**4.1 Position and trajectories of movement for the sodium ions**

Five different temperatures (200K to 373K) have been set as an initial condition but even at the highest temperature 373K, the sodium ions still have not gained enough energy to leave the equilibrium positions for the next available sites in the absence of an electric field. A comparison of the average position for the first and second sodium ions at 200K and 373K is shown in the figure 14. At 373K, the sodium ions have more energy as the fluctuations are bigger compared to the average positions at 200K, but the sodium ions still do not have enough energy to go into the next available site.

Fig. 14. A comparison of the average positions along c-axis for the first and second sodium ions at 200K and 373K as a function of time.

The next part of the research was to introduce the electric field to the hollandite model. The field was applied from 5001th interval to 100000th interval; it was clearly shown in figure 6 that the fluctuations of the sodium ions increase dramatically at 5001th interval. This is due to the extra energy obtained by the sodium ions from the applied field. In figure 6, take the sixth sodium (pink) for example; it gained enough energy from the applied field to hop into the next available site, which is the next cavity. It then vibrates in this new site for about 2000 intervals until it gained enough energy to hop back to the original cavity. This can be seen in figure 15. From the conservation of energy, in a closed system; the total energy is constant although energy may be transferred between kinetic and potential energy and from one group of sodium ions to another. In the case of the hollandite model, the energy is being swapped around between the sodium ions and this is clearly shown in the figure 16. Only three sodium ions have been shown here, the rest of the sodium ions would behave in a similar way.

Five different temperatures (200K to 373K) have been set as an initial condition but even at the highest temperature 373K, the sodium ions still have not gained enough energy to leave the equilibrium positions for the next available sites in the absence of an electric field. A comparison of the average position for the first and second sodium ions at 200K and 373K is shown in the figure 14. At 373K, the sodium ions have more energy as the fluctuations are bigger compared to the average positions at 200K, but the sodium ions still do not have

Fig. 14. A comparison of the average positions along c-axis for the first and second sodium

The next part of the research was to introduce the electric field to the hollandite model. The field was applied from 5001th interval to 100000th interval; it was clearly shown in figure 6 that the fluctuations of the sodium ions increase dramatically at 5001th interval. This is due to the extra energy obtained by the sodium ions from the applied field. In figure 6, take the sixth sodium (pink) for example; it gained enough energy from the applied field to hop into the next available site, which is the next cavity. It then vibrates in this new site for about 2000 intervals until it gained enough energy to hop back to the original cavity. This can be seen in figure 15. From the conservation of energy, in a closed system; the total energy is constant although energy may be transferred between kinetic and potential energy and from one group of sodium ions to another. In the case of the hollandite model, the energy is being swapped around between the sodium ions and this is clearly shown in the figure 16. Only three sodium ions have been shown here, the rest of the sodium ions would behave in a

**4.1 Position and trajectories of movement for the sodium ions** 

enough energy to go into the next available site.

ions at 200K and 373K as a function of time.

similar way.

**4. Discussion** 

Fig. 15. Trajectories of movement in three-dimensions of the 6th sodium ion in the hollandite model with field of 743MV/m.

Fig. 16. A comparison of the position in c-axis for the 5th, 6th and 7th sodium ions with field of 743MV/m. The light blue line represents the position of the cavity.

Molecular Dynamics Simulation and Conductivity

same position as in the results shown in figure 9.

electric field of 743MV/m.

Mechanism in Fast Ionic Crystals Based on Hollandite NaxCrxTi8-xO16 389

likely to relate to a resonance response. The peak frequency for "(f) is in the range of (2.7x1010 – 8.8x1010 Hz). Ions hopping between equilibrium sites usually give a relaxation response, whereas in the case of hollandite model, a resonance response is obtained. What seems to be happening is that the movement of the sodium ions between sites change the vibration of the other sodium ions, so that the vibrations of each sodium ion in a group is coupled together and the hopping of one them to a new site destroys the group motion and acts like a damping effect on the resonance that is due to it, i.e. the motion of the system of dipoles behaves like a damped libration. This gives a good agreement to the prediction by Fröhlich (Fröhlich, 1958) who suggested that the absorption due to displacement of charges bound elastically to an equilibrium position is of resonance character, although in our case the ability of the sodium

In order to fit '(f) an additional resonance of amplitude y0 can be added in equation 6. y0 relates to the isolated higher frequency oscillations which had been removed by the smoothing process. Figure 18 below shows the real and imaginary parts of the susceptibility with a smoothing level of 2000 (that is 2000 points have been used as the number of data points considered to be smoothed at a time). It is clearly shown that adding a higher frequency resonance improves the fit to the data, and hence that oscillations in the form of additional resonance absorptions are present. The lowest resonance absorption was at the

The real and imaginary part of the susceptibility vs Frequency (first 70 intervals) field=743MV/m, 273K, 100000 intervals, time step=10-15s, Smoothing level=2000

Fig. 18. The real and imaginary parts of the susceptibility as a function of frequency with an

A single Lorentzian function does not fit both '(f) and "(f) well. Equation 7 does not fit "(f) well as it gives a symmetrical peak curve whereas "(f) is not symmetrical. The simulated "(f) data shows a steeper gradient at frequencies below the resonance absorption compared to the

ions to displace from site to site adds a visco-elastic dimension to the situation.

Figure 17 shows that when the dc-field was applied at 5001th interval, the sixth sodium ion starts to gain energy to move into the next empty site. With the applied field of 743MV/m, the sodium ion hops into the next cavity almost as soon as the field is applied, whereas with a lower applied field which is 7.43MV/m, about 9x10-12 s is needed for the first hop of the sodium ion. As the electric field increases, the number of hops into the neighbouring sites increases as well.

Fig. 17. A comparison of the position in c-axis for the 6th sodium ion with field of 7.43MV/m and 743MV/m. The light blue line represents the cavity.

This movement of the sodium ion is like a dipole moving around, as the sodium ion represents the positive part and the ions at the lattice site that formed the cage is the negative part. A relaxation behaviour would usually be expected to be obtained as the sodium ions hopping between equilibrium positions are like re-orienting dipoles.

#### **4.2 Frequency dependence of ' and "**

Frequency dependence of ' and " for a range of temperature and a range of field were obtained from the FFT procedure. The values for the frequency '(f) and "(f) obtained in our simulations are too high and this is because the hollandite model used (1 tunnel with 60 layers) is only a small section of the whole crystal. If a bigger model had been considered, the average displacement of the ions would be much smaller because the movement of the sodium ions are affected by the sodium ions in the other tunnels, and a smaller polarisation and smaller value of '(f) and "(f) would be obtained.

From the '(f) obtained (figure 9); it is clearly shown that '(f) is not a relaxation response as part of '(f) gives negative values. Hence the results obtained for '(f) and "(f) would be more

Figure 17 shows that when the dc-field was applied at 5001th interval, the sixth sodium ion starts to gain energy to move into the next empty site. With the applied field of 743MV/m, the sodium ion hops into the next cavity almost as soon as the field is applied, whereas with a lower applied field which is 7.43MV/m, about 9x10-12 s is needed for the first hop of the sodium ion. As the electric field increases, the number of hops into the neighbouring sites

Fig. 17. A comparison of the position in c-axis for the 6th sodium ion with field of 7.43MV/m

This movement of the sodium ion is like a dipole moving around, as the sodium ion represents the positive part and the ions at the lattice site that formed the cage is the negative part. A relaxation behaviour would usually be expected to be obtained as the

Frequency dependence of ' and " for a range of temperature and a range of field were obtained from the FFT procedure. The values for the frequency '(f) and "(f) obtained in our simulations are too high and this is because the hollandite model used (1 tunnel with 60 layers) is only a small section of the whole crystal. If a bigger model had been considered, the average displacement of the ions would be much smaller because the movement of the sodium ions are affected by the sodium ions in the other tunnels, and a smaller polarisation

From the '(f) obtained (figure 9); it is clearly shown that '(f) is not a relaxation response as part of '(f) gives negative values. Hence the results obtained for '(f) and "(f) would be more

sodium ions hopping between equilibrium positions are like re-orienting dipoles.

and 743MV/m. The light blue line represents the cavity.

and smaller value of '(f) and "(f) would be obtained.

**4.2 Frequency dependence of ' and "** 

increases as well.

likely to relate to a resonance response. The peak frequency for "(f) is in the range of (2.7x1010 – 8.8x1010 Hz). Ions hopping between equilibrium sites usually give a relaxation response, whereas in the case of hollandite model, a resonance response is obtained. What seems to be happening is that the movement of the sodium ions between sites change the vibration of the other sodium ions, so that the vibrations of each sodium ion in a group is coupled together and the hopping of one them to a new site destroys the group motion and acts like a damping effect on the resonance that is due to it, i.e. the motion of the system of dipoles behaves like a damped libration. This gives a good agreement to the prediction by Fröhlich (Fröhlich, 1958) who suggested that the absorption due to displacement of charges bound elastically to an equilibrium position is of resonance character, although in our case the ability of the sodium ions to displace from site to site adds a visco-elastic dimension to the situation.

In order to fit '(f) an additional resonance of amplitude y0 can be added in equation 6. y0 relates to the isolated higher frequency oscillations which had been removed by the smoothing process. Figure 18 below shows the real and imaginary parts of the susceptibility with a smoothing level of 2000 (that is 2000 points have been used as the number of data points considered to be smoothed at a time). It is clearly shown that adding a higher frequency resonance improves the fit to the data, and hence that oscillations in the form of additional resonance absorptions are present. The lowest resonance absorption was at the same position as in the results shown in figure 9.

Fig. 18. The real and imaginary parts of the susceptibility as a function of frequency with an electric field of 743MV/m.

A single Lorentzian function does not fit both '(f) and "(f) well. Equation 7 does not fit "(f) well as it gives a symmetrical peak curve whereas "(f) is not symmetrical. The simulated "(f) data shows a steeper gradient at frequencies below the resonance absorption compared to the

Molecular Dynamics Simulation and Conductivity

not the frequency of the group oscillation.

a range of electric field (7.43MV/m – 7.43GV/m).

**4.4 Poley absorption** 

(Coffey et al., 1987).

Mechanism in Fast Ionic Crystals Based on Hollandite NaxCrxTi8-xO16 391

would affect the lifetime of the group oscillation and hence the half-width of the peak but

Fig. 20. The peak of the imaginary part of the susceptibility as a function of temperature for

The imaginary part of the susceptibility has been plotted for a range of electric field (7.43MV/m – 7.43GV/m) at 297K as shown in figure 21 below. When the applied field gets higher, " becomes smaller. This indicates that the higher the field, the less ions are involved in the group site-libration modes and more in damping (i.e. hopping between sites). The hopping of the sodium ions damp the libration of other sodium ions similar to the friction between the disc and annulus damps the libration in the Itinerant Oscillator (IO) model

This IO model does not explain the process that happens in the hollandite model well enough as the IO model is a harmonic model and does not take into account the effect of molecular translations upon the potentials and forces controlling the motion, a factor that our molecular dynamics simulation have shown to be important. The periodic potential model (Vij & Hufnagel, 1985; Praestgaard & van Kampen, 1981) is also not a good approximation for the same reason. On the other hand, the process in the hollandite model has a good agreement with the cluster model presented by Dissado and Alison (Dissado & Alison, 1993), where the cluster model takes into account the translations of the dipole. The displacements (translations) of the solute molecule was affected by the positions of its surrounding solvent molecules (solvent cage) and vice versa to bring the group (cluster) into an equilibrium configuration. This will cause the deformation of the solvent cage so that reorientations of the solute dipole will also involve reorganisation of the solvent cage deformations. A parameter n was defined such that when n=0 the solute did not deform the cage and its dipole reorientation was overdamped giving a relaxation peak with the vibrations of the solvent providing the viscous damping. When n=1 the solute and solvent formed a single cooperative system giving a Poley absorption and no relaxation peak. When

fitted curve (equation 7). Similarly, equation 6 would not fit '(f) as equation 6 would give a symmetrical plot in magnitude where the maximum and the minimum points have the same value in magnitude but different sign. This is understandable as the Lorentzian function is generated when a single oscillator is damped by the surroundings whereas in the hollandite model the movement of the sodium ions between sites depend on the other sodium ions. The hopping of any of the sodium ions would damp the libration of the rest of the sodium ions. Hence, there is no specific frequency, which can be considered as a resonance since the number of sodium ions taking part in a group oscillation can change, with each number involved having a different group-oscillator frequency and hence different resonance frequency. However, the Lorenzian function has been used because there is no general nonlinear frequency dependent expression available and we have to try to find some ways of expressing the results. As shown in figure 18 increasing the number of resonances improves the fit and hence we can expect when enough resonances have been included a good fit will be obtained within the limitations imposed by the time window of the simulation.

#### **4.3 Temperature dependence**

Simulations have been carried out for a range of electric field between 7.43MV/m and 7.43GV/m and temperatures between 200K and 373K. The absorption peak frequency, which is also the resonance frequency, and the resonance peak height, has been plotted as a function of temperature for a range of electric field in figures 19 and 20.

Fig. 19. The resonance frequency as a function of temperature for a range of electric field (7.43MV/m – 7.43GV/m).

From figure 19 the resonance frequency is independent of temperature for all the different electric field applied, and from figure 20 the resonance peak height does not change significantly as the temperature increases. This behaviour is not what would be expected of a dielectric relaxation but is what would be expected from the oscillatory behaviour of groups of sodium atoms whose vibrations are coupled together, where the temperature would affect the lifetime of the group oscillation and hence the half-width of the peak but not the frequency of the group oscillation.

Fig. 20. The peak of the imaginary part of the susceptibility as a function of temperature for a range of electric field (7.43MV/m – 7.43GV/m).

#### **4.4 Poley absorption**

390 Molecular Dynamics – Theoretical Developments and Applications in Nanotechnology and Energy

fitted curve (equation 7). Similarly, equation 6 would not fit '(f) as equation 6 would give a symmetrical plot in magnitude where the maximum and the minimum points have the same value in magnitude but different sign. This is understandable as the Lorentzian function is generated when a single oscillator is damped by the surroundings whereas in the hollandite model the movement of the sodium ions between sites depend on the other sodium ions. The hopping of any of the sodium ions would damp the libration of the rest of the sodium ions. Hence, there is no specific frequency, which can be considered as a resonance since the number of sodium ions taking part in a group oscillation can change, with each number involved having a different group-oscillator frequency and hence different resonance frequency. However, the Lorenzian function has been used because there is no general nonlinear frequency dependent expression available and we have to try to find some ways of expressing the results. As shown in figure 18 increasing the number of resonances improves the fit and hence we can expect when enough resonances have been included a good fit will be

Simulations have been carried out for a range of electric field between 7.43MV/m and 7.43GV/m and temperatures between 200K and 373K. The absorption peak frequency, which is also the resonance frequency, and the resonance peak height, has been plotted as a

Fig. 19. The resonance frequency as a function of temperature for a range of electric field

From figure 19 the resonance frequency is independent of temperature for all the different electric field applied, and from figure 20 the resonance peak height does not change significantly as the temperature increases. This behaviour is not what would be expected of a dielectric relaxation but is what would be expected from the oscillatory behaviour of groups of sodium atoms whose vibrations are coupled together, where the temperature

obtained within the limitations imposed by the time window of the simulation.

function of temperature for a range of electric field in figures 19 and 20.

**4.3 Temperature dependence** 

(7.43MV/m – 7.43GV/m).

The imaginary part of the susceptibility has been plotted for a range of electric field (7.43MV/m – 7.43GV/m) at 297K as shown in figure 21 below. When the applied field gets higher, " becomes smaller. This indicates that the higher the field, the less ions are involved in the group site-libration modes and more in damping (i.e. hopping between sites). The hopping of the sodium ions damp the libration of other sodium ions similar to the friction between the disc and annulus damps the libration in the Itinerant Oscillator (IO) model (Coffey et al., 1987).

This IO model does not explain the process that happens in the hollandite model well enough as the IO model is a harmonic model and does not take into account the effect of molecular translations upon the potentials and forces controlling the motion, a factor that our molecular dynamics simulation have shown to be important. The periodic potential model (Vij & Hufnagel, 1985; Praestgaard & van Kampen, 1981) is also not a good approximation for the same reason. On the other hand, the process in the hollandite model has a good agreement with the cluster model presented by Dissado and Alison (Dissado & Alison, 1993), where the cluster model takes into account the translations of the dipole. The displacements (translations) of the solute molecule was affected by the positions of its surrounding solvent molecules (solvent cage) and vice versa to bring the group (cluster) into an equilibrium configuration. This will cause the deformation of the solvent cage so that reorientations of the solute dipole will also involve reorganisation of the solvent cage deformations. A parameter n was defined such that when n=0 the solute did not deform the cage and its dipole reorientation was overdamped giving a relaxation peak with the vibrations of the solvent providing the viscous damping. When n=1 the solute and solvent formed a single cooperative system giving a Poley absorption and no relaxation peak. When

Molecular Dynamics Simulation and Conductivity

ice clathrate crystal.

between oscillating groups.

**4.5 Group oscillation at very high field** 

Mechanism in Fast Ionic Crystals Based on Hollandite NaxCrxTi8-xO16 393

ions confined to the host structure undergoing displacement under the effect of electric field. The absorptions calculated lie below the resonance frequency of a single sodium ion, which defines the edge of the quantum region. They correspond to coupled displacements of sodium ion groups. It would be reasonable to assume that our calculated absorption corresponds to what would be the Poley absorption for this material. A similar process happens in ice clathrate materials as shown by Johari (Johari, 2002), where an absorption peak was seen in the far-infrared region (7.5x1011 and 1.14x1012 Hz) and was contributed by the rotational oscillation of the tetrahydrofuran molecule, while confined to the cages of the

From figure 19, the resonance frequency is independent of temperature for all the different electric field applied. This is different from the deduction made by Johari (Johari, 2002) on the ice clathrate crystal and Nouskova et al (Novskova et al., 1986) on the model of a restricted rotator where the resonance frequency decreases with the increase in temperature due to the increase of the libration magnitude. From figure 20, the resonance peak height does not change much as the temperature increases. This is also different from the deduction made by both Johari (Johari, 2002) and Nouskova (Novskova et al., 1986). Johari reported that the resonance peak height increases with the increase in temperature whereas Nouskova said that the resonance peak height decreases with the increase in temperature. In our hollandite model, the sodium ion group motions take place in a rigid lattice unlike the experimental situation for ice clathrates. Therefore the libration amplitude of an individual sodium ion dipole is unaffected by temperature and the oscillation frequency of a specific group should not be affected unlike the situation in (Johari, 2002; Novskova et al., 1986). The resonance peak magnitude will be dependent upon the number of sodium ions taking part in a coupled group oscillation at a specific frequency and this will not be affected by temperature. The effect of temperature in our model is expected to lie in the half-width of the absorption peaks, since this is due to the hopping of sodium ions

From figure 13, it is clearly shown that at high electric field, which is 74.3GV/m for the hollandite model, all the sodium ions were driven by the field to move as a group and show a single frequency vibration. The period of the oscillation is 4x10-15s. Since we use specular boundary conditions for the tunnel (i.e. the tunnel ends are reflective) this oscillation period is what would be expected if the sodium ions moved as a whole and were reflected from the boundaries after the first and third time-steps (i.e at 90 degree and 270 degree phases in the cycle). In the case of a smaller field, the sodium ion would just hop between empty sites next to the original site where it belongs. At bigger fields the sodium ions would have enough energy to move to the empty sites located furthest away. Figure 22 shows an example of three sodium ions in a tunnel. When the applied field is small, the sodium ions will hop between the available empty sites next to them. The only available site for Ion1 is the one on its right, whereas Ion2 and Ion3 will have two available sites to go to and this depends on the direction of the force acting on that particular sodium ion at that moment. As shown in Figure 21 increasing the field reduces the number of sodium ions taking part in coupled local group site-motions (reduced peak amplitude) in favour of sodium ion hopping between sites. For this reason a trend towards higher resonance frequencies and broader

0<n<1, the solute motions were partitioned between the Poley absorption and relaxation peak. In our model the sodium ion system forms both solute (with ions able to hop to new sites) and solvent (with ions vibrating around their equilibrium sites). The surrounding lattice is held rigid and cannot be involved in the cage motions.

Fig. 21. The imaginary part of the susceptibility has been plotted for a range of electric field (7.43MV/m – 7.43GV/m) at 297K.

The kinetic energy of the sodium ions (defined by the temperature) would initially cause the librations of the sodium ions under the action of the forces from the ions in the fixed lattice sites. Forces from the environment (which is the other sodium ions with respect to a particular sodium ion, and the ions in the lattice sites) causes the sodium ions themselves to produce a force acting on each other, so that the displacement of each sodium ion would be adjusted according to the new equilibrium configuration at each interval, and hence translations changing the potentials occurs on top of the librations. The cluster model would suggest that since all sodium ion motions act cooperatively the parameter n1 and there would be a Poley peak with either a weak relaxation peak or none at all. Although the simulations have not been carried through to times longer than ~10-10s, this seems to be what is happening, as the ion motions include both hopping and vibration and give just a damped resonance without any evidence of a relaxation peak.

At 297K, the resonance frequency is in the range of about 4.5x1010 – 8.8x1010 Hz for a range of electric field (7.43MV/m – 7.43GV/m). Poley (Poley, 1955) predicted that there is a significant power absorption in dipolar liquids at the ambient temperature in the 1.2 - 70 cm-1 (3.6x1010 – 2.1x1012 Hz) region and Davies (Davies et al, 1969) named the broad peak as 'Poley absorption'. The absorption peaks obtained for the hollandite model lie at the lower end of Poley's prediction range. This absorption is due to the libration of the sodium

0<n<1, the solute motions were partitioned between the Poley absorption and relaxation peak. In our model the sodium ion system forms both solute (with ions able to hop to new sites) and solvent (with ions vibrating around their equilibrium sites). The surrounding

Fig. 21. The imaginary part of the susceptibility has been plotted for a range of electric field

The kinetic energy of the sodium ions (defined by the temperature) would initially cause the librations of the sodium ions under the action of the forces from the ions in the fixed lattice sites. Forces from the environment (which is the other sodium ions with respect to a particular sodium ion, and the ions in the lattice sites) causes the sodium ions themselves to produce a force acting on each other, so that the displacement of each sodium ion would be adjusted according to the new equilibrium configuration at each interval, and hence translations changing the potentials occurs on top of the librations. The cluster model would suggest that since all sodium ion motions act cooperatively the parameter n1 and there would be a Poley peak with either a weak relaxation peak or none at all. Although the simulations have not been carried through to times longer than ~10-10s, this seems to be what is happening, as the ion motions include both hopping and vibration and give just a

At 297K, the resonance frequency is in the range of about 4.5x1010 – 8.8x1010 Hz for a range of electric field (7.43MV/m – 7.43GV/m). Poley (Poley, 1955) predicted that there is a significant power absorption in dipolar liquids at the ambient temperature in the 1.2 - 70 cm-1 (3.6x1010 – 2.1x1012 Hz) region and Davies (Davies et al, 1969) named the broad peak as 'Poley absorption'. The absorption peaks obtained for the hollandite model lie at the lower end of Poley's prediction range. This absorption is due to the libration of the sodium

lattice is held rigid and cannot be involved in the cage motions.

(7.43MV/m – 7.43GV/m) at 297K.

damped resonance without any evidence of a relaxation peak.

ions confined to the host structure undergoing displacement under the effect of electric field. The absorptions calculated lie below the resonance frequency of a single sodium ion, which defines the edge of the quantum region. They correspond to coupled displacements of sodium ion groups. It would be reasonable to assume that our calculated absorption corresponds to what would be the Poley absorption for this material. A similar process happens in ice clathrate materials as shown by Johari (Johari, 2002), where an absorption peak was seen in the far-infrared region (7.5x1011 and 1.14x1012 Hz) and was contributed by the rotational oscillation of the tetrahydrofuran molecule, while confined to the cages of the ice clathrate crystal.

From figure 19, the resonance frequency is independent of temperature for all the different electric field applied. This is different from the deduction made by Johari (Johari, 2002) on the ice clathrate crystal and Nouskova et al (Novskova et al., 1986) on the model of a restricted rotator where the resonance frequency decreases with the increase in temperature due to the increase of the libration magnitude. From figure 20, the resonance peak height does not change much as the temperature increases. This is also different from the deduction made by both Johari (Johari, 2002) and Nouskova (Novskova et al., 1986). Johari reported that the resonance peak height increases with the increase in temperature whereas Nouskova said that the resonance peak height decreases with the increase in temperature. In our hollandite model, the sodium ion group motions take place in a rigid lattice unlike the experimental situation for ice clathrates. Therefore the libration amplitude of an individual sodium ion dipole is unaffected by temperature and the oscillation frequency of a specific group should not be affected unlike the situation in (Johari, 2002; Novskova et al., 1986). The resonance peak magnitude will be dependent upon the number of sodium ions taking part in a coupled group oscillation at a specific frequency and this will not be affected by temperature. The effect of temperature in our model is expected to lie in the half-width of the absorption peaks, since this is due to the hopping of sodium ions between oscillating groups.
