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Glad to see you are underway with your new machine
Proxxon offer the PF230 milling head to fit on the back of your lathe however if you have the space (and money) I would suggest a separate small milling machine. I have a combined lathe and mill which can be a problem when you want to turn a small spacer or mandrel when you are in milling mode.
I would first look for any small ridges or bumps on the taper for the chuck. If there are any carefully remove them with a fine file. Next mark the taper with something (ideally engineers blue but chalk or marker pen will work) and fit it into the socket and turn it round. This should confirm if there is even contact along the length of the taper. If the contact is not even you will need to find or borrow something with a known good MT1 taper to see if the problem is the chuck or the socket.
HSS parting tools are usually better for small lathes as they are sharper than carbide tips. Setting the tool to the correct height is critical. It should be at or just below the centre of the work or it will rub and not cut. Can you post a picture of your parting tool. It is possible it has not been ground correctly. I use HSS parting tools on my PD 150 (one size smaller than yours).
nat......a few observations1. your PD 250E is obviously near new......& has a No2? Morse taper in the tail-stock....drilling any hole with a clean HSS drill bit to the capacity of the chuck [10mm?] should not cause any issue between the tailstock/chuck & the workpieceIf you are experiencing rotational chatter of the chuck arbour. would indicate contamination in the Morse taper M&F components.....after this happens, take the chuck+arbour from the tailstock.......you may well find a small shiny point on the OD of the arbour that is causing this........it is totally acceptable to remove this shiny spot by 1200 W+D paper as needed..... reinstall & repeat exerciseThe Morse taper is designed to hold, never use Loctite Some 50 years ago, a Tradesman machinist taught the apprentice that if a drill arbour slipped in the tailstock, ....take a single leaf of roll your own cigarette tobacco paper & glue it to the arbour [No4 Morse] & that sure eliminated the rotational issue2. the same Tradesman machinist also taught me never to leave the headstock chuck key in a lathe3. I agree with Roger, that HSS tool bits are absolutely preferable for 9/10 tasks in model engineering machining tasksDerek
One thing about using vertical slide on the cross slide of 250E. If it has a cross slide made of aluminum, as my older PD 210 have, it just isn't rigid enough for milling metal, unless you take very light cuts.