XEDS Tools
Solid Angle Calculator
V-2014091607
Download M&M2014 Paper with calculation details
Microsc. Microanal. 20, 1318–1326, 2014
doi:10.1017/S1431927614000956
Circular Cross-section


Radial Detector, Tilt Angle (θ) = 0
Parameter
Value
 units 
Distance to Detector (d) mm
Detector Active Radius (r) mm
Detector Tilt Angle (θ) degrees
Fraction Obstructed (f) 0 < f < 1
Nominal Active Area (A) 0 mm2
Radial Detector Solid Angle (Ω) - sR
Non-Radial Detector Solid Angle (Ω) - sR

Non-Radial Detector, Tilt Angle (θ) > 0
Rectangluar Cross-section

Radial Detector, Tilt Angle (θ) = 0
Parameter
Value
 units 
Distance to Detector (d) mm
Detector Active Height (h) mm
Detector Active Width (w) mm
Detector Tilt Angle (θ) degrees
Fraction Obstructed (f) 0 < f < 1
Nominal Active Area (A) 0 mm2
Radial Detector Solid Angle (Ω) - sR
Non-Radial Detector Solid Angle (Ω) - sR

Non-Radial Detector, Tilt Angle (θ) > 0
Annular Cross-section
-
Parameter
Value
 units 
Distance to Detector (d) mm
Outer Active Radius (ra ) mm
Inner Active Radius (rb ) mm
Fraction Obstructed (f) 0 < f < 1
Nominal Active Area (A) 0 mm2
Calculated Solid Angle (Ω) sR
Calculation of Holder Tilt to Minimize Holder and Detector Penumbra

Parameter
Value
 units 
Holder Side Wall Half-Height (hs ) mm
Holder Half-Width (ws ) mm
Holder Penumbra Angle ( θp )
  degrees

Upper Detector Height (HU ) mm
Lower Detector Height (HL ) mm
Radial Detector Distance (D) mm
Detector Upper Elevation Angle ( θEU )
  degrees
Detector Lower Elevation Angle ( θEL )
  degrees

Minimum Holder Tilt Angle ( α = θp - θEL )
to mitigate penumbra's of holder and/or detector

  degrees

Note: some detector geometries can have
HL which is negative ( i.e. below the specimen plane)
Estimation of Detector Fraction Obstructed ( f )

Moxtek Grid            Holder Penumbra

Moxtek/Holder/Collimator Penumbra Notes:
  • when Ru = R then there is no shadowing by the specimen holder or collimator
  • when Ru = - R then 100% of the detector is shadowed/obstructed
  • when Ru > R then 0% of the detector is shadowed/obstructed
  • a collimator shadow can also create the same effect as the holder penumbra, except it does not change with holder tilt
      Ru ~ R - D tan(θP) if the holder is non-tilted
If the Detector is Windowless and there are no other obstructions then f = 0

If a Detector Window is present and is supported on a Moxtek AP3 Grid then approximately 75% of the window area is open or "not obstructed" hence   fMoxtek ~ 1 - 0. 75 = 0. 25 .

If a Circular Detector is used and is shadowed by the Specimen Holder's Penumbra you can use the following to estimate fHolder Penumbra

Parameter
Value
 units 
Detector Active Radius (R) mm
Unshadowed Radius (Ru )
mm
Fraction Obstructed by Holder Penumbra ( fHolder Penumbra )
Note: you must manually enter f into the parameter space above
  0 < f < 1

If you have both of the above situations, then multiply the corresponding terms appropriately. Here fTotal = fMoxtek * (1- fHolder Penumbra ) . Notice that when you have 2 or more obstructions you must carefully account for their relationships. Here the Moxtek obstruction only affects the "open" or unobstructed area of the detector caused by the holder penumbra. Other geometries are left to the user to calculate.