Sound Attenuation Study on the Bose-Einstein Condensation of Magnons in<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:msub><mml:mrow><mml:mi mathvariant="normal">T</mml:mi><mml:mi mathvariant="normal">l</mml:mi><mml:mi mathvariant="normal">C</mml:mi><mml:mi mathvariant="normal">u</mml:mi><mml:mi mathvariant="normal">C</mml:mi><mml:mi mathvariant="normal">l</mml:mi></mml:mrow><mml:mn>3</mml:mn></mml:msub></mml:math>

We investigate experimentally and theoretically sound attenuation in the quantum spin system TlCuCl3 in magnetic fields at low temperatures. Near the point of Bose-Einstein condensation of magnons a sharp peak in the sound attenuation is observed. The peak demonstrates a hysteresis as a function of the magnetic field pointing to a first-order contribution to the transition. The sound damping has a Drude-like form arising as a result of hard-core magnon-magnon collisions. The strength of the coupling between lattice and magnons is estimated from the experimental data. The puzzling relationship between the transition temperature and the concentration of magnons is explained by their "relativistic" dispersion.

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