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When are we supposed to used:

das Korn, die Körner

and when

der Kern, die Kerne?

Here I mean literal usage rather than idiomatic expressions like "Kern der Sache".

You can give Körner to birds, but you use Sonnenblumenkerne for cooking.

Where is the actual difference?

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The "Kern" (core) is always in the middle, be it of something two- or three-dimensional. As the "Sonnenblumenkern" is in the middle surrounded by flower leaves, it is a "Kern", like the "Atomkern" which is in the center of the atom. A "Korn" (grain) is a tiny, usually spheric item of homogenous nature, e.g. das Samenkorn, das Sandkorn, das Staubkorn, das Maiskorn, das Weizenkorn. As "Weizenkorn" is an extremely important example for a natural "Korn", "Korn" is used synonymously for wheat in general.

However, there is some scheme in using "Körner" or "Kerne" when it comes to what we eat or feed, as "Kerne" usually originate from fruit, nuts, and flowers – it is the nature of those plants to keep their seeds always somewhere in the center –, whereas "Körner" usually originate from cereals like wheat, corn, rice, etc. which openly present their seeds as very strictly arranged and regular portions. That's how you can distinguish those terms on a practical level.

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    Eine Ausnahme ist die Erdbeere, die die Kerne außen hat. – user unknown Feb 27 '15 at 22:24
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Korn ist historisch das alte Wort für Roggen, das frühere Brotgetreide. Kern ist historisch das Wort für Dinkel, der erst geschält werden musste. (Dinkelkerne)

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    Roggen muss man auch schälen - es geht nur einfacher. – tofro Jul 9 '18 at 15:07
1

Kern is more general, apples and melons have Kerne.

Korn is more specific, it's used for grains mostly: Reiskorn, Korn für Mehl, Gerstenkorn.

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