In a previous posting (Electron backscatter at low incident energy), I talked about some of the problems in GamBet calculations that occur when interactions are confined to a thin surface layer. As an example, consider low-energy electron backscatter. At 140 keV, the range in tungsten is only 26 ?m. A customer recently expressed interest in an end-to-end model of an X-ray source, representing multiple generations of backscatter on multiple surfaces. The source assembly had a width of about 10 cm, giving a scale disparity of 4000:1. The code improvement discussed in the previous blog was not sufficient to handle such a calculation.
A practical mesh of the entire system would demand an element size much larger than the depth of electron penetration. In this case, electrons that enter material elements stay extremely close to the surface throughout their complex interactions. As the element size increases, it becomes more difficult for a numerical code to determine whether the particles are inside or outside the material. When the step size between collisions is extremely small, there is a greater chance of an error.
To illustrate, I set up a three-dimensional calculation of 140 keV electrons incident on a shaped tungsten target. With a small element size (0.025 mm), GamBet gave a backscatter ratio of 49.1%, close to the experimental value. When I increased the element depth to 0.2 mm, the backscatter ratio dropped to 38.5% because of numerical errors.
Fortunately, scaling provides a solution to the mesh-disparity problem. The idea is simple. If you want to generate the spatial distribution of low-energy backscattered electrons in a 10 cm system, it doesn’t matter if the particles have an interaction depth in the target of 25 ?m or 250 ?m, as long as everything else scales. If we decrease the density of the tungsten, all interactions will be relatively the same . It’s just that they take place over a greater depth.
As a confirmation, I set the tungsten density in the calculation with 0.2 mm elements to 1.93 gm/cm3 with the following source statement:
* 0.1 density tungsten
Material
Name TungstenLow
Component 74 1.0
Density 1.93
Conductor
End
For this case, GamBet gave a backscatter ratio of 49.0%. The average kinetic energy of backscattered electrons was 109.9 keV for the small element solution and 109.7 keV for the low-density tungsten solution. Most important, the picture below shows that the spatial distributions of backscatter electrons for the two solutions were indistinguishable. The top picture corresponds to the solution with small elements and normal-density tungsten, while the bottom solution has large elements and reduced-density tungsten.
We old-time engineers used scaling all the time, but it’s often difficult to convince present-day customers that the method represents a valid approach to a problem. They think that there is some trick involved and that the resulting calculation isn’t a real simulation. The compelling decision point is whether you want to wait 36 hours while the computer grinds away on a daunting adaptive mesh or whether you want the results in 5 minutes.
