D0 Silicon Upgrade: Thermally Induced Bowing in a 3-CHIP Ladder

The end of the 3 chip ladder, shown below, consists of silicon mounted on a piece of beryllium which is adhered to the cooling channel. Outboard of the cooling channel is a region of ladder composed primarily of silicon/beryllium. Operation and cooling of the ladder results in a change in temperature from the assembly temperature, which will result in deflections due to the difference in expansion coefficients of the two materials, otherwise known as 'bi-metal' bowing. The goal of this note is to present a design of the beryllium plate on the underside of the ladder which reduces the thermally induced bow to a reasonable deflection. This region of ladder will see a fairly large temperature gradient during detector operation due to the heat load of the transceivers on the ladder end. Expected temperatures range between 22 C on the ladder end to 9.5 C near the cooling channel for a coolant temperature of 5 C. The coolant temperature may be as low as -5 C, so we may estimate a lower limit on the ladder temperatures to be 10 C cooler, ranging from 12 C on the ladder end to -0.5 C near the bulkhead (assumes negligible convection from the ladder surface). With a ladder assembly temperature of 23 C we may estimate the end deflection of the ladder based on the assumed temperatures during operation by applying Roark equation 6a. Equation 6a describes end deflection for a cantilever beam under application of a uniform temperature change. The equation is modified to account for a uniform temperature gradient along the bi-metal region. The equation is differentiated twice, the assumed temperature dependence is plugged, and the equation is re-integrated twice. deflection = C*ΔT*L2. The constant C is a function of material thicknesses and moduli. The composite region (beryllium and silicon 'bi-metal' region) is between 21.0 and 25.0 mm in length (the HDI design is still in progress) plus the additional 1.975 mm shown in the ladder drawing below. Hence, the composite region is assumed to extend 27 mm beyond the bulkhead ledge. The silicon extends 4 mm beyond the composite region. Deflection of the ladder end, the silicon, is calculated.