Do the spouses incur sin if they perform a temporary tubal ligation for the wife (temporary sterilization) for the purpose of birth control after trying all other methods and their failure for medical reasons, and what is the expiation for that if sin is incurred?

It is forbidden to permanently cease procreation, as it leads to cutting off and reducing offspring. However, birth control is permissible for a specific period if necessity or need dictates it. If procreation is ceased without necessity or urgent need, one must repent, seek forgiveness, and perform good deeds. There is no specific expiation for this other than repentance.