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Some American friend told me that 'Let me alone' was unnatural and sounded old expression, but I couldn't understand why it was not good. So, I googled and fined out the following answer. Could you tell me whether it is right explanation and whether you agree to it? Thank you.

There is a pretty simple difference between let and leave when used in this context, and it’s quite easy to explain. “Leave alone” means to leave a person all by himself (in solitude).

Please leave me alone, I don’t want company.
Alvin needs to study, so let’s leave him alone in his room.

“Let alone,” on the other hand, means to stop bothering the person.

Stop poking my leg and let me alone, Simon. I’m trying to watch TV.
Theodore wanted his singing coach to let him alone instead of critiquing his singing.

While the definitions above are the traditional uses of both words, it is becoming more common (and more acceptable) to use “leave me alone” in both contexts. But if you want to be a grammar purist, keep them separate.
Comments  
I use "leave" for both.
Let me alone means allow me to be by myself. Leave me alone means go away so I can be by myself.
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I also use "leave" for both.

CJ
BarbaraPAI use "leave" for both.
I do, too.
Then does 'Let me alone' sound unnatural and old to you also?
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Let me be.

An old expression. The Beatles titled a hit song "Let it be."
lucas21cThen does 'Let me alone' sound unnatural and old to you also?
Not necessarily old, but unnatural because it's not, normally used.
  1. I use "let me alone", A little more aggressive than "leave"
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