Why doesn't Black play Bd4?

Question

This puzzle on chesstempo caught my eye (we play as White):
[FEN "4q1k/p4pb/r4npp/4P/n2B/5NNP/5PP/Q4RK b - - 0 1"]

1. Nd7

The move I made was e6, and it was the right one. Yet, the Black's second move was not quite something I foresaw:
[FEN "4q1k/p4pb/r4npp/4P/n2B/5NNP/5PP/Q4RK b - - 0 1"]

1. Nd7 e6
2. Ndc5 Bg7

I believe there is a word for the situation the black bishop finds themself in after 1... e6: no matter what the next move is, it gets captured. As one can see the Black can't play Kg7 on the 2nd board as it is protected by the White queen. Therefore, I would have gone for the White bishop with the "doomed" Black bishop straight away:
[FEN "4q1k/p4pb/r4npp/4P/n2B/5NNP/5PP/Q4RK b - - 0 1"]

1. Nd7 e6
2. Bd4 Nd4

Why does Black choose not to play as I did on the 3rd? Besides, what is the word for the "doomed" pieces, such as the Black bishop here?

Note: chesstempo users went over those moves in the comments under the puzzle, yet the discussion there is quite cumbersome, so I decided to bring this question up here.

Ian Bush · Accepted Answer

Nxd4 is not the best response to Bxd4, instead exd7 counter-attacking the Black Queen wins a piece:
[Result "*"]
[FEN "4q1k1/p4pb1/r4npp/4P3/n2B4/5NNP/5PP1/Q4RK1 b - - 0 1"]

1...Nd7 2.e6 Bxd4 3.exd7 Qxd7 
    ( 3...Bxa1 4.dxe8=Q+ )
4.Nxd4 *

I think the term you are looking for is desperado, even though it doesn't seem quite a typical use to me. Quoting wikipedia (https://en.wikipedia.org/wiki/Desperado_(chess))

In chess, a desperado piece is a piece that is en prise or trapped,
but captures an enemy piece before it is itself captured. This can be
in either a situation where both sides have hanging pieces, "...in
which you use your doomed piece to eat as much material as possible
before it dies";[1] or alternatively a sacrifice that will result in
stalemate if it is captured.

Acccumulation · Answer

After e6, black has two pieces threatened. The knight is threatened with tempo: if exd7 is allowed, the pawn threatens dxe8. And dxe8 simultaneously captures the black queen and promotes to a white queen. Since dxe8 gives white two queens, taking the queen at a1 is insufficient compensation. This means that exd7 is a "free move", as black must spend their next move dealing with the dxe8 threat.
When faced with two pieces being attacked, one with tempo and the other without, it's generally better to save the one attacked with tempo.

Why doesn't Black play Bd4?

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