From episode 20, during her air hockey match with Lahrri. Akari is smiling because she has just realized that she and Kris are together, so everything will be OK. Her smile greatly unsettles Lahrri, who is reminded of Midoh Tomoe's "illogical" smile during competition. (The comment over these cels on the thumbnail page is actually a quote from Lahrri.)

So cool to see air hockey as an important sport--the 3D version looks like fun, too. :-)