"unorganized depth of my files" - That sounds all too familiar to me as well, Martina!
Your photo is of a young Evelyn, wife of the 9th Duke. Her predecessor was Louise, the "Double Duchess," who was married to the Duke of Manchester prior to the Duke of Devonshire. Louise was the hostess of the famous Devonshire House Ball in 1897.
I have also been curious about the emerald tiara which according to the site linked to in Cerys original post does belong to Albion and previously the Devonshires. If that is indeed true, it could have possibly once belonged to Louise. The style of the tiara fits within her lifetime.
She had no children with the Duke of Devonshire and upon her husband's death the title was inherited by his nephew. Louise owned the grand honeysuckle tiara seen in Nellie's post above, but since it was made from Cavendish family stones, it was inherited by the new Duke.
However, as a grand society hostess Louise would have likely owned several tiaras. Her personal property would have been inherited by the children from her first marriage and not the line of the current Dukes of Devonshire. Today Louise's descendants include - among many others - the Duke of Manchester, the Duke of Montrose, the Earl of Gosford, and the Earl of Derby. The emerald tiara could have made its way from some of them to Albion.
Despite the crippling death duties (80%!!!) that had to be paid on the 10th Duke's estate, I have never read of any jewels being sold. I think that if the emerald tiara did belong to a Duchess of Devonshire, it was likely Louise.