The main issue is your question about the samples: provenance, technology or deterioration?
If you like to study historic potteries, you should use some analytical methods such as petrography (microscopy) and elemental and phase analyses suchas XRD, XRF and etc. The method should establish based on your question. More detailed information may help to answer better.
thanks for your answer. I think you propose methods are best suited for proper characterization. The sample consists of handmade ceramic vessels. Neither is complete, but you can rebuild your shape. Will study each of the analytical and the possibilities offered by each.
You could use XRF or ICP for elemental analysis of the ceramics to identify composition of the samples. Also the XRD help you to know phases formed during the manufacturing potteries and join its results with some thermal analyses such as DTA or etc. The Microscopic methods such as petrography and SEM-EDS may use to know about microstructure of potteries, crystals in matrix, deteriorations, some phases and other useful data.