Electron range evaluation and X-ray conversion optimization in tungsten

transmission-type targets with the aid of wide electron beam

Monte Carlo simulations

Andrii Sofiienko1, Chad Jarvis

2, Ådne Voll

3

1University of Bergen, Allegaten 55, PO Box 7803, 5020 Bergen, Norway. E-mail: asofienko@gmail.com

2Christian Michelsen Research AS, Fantoftveien 38, PO Box 6031, NO-5892 Bergen, Norway. E-mail: chad.jarvis@cmr.no

3Visuray AS, Strandbakken 10, 4070 Randaberg, Norway. E-mail: david.ponce@visuray.com

Introduction

X-ray tubes are one of the most common and safest sources of X-rays and are used in many medical

[1-3] and industrial applications [4-6]. Among the many types of X-ray tubes that are available, the

transmission type features the simplest target design [7, 8] and is widely used in X-ray inspection [9,

10]. Despite the widespread use of transmission-type X-ray tubes, there is no detailed study of the

characteristics of the X-ray generation process for wide electron beams (when the beam diameter is

not much smaller than the diameter of the flat target). Additionally, there is no detailed information

about the effect of the geometrical parameters of the electron beam on the angular distribution of the

generated X-rays at accelerating voltages within the range of 250-500 kV, which are interesting for

industrial applications. The general purpose of the presented work was to investigate, using Monte

Carlo simulations, the X-ray generation process in a transmission-type X-ray tube with a wide

electron beam. We determined the following parameters for use in practical applications: approximate

electron range in a tungsten target over an energy range of 250-500 keV, the optimal target thickness,

the angular distribution of generated X-rays and the efficiency coefficient for the transfer of energy

from an electron beam to generated X-rays.

Monte Carlo simulations

MC simulations were performed using the Xenos software suite by Field Precision [12]. The

particular program is called GamBet. GamBet combines Field Precision's technology for finite-

element codes with the package PENELOPE [13]. The solution volume for the mesh created in all

cases had the following range: X-axis [-1.0 mm, 1.0 mm], Y-axis [-1.0 mm, 1.0 mm], Z-axis [0.0

mm, 0.5 mm]. The tungsten target was centred at (0.0 mm, 0.0 mm, 0.4 mm). Outside of the

tungsten target, the voxel size was (25 μm, 25 μm, 10 μm); inside the tungsten target, the voxel size

was (2.0 μm, 2.0 μm, 0.1 μm). The electron source file contained 4.6∙105 electrons with momentum

along the positive Z-axis. MC simulations were generated for target thicknesses equal to 1.0, 2.5,

5.0, 10, 20, 30, 35, 60 and 70 μm and electron source energies equal to 250, 300 and 500 keV. The

electrons were Gaussian-distributed in a regular pattern with diameter De = 0.25 mm perpendicular

to the Z-axis:

The efficiency of X-ray generation for different target thickness

Conclusions

Using Monte Carlo simulations of the X-ray generation process in a transmission-type X-ray tube

with a wide (unfocused) electron beam, several important parameters were determined: the

approximate electron range in tungsten over the energy range of 250-500 keV, the optimal target

thickness for different electron energies, the angular distribution of the flux of generated X-rays and

the efficiency coefficient for the transfer of energy from an electron beam to a generated X-ray flux.

Simple analytical relations were obtained for the electron range in tungsten and for the optimal target

thickness. It was demonstrated that the angular distribution of a flux of generated X-rays in the

forward direction has the same maximum output angle for different acceleration potentials of an X-

ray tube and that the angular distribution is more isotropic at higher energies. The efficiency

coefficient for the transfer of electron energy to a flux of generated X-rays depends on the tungsten

target thickness and compares well with the commonly used empirical relation proposed in [11].

These results can be used in practical applications to design transmission-type X-ray tubes with wide

electron beams to calculate the flux (including the angular dependence) of the generated X-rays.

This work was partially funded by the Research Council of Norway under contract 200888.

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OECD Nuclear Energy Agency, 2011.

2 2

2 2

1, , exp

2 24 4

e e

e e

x yf x y D

D D

Figure 1: A schematic of the MC simulation of the transmission type target (on the left) and the angular distribution of the flux of

generated X-rays for the accelerating potentials of 250 kV and 500 kV (on the right)

The energy distributions of transmitted electrons with energies of 250, 300 and 500 keV were generated

by a MC method for different thicknesses of a tungsten target ranging from 1.0 μm to 70 μm. These

distributions were calculated to investigate the effect of the tungsten target thickness on the energy of

transmission electrons and their intensity behind the target. Integrating the energy distribution of the

transmitted electrons gives the relationship between the number of transmitted electrons and the target

thickness. A simple analytical expression for the estimated electron range in the tungsten target was

derived for the energy range of 250-500 keV: Re(E) = A·EB, where A = (11 ± 1)∙10-3 (μm/keV) and B =

1.38 ± 0.07. An analysis of the dependence of the flux of generated X-rays on the tungsten target

thickness and electron energies produced a function that appoximates the optimal tungsten thickness as

a function of the electron energy as follows: dOptimal(E) = C· EP, where C = (4.8 ± 0.3)∙10-3 (μm/keV)

and P = 1.48 ± 0.06. An analysis of the angular distribution of generated X-rays shows that the angles

with the maximum flux of generated X-rays fall within the same range (400-600 in this case) for

different electron energies. This result may be caused by the beam size and by the electron density

distribution in the beam. However, the probability of X-ray generation in the forward direction varies

with initial electron energy. The obtained efficiency coefficients for the transfer of energy from an

electron beam to a flux of generated X-rays depends on the thickness of tungsten target and compares

well with following commonly used empirical relation: ηX = (8 ± 2)·10-10·Z·eE, where Z is the atomic

number of the target media and eE is the enrgy of the incident electrons.

0 10 20 30 40 50 60 70

0,0

2,5x1015

5,0x1015

7,5x1015

1,0x1016

1,3x1016

1,5x1016

X-r

ay f

lux,

s-1cm

-2

dW

, m

1

2

321 m

17 m

47 m

Figure 2: The flux of generated X-rays behind a tungsten target versus the thickness of the

target for different electron energies: 250 keV (1), 300 keV (2) and 500 keV (3)

10 100

0

1x1014

2x1014

3x1014

7x1014

8x1014

9x1014

1x1015

E, keV

X/

E,

ke

V-1

s-1

cm

-2

1

2

200 500

Figure 3: The energy distribution of the flux of generated X-rays (from MC simulations,

Figure 1) for a transmission-type X-ray tube at different acceleration potentials and target

thicknesses: 250 kV and 20 μm (1) and 500 kV and 60 μm (2)

0 10 20 30 40 50 60 70

0,000

0,005

0,010

0,015

0,020

0,025

0,030

0,035 E

e = 250 keV

Ee = 300 keV

Ee = 500 keV

~ d1.2

W

X,

arb

.un

.

dW

, m

~ dW

1

2

3

Figure 4: The efficiency coefficients for the transfer of energy from an electron beam to a flux of

generated X-rays versus tungsten target thicknesses for different electron energies: 250 keV (1), 300

keV (2) and 500 keV (3)