postion in the middle of the sample to be analyzed.
Applications of this method allow an improvement in the quality of sample analyses, a considerable reduction of costs for analysis and a great enlargement in the field of application for this type of detector. Thus, a new generation of detectors will be born having a better price-to-quality ratio than has existed until now in the domain of analyses of radioactive nuclei in different types of samples.



BRIEF DESCRIPTION OF THE SEVERAL VIEWS OF THE DRAWINGS
FIG. 1 shows a schematic view of the cell of radiation which is represented by the elementary volume element and considered as the sample and analyzed.
FIG. 2 shows a graph illustration of the experimental and intrinsic function of a SSNTD describing the variation of the critical angle (θc) resolving the tracks as a function of the residual energy for conditions enabling chemical resolution and reading of well-defined tracks. This function is from a solid state nuclear track detector that is currently the most used (CR-39: LR-115).
FIG. 3 shows a graph illustration of the variation in the sample space, from (θi) and (θc) as a function of the residual energy and trajectory (X), of an alpha particle of energy Ei and Eindε[Emin, Em] emitted of a radioactive core located at the height h from the surface of the detector.
FIG. 4 show a graph illustration of the trajectory of the alpha particle of energy Ei emitted of a core Ki located at the height h, in the medium to be analyzed and the absorbent medium.
FIG. 5 shows a schematic view of S(θic)∩S(R) and S(θic) in the case of (θic)≧(θ1) and (θic)≦(θ1).



DETAILED DESCRIPTION OF THE INVENTION
II. Methodology and Procedures for Evaluation of Different Parameters
Let (dv) be the volume element of a cylinder having a radius between r and r+dr and a height between h and h+dh inside the sample analyzed, see FIG. II-1 in the drawings section where: (1: SSNTD, 2: plastic capsule, 3: sample analyzed).

dv=2πrdrdh

Let dNi be the number of the alpha particle having energy Ei emitted by the radioactive element having index i contained in the volume dv and for the exposure time te:

dNi=Acite·dv

Where Acite is the alpha activity of the radioactive nuclei of the emitter element i of the alpha particle having energy Ei per unit volume and for the exposure time te.
It is thus necessary to multiply the number dNi of alpha particles emitted in the volume dv by a probability of detection Pi(r,h), [Pi(r,h) is the probability that an alpha particle of energy Ei emitted by the nucleus of index i contained in the volume dv can be detected by the solid state nuclear track detector] and which also represents the efficiency of detection of alpha particles of energy Ei emitted by the nucleus of index i contained in the volume dv.
As a result, for radioactive nuclei of element i contained in the volume dv, the number of tracks developed will be:

dni=Acite·Pi(r,h)·dv

Where: dni is the number of alpha particles having energy Ei emitted by the nuclei of index i contained in the volume dv, for the time of exposure (te) and which are detected in our solid state nuclear track detector.
Actually, the total number of tracks observed due to the radioactive nuclei of index i contained in the total volume is:

nit=2π·Acite·∫∫Pi(r,h)·rdrdh

II-1 Method for Calculating the Probability Pi(r,h).

Let a nucleus of index i denoted Ki, an emitter of alpha particles of energy Ei, be located in a position of coordinates (r,h) (see FIG. II-5 in the drawings section) such that these emitted particles give rise to observable tracks on the solid state nuclear track detector. The particles must satisfy two conditions:

    
    
        1—they are emitted at a well-defined angle of incidence by means of the function of the critical angle of incidence (θic).
        2—they reach the detector.

II-1-1 Identification and Evaluation of the Function (θic).

    
    


A) Identification of the Function (θic).
FIG. II-2 in the drawings section shows the experimental and intrinsic function of a SSNTD describing the variation of the critical angle (θc) resolving the tracks as a function of the residual energy for conditions enabling chemical resolution and reading of well-defined tracks. This function is from a solid state nuclear track detector that is currently the most used (CR-39; LR-115). Three possible cases can be distinguished:

*Eind<Emin, *Eind≦Emin≦Emin, *Eind>Emax

where:

    
    
        Eind is the residual energy at normal incidence on the detector of the alpha particle having energy Ei emitted from a nucleus located at a height h from the plane of the detector.
        Emin is the minimum energy detectable at normal incidence.
        Emax is the maximum energy detectable at normal incidence.
    
    
    Comment: The identification of (θic)=f(h), (θic1)=fa(h), and (θic2)=fb(h) can also be means of the inverse function of the function (θc)=f(Eres):Eres=f(θc) which shows the minimum and maximum energies detectable for each incidence (θi).

A-1—Case of Nuclei Emitters of Alpha Particles Having Energy Ei and with Eind<Emin.



These are nuclei located in the volume for contribution to the tracks defined by hε[Ri−Rmin,Ri] for the case without an absorbent medium, given Ri and the total trajectory in the sample of the alpha particle having energy Ei in the sample, and given Rmin and the total trajectory in the sample of an alpha particle having energy E=Emin. All alpha particles having energy Ei emitted from these nuclei will reach the detector with an energy lower than Emin and as a result will not be detected.
A-2—Case of Nuclei Emitters of Alpha Particles Having Energy Ei and Emin≦Eind≦Emax.
These are nuclei located in the volume for contribution to the tracks defined by hε[Ri−Rmax, Ri−Rmin] for the case without an absorbent medium, given Rmax and the total trajectory in the sample of an alpha particle having energy E=Emax.
For a better explanation, according to FIG. II.2, there are two cases: Eind less than the intermediate energy denoted (Em) and Eind greater than the intermediate energy Em.
1—Case where Eind is Less than Em:
FIG. II-3 in the drawings section shows the variation in the sample space, from (θi) and (θc) as a function of the residual energy and trajectory (X), of an alpha particle of energy Ei and Eindε[Emin, Em] emitted of a radioactive core located at the height h from the surface of the detector. It is observed that:

    
    
        when the residual energy decreases, (θc) decreases and (θi) increases.
        for (θi)=0 (normal incidence for each position h), the corresponding (θc) has the larger value for each position h, and when (θi) increases, (θc) decreases. Thus, (θic) for each position h is none other than the intersection between the angle of incidence of the particle to be detected (θi) and the resolution angle of the detector (θc), because from this value, (θi) is greater than (θc).
    
    


2—Case where Eind is Greater than Em;
At first, in the interval of residual energy [Em, Eind], it is observed that when the residual energy decreases, (θi) increases and (θc) increases.
Secondly, in the interval of residual energy [θ, Em], it is observed that when the residual energy decreases, (θi) increases and (θc) decreases.
For (θi)=0 (normal incidence for each position h), the corresponding angle (θc) is also greater than zero for each position h and when (θi) increases, (θc) increases and then decreases, thus when the function appears as (θc)=F(Eres) {strong phase of decrease of (θc) for Eresε[Em, Ei]}, (θi) will have one and only one intersection with (θc) for each position h, i.e. in the first interval or in the second interval of energy, from this value of intersection, (θi) is greater than (θc).
Thus, in a general manner, when the function appears as (θc)=F(Eres), for each position h where Emin≦Eind≦Emax, (θic) is none other than the intersection between the angle of incidence of the particle to be detected (θi) and the resolution angle of the detector (θc).
A-3—Case of Nuclei Emitters of Alpha Particles Having Energy Ei and for which Eind>Emax.
These are nuclei located in the volume for contribution to the tracks defined by hε[0, Ri−Rmax], for the case without an absorbent medium, where Rmax and the total trajectory in the sample of an alpha particle having energy E=Emax.
In the same manner as above, when the function has the appearance (θc)=F(Eres), there will be either zero intersections, or 2 intersections between (θi) and (θc) for each (h) which will be denoted (θic1) and (θic2) which correspond respectively to the maximum energy and to the one that is the minimum detectable at oblique incidence for each (h).
Thus, for Eind>Emax, only the particles emitted at angles between (θic1) and (θic2) will be detected if they reach the detector.
B) Evaluation of the Function (θic).
In order to determine (θic), (θic1), and (θic2) as a function of h, for the case without an absorbent medium, proceed as follows:

(θic)=f1(Eres)=f2(x), EresεE[0, Ei], xεE[0,Ri]

    
    
        Ri is the total trajectory of the particle of energy Ei in the medium analyzed.
        x: is the trajectory of the particle of energy Ei in the medium analyzed, this particle reaches the detector with an energy Eres, x will be determined from the relation between energy and particle in the medium analyzed [2].

(θi)=arc cos(h/x)=g2(h,x)=g1(h, Eres), xε[h, Ri]

        h is the position of the radioactive nucleus emitter of the particle of energy Ei relative to the surface of the detector.
        x: is the trajectory of the particle of energy Ei in the medium analyzed and varies from h to Ri for each position h.
    
    


In fact, (θic) will be determined by solving the equation: g2(h,x)=f2(x) for the heights hi where it is confirmed that Emin≦Eind≦Emax, (θic1) and (θic2) will be determined by solving the equation: g2(h,x)=f2(x) for the heights hi where it is confirmed that Eind>Emax. Thus, a determination is done of (θic)=f(h) for the values of h for which Emin≦Eind≦Emax, (θic1)=fa(h) and (θic2)=fn(h) for the values of h for which Eind>Emax.
It is possible to convert from a function f1(Eres) to its equivalent f2(x) for each type of alpha particle having energy (Ei) and total trajectory in the sample analyzed (Ri), emitted from a nucleus located in a position (h) within the sample to be analyzed, either by the slope of the function of Linear Energy Transfer (L.E.T.=dE/dx) of the alpha particles in a medium, or by the Energy-Trajectory relation [E=f(R)] of the alpha particles in a medium:

    
    
        For the case without an absorbing medium:
        Using the function of Linear Energy Transfer (dE/dx) in the medium to be analyzed:
    
    






  
    E
    res
  
  =
  
    
      
        ∫
        0
        
          
            R
            i
          
          -
          x
        
      
      ⁢
      
        
      
      ⁢
      
        
          
            d
            E
          
          
            d
            x
          
        
        ·
        
          d
          x
        
      
    
    =
    
      
        E
        i
      
      -
      
        
          ∫
          
            
              R
              i
            
            -
            x
          
          
            R
            i
          
        
        ⁢
        
          
            
              d
              E
            
            
              d
              x
            
          
          ·
          
            
          
          ⁢
          
            d
            x
          
        
      
    
  




    
    
         thus an Eres corresponds for each x and, as a result, the conversion of a function f1(Eres) to its equivalent f2(x) and vice-versa.
        Using the Energy-Trajectory relation, [E=f(R)] in the medium to be analyzed:
    
    


The function E=f(R) is used and the change in the variable (R) is made by (Ri−x), this function becomes: Eres=31 i−x), thus for each x, the corresponding Eres will be determined and as a result, the conversion from one function f1(Eres) to its equivalent f2(x) and vice-versa.

    
    
        For the case with an absorbing medium:
        Using the Energy-Trajectory relation [E=f(R)]:
    
    


According to FIG. II-4 shown in the drawings section, where [(M1): medium to be analyzed. (M2): absorbent medium. (ha): thickness of the absorbent medium. x1 and x2 are, respectively, the trajectory of the alpha particle in the analyzed medium and the one in the absorbent medium.]:
Given:




  
    
      cos
      ⁡
      
        (
        
          θ
          i
        
        )
      
    
    =
    
      
        
          h
          -
          
            h
            a
          
        
        
          x
          1
        
      
      =
      
        
          
            h
            a
          
          
            x
            2
          
        
        =
        
          h
          x
        
      
    
  
  ,
  
    x
    =
    
      
        x
        1
      
      +
      
        x
        2
      
    
  




i.e.:

    
    
        E=f(R), the Energy-Trajectory function of alpha particles in the medium to be analyzed.
        R=g(E), the Energy-Trajectory function of alpha particles in the absorbent medium.
        Eres1 and Eres are, respectively, the residual energy of the incident alpha particle at the input of the absorbant medium and that at the input of the SSNTD.
    
    


Given Eres1=f(Ri−xi), thus Rlab=g(Eres1)=g∘f(Ri−x1). Where Rlab is the total trajectory of the alpha particle having energy Eres1 in the absorbent medium.
As a result, [Rlab−x2]=[g∘f(Ri−x1)−x2]=g(Eres), or




  
    
      x
      2
    
    =
    
      
        
          x
          1
        
        ·
        
          
            h
            a
          
          
            h
            -
            
              h
              a
            
          
        
      
      =
      
        x
        ·
        
          
            h
            a
          
          h
        
      
    
  
  ,




In fact,





  
    
      g
      ·
      
        f
        ⁡
        
          (
          
            
              R
              i
            
            -
            
              x
              ·
              
                
                  h
                  a
                
                
                  h
                  -
                  
                    h
                    a
                  
                
              
            
          
          )
        
      
    
    -
    
      x
      ·
      
        
          h
          a
        
        h
      
    
  
  =
  
    g
    ⁡
    
      (
      
        E
        res
      
      )
    
  




As a result, for each (h), each x corresponds to a residual energy Eres at the input of the SSNTD. Thus, for each position (hi), the function g2(hi,x) will be transformed into its equivalent function g1(hi,Eres) and vice-versa. As a result, the resolution for each (hi) of the equation g1(hi,Eres)=f1(Eres) will make it possible to determine θic(hi) for each

(hi)ε[ha,Ri+ha−f−1∘g−1(ha)].

    
    
        Using the function for Linear Energy Transfer (dE/dx);
    
    


The calculation methodology is the same as the one using the Energy-Trajectory relation, only instead of the functions Energy-Trajectory and Trajectory-Energy in the analyzed medium and the absorbent medium, they are replaced with the concept of the integral of the function for Linear Energy Transfer as described in the case without absorbent medium.
II-1-2—Determination of the Probability Pi(r, h).
II-1-2-a—Case of Nucleus Emitting Alpha Particles of Energy Ei and for which Emin≦Eind≦Emax.
According to FIG. (II-5) shown in the drawings section, the proportion of particles emitted from a nucleus located in a position of coordinates (r, h) and at angles of incidence less than the critical angle of incidence (θic) which reach the detector is the ratio of the surface of intersection {between the surface described by the angle (θic) on the plane of the detector denoted s(θic) and that of the detector denoted S(R)} and the surface described by the angle (θic) on the plane of the detector denoted s(θic). This proportion is described by the following equation:




  {
  
    
      
        S
        ⁡
        
          (
          
            θ
            ic
          
          )
        
      
      ⋂
      
        S
        ⁡
        
          (
          R
          )
        
      
    
    
      S
      ⁡
      
        (
        
          θ
          ic
        
        )
      
    
  
  }




Thus, the probability that a particle of energy Ei emitted by the nucleus having index i contained in the volume dv and confirmed by Emin≦Eind≦Emax can be detected by the solid state nuclear track detector is:




  
    Pi
    ⁡
    
      (
      
        r
        ,
        h
      
      )
    
  
  =
  
    
      {
      
        
          1
          -
          
            cos
            ⁢
            
              
            
            ⁢
            
              θ
              ic
            
          
        
        2
      
      }
    
    ⁢
    
      {
      
        
          
            S
            ⁡
            
              (
              
                θ
                ic
              
              )
            
          
          ⋂
          
            S
            ⁡
            
              (
              R
              )
            
          
        
        
          S
          ⁡
          
            (
            
              θ
              ic
            
            )
          
        
      
      }
    
  




    
    
        Evaluation of
    
    






  
    {
    
      
        
          S
          ⁡
          
            (
            
              θ
              ic
            
            )
          
        
        ⋂
        
          S
          ⁡
          
            (
            R
            )
          
        
      
      
        S
        ⁡
        
          (
          
            θ
            ic
          
          )
        
      
    
    }
  
  ,




    
    
         with S(θic)=πR2c, where Rc=h·tg(θic). (See FIG. II-5)
    
    
    a) for (θic)≧(θ2), S(θic)∩S(R)=πR2, where R: radius of SSNTD
    b) for (θic)≦(θ2): 2 cases are present, (θic)≧(θ1) and (θic)≦(θ1):
    
        if (θic)≧(θ1)
    
    





  
    
      
        S
        ⁡
        
          (
          
            θ
            ic
          
          )
        
      
      ⋂
      
        S
        ⁡
        
          (
          R
          )
        
      
    
    =
    
      2
      ⁡
      
        [
        
          
            
              ∫
              
                r
                -
                Rc
              
              I
            
            ⁢
            
              
                
                  
                    R
                    c
                    2
                  
                  -
                  
                    
                      (
                      
                        x
                        -
                        r
                      
                      )
                    
                    2
                  
                
              
              ·
              
                
              
              ⁢
              
                d
                x
              
            
          
          +
          
            
              ∫
              I
              R
            
            ⁢
            
              
                
                  
                    R
                    2
                  
                  -
                  
                    x
                    2
                  
                
              
              .
              
                
              
              ⁢
              
                d
                x
              
            
          
        
        ]
      
    
  




  where




  
    I
    =
    
      
        
          R
          2
        
        +
        
          r
          2
        
        -
        
          R
          c
          2
        
      
      
        2
        ⁢
        r
      
    
  



    
    
        if (θic)≦(θ1): S(θic)∩S(R)=S(θic)

II-1-2-b—Case of Nuclei Emitting Alpha Particles of Energy Ei and for which Eind>Emax 

    
    


According to FIG. (II-5) shown in the drawings section, the proportion of particles emitted from a nucleus located in a position of coordinates (r, h) and at angles of incidence between (θic1) and (θic2) which reach the detector is the ratio of the surface of intersection {between the surface of the ring described by the angles (θic1) and (θic2) on the plane of the detector denoted s(θic1, θic2) and that of the detector denoted S(R)} and the surface of the ring described by the angles (θic1) and (θic2) on the plane of the detector denoted s(θic1, θic2). This proportion is described by the following equation:




  {
  
    
      
        S
        ⁡
        
          (
          
            
              θ
              
                ic
                ⁢
                
                  
                
                ⁢
                1
              
            
            ,
            
              θ
              
                ic
                ⁢
                
                  
                
                ⁢
                2
              
            
          
          )
        
      
      ⋂
      
        S
        ⁡
        
          (
          R
          )
        
      
    
    
      S
      ⁡
      
        (
        
          
            θ
            
              ic
              ⁢
              
                
              
              ⁢
              1
            
          
          ,
          
            θ
            
              ic
              ⁢
              
                
              
              ⁢
              2
            
          
        
        )
      
    
  
  }




Thus, the probability that a particle of energy Ei emitted by the nucleus having index i contained in the volume dv and confirmed by Eind>Emax can be detected by the solid state nuclear track detector is:



  
    
      Pi
      ⁡
      
        (
        
          r
          ,
          h
        
        )
      
    
    =
    
      
        {
        
          
            
              cos
              ⁢
              
                
              
              ⁢
              
                θ
                
                  ic
                  ⁢
                  
                    
                  
                  ⁢
                  1
                
              
            
            -
            
              cos
              ⁢
              
                
              
              ⁢
              
                θ
                
                  ic
                  ⁢
                  
                    
                  
                  ⁢
                  2
                
              
            
          
          2
        
        }
      
      ⁢
      
        {
        
          
            
              S
              ⁡
              
                (
                
                  
                    θ
                    
                      ic
                      ⁢
                      
                        
                      
                      ⁢
                      1
                    
                  
                  ,
                  
                    θ
                    
                      ic
                      ⁢
                      
                        
                      
                      ⁢
                      2
                    
                  
                
                )
              
            
            ⋂
            
              S
              ⁡
              
                (
                R
                )
              
            
          
          
            S
            ⁡
            
              (
              
                
                  θ
                  
                    ic
                    ⁢
                    
                      
                    
                    ⁢
                    1
                  
                
                ,
                
                  θ
                  
                    ic
                    ⁢
                    
                      
                    
                    ⁢
                    2
                  
                
              
              )
            
          
        
        }
      
    
  




  
    ◆
    ⁢
    
      
    
    ⁢
    Evaluation
    ⁢
    
      
    
    ⁢
    of
    ⁢
    
      
    
    ⁢
    
      {
      
        
          
            S
            ⁡
            
              (
              
                
                  θ
                  
                    ic
                    ⁢
                    
                      
                    
                    ⁢
                    1
                  
                
                ,
                
                  θ
                  
                    ic
                    ⁢
                    
                      
                    
                    ⁢
                    2
                  
                
              
              )
            
          
          ⋂
          
            S
            ⁡
            
              (
              R
              )
            
          
        
        
          S
          ⁡
          
            (
            
              
                θ
                
                  ic
                  ⁢
                  
                    
                  
                  ⁢
                  1
                
              
              ,
              
                θ
                
                  ic
                  ⁢
                  
                    
                  
                  ⁢
                  2
                
              
            
            )
          
        
      
      }
    
  



with: −S(θic1,θic2)=π(Rc22−Rc22−Rc12) where Rc1=h·tg(θic1),Rc2=h·tg(θic2)

    
    
        −S(θic1,θic2)∩S(R)=S(θic2)∩S(R)−S(θic1)∩S(R)
    
    
    a) for (θic1)≧(θ2),S(θic1,θic2)∩S(R)=πR2−πR2=0
    
        where R: radius of the SSNTD
    
    
    b) for (θ1)≦(θic1)≦(θ2): there are 2 cases, (θic2)≧(θ2) and (θic2)≦(θ2)
    
        if (θic2)≧(θ2):
    
    





  
    
      
        S
        ⁡
        
          (
          
            
              θ
              
                ic
                ⁢
                
                  
                
                ⁢
                1
              
            
            ,
            
              θ
              
                ic
                ⁢
                
                  
                
                ⁢
                2
              
            
          
          )
        
      
      ⋂
      
        S
        ⁡
        
          (
          R
          )
        
      
    
    =
    
      
        π
        ⁢
        
          
        
        ⁢
        
          R
          2
        
      
      -
      
        2
        ⁡
        
          [
          
            
              
                ∫
                
                  r
                  -
                  
                    R
                    
                      c
                      ⁢
                      
                        
                      
                      ⁢
                      1
                    
                  
                
                I
              
              ⁢
              
                
                  
                    
                      R
                      
                        c
                        ⁢
                        
                          
                        
                        ⁢
                        1
                      
                      2
                    
                    -
                    
                      
                        (
                        
                          x
                          -
                          r
                        
                        )
                      
                      2
                    
                  
                
                ·
                
                  
                
                ⁢
                
                  d
                  x
                
              
            
            +
            
              
                ∫
                
                  I
                  1
                
                R
              
              ⁢
              
                
                  
                    
                      R
                      2
                    
                    -
                    
                      x
                      2
                    
                  
                
                ·
                
                  
                
                ⁢
                
                  d
                  x
                
              
            
          
          ]
        
      
    
  




  
    
      where
      ⁢
      
        
      
      ⁢
      
        I
        1
      
    
    =
    
      
        
          R
          2
        
        +
        
          r
          2
        
        -
        
          R
          
            c
            ⁢
            
              
            
            ⁢
            1
          
          2
        
      
      
        2
        ⁢
        r
      
    
  




  
    
      
        ◆if
        ⁢
        
          
        
        ⁢
        
          (
          
            θ
            
              ic
              ⁢
              
                
              
              ⁢
              2
            
          
          )
        
      
      ≤
      
        
          
            (
            
              θ
              2
            
            )
          
          ⁢
          
            :
          
          ⁢
          
            S
            ⁡
            
              (
              
                
                  θ
                  
                    ic
                    ⁢
                    
                      
                    
                    ⁢
                    1
                  
                
                ,
                
                  θ
                  
                    ic
                    ⁢
                    
                      
                    
                    ⁢
                    2
                  
                
              
              )
            
          
        
        ⋂
        
          S
          ⁡
          
            (
            R
            )
          
        
      
    
    =
    
      
        S
        ⁡
        
          (
          
            θ
            
              ic
              ⁢
              
                
              
              ⁢
              2
            
          
          )
        
      
      ⋂
      
        S
        ⁡
        
          (
          R
          )
        
      
      -
      
        S
        ⁡
        
          (
          
            θ
            
              ic
              ⁢
              
                
              
              ⁢
              1
            
          
          )
        
      
      ⋂
      
        S
        ⁡
        
          (
          R
          )
        
      
    
  




  
    where
    ⁢
    
      :
    
  




  
    
      
        S
        ⁡
        
          (
          
            θ
            
              ic
              ⁢
              
                
              
              ⁢
              2
            
          
          )
        
      
      ⋂
      
        S
        ⁡
        
          (
          R
          )
        
      
    
    =
    
      2
      ⁡
      
        [
        
          
            
              ∫
              
                r
                -
                
                  R
                  
                    c
                    ⁢
                    
                      
                    
                    ⁢
                    2
                  
                
              
              
                I
                2
              
            
            ⁢
            
              
                
                  
                    R
                    
                      c
                      ⁢
                      
                        
                      
                      ⁢
                      2
                    
                    2
                  
                  -
                  
                    
                      (
                      
                        x
                        -
                        r
                      
                      )
                    
                    2
                  
                
              
              ·
              
                
              
              ⁢
              
                d
                x
              
            
          
          +
          
            
              ∫
              
                I
                2
              
              R
            
            ⁢
            
              
                
                  
                    R
                    2
                  
                  -
                  
                    x
                    2
                  
                
              
              ·
              
                
              
              ⁢
              
                d
                x
              
            
          
        
        ]
      
    
  




  
    
      I
      2
    
    =
    
      
        R
        2
      
      +
      
        r
        2
      
      -
      
        
          R
          
            c
            ⁢
            
              
            
            ⁢
            2
          
          2
        
        ⁢
        
          
        
        [
        sic
        ]
      
    
  




  and




  
    
      
        S
        ⁡
        
          (
          
            θ
            
              ic
              ⁢
              
                
              
              ⁢
              1
            
          
          )
        
      
      ⋂
      
        S
        ⁡
        
          (
          R
          )
        
      
    
    =
    
      2
      ⁡
      
        [
        
          
            
              ∫
              
                r
                -
                
                  R
                  
                    c
                    ⁢
                    
                      
                    
                    ⁢
                    1
                  
                
              
              
                I
                1
              
            
            ⁢
            
              
                
                  
                    R
                    
                      c
                      ⁢
                      
                        
                      
                      ⁢
                      1
                    
                    2
                  
                  -
                  
                    
                      (
                      
                        x
                        -
                        r
                      
                      )
                    
                    2
                  
                
              
              ·
              
                
              
              ⁢
              
                d
                x
              
            
          
          +
          
            
              ∫
              
                I
                1
              
              R
            
            ⁢
            
              
                
                  
                    R
                    2
                  
                  -
                  
                    x
                    2
                  
                
              
              ·
              
                
              
              ⁢
              
                d
                x
              
            
          
        
        ]
      
    
  




  
    
      I
      1
    
    =
    
      
        
          R
          2
        
        +
        
          r
          2
        
        -
        
          R
          
            c
            ⁢
            
              
            
            ⁢
            1
          
          2
        
      
      
        2
        ⁢
        r
      
    
  



    c) for (θic1)≦(θi): there are 2 cases, (θic2)≧(θ1) and (θic2)≦(θ1):
    1—for (θic2)≧(θ1): there are 2 cases, (θic2)≧(θ2) and (θic2)≦(θ2):
    
        if (θic2)≧(θ2): S(θic1, θic2)∩S(R)=πR2−πRc12
    
    





  
    
      
        ◆if
        ⁢
        
          
        
        ⁢
        
          (
          
            θ
            
              ic
              ⁢
              
                
              
              ⁢
              2
            
          
          )
        
      
      ≤
      
        
          
            (
            
              θ
              2
            
            )
          
          ⁢
          
            :
          
          ⁢
          
            S
            ⁡
            
              (
              
                
                  θ
                  
                    ic
                    ⁢
                    
                      
                    
                    ⁢
                    1
                  
                
                ,
                
                  θ
                  
                    ic
                    ⁢
                    
                      
                    
                    ⁢
                    2
                  
                
              
              )
            
          
        
        ⋂
        
          S
          ⁡
          
            (
            R
            )
          
        
      
    
    =
    
      
        2
        ⁡
        
          [
          
            
              
                ∫
                
                  r
                  -
                  
                    R
                    
                      c
                      ⁢
                      
                        
                      
                      ⁢
                      2
                    
                  
                
                
                  I
                  2
                
              
              ⁢
              
                
                  
                    
                      R
                      
                        c
                        ⁢
                        
                          
                        
                        ⁢
                        2
                      
                      2
                    
                    -
                    
                      
                        (
                        
                          x
                          -
                          r
                        
                        )
                      
                      2
                    
                  
                
                ·
                
                  
                
                ⁢
                
                  d
                  x
                
              
            
            +
            
              
                ∫
                
                  I
                  2
                
                R
              
              ⁢
              
                
                  
                    
                      R
                      2
                    
                    -
                    
                      x
                      2
                    
                  
                
                ·
                
                  
                
                ⁢
                
                  d
                  x
                
              
            
          
          ]
        
      
      -
      
        πR
        
          c
          ⁢
          
            
          
          ⁢
          1
        
        2
      
    
  




  
    
      
        
          2
          ⁢
          
            -
          
          ⁢
          
            
          
          ⁢
          for
          ⁢
          
            
          
          ⁢
          
            (
            
              θ
              
                ic
                ⁢
                
                  
                
                ⁢
                2
              
            
            )
          
        
        ≤
        
          
            
              (
              
                θ
                1
              
              )
            
            ⁢
            
              :
            
            ⁢
            
              S
              ⁡
              
                (
                
                  
                    θ
                    
                      ic
                      ⁢
                      
                        
                      
                      ⁢
                      1
                    
                  
                  ,
                  
                    θ
                    
                      ic
                      ⁢
                      
                        
                      
                      ⁢
                      2
                    
                  
                
                )
              
            
          
          ⋂
          
            S
            ⁡
            
              (
              R
              )
            
          
        
      
      =
      
        
          S
          ⁡
          
            (
            
              
                θ
                
                  ic
                  ⁢
                  
                    
                  
                  ⁢
                  1
                
              
              ,
              
                θ
                
                  ic
                  ⁢
                  
                    
                  
                  ⁢
                  2
                
              
            
            )
          
        
        =
        
          
            π
            ⁢
            
              
            
            ⁢
            
              R
              
                c
                ⁢
                
                  
                
                ⁢
                2
              
              2
            
          
          -
          
            π
            ⁢
            
              
            
            ⁢
            
              R
              
                c
                ⁢
                
                  
                
                ⁢
                1
              
              2
            
          
        
      
    
    ⁢
    
      
    
  



II-2—Generalization for the Total Contribution Volume for Resolution of Tracks:

According to the above, the total number of tracks observed and which are due to the radioactive nuclei of index i contained in the total volume:

nit=2π·Acite·∫∫Pi(r,h)·rdrdh

For the case Emin≦Eind≦Emax, a comparison of (θic) with (θ1) and (θ2) allows to determine respectively r1(h) and r2(h) where these latter are characteristics of respectively the radii defining the interval of the volume where (θic)≦(θ1) from the one where (θic)≧(θ1) and the ones defining the interval of the volume where (θic)≦(θ2) from the one where (θic)≧(θ2).
For the case Eind>Emax, a comparison of (θic1) with (θ1) and (θ2) allows to determine respectively r1,1(h) and r2,1(h) where these latter are characteristics of respectively the radii defining the interval of the volume where (θic)≦(θ1) [illegible] (θic)≧(θ1) and the ones defining the interval of the volume where (θic1)≦(θ2) [illegible] (θic1)≧(θ2). And a comparison of (θic2) with (θ1) and (θ2) allows us to determine respectively r1,2(h) and r2,2(h) where these latter are characteristics of respectively the radii defining the interval of the volume where (θic2)≦(θ1) from the one where (θic2)≧(θ1) and the ones defining the interval of the volume where (θic2)≦(θ2) from the one where (θic2)≧(θ2).
The next part of the procedure is an integration by parts of the interval over r and h.
To determine r1(h), r2(h), r1,1(h), r2,1(h), r1,2(h), and r2,2(h), it is simply a matter of solving the following equations:

It has: −(θic)=f(h); (θic1)=fa(h); (θic2)=fb(h)





  
    
      -
      
        (
        
          θ
          1
        
        )
      
    
    =
    
      arc
      ⁢
      
        
      
      ⁢
      
        cos
        ⁢
        
          
        
        [
        
          h
          
            
              
                h
                2
              
              +
              
                
                  (
                  
                    R
                    -
                    r
                  
                  )
                
                2
              
            
          
        
        ]
      
    
  
  ;
  
    
      (
      
        θ
        2
      
      )
    
    =
    
      arc
      ⁢
      
        
      
      ⁢
      
        cos
        [
        
          h
          
            
              
                h
                2
              
              +
              
                
                  (
                  
                    R
                    +
                    r
                  
                  )
                
                2
              
            
          
        
        ]
      
    
  




thus:

*(θic)=(θ1)r1(h); *(θic)=(θ2)r2(h)

*(θic1)=(θ1)r1,1(h); *(θic1)=(θ2)r2,1(h)

*(θic2)=(θ1)r1,2(h); *(θic2)=(θ2)r2,2(h)

Actually, the total volume of the sample contributing to the tracks (Vt) will be the total of a series of the volume Vf (Vt=ΣVj), where each Vj itself corresponds to a probability Pi,j(r,h).
In conclusion and in a general manner, the total number of tracks observed and which are due to the radioactive nuclei of index i contained in the total volume:




  
    n
    it
  
  =
  
    
      A
      ci
      te
    
    ·
    
      
        ∑
        j
      
      ⁢
      
        ∫
        
          ∫
          
            ∫
            
              
                
                  P
                  ij
                
                ⁡
                
                  (
                  
                    r
                    ,
                    h
                  
                  )
                
              
              ·
              
                d
                
                  v
                  j
                
              
            
          
        
      
    
  




where: Pij(r,h) and vj will be defined as a function of r1(h), r2(h), r1,1(h), r2,1(h), r1,2(h), and r2,2(h)

with:




  
    
      
        Pi
        ⁡
        
          (
          
            r
            ,
            h
          
          )
        
      
      =
      
        
          (
          
            
              1
              -
              
                cos
                ⁢
                
                  
                
                ⁢
                
                  θ
                  ic
                
              
            
            2
          
          )
        
        ⁢
        
          {
          
            
              
                S
                ⁡
                
                  (
                  
                    θ
                    ic
                  
                  )
                
              
              ⋂
              
                S
                ⁡
                
                  (
                  R
                  )
                
              
            
            
              S
              ⁡
              
                (
                
                  θ
                  ic
                
                )
              
            
          
          }
        
      
    
    ;
    
      
        for
        ⁢
        
          
        
        ⁢
        E
        ⁢
        
          
        
        ⁢
        min
      
      ≤
      
        E
        ⁢
        
          
        
        ⁢
        ind
      
      ≤
      
        E
        ⁢
        
          
        
        ⁢
        max
      
    
  




  
    
      
        Pi
        ⁡
        
          (
          
            r
            ,
            h
          
          )
        
      
      =
      
        
          (
          
            
              
                cos
                ⁢
                
                  
                
                ⁢
                
                  θ
                  
                    ic
                    ⁢
                    
                      
                    
                    ⁢
                    1
                  
                
              
              -
              
                cos
                ⁢
                
                  
                
                ⁢
                
                  θ
                  
                    ic
                    ⁢
                    
                      
                    
                    ⁢
                    2
                  
                
              
            
            2
          
          )
        
        ⁢
        
          {
          
            
              
                S
                ⁡
                
                  (
                  
                    
                      θ
                      
                        ic
                        ⁢
                        
                          
                        
                        ⁢
                        1
                      
                    
                    ,
                    
                      θ
                      
                        ic
                        ⁢
                        
                          
                        
                        ⁢
                        2
                      
                    
                  
                  )
                
              
              ⋂
              
                S
                ⁡
                
                  (
                  R
                  )
                
              
            
            
              S
              ⁡
              
                (
                
                  
                    θ
                    
                      ic
                      ⁢
                      
                        
                      
                      ⁢
                      1
                    
                  
                  ,
                  
                    θ
                    
                      ic
                      ⁢
                      
                        
                      
                      ⁢
                      2
                    
                  
                
                )
              
            
          
          }
        
      
    
    ;
    
      
        for
        ⁢
        
          
        
        ⁢
        E
        ⁢
        
          
        
        ⁢
        ind
      
      >
      
        E
        ⁢
        
          
        
        ⁢
        max
      
    
  



where:





  
    {
    
      
        
          S
          ⁡
          
            (
            
              θ
              ic
            
            )
          
        
        ⋂
        
          S
          ⁡
          
            (
            R
            )
          
        
      
      
        S
        ⁡
        
          (
          
            θ
            ic
          
          )
        
      
    
    }
  
  ⁢
  
    
  
  ⁢
  and
  ⁢
  
    
  
  ⁢
  
    {
    
      
        
          S
          ⁡
          
            (
            
              
                θ
                
                  ic
                  ⁢
                  
                    
                  
                  ⁢
                  1
                
              
              ,
              
                θ
                
                  ic
                  ⁢
                  
                    
                  
                  ⁢
                  2
                
              
            
            )
          
        
        ⋂
        
          S
          ⁡
          
            (
            R
            )
          
        
      
      
        S
        ⁡
        
          (
          
            
              θ
              
                ic
                ⁢
                
                  
                
                ⁢
                1
              
            
            ,
            
              θ
              
                ic
                ⁢
                
                  
                
                ⁢
                2
              
            
          
          )
        
      
    
    }
  




are well defined for c [sic]

III—Application Example:

Taking the simplest case, involving the media where the total trajectory of the particle αi of energy Ei is much less than the radius of the detector (Ri<<R) {case most reported due to the fact that this condition is verifiable for alpha particles in almost all liquid and solid samples} and a detector having a response function for conditions of developing and reading the well-defined tracks such as Emin=0 and Emax>Ej.
Following the above gives:




  
    cos
    ⁢
    
      
    
    ⁢
    
      (
      
        θ
        2
      
      )
    
  
  =
  
    h
    
      
        
          h
          2
        
        +
        
          
            (
            
              R
              +
              r
            
            )
          
          2
        
      
    
  




Thus, (θ2min) for (Ri<<R) approaches





  
    π
    2
  
  ,




and as a result, (θic) is always less than (θ2)

Actually, a comparison of (θic) with (θ1) allows us to determine r1(h) and, as a result:



  
    
      n
      it
    
    =
    
      2
      ⁢
      
        
      
      ⁢
      
        π
        ·
        
          A
          ci
          te
        
        ·
        
          {
          
            
              
                ∫
                0
                
                  R
                  
                    
                      1
                      ′
                    
                    ⁢
                    i
                  
                
              
              ⁢
              
                
                  ∫
                  0
                  
                    (
                    h
                    )
                  
                
                ⁢
                
                  
                    
                      1
                      -
                      
                        cos
                        ⁢
                        
                          
                        
                        ⁢
                        
                          θ
                          ic
                        
                      
                    
                    2
                  
                  ⁢
                  r
                  ⁢
                  
                    
                  
                  ⁢
                  
                    d
                    r
                  
                  ⁢
                  
                    
                  
                  ⁢
                  
                    d
                    h
                  
                
              
            
            +
            
              
                ∫
                0
                
                  R
                  1
                
              
              ⁢
              
                
                  ∫
                  
                    
                      r
                      1
                    
                    ⁡
                    
                      (
                      h
                      )
                    
                  
                  R
                
                ⁢
                
                  
                    
                      1
                      -
                      
                        cos
                        ⁢
                        
                          
                        
                        ⁢
                        
                          θ
                          ic
                        
                      
                    
                    2
                  
                  ⁢
                  
                    (
                    
                      
                        
                          S
                          ⁡
                          
                            (
                            
                              θ
                              ic
                            
                            )
                          
                        
                        ⋂
                        
                          S
                          ⁡
                          
                            (
                            R
                            )
                          
                        
                      
                      
                        S
                        ⁡
                        
                          (
                          
                            θ
                            ic
                          
                          )
                        
                      
                    
                    )
                  
                  ⁢
                  r
                  ⁢
                  
                    
                  
                  ⁢
                  
                    d
                    r
                  
                  ⁢
                  
                    
                  
                  ⁢
                  
                    d
                    h
                  
                
              
            
          
          }
        
      
    
  




  
    where
    ⁢
    
      :
    
  




  
    
      
        _S
        ⁢
        
          (
          
            θ
            ic
          
          )
        
      
      =
      
        π
        ⁢
        
          
        
        ⁢
        
          R
          c
          2
        
      
    
    ,
    
      
        R
        c
      
      =
      
        h
        ·
        
          tg
          ⁡
          
            (
            
              θ
              ic
            
            )
          
        
      
    
  




  
    
      
        _S
        ⁢
        
          (
          
            θ
            ic
          
          )
        
      
      ⋂
      
        S
        ⁡
        
          (
          R
          )
        
      
    
    =
    
      2
      ⁡
      
        [
        
          
            
              ∫
              
                r
                -
                
                  R
                  
                    c
                    ⁢
                    
                      
                    
                  
                
              
              I
            
            ⁢
            
              
                
                  
                    R
                    
                      c
                      ⁢
                      
                        
                      
                    
                    2
                  
                  -
                  
                    
                      (
                      
                        x
                        -
                        r
                      
                      )
                    
                    2
                  
                
              
              ·
              
                
              
              ⁢
              
                d
                x
              
            
          
          +
          
            
              ∫
              I
              R
            
            ⁢
            
              
                
                  
                    R
                    2
                  
                  -
                  
                    x
                    2
                  
                
              
              ·
              
                
              
              ⁢
              
                d
                x
              
            
          
        
        ]
      
    
  




  
    I
    =
    
      
        
          R
          2
        
        +
        
          r
          2
        
        -
        
          R
          c
          2
        
      
      
        2
        ⁢
        r
      
    
  



IV—Solution to the Problem Involving Media Having Several Types of Nuclei Emitting Particles with Different Energies and Activities.

A technique for extracting energy is used whereby an absorbent medium must be installed in contact with the solid state nuclear track detector in a manner such that only those particles having the largest energy can reach the detector and the other energies follow in sequence.
And if it appears that for a given medium, the ability to separate the energies by the SSNTD using an absorbent medium is not feasible, then an absorbent medium of thickness (ha) is installed in contact with the detector with different thicknesses (ha) of the absorbent medium, in a manner such that the number of cases of thicknesses (ha) of the absorbent medium coincides with the total number of different alpha emitters of the medium to be analyzed. As a result, one has the number of equations linking the total number of tracks observed for each (ha) to the activities of the different alpha emitters, [illegible] of the alpha activities of the different emitters (the unknowns). Thus, [illegible] alpha activities of the different emitters.
For the determination of (θic(h)), (θic1(h)), (θic2(h)) of each type of nucleus in the presence of an absorbent medium, the method is to use two equations (θc=f(Eres)) and (θi=f(Eres)) (see part B of paragraph II-1-1) and the calculation methodology is the same while taking into account the presence of the absorbing medium.
V—Conclusion:
This direct technique allowing the dosimetry of the radioactive nucleii in different types of samples has been developed at home and in an individual manner. It is based on the use of a novel method adapted to the experimental conditions using the solid state nuclear track detectors.
It consists in calculating the probability Pi(r,h) that an alpha particle emitted by a radioactive nucleus contained in the volume dv of a sample can be detected by a solid state nuclear track detector, where (dv) is the volume element of a cylinder having a radius between r and r+dr and a height between h and h+dh inside the sample to be analyzed.
This method is easy to implement even though its formalism appears quite complex. It is precise, absolute, and not costly.
Solely for reasons of simplification of the calculations the radius is set of the detector equal to the radius of the irradiation cell, the methodology of calculation is the same for a radius of the detector less than the radius of the irradiation cell with small modifications at the step of evaluation of the surfaces of intersection {S(θic)∩S(R)} and {S(θic1,θic2)∩S(R)}, this time, the calculation must be made of the surfaces of intersection for r (characterizing the position of the emitting nucleus) less than the radius of the detector and for r greater than the radius of the detector.
In a general manner, regardless of the appearance of the experimental function of the intrinsic critical angle of a detector θc=F(Eres) and for the position (hi), more than 2 intersections with θi(hi) are present, the efficiency of the detection for these positions (hi) will be a series of combinations between the two general formulas for efficiencies of detection described in this paper, the one corresponding to a single intersection and the other corresponding to two intersections, where the nature of this combination depends on the number of intersections.
REFERENCES


    [1] G. Somogyi, Nucl. Instr. Meth., 173, 21 (1980).
    [2] J. F. Ziegler, Programme TRIM.

## Claims

1. A process for defining a probability of detection by a detector of traces of nuclear particles for nuclei of different types, the process comprising:
identifying angles of incidence (θic), (θic1) and (θic2) of alpha particles of energy (Ei) emitted by the nucleus and reaching the detector;
detecting the angles of revelation of the detector;
determining a relation between the angle of incidence (θi) of the particles on the detector and the angle of revelation (θc) of the detector for values (h) characterizing an intersection between the angle of incidence of the particular on the detector and the angle of revelation of the detector;
obtaining either zero intersections or two intersections for each of the values (h) and which correspond to a maximum energy (Emax) and a minimum energy (Emin) that are detectable in oblique incidence of the particles for each of the values (h);
identifying and evaluating intersection surfaces {S(θic)∩S(R)} and {S(θic1,θic2)∩S(R) } representing respectively the intersection surfaces between the surface described by the angle (θic) on a plane of the detector and that of the detector at a radius (R) and the intersection surface between a surface described by a ring limited by the angles (θic1) and (θic2) on the plane of the detector and that of the detector at the radius (R), a residual energy of the alpha particle of energy Ei emitted by the nucleus at a height (h) of the plane of the detector is in an interval Emin≦Eind≧Emax such that particles are emitted under angles between (θic1) and (θic2) for which Eind>Emax, a proportion of the particles emitted by a nucleus located in a position of coordinates (r,h) and under angles of incidence below a critical angle of incidence (θic) which reach the detector is a ratio between a surface defined by the surface S(θic) described by the angle (θic) on the plane of the detector and the surface S(θic) described by the angle (θic) on the plane of the detector defined by





  
    
      {
      
        
          
            S
            ⁡
            
              (
              
                θ
                ic
              
              )
            
          
          ⁢
          
            S
            ⁡
            
              (
              
                θ
                ic
              
              )
            
          
          ⁢
          
            S
            ⁡
            
              (
              R
              )
            
          
        
        
          S
          ⁡
          
            (
            
              θ
              ic
            
            )
          
        
      
      }
    
    ⁢
    
      
    
    ⁢
    and
  
  ;



determining a function Pi(r,h) expressing the probability of detection by the detector of traces of nuclear particles for each type of nucleus emitting the alpha particles of energy Ei for Emin≦Eind≦Emax in which





  
    
      
        
          
            
              P
              i
            
            ⁡
            
              (
              
                r
                ,
                h
              
              )
            
          
          =
          
            1
            -
            
              cos
              ⁢
              
                
              
              ⁢
              
                θ
                ic
              
            
          
        
      
      
        
          (
          
            
              1
              -
              
                cos
                ⁢
                
                  
                
                ⁢
                
                  θ
                  ic
                
              
            
            2
          
          )
        
      
    
  
  ⁢
  
    {
    
      
        
          S
          ⁡
          
            (
            
              θ
              ic
            
            )
          
        
        ⋂
        
          S
          ⁡
          
            (
            R
            )
          
        
      
      
        S
        ⁡
        
          (
          
            θ
            ic
          
          )
        
      
    
    }
  



 in which


S(θic)=πRc2,Rc=h·tg(θic).

2. The process of claim 1, further comprising:
using a position for Eind>Emax, a proportion of particles emitted by a nucleus located in a position of coordinates (r,h) and under angles of incidence between (θic1) and (θic2) and reaching the detector being a ratio between the intersection surface defined by surface S (θic,θic2) described by angles (θic1) and (θic2) on the plane of the detector and the surface S(R) of the detector and the surface S(θic1,θic2) of the ring described by angles (θic1) and (θic2) on the plane of the detector in accordance with





  
    {
    
      
        
          S
          ⁡
          
            (
            
              
                θ
                
                  ic
                  ⁢
                  
                    
                  
                  ⁢
                  1
                
              
              ,
              
                θ
                
                  ic
                  ⁢
                  
                    
                  
                  ⁢
                  2
                
              
            
            )
          
        
        ⋂
        
          S
          ⁡
          
            (
            R
            )
          
        
      
      
        S
        ⁡
        
          (
          
            
              θ
              
                ic
                ⁢
                
                  
                
                ⁢
                1
              
            
            ,
            
              θ
              
                ic
                ⁢
                
                  
                
                ⁢
                2
              
            
          
          )
        
      
    
    }
  
  ;
  and



determining a function Pi(r,h) of a probability of detection by the detector of nuclear particles for each type of nucleus emitting alpha particles of Ei in which the function Pi(r,h) is in accordance with





  
    
      P
      i
    
    ⁡
    
      (
      
        r
        ,
        h
      
      )
    
  
  =
  
    
      (
      
        
          
            cos
            ⁢
            
              
            
            ⁢
            
              θ
              
                ic
                ⁢
                
                  
                
                ⁢
                1
              
            
          
          -
          
            cos
            ⁢
            
              
            
            ⁢
            
              θ
              
                ic
                ⁢
                
                  
                
                ⁢
                2
              
            
          
        
        2
      
      )
    
    ⁢
    
      {
      
        
          
            S
            ⁡
            
              (
              
                
                  θ
                  
                    ic
                    ⁢
                    
                      
                    
                    ⁢
                    1
                  
                
                ,
                
                  θ
                  
                    ic
                    ⁢
                    
                      
                    
                    ⁢
                    2
                  
                
              
              )
            
          
          ⋂
          
            S
            ⁡
            
              (
              R
              )
            
          
        
        
          S
          ⁡
          
            (
            
              
                θ
                
                  ic
                  ⁢
                  
                    
                  
                  ⁢
                  1
                
              
              ,
              
                θ
                
                  ic
                  ⁢
                  
                    
                  
                  ⁢
                  2
                
              
            
            )
          
        
      
      }
    
  



 where





  (
  
    
      S
      ⁡
      
        (
        
          
            θ
            
              ic
              ⁢
              
                
              
              ⁢
              1
            
          
          ,
          
            θ
            
              ic
              ⁢
              
                
              
              ⁢
              2
            
          
        
        )
      
    
    =
    
      π
      ⁡
      
        (
        
          
            R
            ⁢
            
              2
              
                
                   
                  c
                
                ⁢
                2
              
            
          
          -
          
            R
            
              c
              ⁢
              
                
              
              ⁢
              1
            
            2
          
        
        )
      
    
  



 and


Rc1=h·tg(θic1) and Rc2=h·tg(θic2).

3. The process of claim 1, further comprising:
distributing a total volume (Vt) of contribution to the traces in a series of volumes Vj(Vt=ΣVj) where each Vj corresponds to a function representing an efficacy of detection Pij(r,h).