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Submitted on 1 Jan 1981
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PHONON PULSES FRQM A RELAXING SYSTEM
R. Englman
To cite this version:
R. Englman. PHONON PULSES FRQM A RELAXING SYSTEM. Journal de Physique Colloques,
1981, 42 (C6), pp.C6-241-C6-243. �10.1051/jphyscol:1981669�. �jpa-00221605�
JOURNAL DE PHYSIQUE
CoZZoque C6, suppZBment au n '1 2, Tome 42, dgcembre 1981 page C6-241
PHONON PULSES FROM A R E L A X I N G SYSTEM
R. Englman
Israel Atomic Energy Commission, Soreq NucZear Research Center, Yavne, Israel
Abstract.- The spread of polychromatic phonon pulses, following optical absorption by a defect, is calculated up to %lo2 lattice-spacings far.
Phonon dispersion attenuates the pulse. The angular variation of the pulse peak position reflects the spectral anisotropy.
1. Introduction.- Recent detection of monochromatic phonon pulses explored bal- listic propagation and angular dependences in the phonon spectrum. y 2, Phonon pulses arising from a relaxing excited impurity have been described theoretical- l~.(~) These span a broad frequency range, are easy to generate and may with some sophistication be detected (a) macroscopically, namely at a distance several times the illuminated region, or (b) microscopically, by coherent measurement of signals produced by secondary absorbers near the impurity.
2. Results.
-
As noted previously, (3) the pulse-shape depends on the frequency spectrum w(k,j) as function of the wave vectork
and branch index j. In Fig. 1 the pulse, which is initially localized near the impurity, maintains its shape and height for a Debye spectrum, w a k, as far as 30 lattice spacings (a) away but distorts and attenuates strongly by dispersion w a sinak12.The distance
(r)
and time (t) dependence of the pulse isThe amplitude q of the initial excitation is expected to follow asymptotically for small k theorelationship
%
a k-2 (or k-' in 2D)(4). The polarization vector included in%
has a complicated behaviour especially for degenerate modes. (5) We assume an isotropic form.Then the angular variation of the pulse intensity arises from the anisotropy of w(&). Assuming a cubic form like
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1981669
JOURNAL DE PHYSIQUE
w2 a :k
+
k4+
k: (or a kX 4+
k4 in 20)Y Y
the outward (radial) pulse-height rq (6,t) was computed as function of the azimuthal angle 9 in the (001)-plane (or of the polar angle in a cylindrical ZD-geometry)
.
The location of the main maximum i n rq(+,t) at each instant t can be derived by assuming that the main contribution to the sum in (1) comes from points in the Brillouin-zone where the group velocity satisfies
The maximum occurs for values of r,t where the phase in (1) w(k, j) t
- hr
= 2r x integerIt can then be shown that the maximum amplitude is at points rM (9,t) where
The rhs of (2) is shown in Fig. 2. The values of rx($,t) obtained from computa- tion of (1) at a time t = av-I (vs = velocity of sound) resemble the functional form of (2) in both 20 and 3D. [~ote however from the figure that only a little later the computed maxima are distorted with respect to the simple prediction (2)]. 3. Conclusion.- It follows that with suitable experimental means of detection spontaneous phonon pulses from excited, relaxing impurities can yield useful though limited information about lattice phonons and the impurity-lattice interaction.
4. References
(1) G.A. Northrop and J.P. Wolfe, Phys. Rev. B 22, 6196 (1981)
(2) P. Hu, V. Narayanamurti and M.A. Chin, Phys. Rev. Letters
3,
192 (1980) (3) R. Englman, Chem. Phys.58,
227 (1981)(4) R.I. Elliott and J.B. Parkinson, Proc. Phys. Soc. Lond.
92,
1024 (1967) (5) B. Halperin and R. Englman, J. Phys. C8,
3975 (1975).Fig. 1.- rq vs r
Radial, weiEhted phonon amplitudes as function of distance in units of lattice spacing for isotropic spectra at different times. Initially the pulse is
localized at the centre, later it propagates sharply outwards for a non-dispersive spectrum (w = k ) but gets attenuated and distorted for a dispersive spectrum.
Time is in units of ax(ve1ocity of sound)-1.
0.8 -
0" 15" 30" 45"
Angle )
Fig. 2 . - Anisotropy effect
The position r~ of the maln maximum as function of the angle of orientation@
(in a [001]-plane) as obtained heuristically [eq. (2)1, by computation in 3D and 2D at a time = 5av -1. The dots show computed results in 2D at a time = 7avs-l.
