Beryllium Window Thickness for a 3 He Target

Transcription

1 Beryllium Window Thickness for a 3 He Target Maduka Kaluarachch 1 and Al Tobias 1 1 University of Virginia August 16, 2012 Abstract For all JLab polarized 3 He experiments, we have been using GE-180 or C1720 glass as the target cell material including the thin windows where the electron beam enters and exits the cell. For the upcoming A n 1 experiment, higher beam current will be used and so we propose to introduce beryllium windows via a glass to metal seal. In this note, we show the foil stress and radiation length calculations in order to determine the appropriate thickness and aperture of a Be window. 1 Design The current design for incorporating metal windows into the glass cell is to create metal endcaps on the target chamber. We are considering glass to metal seals as supplied by Larson Electronic Glass in Redwood City, CA. Two possibilities include: 1) an indirect seal of OFHC copper to GE180 glass via a small 7052 glass transition and 2) a direct seal of stainless-316 to GE180 glass. In both cases, care is taken to make sure the tubing used is well polished. Figure 1: Be Window attached to glass-copper seal Onto these endcaps one can seal beryllium windows. This is acheived by a process carried out by Materion Electrofusion in Fremont, CA. They supply 1

2 beryllium foil of various thicknesses. They would create a copper (or stainless) frame onto which a beryllium foil of appropriate thickness and aperture is silver brazed. Then this frame is electron beam welded to the metal seal tube. The Be foil design characteristics must maximize strength while minimizing electron beam energy loss through the window. The target cells are filled with 3 He gas to high pressure, on order of 10 atm. During operation when the pumping chambers are heated to over 230C, the pressure in the cell can increase to pressures as high as 13 atm. Be window strength is critical to avoid rupture of the window. We can calculate the stress imposed on Be foil of various thicknesses and aperture given this 13 atm pressure on the walls. As for energy loss through the Be foil, we would like to keep it at or below the amount the traditional GE180 glass thickness of inches would impose. 2 Stress According to Materion Electrofusion specifications [2], the suggested design strength for a Be window is 40,000 psi. Therefore, one needs to avoid any stress above this number on the foil. The procedure to determine deflection and stress at any given foil thickness involves solving equation (1) to find deflection y and then use that value to solve for maximum stress σ in the equation (2). qr 4 Et 4 = K y ( y ) 3 1 t + K 2 (1) t σr 2 Et 2 = K y ( y ) 2 3 t + K 4 (2) t solve for: y Maximum deflection σ Maximum stress due to bending and tension variables: t Beryllium thickness r Radius of aperture q Unit lateral pressure, typically 15 psi constants: 2

3 ν Poisson s Ratio for Be = E Young s modulus for Be = psi K 1 = 5.33/(1 ν 2 ) K 2 = 2.6/(1 ν 2 ) K 3 = 2/(1 ν) {at center} K 4 = {at center} K 3 = 4/(1 ν 2 ) {at edge} K 4 = {at edge} Used values q =176.4 psi (13 atm cell) ν = E = 44MSI Calculations for various thicknesses are shown in table 1. Thickness Diameter Radius Deflection Max Stress center Max Stress edge t (in) d (cm) r (in) y (in) (psi) (psi) Table 1: Be Deflection & Stress All of the above Be foils with thicknesses inch and inch are under the threshold stress limit. 3 Radiation Loss We want to find the maximum thickness of Be that will reduce the electron beam s energy loss by the same or less than 5 inch thick GE-180 glass. In order to do that, we find the radiation lengths for GE-180 and Be, and compare. 3

4 3.1 Radiation Length Definition The radiation length of a material is the mean length (in cm) to reduce the energy of an electron by the factor 1/e. Physical Explanation An electron arriving in the vicinity of an atom will be affected by the electromagnetic field produced by the electrons of this atom. Because of this interaction, the electron will emit photons which will reduce its energy. This is called the Bremsstrahlung radiation. It is clear that this interaction will depend on the number of electrons of the atom (atomic number Z) but also the size of the atom, represented by its atomic weight A. We can use following equations to roughly estimate radiation length of an element.[1] L rad radiation length Z atomic number α 1/137 1 L rad [4Z(Z + 1)ρ N a A ]r2 eα[ln(183z 1/3 ) f(z)] (3) r e classical electron radius ( cm) N a Avagadro s number ( mol 1 ) A atomic weight of absorbing material ρ density of absorbing material f(z) =a 2 [(1 + a 2 ) a a a 6 ] a = Z/137 units of L rad cm or g/cm 2 (if ρ is not included) For compounds and mixtures we can use the following equation to find the combined radiation length = ω 1 ( ) 1 + ω 2 ( ) (4) L rad L rad L rad ω 1,ω 2 fractions by weight of each element in the mixture ω i = aiai A m a i # of atoms of i th element in the molecule 4

5 A i atomic weight of i th element A m = n i=1 a ia i 3.2 Beryllium vs. GE-180 Be is a pure element whereas GE-180 is an aluminosilicate glass made up of several compounds. Table 2 summarizes radiation lengths of the elements which these compounds are made of, calculated using equation (3). Element Z A (g/mol 1 ) ρ (g/cm 3 ) Radiaton Length (g/cm 2 ) Be Si O Ba Al Ca Sr Table 2: Radiation Lengths - Elements Using the compositions given in table 3, radiation lengths were calculated for compounds and total GE-180 using equation (4). Compound Composition Radiation Length (g/cm 2 ) SiO % BaO 18.2% Al 2 O % CaO 6.5% SrO 0.25% GE Table 3: Radiation Lengths - Compounds Having these two calculated values, we are able to compare the radiation lengths. For a given thickness of GE-180 we would like to calculate the thickness of Be which gives the same radiation loss. d Be = d GE180 (5) L rad(be) L rad(ge180) 5

6 Earlier targets used inch ( cm) thick GE-180, so we find inches of Be gives the same radiation loss as inches of GE Conclusion Taking stress and radiation loss factors into account, we propose a Be thickness of inches would be a good selection. References [1] W. R. Leo. Techniques for Nuclear and Particle Physics Experiments. Springer, [2] Materion Corporation, Materion Electrofusion Beryllium Window Thickness Guidelines, eqf /2011 edition. 6