Improvements of birefringence imaging techniques to observe stress-induced ferroelectricity in SrTiO3 based on K-means clustering with circular statistics

ABSTRACT Optical birefringence imaging techniques have enabled quantitative evaluation of macroscopic structures, e.g. domains and grain boundaries. With inhomogeneous samples, the selection of regions for analysis can significantly affect the conclusions; thus, arbitrary selection can lead to inaccurate findings. Thus, in this study, we present a method to cluster all birefringence imaging data using K-means multivariate clustering on a pixel-by-pixel basis to eliminate arbitrariness in the region selection process. Linear statistics cannot be applied to the polarization states of light described by angles and their periodicity; thus, circular statistics are used for clustering. By applying this approach to a 42,280-pixel image comprising 12 explanatory variables of stress-induced ferroelectricity in SrTiO3, we were able to select a region of locally developed spontaneous polarization. This region covers only 1.9% of the total area, where the stress and/or strain is concentrated, thereby resulting in a higher ferroelectric phase transition temperature and larger spontaneous polarization than in the other regions. The K-means multivariate clustering with circular statistics is shown to be a powerful tool to eliminate arbitrariness. The proposed method is a significant analysis technique that can be applied to images using the polarization of light, azimuthal angle of crystals, scattering angle. Graphical abstract IMPACT STATEMENT Multimodal birefringence imaging data represented by angles and their periodicities were clustered by K-means method using circular statistics. We successfully selected large spontaneous polarization area without arbitrariness in analysis process.


Introduction
Enormous scientific progress has been made in terms of controlling the arrangement of atoms and molecules as many researchers continue to explore various physical phenomena at the nanoscale [1].However, the formation of macroscopic structures, e.g.domains and grain boundaries can significantly affect the performance of devices based on these phenomena.Scanning probe microscopy has been developed to understand and control macroscopic structures [2,3].Note that certain methods can realize nanoscale spatial resolution [4,5]; however, the processing time of such methods is an outstanding issue that should be addressed.The polarized light microscopy method is popular due to its simplicity despite demonstrating degradation in the spatial resolution.In this method, the intensity of transmitted light varies nonlinearly depending on the relative relationship between the linearly polarized incident light and the optical principal axis derived from a crystal axis, i.e.Malus' law [6,7].Thus, in inhomogeneous samples, evaluating the distribution of polarization states quantitatively is difficult due to the inevitable misalignment of the crystal axis.To address this problem, we have been developed birefringence imaging techniques that facilitate quantitative measurements of the distribution of polarization states on a pixel-by-pixel basis.
Accurate evaluation of inhomogeneous samples is essential to understand and control macroscopic structures comprehensively.However, the selection of regions for analysis has a significant impact on the acquired results; thus, arbitrary selection can lead to inaccurate findings.In many cases, the spatial distribution of the macroscopic structures is evaluated qualitatively from a limited number of images, and researchers may deliberately select certain regions to perform quantitative evaluations based on the average physical quantities in each region.However, this technique may not eliminate arbitrariness in the analysis as the population distribution becomes multimodal.Therefore, we propose a method to cluster all of the corresponding birefringence imaging data using the K-means clustering method, which effectively eliminates arbitrariness in the region selection process [8].Although the K-means clustering method is well established, its application to the analysis of the polarization state of light causes a serious problem in the treatments of angles and their periodicity.For example, when calculating the relationship between 10 ° and 170 °, the Euclidean distance should be far in the case of a 360 ° period, but close in the case of a 180 ° period (see Supplementary material).To address this problem, circular rather than linear statistics should be utilized [9]; however, to the best of our knowledge, not many studies have reported on machine learning applications based on circular statistics.Thus, the goal of this study is to demonstrate that the K-means clustering method can be applied to the analysis of the polarization state based on circular statistics.The proposed method facilitates comprehensive understanding of the characteristics of the physical quantities for each cluster, thereby providing more accurate and reliable ways to evaluate the macroscopic structures in inhomogeneous samples.We believe that our findings will have a positive impact, potentially changing the approach to analyzing and understanding macroscopic structures observed by not only polarization of light, but also azimuthal angle of crystals, scattering angle.

Birefringence images
In the proposed method, the K-means clustering is applied to the stress-induced ferroelectricity in a SrTiO 3 ð110Þ substrate.Under stress-free conditions, at 105 K SrTiO 3 undergoes a structural phase transition from the cubic paraelectric state to the tetragonal quantum paraelectric state, where the ferroelectric state is suppressed by the quantum fluctuation.When an external force is applied along ½001� c , whose direction corresponds to the cubic phase, the stress-induced ferroelectric-phase transition occurs below 30 K due to suppression of the quantum fluctuation, and the spontaneous polarization appears along ½010� c [10][11][12].A previous study reported that the crystal symmetry at the lowest temperature under an external force changes from 4=mmm in the tetragonal quantum paraelectric state through mmm in the intermediate state to mm2 in the stress-induced ferroelectric state [11].Recently, our research group reported that the birefringence anomalies indicative of the ferroelectric-phase transition were found at ~20 K under an external force of 231 MPa along ½001� c [13,14].However, due to the inhomogeneous spreading of internal force (stress) and/or strain caused by the applied external force, the birefringence also spread in an inhomogeneous manner.
The refractive indices n 1 (for the slow axis) and n 2 (for the fast axis) with n 1 � n 2 > 1 are defined as the directions of the optical principal axes of the refractive ellipsoid.Generally, birefringence is defined as Δn;jn 1 À n 2 j, and retardance is calculated as Δn � t, where t is the sample thickness.The temperature dependence of retardance allows us to evaluate the development of spontaneous polarization quantitatively due to the Pockels effect and the distributions of the stress and/or strain due to the photoelastic effect [15][16][17].In addition, the temperature dependence of the fast-axis direction allows for highly sensitive detection of changes in the crystal symmetry [18,19].Domain structures can be identified from the distributions of the fast-axis direction [19][20][21].As reported in the literature [13], Figure 1 shows images of the retardance (t ¼ 0:333 mm), as well as the fast-axis direction at 30.0 K (the paraelectric state) and 14.1 K (the ferroelectric state).Note that these images are converted at the incident light wavelength λ ¼ 543 nm.Stripe patterns due to slip planes appear even at 300.0 K, and their directions coincide with the directions of the projection of h111i c onto ð110Þ.Thus, the birefringence images at 14.1 K could not distinguish between the ferroelectric domains and the dislocations.Focusing on the three areas (A1, A2, and A3) shown in Figures 1(a) and 2 shows the temperature dependence of the retardance and the fast-axis direction.Here, the inflection point of each retardance curve, which was obtained as the intersection of the two extrapolated lines, was found to correspond to the ferroelectric-phase transition temperature (T F ) [13].In Ref. 13, when the sample region of the images was divided into 90 meshes, the temperature dependence was calculated for each mesh and the values of T F were found to be distributed from 17.0 to 24.9 K due to the stress and/or strain distribution.These results indicate that the ferroelectric states are distributed inhomogeneously.In contrast, the fast-axis direction in Figures 2(d-f) shows nearly 90 ° and rotates slightly below T F .However, the relationship between the fast-axis direction and the ferroelectric domain remains unclear because the stripe patterns caused by the dislocations cover the substrate entirely.Therefore, in this study, we developed a method to observe the ferroelectric domain by preprocessing the birefringence images  into the changes of the polarization state before and after the ferroelectric-phase transition.

Image preprocessing
Without understanding the principles of the birefringence imaging techniques, the raw data cannot be preprocessed into a format that is suitable for Kmeans clustering.In this study, the sample was irradiated with right-handed circularly polarized light using a white light emitting diode light source (see Supplementary material).Thus, with our equipment, the polarization state of the transmitted light can be obtained quantitatively on a pixel-by-pixel basis (a total of 384�288 pixels) without being affected by crystal axis misalignment.The spatial resolution in optical microscopy, i.e. the limit of discrimination when the light emitted from two objects overlaps, is typically approximately λ=2.In addition, in polarized light microscopy, the overlap of the light causes a problem, i.e. superposition of the polarization states.According to Ref. 22, the spatial resolution in polarization measurements is typically degraded to λ.Under the experimental conditions implemented in this study, the effect of the superposition is expected to be small because the distance of the adjacent pixels is considerably greater than λ [13,14].However, the effect of the shot noise may not be negligible when the analysis is performed for each pixel.Thus, we averaged over a 3�3-pixel area using the common moving average method.By limiting the analysis area to only the sample, the K-means clustering method of the SrTiO 3 ð110Þ substrate was performed on a total of 42,280 pixels (302�140 pixels) in the images.
To clarify the large retardance exceeding λ=2, the transmitted light was separated into three different λ values (523 nm, 543 nm, and 575 nm) using bandpass filters.As shown schematically in Figure 3(a), the retardance corresponds to the length of the arc on the Poincaré sphere (see Supplementary material).One rotation around the Poincaré sphere results in the retardance of 360 � that is equal to λ.Note that the degree of freedom in the rotation direction and the number of the rotations cannot be determined when the retardance exceeds λ=2.
To solve this problem, the relative rotation angles using different λ values are considered because a shorter λ leads to a larger rotation angle.This procedure has been established in the literature [23]; however, calculating the relative rotation angles on a pixel-by-pixel basis is difficult in terms of numerical cost because the λ-dependence of the retardance cannot be ignored in practice.To reduce the numerical cost, the rotation angle is defined as θ, and the shortest path is always selected, as shown in Figure 3(a).Although the periodicity of θ is 360 � , the condition θ½0 � ; 180 � � is satisfied and the retardance is expressed as jl � θ � 360 � j � λ nm, (l ¼ 0; 1; 2; � � �).A characteristic distribution is expected to appear at such a foldback at 180 � and/or 0 � if the retardance exceeds λ=2.In this study, we attempted to establish the K-means multivariate clustering methods using the θ values at the three different λ values without identifying the coefficients 'l' and '�' of the retardance.
Figure 3(b) shows the Stokes parameters (S 1 ; S 2 ; S 3 ) on the Poincaré sphere (see Supplementary material).In our experiments, the horizontal and vertical directions of the birefringence images correspond to S 1 ¼ 1 and À 1, respectively.The circle of radius 1 on the Poincaré sphere is formed by the polarization states of the incident light (S 3 ' À 1) and the transmitted light.The circle passing through these polarization states corresponds to the circle in Figure 3(a) normalized to radius 1.Here, the rotation axis of this circle passes through the origin of the coordinate system in the S 1 S 2 -plane.The direction is defined as the left (or right) screw direction of the rotation from the incident light to the transmitted light.Then, the intersection of the rotation axis and the equator of the Poincaré sphere is determined.The direction of the linear polarization at the intersection corresponds to the fast-axis (or slow-axis) direction.Since the rotation direction is not identified, distinguishing between the fast-axis and slow-axis directions is difficult; thus, the axial direction on the Poincaré sphere is used instead.As shown in Figure 3(b), the azimuth of the axial direction ϕ is defined as an angle measured counterclockwise from the þS 1 -axis to the intersection.The angle ϕ has a periodicity of 180 � such that only the intersection in the first and second quadrants of the S 1 S 2 -plane, i.e. ϕ½0 � ; 180 � Þ, should be considered.The optical principal axis directions, i.e. the fast-axis and slow-axis directions, of the refractive ellipsoid are represented by ðϕ � 90 � Þ=2.

Circular statistics
To discuss the polarization state on the Poincaré sphere, it is necessary to apply the angle θ with the 360 � period and ϕ with the 180 � period to the circular statistics as similar to the analysis of arguments with Euler diagrams (see Supplementary material) [9].The circular normal distribution, also referred to as the von Mises distribution, is expressed as follows: where μ and κ are the mean direction and the concentration parameter, respectively.I 0 ðκÞ is the modified Bessel function of the first kind of order zero.
Comparing Equation (1) to the normal distribution of the linear statistics, i.e. the Gaussian distribution, μ and 1=κ resemble the mean and the variance (but are not equivalent), respectively [9].Thus, the maximum likelihood estimatation of μ is the mean value of x expressed as x.By transforming the mean values of angle θ and ϕ, i.e. θ and ϕ, into those utilized in the linear statistics, we obtain the following: ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffiffi where n is the number of samples, and R is the mean resultant length.The value of R varies between 0 and 1, where values closer to 1 indicate that the distribution is well aligned in one direction.According to the literature [9], when the R value is large, the approximate formula for the 95% interval estimation of the angle data is expressed as follows: x � arcsin 1:96 ffi ffi ffi ffi ffi ffi ffi ffi nRκ The value of κ can be estimated numerically as follows: where I 1 ðκÞ is the modified Bessel function of the first kind of order one, and the maximum likelihood estimatation of κ is κ ¼ I 0 ðRÞ=I 1 ðRÞ [9,24].To obtain the θ and ϕ values, the variables cos θ i =R θ , sin θ i =R θ , cos 2ϕ i =R ϕ , and sin 2ϕ i =R ϕ are calculated from θ i and ϕ i obtained in the i-th measurement.These values are then averaged over the total number n.Using the four variables, incorporating the circular statistics, and treating the results as the linear statistics is feasible.
Here, the experimental results are obtained using the three different λ values; thus, 12 explanatory variables are considered in the K-means multivariate clustering method.

Data analysis procedure
In terms of clustering studies in the machine learning field, certain pioneering studies have applied reflection high-energy electron diffraction images in clustering to observe different crystal growths separately [25][26][27][28].
Based on the results of these studies, we adopt the Kmeans multivariate clustering method on the Euclidean distance with 42,280�12 matrix data because this method has a numerical cost advantage over hierarchical clustering methods [8].The computation was performed using the R software (v.4.2.1), and the computer was equipped with an Intel(R) Core(TM) i5-12400 CPU at 2.5 GHz with 64 GB memory.The data were standardized in the preprocessing step to each explanatory variable for the 42,280 data prior to performing clustering.Hyperparameter k (i.e. the number of clusters) was evaluated using the NbClust package, which searches for the optimal k value using 30 different indices.Then, the optimal k value was determined by majority vote.Here, in the case of a tie, the lowest k value was selected.In addition, the groups obtained by the K-means multivariate clustering method were tested for significant differences at a 5% level which is a lower level of statistical significance by applying the multivariate analysis of variance (MANOVA) method to the 12 explanatory variables.

Results and discussion
Figures 4 and 5 show the distributions of θ and ϕ at 30.0 K and 14.1 K, respectively.Although the θ and ϕ values differ for each λ, these distributions are very similar among the considered λ values.Similar to the images shown in Figure 1, the stripe patterns appear due to the dislocations.Figures 6(a,b) show the K-means clustering results at 30.0 K and 14.1 K, respectively, where k ¼ 2 is the optimal value in both cases.The results of the MANOVA method confirmed significant differences between clusters B1 and B2 at 30.0 K and between clusters C1 and C2 at 14.1 K.The corresponding angular histograms of θ and ϕ for each cluster are shown in Figures 7 and 8, respectively.In addition, Tables 1 and  2 show the θ, R θ , ϕ, and R ϕ values at 30.0 K and 14.1 K for each λ.Note that the clustering results in Figure 6 show that the stripe patterns are not as clear as those shown in Figures 4 and 5. From Tables 1 and 2, the θ values are smaller in cluster B1 than in B2, and in C1 than in C2.In addition, the ϕ values are nearly the same across all clusters; however, the R ϕ values become smaller in B1 than in B2 and in C1 than in C2.At each temperature, θ becomes smaller as λ becomes shorter.From the definition of θ illustrated in Figure 3(a), this tendency indicates a foldback at 180 °.Thus, when the temperature decreases from 30.0 K to 14.1 K, the reduction of θ in the total distribution agrees well with increased retardance shown in Figure 2.Even when the retardance exceeds λ=2, the effectiveness of the K-means multivariate clustering with the 12 explanatory variables is confirmed.For clusters B1 and C1, the small R ϕ value indicates that the crystal axis is highly disordered according to the stress  and/or strain concentration.In addition, the θ and R ϕ values appear to be smaller in C1 than in B1, which may be due to the ferroelectric-phase transition, even though the clustering results in Figure 6 were obtained by performing at each temperature.
To clarify the ferroelectric domains, the temperature dependence of the 12 explanatory variables for each pixel should be considered as multivariate time-series data.However, the amount of time-series data becomes exceedingly large; thus, in this study, the ferroelectric domains were deduced by subtracting from the 14.1 K data using the 30.0K data as a reference.Note that discussing the spontaneous polarization direction at 14.1 K is difficult because the fast-axis direction is misaligned even at 30.0 K due to the dislocations.To reduce numerical costs, the absolute values of the difference data are used, which facilitates easier understanding of the clustering results.
For the subtraction involving θ defined as Δθ;jθð14:1KÞ À θð30:0KÞj, Figures 9(a-c) show the distributions of Δθ, and Figures 9(d-f) show the distributions of Δϕ defined as Δϕ;jϕð14:1KÞ À ϕð30:0KÞj.These calculations are valid when the polarization states of the incident light at 30.0 K and 14.1 K are the same.In this study, from the images at 30.0 K and 14.1 K, we confirmed that the polarization states in the region excluding the sample and the apparatus are the same within the accuracy of the measurements [29].The distributions of Δθ and Δϕ are very similar among the considered λ values.Figure 10 shows the K-means multivariate clustering results where k ¼ 3 is the optimal value.From this result, since clusters D1 and D2 generally correspond to B1 at 30.0 K and to C1 at 14.1 K, cluster D3 is expected to be a region where the stress and/or strain generates uniformly and the crystal axis is nearly aligned.Figure 11 shows the angular histograms of Δθ and Δϕ for each cluster, and Table 3 shows the Δθ, R Δθ , Δϕ, and R Δϕ values for each λ.The results of the MANOVA method confirmed significant differences between clusters D1, D2 and D3.The angular histograms were found to separate the multimodal data successfully.For Δϕ, the distribution on the high angle side is roughly explained by D1, and R Δϕ in D1 is the smallest among the clusters for each λ because the crystal axis becomes highly disordered through the ferroelectric-  phase transition.In addition, a large Δθ in D1 clearly indicates that a large spontaneous polarization is induced locally by the stress and/or strain concentration.In both D2 and D3, which occupy most of the observed field, the ferroelectric state develops nearly uniformly because the Δϕ and R Δϕ values are nearly the same for both regions.However, from the difference of Δθ in D2 and D3, we conclude that D2 is a location of relatively larger spontaneous polarization.
The Δθ and Δϕ values exhibit slightly different λdependence.Note that as θ approaches 0 °, the measurement accuracy of both θ and ϕ will become worse in principle.Since θ varies with temperature, we focus on the temperature dependence in each cluster, similar to Figure 2. Figure 12 shows the temperature dependence of the retardance and the fast-axis direction obtained by averaging the Stokes parameters (S 1 ; S 2 ; S 3 ) on the Poincaré sphere in clusters D1, D2, and D3.Based on the results shown in Figure 2, the retardance is calculated as ð1 À θ=360 � Þ � λ due to the foldback at 180 °, and the fast-axis direction is selected to be close to 90 ° in ðϕ � 90Þ=2.Since the retardance in D1 at 523 nm  increases to close to 523 nm, i.e. the θ value is reduced and becomes close to 0 � with decreasing temperature, as shown in Figure 12(a), the measurement accuracy for 523 nm is reduced.From the angular histograms of θ at 14.1 K shown in Figure 8, the total distribution for 523 nm appears to be the foldback at 0 °, while that for 575 nm is not.Thus, we focus on the data for 575 nm.As shown in Figure 12(c), the retardance becomes larger in D1 than in both D2 and D3, thereby resulting in the larger spontaneous polarization.The T F value is obtained as the intersection of the two extrapolated lines.As a result, the value of T F in D1 is comparable to the highest temperature identified in the previous reports [10][11][12][13].
Although the area of D1 is only 1.9% of the total area, the region with the higher T F and the larger spontaneous polarization due to the stress and/or strain concentration was selected successfully on a pixel-by-pixel basis without arbitrariness.Below T F , the difference in the retardance between D2 and D3 becomes large, which suggests that the spontaneous polarization becomes larger in D2 than in D3.In the fast-axis direction, D1 differs from both D2 and D3 due to the stress and/or strain concentration.As shown in Figure 12(f), the fast-axis direction rotates slightly below T F .Even if the spontaneous polarization appears along ½100� c and/or ½010� c [11], the rotation of the fast-axis direction in ð110Þ is difficult to explain.The small rotation is likely due to the crystal axis misalignment in ½1 � 10� c .However, this is difficult to evaluate quantitatively because several layers of domains overlap along the direction of light propagation and we observed them on average, as reported in the literature [13].From the above discussion, we have demonstrated that the distribution of the ferroelectric states in the substrate can be identified more accurately by evaluating the temperature dependence of the retardance and the fast-axis direction based on the clustering results.We conclude   that the K-means multivariate clustering results of the birefringence images successfully reveal the distribution of the spontaneous polarization in the SrTiO 3 ð110Þ substrate depending on the stress and/or strain distribution.

Conclusion
In this study, we have successfully identified the distribution of the stress-induced ferroelectric domains in the SrTiO 3 ð110Þ substrate by utilizing the K-means multivariate clustering method applied to the birefringence images in combination with the circular statistics.At 30.0 K and 14.1 K, even when the retardance exceeds λ=2, we have shown that 42,280 pixels can be clustered using the 12 explanatory variables.The analysis of the circular histograms of θ and ϕ confirmed that the clustering was successful in terms of the stress and/ or strain concentrations.To evaluate the crystal axis disorder, the R ϕ value is useful.By subtracting the data obtained at 30.0 K from that obtained at 14.1 K, the locations where the spontaneous polarization becomes locally large and the crystal axis highly disorders due to the stress and/or strain concentration could be selected successfully on a pixel-by-pixel basis without the arbitrariness.In this region, the T F value could be better estimated from the temperature dependence of the retardance.This clustering method is a significant analysis technique that can be applied to images using not only the polarization of light, but also the azimuthal angle of crystals, scattering angle.

Figure 1 .
Figure 1.Results of birefringence imaging measurements with an external force of 231 MPa applied along ½001� c on a 0.333 mm thick SrTiO 3 (110) substrate.(a) retardance and (b) fast-axis direction images at 30.0 K when we converted at the incident light wavelength λ ¼ 543 nm.(c) retardance and (a) fast-axis direction images at 14.1 K.The large open arrows schematically indicate the direction of the external force.In (a), the 15�30-pixel and 30�30-pixel rectangles indicate the sample-analysis areas A1, A2, and A3.

Figure 2 .
Figure 2. Temperature dependence of the retardance for (a) λ ¼ 523 nm, (b) 543 nm, and (c) 575 nm, and the fast-axis direction for (d) 523 nm, (e) 543 nm, and (f) 575 nm.The analysis areas A1, A2, and A3 are shown in Figure 1(a).In (c), ferroelectric phase transition temperature T F is obtained as the intersection of two extrapolated dashed lines.In (f), the arrows indicate T F obtained in (c).

Figure 3 .
Figure 3. (a) schematic of a cross-section of the Poincaré sphere.The polarization conditions of the incident and transmitted light are located at the labels In and Out, respectively.(b) schematic of the relationship between the rotation axis and fast-axis direction on the Poincaré sphere determined by the left screw direction.The crosses indicate the intersections of the rotation axis and the equator of the Poincaré sphere, and the direction of the linear polarization on the intersection corresponds to the fast-axis direction.

Figure 10 .
Figure 10.Clustering result obtained from the Δθ and Δϕ images.

Figure 12 .
Figure 12.Temperature dependence of the retardance for (a) λ ¼ 523 nm, (b) 543 nm, and (c) 575 nm and the fast-axis direction for (d) 523 nm, (e) 543 nm, and (f) 575 nm.In (a), the dashed line is level at which the retardance corresponds to 523 nm, i.e. θ¼ 0 � .In (c), the ferroelectric phase transition temperature T F is obtained as the intersection of two extrapolated dashed lines.In (d) and (e), the fast-axis directions in D2 and D3 nearly overlap for each λ value.In (f), the arrows indicate T F obtained in (c).

Table 1 .
Mean directions θ and ϕ and mean resultant lengths R θ and R ϕ for each λ at 30.0 K.

Table 2 .
Mean directions θ and ϕ and mean resultant lengths R θ and R ϕ for each λ at 14.1 K.

Table 3 .
Mean directions Δθ and Δϕ and mean resultant lengths R Δθ and R Δϕ for each λ.