Basically the same way you got the first. Go to the excavation site after 9:00pm with your hammer equipped. You should meet Carter there and he will ask you to use your hammer to blast another hole in the wall, thus openning the second mine. Note that you must have reached the bottom floor of the first mine (10th floor) and have defeated the 'dark chick'.
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You can mine it in the 2nd mine at level 7-10
On Harvest Moon: DS and Harvest Moon: DS Cute, you get the third mine by reaching the bottom of the second mine, which is 255 floors deep.
it is in the second mine to get the second mine you have to complete the first mine
go into the mine after 8:00 pm
You can find Alexandrite on the first floor in the third mine.