GATE 2025 PH – Question 34
Consider a monatomic chain of length 30 cm. The phonon density of states is $1.2\times10^{-4}$ s. Assuming the Debye model, the velocity of sound in m/s (rounded off to one decimal place) is _____
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Correct answer: 791.82 to 799.78
Explanation
**Debye model in one dimension.** For a chain of length $L$ with sound velocity $v$, the allowed wave vectors are spaced by $2\pi/L$. The number of modes between $\omega$ and $\omega+d\omega$ (counting both directions of propagation) gives the density of states per unit angular frequency
$$g(\omega)=\frac{L}{\pi v},$$
which is **constant** in the Debye model.
**Solve for the velocity:**
$$v=\frac{L}{\pi\,g(\omega)}=\frac{0.30\text{ m}}{\pi\times1.2\times10^{-4}\text{ s}}=795.8\text{ m/s}.$$
The speed of sound is **795.8 m/s**.