Page 32 - Science at the nanoscale

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RPS: PSP0007 - Science-at-Nanoscale
June 9, 2009
Classical Physics at the Nanoscale
22
wave velocity, the frequency can be written as:
v
2π

(2.7)
(nodes at both ends)
= π
ω =
t
L
v
2π

(node at one end)
(2.8)
= π
ω =
t
2L
From the physics of waves, the wave velocity v can be expressed
as:
s
T
(2.9)
v =
ρ
where T is the tension of the stretched string and ρ is its mass per
unit length. For a three dimensional solid material, we can write
the wave velocity in (2.8) in terms of the Young’s modulus Y of the
material (Y = force per unit area per fractional deformation [Pa],
−3
):
or stress/strain), and the material density ρ (kgm
s
Y
v =
ρ
From Table 2.1, for a silicon cantilever with Y = 182 GPa and
−3
ρ = 2300 kgm
, we can calculate the speed of sound in silicon to
−1
be v = 8900 ms
. From equation (2.8), the resonant frequency of
a 1 m long silicon cantilever is ω = 14 kHz. If we reduce the length
of the silicon cantilever to 1 cm, its resonant frequency will be
about 1400 kHz. A typical silicon AFM cantilever with k between
0.01–100 N/m has a resonant frequency ω of 10–200kHz. It can be
Table 2.1 Elastic properties of selected engineering materials. (2.10) ch02
3
Material Density (kg/m ) Young’s Modulus (GPa)
Diamond 1800 1050
Silicon nitride 2200 285
Steel 7860 200
Silicon 2300 182
Aluminum 2710 70
Glass 2190 65
Polystyrene 1050 3