Re: Can I use the battery 7.70Wh from my Samsung core prime in an S3 which normally runs a 5.50Wh
Check the terminals. If they're marked the same, it may work - if the battery fits. If they're not both marked the same, using the Prime battery in the S3 could turn the S3 into a useless piece of plastic. (If, for instance, the polarity is reversed, you'll blow most of the main board.) And the terminals may not be in the same place, so they won't make contact, and the battery won't be connected to the phone.
Unless you've had at least some experience as a hardware tech, it's a bad idea to even try.