The Royal Scots Officier badge (for glengarry)

[Fante]

Here is my first request for help ..
I'm trying to find a Royal Scots badge of the specific model for the glengarry.
From my web search (I still don't have any texts or publications to help me) I realized that every Regiment Battalion had its own badges, different in graphics from that of others.
I ask you to kindly post pictures of the officer badges commonly in use on this hat and tell me about any differences (in size) compared to those for the cap.
Of course it would be useful (unless the drawing has remained the same until now) to check the type and size of the badges used during WW2.
Thank you all!!

Note:

I found this on the web (2nd Battalion?), size cm 5,3 x 4,7...Is it a glenfarry or cap badge?