Dug into my "archives", (okay, actually just an old looseleaf binder), for some frequency response charts for some old models of Audax 10 mm and 14 mm tweeters. Although Audax now has different models, the frequency response, both on-and-off axis, should be similar.
First, the 10 mm. models, Audax recommends you cross over at 5,000 Hz.
As you can see, at 15,000 Hz, (15 K Hz), there is hardly even 1 dB difference between on-axis and off-axis response. That is truly excellent dispersion.
Incidentally, some of the currently available Audax 10 mm and 14 mm models are available for less than $10-very inexpensive.
If any other poster knows of any other manufacturer making tweeters around half an inch or so, please feel free to join right in. I know there are a few manufactuers making 3/4 inch tweeters as well.