Copyright 2004 by Marcel Dekker, Inc. All Rights Reserved.

In this model, the electrons couple to the phonon modes in a way that dependson their temperature difference [40]:C e ðT e Þ AT e¼ gðT e T l Þ þ E 0 t 2 pffiffiffiexpAtp rL r 2 ð5aÞLandC 1AT lAt ¼ gðT eT l ÞT l 298s sð5bÞIn these equations, g is the electron–phonon coupling constant, C e (T e ) = gT eand C l are the heat capacities of the electrons and the lattice, respectively, ands s is the timescale for energy transfer to the solvent. The last term in Eq. (5a)represents the laser pulse: E 0 corresponds to the amount of energy absorbedby the sample and j L gives the pulse width [36]. Equations (5a) and (5b) can besolved numerically if E 0 and s s are specified (all the other parameters areknown).Once T e and T l have been determined, the response of the system toheating can be calculated by the following harmonic oscillator equation:d 2 Rdt 2þ 2 dRs d dt þ 2p 2½R R 0 Š ¼ F e ðT e Þ þ F l ðT l Þ ð6ÞTwhere T is the period the phonon mode that correlates with expansion, R 0 isthe initial radius, s d is a phenomenological constant that accounts for thedecay of the modulations, and F e (T e ) and F l (T l ) are the forces due to heatingthe electrons and the lattice, which are proportional to j e and j l , respectively.The results of these calculations are shown as the solid line in Fig. 3. Thevalues of T, R 0 , s d , and s s were chosen to match the data in Figs. 1 and 3. Thevalue of E 0 (the absorbed energy) was determined from the experimentalconditions: the laser spot size, the energy per pulse, and the absorbance of thesample at 400 nm. For our experiments E 0 = (7 F 1) 10 3 J/mol, whichproduces an overall temperature rise in the lattice of DT l = 290jC just afterthe electrons and phonons have reached equilibrium (i.e., before any heatdissipaton to the solvent). Note that the only adjustable parameters in thesecalculations are s d and s s .There are several points to note about the data in Fig. 3. First, theamplitude (the amount of overexpansion and compression) and the phase ofthe oscillations are in excellent agreement for the calculations and theexperiments. The amplitude is mainly determined by the lattice heatingcontribution to the driving force for the expansion. Thus, the agreementbetween the calculated and experimental amplitudes shows that all of theabsorbed energy goes into expansion. In contrast, the phase is sensitive toCopyright 2004 by Marcel Dekker, Inc. All Rights Reserved.