I'd better begin with the easiest part - the required translations.
- He studies a PhD. - Он учится в аспирантуре.
- He's a PhD student. - Он аспирант.
- She has a PhD. - Она кандидат наук.
- I write a PhD thesis. - Я пишу кандидатскую диссертацию.
- I write a PhD thesis. - Я пишу кандидатскую.
- They are studying a PhD. - Они учатся в аспирантуре.
- He has a DPhil. - Он доктор философских наук.
As far as I know, "доктор наук" doesn't have an equivalent in English, but it is one level higher than PhD. In specific fields of studies, however, it has a translation:
- DPhil - доктор философии
- MUDr - доктор медицинских наук
- D.Litt - доктор словесных наук
- D.Sc - доктор естественных наук
- LL.D - доктор права
Analogously with "аспирантура", there exists a word "докторантура", and the student studying for their Doctor of Science degree is called "докторант" (note how alike it is to "аспирант").
Basically, the word "диссертация" can refer either to PhD or Doctor of Science thesis. If it is not clear from the context, you can add adjectives:
- Он защитил докторскую диссертацию / кандидатскую диссертацию.
Often those adjectives are used as substantives :
- Он защитил (что?) докторскую / кандидатскую.
The process of thesis defense is called защита докторской / кандидатской.
(There is also a sort of sausage called "докторская", feel free to delete this picture ˆ__ˆ ).
Well, let's continue. The difference between those translations is that PhD can refer to :
- current studies at University - then it means "аспирантура";
- an academic degree - then it means "кандидат наук";
- a person who is capped with that degree is also "кандидат наук";
- a person that is currently in the middle of PhD studies - "аспирант";
- a PhD thesis - "кандидатская диссертация";
And last - some notes on usage of those words. Don't be surprised to hear
"Он получил кандидата наук" or "Он защитил кандидата наук".
It doesn't mean that he received a person by post or that he protected a poor PhD student from bullies. It only means that he was finally assigned PhD degree ˆ_ˆ .
All degrees are used with Genitiv, so we don't say "степень кандидат наук",
but "степень (кого? чего?) кандидата наук".