Some people work full time, others part time.

Are there any acceptable alternatives to "part time", such as

a) He works half time.

b) He works half day.

c) He works half a day.

d) He works half the day.

To me, they all sound quite odd, but maybe some of them are okay. Deepl uses some of these terms, dict.cc, too.

Asterixa) He works half time.

That is more specific than "part time."

Asterixb) He works half days.

The plural is needed. Again it is more specific.

The other two are not idiomatic.