Determination of the scale and the incidence angle

In order to be able to simulate the layer lines several parameters have to be known.

On the one hand the chiral indices (n, m) for the candidates which are given by all means. On the other hand the scale, that connects the units in the simulation to sizes in pixels, and the angle γ between the tube axis and the incoming electron beam. These two parameters cannot be determined, respectively not with sufficient precision, by the experimenter in the TEM and therefore have to be obtained in the diffraction image.

As the distances of the layer lines to the equatorial line are already known, it is favorable to use them to determine the scale. Following the underlying theory the distanceD in the reciprocal space of the first layer line and perpendicular incident is given by (cf. equation 2.82):

D₁ =a^∗cosα (4.25)

where a^∗ denotes the length of the reciprocal basis vector and α the helical angle.

The distance for the second layer line is (cf. equation 2.83)

D₂ =a^∗sin(30^◦+α) (4.26)

and for the third (cf. equation 2.84)

D₃ =a^∗cos(30^◦−α) (4.27) The distances increase for non-perpendicular incidence (cf. section 2.4):

D_i^γ =D⁹⁰_i ^◦ 1

sinγ (4.28)

where γ denotes the angle between the incoming electron beam and the tube axis and D_i^90◦ the distance of the ith layer line to the equatorial line for perpendicular incidence (i.e. γ = 90^◦).

The scalesof the image is therefore related to the incidence angleγ by the following formula:

s= D⁹⁰_i ^◦

sin(γ)d_i (4.29)

where d_i is the distance of the ith layer line to the equatorial line in the image in pixels, which was measured in the previous section.

However the angle is not known yet. But a second quantity is apparent in the images, which can be measured. The distance of the first maximum of a layer line to the tube axis can be obtained easily. Figure 4.20 pictures the intensity distribution in relation to the distance to the tube axis for the examined example, whereas it is averaged over the four quadrants, for all three layer lines.

Contrary to the measurement of the distances of the layer lines in this case a smooth-ing with a Gaussian filter is applicable to reduce noise. It can be done because the risk of maxima disappearing is only small, because the interval between the max-ima on the layer lines is generally quite large. The filtering was accomplished by a Gaussian filter withσ = 7.

The distances of the said maxima to the tube axis moreover are dependent on the angle between the incident electron beam and the tube axis (cf. section 2.4). Figure 4.21 shows a simulated layer line for various angles.

For every given candidate one has two wanted parameters s and γ and for every layer line two measured parameters d_i and k_i, while k_i denotes the distance of the first maximum to the tube axis of theith layer line, whereas the first maximum also constitutes the point of the highest intensity of the distribution.

For every layer line i the values for s_i and γ_i are numerically sought after in the following with the constraint, that the distances of the layer lines and the positions of the first maxima in the simulation have to concur with the measurement.

To be able to estimate the error of k_i one proceeds analogously to the error of d_i.

4.7. COMPARISON OF THE CANDIDATES TO THE SIMULATION 77

Distance to tube axis [pixels]

layer line 1 layer line 2 layer line 3

Figure 4.20: The intensities related to the distance to the tube axis on the layer lines measured in the preprocessed image averaged over the four quadrants. The curves were smoothened with a Gaussian filter withσ = 7.

0

Distance to tube axis [a.u.]

90°

85°

80°

75°

70°

Figure 4.21: Simulated intensities on the first layer line of a (15,8) nanotube for various angles γ between the incoming electron beam and the tube axis.

The intensity distributions are measured separately in the four quadrants and the distances are determined in every quadrant. The error ∆k_i equals then the largest discrepancy of one quadrant to the obtained position with the help of the averaged distribution. Figure 4.22 pictures the intensity curve of the second layer line in all four quadrants and the averaged graph.

0 5000 10000 15000 20000 25000 30000 35000

0 50 100 150 200 250 300 350 400 450 500

Intensity

Distance to tube axis [pixels]

averaged 1. quadrant 2. quadrant 3. quadrant 4. quadrant

Figure 4.22: Intensity related to the distance to the tube axis on the second layer line measured in the four quadrants and the averaged curve. The curves were filtered with a Gaussian filter withσ= 7. The zero values in the second quadrant are caused by the beamstopper.

A parabola is fitted into the neighborhood of the highest point, to determine the maxima as precisely as possible. Table 4.4 shows the measured values for the dis-tances of the diffraction spots on the layer lines to the tube axis with their errors.

k₁ ∆k₁ k₂ ∆k₂ k₃ ∆k₃ 117.0 1.1 211.5 0.9 315.9 2.1

Table 4.4: Measured distances to the tube axis and their errors for the simulated example image.

For every layer line the errors ∆s_i and ∆γ_i are estimated by calculating the angle γ and the scale s for d_i+ ∆d_i and k_i−∆k_i and d_i −∆d_i and k_i+ ∆k_i. The error is the largest divergence to the values calculated ford_i andk_i. One has to take into

4.7. COMPARISON OF THE CANDIDATES TO THE SIMULATION 79 account, that the scale s, the angle γ and also their errors are different for every candidate.

In order to evaluate the incidence angle for a candidate, the mean value is computed with the help of a weighting function. With the weights

w_i^γ = 1

where i runs through all located layer lines. The error is then given by

∆γ =

s 1 P

iw^γ_i (4.32)

Analogously the weights for the scale are defined:

w^s_i = 1

(∆s_i)² (4.33)

The scale is given by

s=

Table 4.5 shows the estimated incidence angles and the scales and their errors for the candidates.

The specification of the error of the averaged angle and scale is only given to provide the user with information. It is not used in the following analysis.