**3.4 Fast Fourier Transform (FFT) – Susceptibility**

The Fast Fourier Transform (FFT) is performed in order to calculate the behaviour of the hollandite model in the frequency domain. The continuous average of the polarisation is imported to "Origin" program. A graph of polarisation versus time is plotted. Only the data between the 5001th and 100000th gives the polarisation since the electric field is switched on at the 5000th interval. The time derivative of the polarisation is then obtained via the program, and is plotted. The one-sided Fourier Transform of / / <sup>0</sup> *dP dt Ee* gives the frequency dependent dielectric susceptibility for comparison with experiment. The easier way to carry out the FFT in "Origin" software is by performing FFT on the *dP dt* / , the results obtained are then divided by electric field, E and permittivity of free space, 0 to give the real and imaginary parts of the susceptibility. The FFT mathematical description is shown as follows:

$$X[k] = \sum\_{n=0}^{N-1} x[n] \exp(-i2\pi F\_k n) \tag{5}$$

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function of frequency (f) as shown in figure 9.

electric field of 743MV/m.

carried out.

**3.5 Depolarisation** 

Mechanism in Fast Ionic Crystals Based on Hollandite NaxCrxTi8-xO16 381

permittivity of free space. The ' (f) and "(f), obtained from the FFT, are plotted as a

Fig. 9. The real and imaginary parts of the susceptibility as a function of frequency with an

The simulation was run for 100ps (100000 x 10-15s), which correspond to 1010 Hz. The susceptibility at frequencies below 4x1010Hz is unreliable as it relates to FFT extrapolations to regions, which are not consistent with the largest time (10-10s) reached by the computation. At the higher frequency range, the smoothing process has removed the frequency higher than 2x1011 Hz; hence, frequency higher than 2x1011 Hz is similarly unreliable. Figure 9 shows '(f) and "(f) as a function of frequency. '(f) goes to a positive value, then drops to a negative value and then starts to fluctuate around a smaller negative value. "(f) shows a peak at about 1x1011 Hz and the frequency of the peak lies about half way along the slope of the '(f). From the overall non-monotonic behaviour of '(f), it is clear that the response is not that of a dielectric relaxation; hence the curve fitting to the Lorenzian function which is the resonance response would have to be

An electric field was applied to the hollandite model at the start of the simulation for 5000 intervals. Then the field was switched off at 5001th interval. The depolarisation, continuous average for the depolarisation, FFT and curve fitting were obtained similarly to the procedures outlined above. Depolarisation was carried out with initial conditions of temperature=297K, time step=10-15s, 100000 intervals and electric field=7.43GV/m. The results of the curve fitting to equations 6 and 7 for '(f) and "(f) (obtained from the depolarisation) respectively as a

function of frequency are shown in the figure 10 and 11 below.

**The real and imaginary part of the susceptibility vs Frequency (first 50 intervals) field=743MV/m, 273K, 100000 intervals, time step=10-15s**

with *F kN <sup>k</sup>* / , where FFT transforms x[n] into X[k]. The input data set is x[n] with index n in the range 0 1 *n N* . It is easy to convert the index into "time" *t n* , where is the (time) interval, and *F kN <sup>k</sup>* / into "frequency" *f kN <sup>k</sup>* / .

Curve fitting is important, as it will give the best curve fit to the graph obtained. Firstly, the Real and Imaginary part of the susceptibility (the results generated by the FFT from the previous section) are plotted against frequency in two different graphs. This is to ease the fitting procedure as the two graphs are of different shape hence two different fitting functions are defined. If the results exhibit relaxation behaviour, then we would expect the loss peak to be characterised by power law frequency dependencies above and below the peak frequency.

If the results show a resonance character then we have to fit them to an appropriate expression. Two types of expressions are commonly used.


The real part of the susceptibility given in term of x (frequency) and w (damping factor) is shown in equation 6:

$$y = \frac{4A}{\pi} \frac{\mathbf{x}\_c - \mathbf{x}}{4(\mathbf{x}\_c - \mathbf{x})^2 + w^2} \tag{6}$$

The imaginary part of the susceptibility given in term of x and w is shown in equation 7:

$$y = \frac{2A}{\pi} \frac{w}{4(\chi\_c - \chi)^2 + w^2} \tag{7}$$

*xc, w* and *A* are the parameters used for the resonant frequency, full width at ½ maximum and amplitude factor respectively.

The real part of the susceptibility, '(f) and the imaginary part of the susceptibility, "(f) were then obtained by dividing the results generated from the FFT by the applied field and

results obtained are then divided by electric field, E and permittivity of free space, 0 to give the real and imaginary parts of the susceptibility. The FFT mathematical description is

[ ] [ ]exp( 2 )

with *F kN <sup>k</sup>* / , where FFT transforms x[n] into X[k]. The input data set is x[n] with index

Curve fitting is important, as it will give the best curve fit to the graph obtained. Firstly, the Real and Imaginary part of the susceptibility (the results generated by the FFT from the previous section) are plotted against frequency in two different graphs. This is to ease the fitting procedure as the two graphs are of different shape hence two different fitting functions are defined. If the results exhibit relaxation behaviour, then we would expect the loss peak to be characterised by power law frequency dependencies above and below the peak frequency. If the results show a resonance character then we have to fit them to an appropriate

1. A Gaussian absorption function. This relates to a superposition of oscillator resonances with each oscillator vibrating independently of one another at a specific frequency, with the probability of a given frequency being defined by the Gaussian function. In our case the vibrations are those of the sodium ions, and these are not independent of one

2. A Lorentzian function. This relates to an oscillator whose oscillations are damped by interaction with its surroundings. In our case we can think of a chosen sodium ion oscillator as being damped by its interaction with the other sodium ions. So the function may approximate to our situation. However, the other sodium ions also contribute to the response via group motions, so the Lorentzian is at the best an approximation to our hollandite model. Nonetheless the Lorentzian will be fitted to the data to see how good

The real part of the susceptibility given in term of x (frequency) and w (damping factor) is

4( ) *c c A x x*

4( ) *<sup>c</sup> A w*

*xx w*

*xc, w* and *A* are the parameters used for the resonant frequency, full width at ½ maximum

The real part of the susceptibility, '(f) and the imaginary part of the susceptibility, "(f) were then obtained by dividing the results generated from the FFT by the applied field and

The imaginary part of the susceptibility given in term of x and w is shown in equation 7:

 *xx w* 

2 2

2 2

*Xk xn i Fn*

*k*

.

(5)

, where

(6)

(7)

is the

1

*N*

*n*

n in the range 0 1 *n N* . It is easy to convert the index into "time" *t n*

(time) interval, and *F kN <sup>k</sup>* / into "frequency" *f kN <sup>k</sup>* /

expression. Two types of expressions are commonly used.

another, so the Gaussian form should not apply.

an approximation it is, and to determine in what respect it fails.

*y*

*y*

4

2

0

shown as follows:

shown in equation 6:

and amplitude factor respectively.

permittivity of free space. The ' (f) and "(f), obtained from the FFT, are plotted as a function of frequency (f) as shown in figure 9.

Fig. 9. The real and imaginary parts of the susceptibility as a function of frequency with an electric field of 743MV/m.

The simulation was run for 100ps (100000 x 10-15s), which correspond to 1010 Hz. The susceptibility at frequencies below 4x1010Hz is unreliable as it relates to FFT extrapolations to regions, which are not consistent with the largest time (10-10s) reached by the computation. At the higher frequency range, the smoothing process has removed the frequency higher than 2x1011 Hz; hence, frequency higher than 2x1011 Hz is similarly unreliable. Figure 9 shows '(f) and "(f) as a function of frequency. '(f) goes to a positive value, then drops to a negative value and then starts to fluctuate around a smaller negative value. "(f) shows a peak at about 1x1011 Hz and the frequency of the peak lies about half way along the slope of the '(f). From the overall non-monotonic behaviour of '(f), it is clear that the response is not that of a dielectric relaxation; hence the curve fitting to the Lorenzian function which is the resonance response would have to be carried out.

#### **3.5 Depolarisation**

An electric field was applied to the hollandite model at the start of the simulation for 5000 intervals. Then the field was switched off at 5001th interval. The depolarisation, continuous average for the depolarisation, FFT and curve fitting were obtained similarly to the procedures outlined above. Depolarisation was carried out with initial conditions of temperature=297K, time step=10-15s, 100000 intervals and electric field=7.43GV/m. The results of the curve fitting to equations 6 and 7 for '(f) and "(f) (obtained from the depolarisation) respectively as a function of frequency are shown in the figure 10 and 11 below.

Molecular Dynamics Simulation and Conductivity

frequency.

Mechanism in Fast Ionic Crystals Based on Hollandite NaxCrxTi8-xO16 383

Fig. 11. The "(f) plot with the best curve fitting (red) to equation 7 as a function of

Fig. 12. The results of the curve fitting to equation 6 and 7 for both the '(f) and the "(f) of polarisation and the depolarisation respectively with an electric field of 7.43GV/m.

Fig. 10. The '(f) plot with the best fitting curve (red) to equation 6 as a function of frequency.

From figure 10, it is clearly shown that the equation 6 does not fit well to '(f) at all. For a resonance response, the magnitude for the lower frequency range should not be lower than the magnitude in the higher frequency range. Figure 11 shows that the equation 7 does not fit well with the simulation values of "(f) either. The fitted curve fits nicely in the higher frequency range but in the lower frequency range, the gradient of "(f) is much bigger than the fitting curve. It seems that the hollandite model had not reached an equilibrium state when the electric field was taken off. From the figure 12, the '(f) and the "(f) for the polarisation and the depolarisation were different although they should be identical for a linear response. Both the magnitude of the '(f) and "(f) for the depolarisation are bigger than the '(f) and "(f) of the polarisation. The resonance frequency of "(f) for the polarisation is lower compared to the resonance frequency of "(f) for the depolarisation.

#### **3.6 Results obtained at the temperatures of 200K to 373K and with electric fields between 7.43MV/m and 74.3GV/m**

The simulation has also been carried out with initial conditions of temperature between 200K and 373K, which were 200K, 250K, 273K, 297K and 373K. At each temperature, six different electric fields were investigated between 7.43MV/m and 74.3GV/m, which were 7.43MV/m, 74.3MV/m, 371.5MV/m, 743MV/m, 7.43GV/m and 74.3GV/m. On each of the results the procedures described in section 3.4 were carried out for the data analysis. The

Fig. 10. The '(f) plot with the best fitting curve (red) to equation 6 as a function of

From figure 10, it is clearly shown that the equation 6 does not fit well to '(f) at all. For a resonance response, the magnitude for the lower frequency range should not be lower than the magnitude in the higher frequency range. Figure 11 shows that the equation 7 does not fit well with the simulation values of "(f) either. The fitted curve fits nicely in the higher frequency range but in the lower frequency range, the gradient of "(f) is much bigger than the fitting curve. It seems that the hollandite model had not reached an equilibrium state when the electric field was taken off. From the figure 12, the '(f) and the "(f) for the polarisation and the depolarisation were different although they should be identical for a linear response. Both the magnitude of the '(f) and "(f) for the depolarisation are bigger than the '(f) and "(f) of the polarisation. The resonance frequency of "(f) for the polarisation is lower compared to the resonance frequency of "(f) for the depolarisation.

**3.6 Results obtained at the temperatures of 200K to 373K and with electric fields** 

The simulation has also been carried out with initial conditions of temperature between 200K and 373K, which were 200K, 250K, 273K, 297K and 373K. At each temperature, six different electric fields were investigated between 7.43MV/m and 74.3GV/m, which were 7.43MV/m, 74.3MV/m, 371.5MV/m, 743MV/m, 7.43GV/m and 74.3GV/m. On each of the results the procedures described in section 3.4 were carried out for the data analysis. The

frequency.

**between 7.43MV/m and 74.3GV/m** 

Fig. 11. The "(f) plot with the best curve fitting (red) to equation 7 as a function of frequency.

Fig. 12. The results of the curve fitting to equation 6 and 7 for both the '(f) and the "(f) of polarisation and the depolarisation respectively with an electric field of 7.43GV/m.

Molecular Dynamics Simulation and Conductivity

Mechanism in Fast Ionic Crystals Based on Hollandite NaxCrxTi8-xO16 385

Table 1. The values for the parameters used (*xc, w* and *A*) for the curve fitting for the

imaginary part of the susceptibility.

real part of the susceptibility does not fit well with equation 6. Hence, only the values for the parameters used (*xc, w, A0*) for the imaginary parts of the susceptibility are shown in table 1.

For all five temperatures (200K to 373K), the imaginary parts of the susceptibility (fitted to equation 7) for the electric field in the range of 7.43MV/m to 7.43GV/m show an absorption peak. The magnitude of the absorption peak differs with temperature and electric field. At an electric field of 74.3GV/m the polarisation was found to oscillate between two values. The time period can be determined from the polarisation plot. An example of the polarisation plot at 200 K is given in figure 13 which is just a part of the polarisation plot (5500th-5530th intervals) with initial conditions of 200K, 74.3GV/m and time step=10-15s. The polarisation plot of 100000 intervals is the replication of the plot shown above. There are two and a half oscillations highlighted between the two red lines in the figure and the period of the oscillation is 4x10-15s. The time step for the simulation is 10-15s, and the period of the oscillation is four times the time step. Therefore, at high electric field, all the sodium ions were driven by this force field to move as a group and they show a single frequency vibration caused by reflections from the tunnel boundaries according to the input boundary conditions.

Fig. 13. The polarisation in c-axis (5500th-5530th intervals) with an applied field of 74.3GV/m and at temperature=200K.

real part of the susceptibility does not fit well with equation 6. Hence, only the values for the parameters used (*xc, w, A0*) for the imaginary parts of the susceptibility are shown in table 1. For all five temperatures (200K to 373K), the imaginary parts of the susceptibility (fitted to equation 7) for the electric field in the range of 7.43MV/m to 7.43GV/m show an absorption peak. The magnitude of the absorption peak differs with temperature and electric field. At an electric field of 74.3GV/m the polarisation was found to oscillate between two values. The time period can be determined from the polarisation plot. An example of the polarisation plot at 200 K is given in figure 13 which is just a part of the polarisation plot (5500th-5530th intervals) with initial conditions of 200K, 74.3GV/m and time step=10-15s. The polarisation plot of 100000 intervals is the replication of the plot shown above. There are two and a half oscillations highlighted between the two red lines in the figure and the period of the oscillation is 4x10-15s. The time step for the simulation is 10-15s, and the period of the oscillation is four times the time step. Therefore, at high electric field, all the sodium ions were driven by this force field to move as a group and they show a single frequency vibration caused by reflections from the tunnel boundaries according to the input boundary

Fig. 13. The polarisation in c-axis (5500th-5530th intervals) with an applied field of 74.3GV/m

conditions.

and at temperature=200K.


Table 1. The values for the parameters used (*xc, w* and *A*) for the curve fitting for the imaginary part of the susceptibility.

Molecular Dynamics Simulation and Conductivity

model with field of 743MV/m.

Mechanism in Fast Ionic Crystals Based on Hollandite NaxCrxTi8-xO16 387

Fig. 15. Trajectories of movement in three-dimensions of the 6th sodium ion in the hollandite

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.
