Question:
I see many people raising their hands when reciting the dua after athan.  I have also seen some scholars not raising their hands? What is the correct procedure?

Answer:

In the Name of Allah, the Most Gracious, the Most Merciful.

As-salāmu ‘alaykum wa-rahmatullāhi wa-barakātuh.

In essence there are two types of du῾ā’s: 

1)     Those which do not consist of any particular wording for any given situation.  For example, a person makes his own du῾ā’ for a need he has after performing two rak῾at of voluntary prayer.

2)     Those du῾ā which have specific wordings and phrases at particular times and instances.  For example, the du῾ā’s after eating, after sleeping etc.

The Ahādith pertaining to raising the hands for du῾ā’ are in relation to the first category and not the second category.  The second category of du῾ā’s are considered as athkār (remembrance of Allah) and thus, the hands should not be raised for such utterances. 

The du῾ā’ after adhān falls under category two, therefore, the hands should not be raised when reciting the du῾ā’ after adhān.[1]
 
And Allah Ta’āla Knows Best
Mufti Faraz ibn Adam al-Mahmudi,
www.darulfiqh.com