I think you probably have an ER11 or ER20 collet on the end of that spindle. Extra collets aren't very expensive (take a look at Arc Euro Trade, for example) and it's well worth using the right ones. In theory, one collet will handle nominal size shanks or up to 1mm smaller but especially in the smaller sizes they don't like compressing that easily so I also use a mix of metric and imperial size collets. 6mm shank cutters go in a 6mm collet and the 1/4" cutters (nominal 6.35mm) go in a 1/4" collet rather than using the larger collet for both. You only have to save one broken cutter to make up the cost of a collet!