By means of thermal expansion and specific heat measurements on the high-pressure phase of (VO)(2)P(2)O(7), the effects of two energy scales of the weakly dimerized antiferromagnetic S=1/2 Heisenberg chain are explored. The low-energy scale, given by the spin gap Delta, is found to manifest itself in a pronounced thermal expansion anomaly. A quantitative analysis, employing the density-matrix renormalization-group approach for transfer matrices calculations, shows that this feature originates from changes in the magnetic entropy with respect to Delta, partial differentialS(m)/partial differentialDelta. This term, inaccessible by specific heat, is visible only in the weak-dimerization limit, where it reflects peculiarities of the excitation spectrum and its sensitivity to variations in Delta.
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