Created: April 10, 2004    Last updated: April 13, 2004   
B. Confined Vibrations of a PMI

   The 51 atoms in the PMI have average positions roj and instantaneous positions rj(t). Consider a harmonic vibration of the PMI with frequency ωp. This is understood to correspond to a confined phonon mode. The PMI will actually have many normal modes of vibration. We are considering inelastic X-ray scattering from one of them, mode p. Let p be an index for the PMI's normal modes.
   A microscopic calculation of the phonon modes within a PMI is not available. The approximation adopted here is to treat the PMI as a continuum elastic sphere with a free surface. This is called the Free Sphere Model (FSM). It is a classic problem of elastic mechanics first solved in 1882 by Lamb. The assumptions made about the PMI in applying FSM are as follows:

- The material in the PMI is continuous rather than discrete
- The mass density of the PMI is homogenneous (i.e. same at all positions)
- The elastic constants of the PMI are hhomogeneous
- The elastic constants are isotropic (nno direction dependence)
- The PMI has a well-defined surface whiich is a sphere of diameter 1.0 nm
- The surface of the PMI has zero tractiion forces (i.e. zero surface stress)
- The mass density of the PMI is ρ, equal to its mass over its volume
- The elastic constants correspond to thhe bulk longitudinal and transverse speeds of sound in AlPdMn (vL = 6500 m/s and vT = 3500 m/s)

   Bulk AlPdMn is an elastically isotropic material with longitudinal speed of sound being measured as 6512 m/s and 6530 m/s along differing axes, and transverse speed of sounds was 3595 m/s and 3590 m/s. [ Amazit, de Boissieu and Zarembowitch 1992 ]

   FSM is a crude model of a PMI. The mass distribution inside a PMI is not very homogeneous. In particular, the mass in a PMI is concentrated mostly in the 42 atoms at its outer surface. Only 9 atoms are in the interior. Therefore, only the most low-lying vibrational modes could conceivably be well approximated by FSM. On the other hand the icosahedral symmetry of the PMI means that the PMI is approximately spherical. For simple vibrational modes of a sphere such as breathing, dipolar vibration or quadrupolar vibration, an icosahedron "looks" spherical. Only higher order vibrational modes would require detailed angular dependence of the PMI's material properties. Therefore, the spherical symmetry of FSM is quite suitable, and homogeneity makes sense for θ and φ. It may be that r-dependence of density and the elastic constants are important. Conveniently, an extension of FSM with r-dependent material parameters is possible. In the simplest version, the core-shell model, the sphere is separated into an inner core and outer shell, each homogeneous.

   FSM normal modes are classified by four integer labels:
q - mode "class" (1=torsional (TOR) or 2=spheroidal(SPH)) (This follows the terminology of the original 1882 Lamb paper)
l  - angular momentum (0, 1, 2, 3...)
m - angular momentum z-component (-l ... 0 ... l )
n  - mode index (0, 1, 2, 3...)
   In the FSM approximation these four indices (q,l,m,n) replace phonon mode index p. The angular frequencies of these modes are denoted by ωqln. There is no m dependence. When talking about the modes themselves it is convenient to write (TOR,l,m,n) or (SPH,l,m,n).

   It is useful to distinguish between material coordinates R and real-space coordinates r. A given material point of a vibrating elastic body has fixed material coordinate R. The position in space of that material point r(t) will vary with time due to the vibration. The displacement field is defined by
u(R,t) = r(t) - R
(B1)
If the object is not displaced from its equilibrium configuration then r = R for all points.
   The frequency of the vibration is ωqln. The speeds of sound are vL and vT The wavevectors of longitudinal and transverse waves are kL = ωqln/vL and kT = ωqln/vT
   For torsional modes (q=1), the displacement field is of the form
u = ∇ × ( r χ )
(B2)
where χ is a scalar field satisfying the scalar Helmholtz equation:
(∇2 + kT2) χ = 0
(B3)
The form of χ for the m = 0 (φ-independent) case is
χ = G jl(kT r) Pl(cos θ)
(B4)
where G is a normalizing constant, jl are spherical Bessel functions and Pl are Legendre polynomials.
   The spheroidal mode with l=0 is a special case. It is the breathing mode. The displacement field is of the form
u = ∇ Φ
(B5)
where Φ is a scalar field satisfying
(∇2 + kL2) Φ = 0
(B6)
where Φ for the l = 0 case is of the form
Φ = H j0(kL r) / kL
(B7)
where H is a normalizing constant, and the factor of kL is included for dimensional convenience. Note that j0(x) = sin(x)/x.
   Spheroidal modes with l ≠ 0 are more complicated. The displacement field is of the form
u  =  ∇ Φ  +  ∇ × ∇ × ( r ψ )
(B8)
where Φ satisfies Eq. (B6) and is of the form (for the m=0 case):
Φ = H jl(kL r) Pl(cos θ) / kL
(B9)
and ψ in Eq. (B8) satisfies Eq. (B3) and is of the form (for the m=0 case)
ψ = I jl(kT r) Pl(cos θ) / kT
(B10)
However, the two terms in Eq. (B8) must be combined with the correct amplitude so as to satisfy the zero traction force boundary condition at all points on the surface of the sphere.
   To determine the normalization constants G, H and I for each mode (q,l,m,n), the displacement field of a given mode is normalized according to [Murray and Saviot 2004]
M = ∫ ρ(r) uqlmn · uqlmn d3r
(B11)
where M = ∫ ρ(r) d3r is the mass of the PMI and where the volume of the integration is the interior of the 1.0 nm diameter sphere that approximates the PMI. In this way, the normalized displacement fields are dimensionless. The normalized displacement fields are to be multiplied by a coefficient xqlmn which is in metres, so as to get the actual displacement to be used:
u = xqlmn uqlmn
(B12)
The mechanical energy associated with the mode is [Murray and Saviot 2004]
E = (1/2) M (ωqln)2 |xqlmn|2
(B13)
where M is the total mass of the PMI. A natural assumption would be that E = kB T where kB is Boltzmann's constant. This is valid as long as temperature T is high enough so that mode (q,l,m,n) has a significant occupation number, which is true for low-order modes at room temperature where kB T = 26 meV. AlPdMn quasicrystal optical modes have energies starting at 7 meV, so a classical treatment is plausible. Otherwise a quantum mechanical treatment would have been required.

   In order to apply this to the inelastic X-ray scattering problem,
rj(t) = roj + u(roj,t)
(B14)
The detected intensity F is then calculated from Eq. (A10).

Next page:
C. Simple Examples

Go back to previous page:
A. Inelastic scattering from a vibrating PMI
See animations of a vibrating PMI
References

D. B. Murray and L. Saviot "Phonons in an inhomogeneous continuum: Vibrations of an embedded nanoparticle" Phys. Rev. B 69 094305 (2004)    cond-mat/0310099

Y. Amazit, M. de Boissieu and A. Zarembowitch, Europhys. Lett. 20 703 1992

Daniel Murray
Associate Professor
Math, Stats & Physics Unit
University of British Columbia - Okanagan
Kelowna, BC, Canada
daniel "dot" murray "at" ubc "dot" ca

For a list of related articles click on the link.
Hosted by www.Geocities.ws

1