Hi Jesper,
Steve is correct that the Pinion mate cross shafts should be able to move in operation, but they should not be loose enough to rattle excessively around when stationary. The Jaguar E Type Service Manual states “clearance in the differential case permits slight peripheral movement at the ends of the shafts.” Suitably ambiguous, but once everyone is coated in oil on assembly, any movement should be minimal. The Service manual also states “Due to the lateral movement of the pinion shafts in the differential case, a little more backlash may be apparent in a limited slip rear axle”. Your photo shows the components very clean and oil free, so once they are suitably lubricated, this should stop the present noises you are experiencing.
When you fitted the clutch packs each side, there should have been 2 friction discs with internal teeth which fit onto the bevel side gear splines, plus 3 friction plates with external “ears” which go into the differential case. The outermost “eared” friction plate is dished as it is a Belleville spring which when the two sides of the casing are bolted together provides a certain preload onto the side gear rings.
When the Powr-Lok Limited Slip Differential operates, the angled ends of cross shafts move up corresponding ramps in the differential case to force the cross shafts out, moving the bevel side gears out and locking the friction discs to the friction plates - making the whole of that side of the differential locked.
Just checking that when you assembled the clutch pack, were all 5 disc and plates fitted and was the outer plate the curved Belleville spring ? If the Belleville spring is cracked, it will not function correctly.
Also however the differential is rebuilt, the friction plates need to be perfect clean and given a coating of suitable LSD as new plates will “chatter” until suitably bedded in and oil soaked. This is what I use :-
A common practise is to drive a newly rebuilt LSD differential to an open area and drive a series of figure 8s to get the clutch plates operating and reduce potential chatter.
Trust this helps,
Regards,
Dave