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The important feature of Lithium ion batteries is the ability to intercalate and deintercalate lithium ions in anode and cathode . However when we use half cell, using lithium metal foil as anode, intercalation and deintercalation process occurs in this case too. How the lithium ion moves and what process cause the current to flow in discharge?

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When you're using a lithium metal anode there is no intercalation really, just reduction of the lithium ions to lithium metal:

$$\ce{Li^+ + e^- -> Li^0_{(s)}} $$

The cathode reaction is still the same.

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