When an electron in an excited atom jumps from the $L$ shell to the $K$ shell,$X$-rays are emitted. These $X$-rays undergo first-order diffraction at an angle of $7.75^{\circ}$ by a crystal with an interplanar spacing of $2.64 \ \mathring{A}$. Calculate the energy difference between the $K$ shell and the $L$ shell. $(\sin \ 7.75^{\circ} = 0.1349)$

  • A
    $2.788 \times 10^{-13} \ J$
  • B
    $2.788 \times 10^{-15} \ J$
  • C
    $3.586 \times 10^{12} \ J$
  • D
    $1.075 \times 10^{13} \ J$

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