This can be accomplished with an
Intersect, followed by a
Field Calculate, and then finally a
Summary Statistics. Make sure that your buffer feature class has a unique ID field.
Before getting started, you will need to add a field (name
Double) to your population polygon feature class, and then field calculate it, using
Shape_Area as the input. This will create a field with your areas that can be carried over in geoprocessing.
Now perform an
intersect with your population polygons and your buffers as inputs.
A new feature class is created. Add a new field to it - maybe
Field Calculate this new field in your intersect feature class:
Population field * (Shape_Area / Polygon_Areas)
Finally, you can perform a summary statistics on your intersect feature class. Use the
Area_Pop as your statistics field and chose
SUM as your type. Add the unique ID field for your buffer feature class as your Case field.
Voila. You can perform a Tabular Join from your new table back to your original buffer feature class and add the
SUM_Area_Pop field to a new field with a field calculate.