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The exponential decay of the amplitude envelope of an underdamped harmonic oscillator leads to a similar exponential envelope for the kinetic and potential energies. The total energy does not decay smoothly, but includes considerable ripple, even under the conditions of small damping where the decay constant is less than 1% of the undamped natural frequency. The treatment of this varies considerably among the current textbooks and a clear demonstration of the phenomenon is presented here.

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Energy transfer and upconversion studies in yttrium-aluminum-garnet crystal doped with thulium and holmium under near-infrared laser excitationThe optical absorption spectra of single-crystal Tm2(SO.la"8H,O and Tm H in Y,(SO')3"8H20 have been studied in the range from 2500 to 8000 A at 4.2 and 77°K. Nine [S LJ-J levels were identified including 3 P'.1.0, 1/6, I D2, IG., 3 F2.a". The vibronic, electronic, and temperature-dependent absorption lines of these multiplets were identified and tabulated. The spectra were studied as a function of magnetic field strength and as a function of orientation in the magnetic field. Zeeman absorption, paraelectric resonance, and zero-magnetic field optical absorption spectra all indicate a nearly accidental degeneracy in the ground state of tripositive thulium in the octahydrated sulfate salts. The magnetic moment was found to lie in the rectangular plane formed by the crystal's macroscopic habit and inclined at an angle of 46° with respect to the crystal's major habit direction. The crystalline lattice of these salts was found to possess magnetically inequivalent sites. The optical spectra arising from thulium ions in both sites are indistinguishable, but the application of an external magnetic field revealed that the local ionic magnetic moments in these two kinds of sites are inclined at a relative angle of approximately 60°. The observed splitting factor for thulium ions in both sites was found to be 13.4 Lorentz units. The principle features of the magnetic properties of these salts were confirmed by anisotropic paramagnetic susceptibility measurements.

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