Beryllium X-Ray Window Thickness Guidelines
Beryllium x-ray windows frequently operate with atmospheric pressure on one side and vacuum on the other side. The following equations are suggested to compute the stress on the x-ray window for typical conditions.
Beryllium foil is not characterized for mechanical properties, but a design strength of 40,000 PSI or 275 MPa is typical. An appropriate safety factor should also be employed. The procedure to determine deflection and stress at any given foil thickness is to use a math program or spreadsheet to solve for deflection (“y”) in the first equation. Then, use that value to solve for maximum stress (“s”) in the second equation. This analysis considers the x-ray window to be a flat plate with the edges held and fixed, as would be the case for a diffusion bonded assembly. This formula takes into account the shear stress and diaphragm stress, which is the stress in the material carried as tension, so it is valid for windows even if they undergo a deflection larger than half the material thickness. This equation is intended for use on a circular aperture. If the application requires a rectangular aperture, a practical approximation is to use this guideline, substituting the small dimension of the rectangle for the diameter of the aperture. The variables and formulas are:
Poisson’s Ratio (ν) for Be = 0.03 – 0.08
Young’s Modulus (E) for Be = 44MSI or (44 x 106 PSI) or 303 GPa
K1 = 5.33/(1- ν2)
K2 = 2.6/(1- ν2)
K3 = 2/(1- ν) {at the center}
K4 = 0.976 {at the center}
K3 = 4/(1- ν2) {at the edge}
K4 = 0.476 {at the edge}
q = unit lateral pressure
@ 1ATM, q = 15psi or 101kPa
t = beryllium thickness
r = radius of aperture (inches or meters)
s = maximum stress due to bending and tension
y = maximum deflection (solve for this first)
Deflection Equation:
Maximum Stress Equation:
These equations are from pages 477 & 478, Roark’s Formulas for Stress & Strain by Warren C. Young, 6th Edition, (1989) McGraw-Hill.
