Is the wife considered divorced if her husband said to her in a moment of anger: "You are unlawful to me if your foot crosses the doorstep," and she then left the house? And what is the expiation for that?

The preponderant opinion is that intention is considered in oaths. If intention is absent, then reference is made to the cause of the oath. If the cause is absent, then reference is made to the apparent meaning of the لفظ (wording).

So, if you intended by your oath to prevent your wife from going out absolutely, then you have broken your oath by her going out. If you intended to prevent her for a specific period, then there is no breaking of the oath by her going out after that period. Or if you intended to prevent her from going out in a specific manner, then there is no breaking of the oath by her going out in another manner, because intention specifies the general and restricts the absolute.

If you had no intention, then reference is made to the cause of the oath. If the cause necessitates preventing her at a certain time and not another, or for a certain purpose and not another, then your oath is restricted by the cause.

If you break your oath, what results from your breaking it varies depending on your intention by the prohibition. If you intended ẓihār (a type of divorce), it is ẓihār. If you intended ṭalāq (divorce), it is ṭalāq. If you intended an oath, or did not intend anything specific, then it is an oath for which you must offer expiation by feeding ten needy persons or clothing them. We advise presenting the matter to trustworthy scholars.