What is the expiation for a mother spending the night of mourning outside her home during her waiting period, while being ignorant of the rulings of the waiting period?

An 'iddah observer must spend the night in her home. It is not permissible for her to spend the night elsewhere unless there is a necessity. If she leaves due to her ignorance of this ruling, then she should seek Allah's forgiveness, and there is nothing (further) upon her.