corresponding frequencies being v max ¼ c= 2a
ð Þ and v min ¼ c= 2Na
ð
Þ, where c is the
speed of elastic waves in the material. Each one of these vibrations is connected to an
energy hn, where h is Planck’s constant. In order to derive the energy of a crystal, all
energies of the lattice vibrations must be summed; hence, the thermal energy E due
to lattice vibrations of a crystal is described by:
E ¼
X
i
n i v i h
ð7:2Þ
The number n i of vibrations with frequency n i is calculated using Bose–Einstein
statistics. The possible frequencies n i are a function of the particle size and the
following wavelengths are possible:
l ¼
2Na
1
;
2Na
2
;
2Na
3
; . . . ;
2Na
N
or l ¼
2L
1
;
2L
2
;
2L
3
; . . . ;
2L
N
ð7:3aÞ
This leads to the allowed frequencies:
n ¼ 1
c
2Na
; 2
c
2Na
; 3
c
2Na
; . . . ; N
c
2Na
or n ¼ 1
c
2L
; 2
c
2L
; 3
c
2L
; . . . ; N
c
2L
ð7:3bÞ
In Eq. (7.2) the only temperature-dependent term is the number of vibrations n i of
the frequency n i . From Eqs. (7.3a) and (7.3b) it is clear that, by reducing the particle
size, the energy of the vibrations with the longest wavelength l max ¼ 2Na ¼ 2L (the
one with the lowest energy) increases. As these vibrations are excited primarily at
low temperatures, a reduction of the heat capacity at low temperatures may be
expected.
This simple model does not take into account the increased degrees of freedom
for vibrations of the atoms at the surface. In fact, the large number surface atoms
(see Chapter 2) may make a significant contribution to the heat capacity, provided
that the particles are sufficiently small. A precise and detailed theory of heat
capacity as a function of the particle size is provided by Malinovskaya and Sachkov
[1]. Although this theory leads, as expected, to a decrease in heat capacity with
decreasing particle size, in the case of extremely small particles – when the
particle virtually now consists only of surface – an increased heat capacity is
predicted. The results of detailed calculations for the heat capacity at 298 K are
shown in Figure 7.3, for In 2 O 3 , where there is a remarkable and sudden increase
in C V at particle sizes below about 1.2 nm. Clearly, for these sizes the calculations
showed increased degrees of freedom for almost all atoms. When used as a simple
model, it is possible to compare the degrees of freedom for the vibration of atoms
at the surface with those of a liquid.
The results of this plausible model are not reproduced directly by experimental data. As an example, the heat capacity of nanocrystalline and coarsegrained copper and palladium is depicted in Figure 7.4a and b. For both the
metals, a larger heat capacity is found for the nanocrystalline material as
compared to the coarse-grained counterpart. The material used for the measurements depicted in Figure 7.4a and b was sintered, and as sintered material
7.2 Heat Capacity of Nanoparticles j137
ð Þ and v min ¼ c= 2Na
ð
Þ, where c is the
speed of elastic waves in the material. Each one of these vibrations is connected to an
energy hn, where h is Planck’s constant. In order to derive the energy of a crystal, all
energies of the lattice vibrations must be summed; hence, the thermal energy E due
to lattice vibrations of a crystal is described by:
E ¼
X
i
n i v i h
ð7:2Þ
The number n i of vibrations with frequency n i is calculated using Bose–Einstein
statistics. The possible frequencies n i are a function of the particle size and the
following wavelengths are possible:
l ¼
2Na
1
;
2Na
2
;
2Na
3
; . . . ;
2Na
N
or l ¼
2L
1
;
2L
2
;
2L
3
; . . . ;
2L
N
ð7:3aÞ
This leads to the allowed frequencies:
n ¼ 1
c
2Na
; 2
c
2Na
; 3
c
2Na
; . . . ; N
c
2Na
or n ¼ 1
c
2L
; 2
c
2L
; 3
c
2L
; . . . ; N
c
2L
ð7:3bÞ
In Eq. (7.2) the only temperature-dependent term is the number of vibrations n i of
the frequency n i . From Eqs. (7.3a) and (7.3b) it is clear that, by reducing the particle
size, the energy of the vibrations with the longest wavelength l max ¼ 2Na ¼ 2L (the
one with the lowest energy) increases. As these vibrations are excited primarily at
low temperatures, a reduction of the heat capacity at low temperatures may be
expected.
This simple model does not take into account the increased degrees of freedom
for vibrations of the atoms at the surface. In fact, the large number surface atoms
(see Chapter 2) may make a significant contribution to the heat capacity, provided
that the particles are sufficiently small. A precise and detailed theory of heat
capacity as a function of the particle size is provided by Malinovskaya and Sachkov
[1]. Although this theory leads, as expected, to a decrease in heat capacity with
decreasing particle size, in the case of extremely small particles – when the
particle virtually now consists only of surface – an increased heat capacity is
predicted. The results of detailed calculations for the heat capacity at 298 K are
shown in Figure 7.3, for In 2 O 3 , where there is a remarkable and sudden increase
in C V at particle sizes below about 1.2 nm. Clearly, for these sizes the calculations
showed increased degrees of freedom for almost all atoms. When used as a simple
model, it is possible to compare the degrees of freedom for the vibration of atoms
at the surface with those of a liquid.
The results of this plausible model are not reproduced directly by experimental data. As an example, the heat capacity of nanocrystalline and coarsegrained copper and palladium is depicted in Figure 7.4a and b. For both the
metals, a larger heat capacity is found for the nanocrystalline material as
compared to the coarse-grained counterpart. The material used for the measurements depicted in Figure 7.4a and b was sintered, and as sintered material
7.2 Heat Capacity of Nanoparticles j137