Crossbows in Europe, Asia Minor or North Africa c 1000 AD

I am attempting to write a novel set in England in the early years of the 11th century AD and am thinking of including a crossbowman as a character. I'm aware that in western Europe crossbows were not generally used in battle at that time, although it appears that William the Bastard used crossbowmen at the Battle of Hastings not much later. Perhaps the Normans - or the French or the Spaniards - were already experimenting with them during my period. Or the Arabs of southern Spain and North ...