Investigation of Bending and Shearing in the Diamond Radiator. Fluctuations in beam quality have been seen during JLab test runs, and a proposed explanation was deformation of the diamond radiator by its mount.
Fluctuations in beam quality have been seen during JLab test runs, and a proposed explanation was deformation of the diamond radiator by its mount.
The 10m-thick crystal is mounted on two parallel tungsten wires with 25mdiameters. If the wires had torsions when the diamond was glued on, they would exert torques on the crystal. Our investigation determined that wire torsions of a few radians could produce significant warping of the crystal.
To see an applet simulation of the warping, go to http://zeus.phys.uconn.edu/~bevans/physlets/bend.html.
The Diamond Mount
25m dia. tungsten wire
1 cm x 1cm x 10mdiamond wafer
There are two contributions to torque in the tungsten wires: shear, where the fibers move parallel to each other but do not change length, and twist, where the fibers do not move parallel to each other and do change length.
The shearing torque: t = hfR4p/2L,
where e is the shear modulus, f the angular torsion, R the wire’s radius and L the wire’s length.
The twisting torque: t = pYf3R6/24L3
where Y is Young’s modulus.
Since the radius is small, we ignore the twisting torque.
Warping Effects in the Crystal Radiator
Next, we calculated the vertical displacement of the crystal at a position x as a function of the torques exerted by each wire. As with the wires, there is shearing, where fibers stay parallel, and bending, where they do not. Let L be the extent of the crystal in x, d its extent in z, h its thickness, h its shearing modulus, and Y its Young’s modulus. Then the displacement is
y = (t1 + t2)/Lhdh + 12[t1x2/2 - (t1 + t2)x3/6L]/Ydh3
Now consider the case where the torsions on the wire are equal and opposite in sign, so that the crystal deforms to an arc, not an S shape. The maximum vertical displacement occurs at x = L/2, and is, as a function of wire torsion,
Ymax = 3Ldia2hwirefRwire4p/4LwireYdiadh3
Ymax is about 190m for 10-radian torsions.