Monte Carlo Simulation for Solid Angle Calculations in Alpha Particle Spectrometry

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1 Monte Carlo Simulation for Solid Angle Calculations in Alpha Particle Spectrometry John Keightley NSUF 18 May 2010 Acknowledgement: Several discussions with Stefaan Pommé, IRMM, EC-JRC, Geel Some diagrams reproduced with permission

2010 Acknowledgement: Several discussions with Stefaan

2 Brief Overview Why is NPL interested in this topic New Defined Solid Angle (DSA) counting system Work in progress I have noticed that some laboratories are using questionable formulae for simple disc-disc solid angle calculation. I have written some simple computer code, to calculate solid angles using Monte Carlo simulation techniques Comparisons with published formulae ongoing

for simple disc-disc solid angle calculation.

3 Introduction Defined Solid Angle (DSA) Counting NPL s interest in this is to set up a new Defined Solid Angle (DSA) counting system for the Primary Standardisation of certain radionuclides. DSA counting is an excellent primary standardisation technique, only suitable for particles that: travel in straight lines (are not readily scattered) are heavily absorbed in detector, source substrate, diaphragm i.e.: alpha particles (several MeV) Low energy photons, less than 80 kev

DSA counting is an excellent primary standardisation technique, only suitable for particles that: travel in straight

4 Introduction Defined Solid Angle (DSA) Counting The DSA counting system at IRMM >>>>>> NPL are in process of building a similar system Work on the analysis software has begun Monte Carlo

5 Definition of Solid Angle (steradian) The three dimensional angle formed at the vertex of a cone When this vertex is the centre of a sphere of radius r and the base of the cone cuts out an area s on the surface of the sphere, the solid angle in steradians is defined as Ω =s/r 2 4π steradians in a complete sphere

and the base of the cone cuts out an area s on the surface of the sphere, the

6 Geometry Factor The Geometry Factor (G) is equal to: the ratio of the solid angle to 4 π steradians G =Ω/4π the fraction of alpha particles emitted directly towards the detector the counting efficiency, if detection efficiency is 100%

$fraction of alpha particles emitted directly towards the$

7 Spherical Coordinate System Z r sinθ dϕ Integration over angular degrees of freedom r dθ 0 θ π 0 ϕ 2π ϕ θ r Y dϕ X ds= r 2 sinθ dϕ dθ dω= sinθ dϕ dθ

8 Point Source on axis of symmetry of a circular detector (diaphragm) The ONLY geometry with a simple solution for Ω Ω = 2π(1 cos θ) circular aperture diaphragm/detector 0<ϕ<2π The maximum value of θ for a hit is : θ max = atan( /d) d 0<θ<atan( /d) coaxial point source

2π(1 cos θ) circular aperture diaphragm/detector 0<ϕ<2π The

9 Geometric Considerations affecting precision In reality : inhomogeneous, off-axis, non-circular source unknown exact detector size diaphragm edge has finite thickness inhomogeneous activity distribution Require some means of precise solid angle calculation and uncertainty estimation Monte Carlo technique Numerical Integration

finite thickness inhomogeneous activity distribution Require some means of

10 Simulation Software Random Number Generation Requires good random number generator B.A. Wichmann and I.D. Hill Generating good pseudo-random numbers Computational Statistics and Data Analysis, 51 (3), Combination of four linear congruence generators Period of generator is approx 2.6 E+36 Passes Big Crush test 2.6 E+6 calls per second (in my C/C++ implementation)

Hill Generating good pseudo-random numbers Computational Statistics and Data Analysis, 51 (3),

11 Wichman-Hill Random Number Generator (in C/C++)

12 Simulation of a point on a disk source of radius R [ ] LET : U 0,1 = random() NOT simply: angle = U.2π distance from centre = U.R points are clustered near the centre FOR EQUAL AREAS : EQUAL ACTIVITY angle = U.2π distance from centre = U.R

13 Simulation of point on source annulus Outer radius Ro Inner radius Ri angle = U.2π distance from centre = Ri + U.( Ro Ri ) 6 5 weighing factor radius (mm)

14 Simulation of particle direction ϕ : azimuthal angle ϕ = U.2π 0<ϕ<2π θ : elevation angle θ = U.π d But, we may wish to limit angle for simulations (θ 1 < θ < θ 2 ) to avoid wastage of random numbers 0<θ<atan( /d) ( ) ( ) U ( ) ( ) ( ) θ = acos cos θ cos θ cos θ

15 Eccentric Point Source with circular diaphragm/detector ϕ ϕ 2 ϕ 1 r θ 1 2 Ω = R + r = a tan D d d r 2 π (1 cos θ 1) for r < R 0 for r R D D ϕ( θ) θ 1 eccentric R = a tan θ 2 1 r d D Ω 2 = 2( π ϕ( θ)) sinθ dθ θ R = a cos 2 D {r 2 + d 2 2r d tan θ tan 2 d θ} r θ θ 2 1 R + r = a tan D d r R = a tan d eccentric Ω = Ω 1 + Ω2 D

R = a tan θ 2 1 r d D Ω 2 = 2( π ϕ( θ)) sinθ dθ θ R = a cos 2 D {r 2 + d 2 2r d

16 Eccentric Point Source with circular diaphragm/detector (top view) d tanθ d tanθ ϕ 2 ϕ 1 ϕ 1 ϕ 2 r source Ω 1 r source Ω 2 Ω 2 Ω 1 =0 circular diaphragm/detector r< r> a" hit " is defined as : 2 2 D ( tanθ ) 2 2 ( tanθ ) cos( π ϕ ) R > r + d r d 1

Ω 2 Ω 1 =0 circular diaphragm/detector r< r> a" hit " is

17 Coaxial circular source ψ= R S /d Attempts to derive formula: Ruby NIM 58 (1968), 345 NIMA, 337, (1994), 531 Expressions as integrals as products of Bessel Functions ω = /d r ϕ d tanθ Tsoulfanidis Measurement and Detection of Radiation, Hemisphere (1983). Chapter 8. Simple algebraic approximation to 8 th order in ψ and ω d θ Serial Expansions by Pommé and Conway Appear to derail for large values of ψ and ω R S r

Measurement and Detection of Radiation, Hemisphere (1983). Chapter 8.

18 Eccentric circular source ϕ d tanθ ϕ d tanθ d θ d θ R S +r R S +r r R r S R S -r R S

19 Eccentric Sources Useful References on expressions Pommé NIMA, 505 (2003), NIMA, 579 (2007), ) Conway NIMA, 562 (2006), NIMA, 583, (2007), NIMA, 589 (2008), NIMA, 614 (2010), 17-27

272-274) Conway NIMA, 562 (2006), 146-153 NIMA, 583,

20 Ongoing work 2 Incorporate Realistic Diaphragms: (I.e.: two theoretical diaphragms) d h Continue verification of formulae (serial expansions) of Pommé and Conway

21 Conclusions Depending on the source detector geometrical configuration, the derivations of expressions for solid angles can be COMPLICATED! Numerical integration for solid angle calculations from can be difficult to program. Rather simpler to use Monte Carlo technique Acts as useful validation tool for expressions Code is still under development, but is in use for rough calculations, as further validation is required

Calculation of Source-detector Solid Angle, Using Monte Carlo Method, for Radioactive Sources with Various Geometries and Cylindrical Detector

International Journal of Pure and Applied Physics ISSN 0973-1776 Volume 3, Number 2 (2007), pp. 201 208 Research India Publications http://www.ripublication.com/ijpap.htm Calculation of Source-detector