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## Creator

[Katsuaki Nakazawa](https://orcid.org/0000-0002-6056-5615), [Kazutaka Mitsuishi](https://orcid.org/0000-0002-9361-4057)

## Rights

This is a pre-copyedited, author-produced version of an article accepted for publication in Microscopy following peer review. The version of record Katsuaki Nakazawa, Kazutaka Mitsuishi, Relaxation time measurement: correlating diffraction patterns, Microscopy, 2025;, dfaf048 is available online at: https://doi.org/10.1093/jmicro/dfaf048.[In Copyright](http://rightsstatements.org/vocab/InC/1.0/)

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[Relaxation time measurement: correlating diffraction patterns](https://mdr.nims.go.jp/datasets/a55c6d26-e1cc-4e0c-8698-823657b897b0)

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1  Title 1 Relaxation Time Measurement: Correlating Diffraction Patterns 2  3 Mini-abstract 4 This study compares one-time and two-time correlation functions for analyzing temporal 5 changes in diffraction patterns. It finds that two-time correlation functions require shorter 6 observation times and show small sensitivity to radiation dose, highlighting the advantage of 7 two-time correlation functions in assessing dynamics in liquids and glasses via electron 8 diffraction. 9  10 Abstract 11 Dynamics in liquids and glasses can be assessed using X-ray photon correlation spectroscopy 12 or electron correlation microscopy, which involves measuring the temporal changes in 13 diffraction patterns. Two methods are commonly used to evaluate these temporal changes: one-14 time correlation function or two-time correlation function. However, the specific characteristics 15 of these methods have not been thoroughly studied. In this study, we investigated the 16 differences between these methods and found that the two-time correlation function can 17 measure dynamics for longer periods than the method relying on the one-time correlation 18 function. Additionally, we demonstrated that the two-time correlation function exhibits a weak 19 2  dependence on the amount of dose applied. 20  21 Introduction 22 X-ray Photon Correlation Spectroscopy (XPCS) and Electron Correlation Microscopy (ECM) 23 have been utilized to assess the dynamics of atoms or particles in non-crystalline materials such 24 as liquids and glasses 1,2. These techniques gauge dynamics through temporal diffraction 25 patterns. In these patterns, high-intensity spots, referred to as speckles, appear when the atomic 26 structures exhibit local orders like short-range or medium-range order 3. Additionally, the well-27 known diffraction pattern of glasses, known as the halo pattern, is formed by accumulating 28 speckles from numerous local structures when X-rays or electron beams irradiate a wide area. 29 This relationship is similar to how the diffraction pattern of polycrystalline materials forms 30 Debye-Scherrer rings. Since speckles reflect the local atomic structure of the material, 31 fluctuations in speckle intensity indicate changes in the local atomic structure. In XPCS and 32 ECM, dynamics are gauged from the autocorrelation of intensity fluctuation and assessed by 33 the relaxation time, which indicates the time for correlation attenuation; longer relaxation times 34 signify slower dynamics and vice versa. In other words, these methods evaluate atomic motion 35 in the amorphous atomic structure through the changes in the speckle pattern that arise from 36 variations in the atomic structures. 37 Since an electron beam can converge down to sub-nm scales, unlike XPCS, ECM can measure 38 3  local dynamics4. However, this measurement is confined to a fixed point, making it challenging 39 to measure the spatial distribution of relaxation time. Tilted Dark Field (tDF) microscopy has 40 overcome this limitation, enabling the measurement of spatial distributions of relaxation time 41 at the nanoscale5. This method uses a portion of the diffraction pattern to generate real-space 42 dark-field images and measures the relaxation time through the temporal change of intensities 43 at each pixel in the real-space image. Furthermore, the five-dimensional scanning transmission 44 electron microscopy (5D-STEM) method has recently been proposed and applied to measure 45 the spatial distribution of relaxation times 6–8. 5D-STEM captures the spatiotemporal 46 distribution of diffraction patterns, allowing the measurement of relaxation times from time-47 series diffraction patterns at each scanning point. Compared to tDF, which sacrifices the 48 acquisition of the diffraction pattern itself to obtain the spatial distribution of relaxation times, 49 5D-STEM acquires the whole spatiotemporal distribution of diffraction patterns. This 50 difference enables 5D-STEM to simultaneously analyze structure and relaxation time, making 51 it distinct in the method of measuring relaxation time. In this paper, we investigate the 52 differences in methods for measuring relaxation times. 53 Two types of relaxation time measurement methods are applicable to a 5D-STEM data set: 54 one measures relaxation time from intensity modulation, similar to XCPS, ECM, and tDF 4,9, 55 while the other obtains relaxation time from the correlation of diffraction patterns, as 56 demonstrated by Nakazawa et al. 6,8. The former is called one-time correlation function (OTCF), 57 4  and the latter is called two-time correlation function (TTCF). The differences between these 58 measuring methods have not been thoroughly investigated. Additionally, while the influence 59 of electron beam irradiation on the OTCF method has been studied 10,11, it has not been explored 60 for the TTCF method. In this study, we examine the differences between the OTCF and TTCF 61 methods by comparing the relaxation time measurement results derived from the same 5D-62 STEM data set. We also present the influence of electron dose on the correlation-based method. 63  64 Methods 65 In this experiment, we selected Zr50Cu40Al10 metallic glass as the sample (purchased from 66 DIAVAC Ltd.). The TEM sample was prepared using the Focused Ion Beam (JEOL JIB-4000). 67 The sample was heated to 653 K using the Fusion heating holder (Protochips Inc.). 5D-STEM 68 observation was performed using the JEM-ARM200F (JEOL Ltd.) and 4DCanvas (JEOL Ltd.) 69 with an acceleration voltage of 200 kV, a convergence semi-angle of 1.6 mrad, 60×180 scan 70 points, and a scan step of 0.27 nm. The diameter of the electron beam in real space was 0.78 71 nm (full width at half maximum) with this setting. The dwell time was 133 μs, and the number 72 of pixels in the diffraction pattern was 264×66 pixels (hardware binned by 4 along the 𝑘𝑦 73 direction). The total number of measured diffraction patterns was 8388608, and the observation 74 time was 1115.6 s. A beam stopper was used to prevent blooming caused by the transmitted 75 beam. Following the measurements, we binned the diffraction pattern by 4 along the 𝑘𝑥 76 5  direction to correct the aspect ratio. Consequently, the number of pixels in the diffraction 77 pattern and the pixel size became 66×66 and 0.23 nm−1, respectively. Approximately 600 78 diffraction patterns were obtained at each scanning point. We conducted this measurement 79 using three different dose amounts: 11.5 pA, 5.8 pA, and 2.8 pA. The sample thickness 80 measured by EELS was approximately 50 nm (See supplemental Fig. S1) and does not 81 significantly affect the measurement of the relaxation time (See supplemental Fig. S2). To 82 avoid abrupt changes in relaxation time by aging during the 5D-STEM measurement, we 83 started measurements 3 hours after heating12. In supplemental, we discussed the effect of spatial 84 resolution in real space and angular resolution in diffraction (supplemental Fig. S3) and showed 85 the obtained diffraction pattern under 11.5 pA electron dose (supplemental Fig. S4). 86  87 Results and Discussions 88 Difference between one-time correlation function and two-time correlation function 89 In this section, we delineate the disparity between the OTCF and TTCF methods. Both 90 methods gauge relaxation time through time-series diffraction patterns using two steps: 1) 91 measuring temporal decay due to atomic displacements and 2) fitting a decay function such as 92 the Kohlrausch-Williams-Watts (KWW) function 𝑓(𝑡) = exp (−2 (𝑡𝜏)𝛾)  to ascertain 93 relaxation time 𝜏. 𝛾 is a fitting parameter known as the stretching exponent, and 𝑡 is time. 94 The distinction between the OTCF and TTCF solely arises in step 1, with no variation in step 95 6  2. It is important to note that neither method directly gauges dynamics from atomic 96 displacements. Here, we elucidate the contrast between these approaches. The schematics of 97 both methods are depicted in Figure 1. The OTCF estimates dynamics through the temporal 98 change of intensity, while the TTCF estimates dynamics through changes in the correlation of 99 temporal diffraction patterns. Using reciprocal vertical and horizontal positions defined as 𝒌 =100 (𝑘𝑥, 𝑘𝑦) and time as 𝑡, the intensity of a diffraction pattern at 𝒌 and 𝑡 is denoted as 𝐼(𝒌, 𝑡). 101 Time-series diffraction patterns are recorded with a fixed interval time Δ𝑡 . The OTCF 102 calculates the autocorrelation of intensity fluctuations 𝑔𝑜𝑛𝑒(𝒌, 𝑡) at a specific position 𝒌 in 103 the diffraction pattern5 by the following equation.  104 [Figure 1 here] 105 𝑔𝑜𝑛𝑒(𝒌, 𝑗Δ𝑡) =(𝑁 − 𝑗) ∑ 𝐼(𝒌, 𝑡𝑖)𝐼(𝒌, 𝑡𝑖+𝑗)𝑁−𝑗𝑖=1[∑ 𝐼(𝒌, 𝑡𝑖)𝑁−𝑗𝑖=1][∑ 𝐼(𝒌, 𝑡𝑖+𝑗)𝑁−𝑗𝑖=1]. (1) 106 𝑁  denotes the total number of diffraction patterns in time series, 𝑖  and 𝑗  are the natural 107 numbers and represent the index of the series. Then, 𝑡𝑖  denote the time at which the 𝑖 -th 108 diffraction pattern was acquired. In tDF, 𝒌  is determined by the position of the objective 109 aperture. Meanwhile, XPCS and ECM methods occasionally take azimuthal average over 𝒌 110 values that share the same radius from the center. In this research, we use azimuthal average of 111 𝑔𝑜𝑛𝑒(𝒌, 𝑡) to compare with TTCF. We use 〈⋯ 〉𝜙 for ensemble average over pixels 𝜙 whose 112 |𝒌| are in a certain spatial frequency range (or azimuthal average), ⋯̅ for time averages. For 113 7  simplicity, assuming that the difference between ∑ 𝐼(𝒌, 𝑡𝑖)𝑁−𝑗𝑖=1  and ∑ 𝐼(𝒌, 𝑡𝑖+𝑗)𝑁−𝑗𝑖=1  is small, 114 after averaging over the azimuthal direction, equation 1 can be modified to 115 𝑔𝑜𝑛𝑒(𝑘, 𝑗Δ𝑡) = 〈(𝑁 − 𝑗) ∑ 𝐼(𝒌, 𝑡𝑖)𝐼(𝒌, 𝑡𝑖+𝑗)𝑁−𝑗𝑖=1[∑ 𝐼(𝒌, 𝑡𝑖)𝑁−𝑗𝑖=1][∑ 𝐼(𝒌, 𝑡𝑖)𝑁−𝑗𝑖=1]〉𝜙=(𝑁 − 𝑗)(𝑁 − 𝑗) 〈𝐼(𝒌, 𝑡𝑖)𝐼(𝒌, 𝑡𝑖+𝑗)̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ 〉𝜙(𝑁 − 𝑗)(𝑁 − 𝑗)〈𝐼(𝒌, 𝑡𝑖)̅̅ ̅̅ ̅̅ ̅̅ ̅ 𝐼(𝒌, 𝑡𝑖)̅̅ ̅̅ ̅̅ ̅̅ ̅〉𝜙=〈𝐼(𝒌, 𝑡𝑖)𝐼(𝒌, 𝑡𝑖+𝑗)̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ 〉𝜙〈𝐼(𝒌, 𝑡𝑖)̅̅ ̅̅ ̅̅ ̅̅ ̅2〉𝜙. (2) 116 Conversely, the TTCF assesses the temporal decay of correlation between diffraction patterns 117 at different times. The correlation between two diffraction patterns is measured using the 118 following equation 13. 119 𝐺(𝑘,  𝑡𝑖,  𝑡𝑖+𝑗) =⟨𝐼(𝒌,  𝑡𝑖)𝐼(𝒌,  𝑡𝑖+𝑗)⟩𝜙⟨𝐼(𝒌,  𝑡𝑖)⟩𝜙⟨𝐼(𝒌,  𝑡𝑖+𝑗)⟩𝜙. (3) 120 During this calculation, averaging occurs along the azimuthal direction, leading to a reduction 121 in azimuthal resolution. By averaging over a fixed delay time 𝑗Δ𝑡 = 𝑡𝑖+𝑗 − 𝑡𝑖, we can derive 122 the temporal decay function 𝑔𝑡𝑤𝑜. 123 𝑔𝑡𝑤𝑜(𝑘,  𝑗Δ𝑡) = 𝐺(𝑘,  𝑡𝑖,  𝑡𝑖+𝑗)̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅. (4) 124 Hereafter, when referring to both 𝑔𝑜𝑛𝑒(𝑘, 𝑡) and 𝑔𝑡𝑤𝑜(𝑘,  𝑡), they will be collectively denoted 125 as 𝑔(𝑘, 𝑡) . In this article, for the calculation of 𝑔(𝑘, 𝑡) , the region where the length of 𝒌 126 ranges from 3.9 to 4.1 nm−1 (corresponding to a thickness of one pixel) where the peak of halo 127 pattern appeared was used. Thus, the number of pixels used for calculating 𝑔𝑜𝑛𝑒(𝑘, 𝑡) and 128 𝑔𝑡𝑤𝑜(𝑘,  𝑡) are the same. Since we are not focusing on the 𝑘 dependency of 𝑔(𝑘, 𝑡), we will 129 8  omit 𝑘 from hereout. 130 The difference between these methods becomes significant when the relaxation time is much 131 longer than the observation time. Figure 2 depicts the dependence of 𝑔(𝑡) on the length of the 132 observation time, as measured by OTCF and TTCF. These 𝑔(𝑡) values are derived from the 133 same experimental data, with varying lengths of data used to calculate 𝑔(𝑡)  to simulate 134 various observation time windows. This includes using the entire time domain, as well as only 135 half, one-third, and one-quarter of the acquired time domain. It is evident that the 𝑔𝑜𝑛𝑒(𝑡) 136 measured by OTCF is strongly influenced by the length of the observation time. 𝑔𝑜𝑛𝑒(𝑡) 137 decays faster with shorter observation times. Conversely, this effect is not observed in 𝑔𝑡𝑤𝑜(𝑡) 138 measured by TTCF. Irrespective of the observation time's length, it exhibits the same time 139 dependence. 140 [Figure 2 here] 141 This difference arises from the calculation methods of OTCF and TTCF. If we define the 𝐼(𝑡𝑖)̅̅ ̅̅ ̅̅  142 as 𝐼𝑎𝑣𝑒  and the deviation from the average as 𝛿𝐼(𝑡𝑖), equation 2 can be modified to 143 𝑔𝑜𝑛𝑒( 𝑗Δ𝑡) =〈(𝐼𝑎𝑣𝑒 + 𝛿𝐼(𝑡𝑖)) (𝐼𝑎𝑣𝑒 + 𝛿𝐼(𝑡𝑖+𝑗))̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅〉 𝜙〈𝐼𝑎𝑣𝑒2 〉𝜙= 1 +〈𝛿𝐼(𝑡𝑖)𝛿𝐼(𝑡𝑖+𝑗)̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ 〉𝜙〈𝐼𝑎𝑣𝑒2 〉𝜙 . (5) 144 Here, due to the property of average, 𝛿𝐼(𝑡𝑖)̅̅ ̅̅ ̅̅ ̅̅   become 0. Given that 𝛿𝐼(𝑡𝑖)  represents the 145 deviation from the temporal average, 𝛿𝐼(𝑡𝑖) and 𝛿𝐼(𝑡𝑖)𝛿𝐼(𝑡𝑖+𝑗)̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅  must exhibit negative values 146 9  at certain 𝑡𝑖  and 𝑗 , except when all 𝛿𝐼(𝑡𝑖)  = 0. (Detailed discussion is provided in the 147 supplemental and fig. S5.) This result demonstrates that, irrespective of the actual change of 148 the atomic structure, 𝑔𝑜𝑛𝑒(𝑡) − 1  can take negative values in the observation time. This 149 discussion is conducted under general conditions without imposing constraints on the 150 𝛿𝐼(𝑡𝑖) (except not all 𝛿𝐼(𝑡𝑖)  = 0). In reality, however, 𝛿𝐼(𝑡𝑖)  often exhibits temporal 151 correlations, with closer time intervals yield more similar values. Such behavior is particularly 152 common for atomic structural changes and the corresponding variations in diffraction patterns. 153 In these cases, the 𝑔𝑜𝑛𝑒 typically decays gradually, beginning at time shorter than the 𝑗Δ𝑡. As 154 a result, this 𝑗Δ𝑡  often becomes the effective upper limit of relaxation times that can be 155 reliably measured within the length of observation time. Consequently, 𝑔𝑜𝑛𝑒(𝑡)  is 156 significantly influenced by the length of the observation time, decaying faster when the 157 observation time is shorter. This poses challenges in measuring relaxation times longer than 158 the observation time, as 𝑔𝑜𝑛𝑒(𝑡) is compelled to converge to 1 within the observation time. 159 In practice, accurately measurable relaxation times are typically smaller than 1/20 or 1/400 of 160 the observation time 5,14.  161 In contrast, the TTCF directly measures the correlation between two diffraction patterns 162 through equation (3). When calculating 𝑔𝑡𝑤𝑜(𝑡)  using equation (3), the length of the 163 observation time primarily affects the number of 𝐺(𝑘,  𝑡𝑖,  𝑡𝑖+𝑗)  for averaging to calculate 164 𝑔𝑡𝑤𝑜(𝑡) rather than significantly altering the shape of 𝑔𝑡𝑤𝑜(𝑡). Thus, the TTCF method is 165 10  minimally impacted by the length of observation time and does not artificially converge to 1. 166 This characteristic suggests that TTCF can potentially measure relaxation times longer than the 167 observation time and capture dynamics that do not converge to 1, i.e., dynamics that do not 168 completely eliminate the correlation of diffraction patterns. 169  170 Accuracy of measurement of relaxation time and 𝜸 171 In this section, we investigate the impact of varying observation times on the accuracy of 172 fitting by simulation. In the previous section, it was suggested that the TTCF method could 173 potentially measure relaxation times longer than the observation time through 𝑔𝑡𝑤𝑜(𝑡) . 174 However, there should be a limitation regarding the ratio between relaxation time and 175 observation time for accurately measuring relaxation time. This limitation arises because when 176 the relaxation time significantly exceeds the observation time, only the initial decay of 𝑔𝑡𝑤𝑜(𝑡) 177 is observable, leading to reduced robustness against noise and decreased fitting accuracy. Here, 178 we specifically examine how the reliability of the measured relaxation time depends on this 179 time ratio. We utilized the following KWW equation with background intensity for fitting, as 180 TTCF has the potential to capture dynamics that do not converge to 1. 181 (𝑔𝑡𝑤𝑜(𝑡) − 1)𝛽= (1 − 𝑏𝑔) × exp (−2 (𝑡𝜏)𝛾 ) + 𝑏𝑔 (6) 182 Here, 𝛽 represents the normalization factor, τ represents the relaxation time, 𝛾 represents the 183 stretching factor, and 𝑏𝑔 represents the background. Figure 3 illustrates examples of equation 184 11  (6) for various values of 𝜏 , 𝛾 , and 𝑏𝑔 . (𝑔𝑡𝑤𝑜(𝑡) − 1)/𝛽  shifts in the time direction with 185 changes in 𝜏 . While altering only 𝛾  still passes through the same points, the point is the 186 relaxation time, and as 𝛾 increases, the change around the relaxation time becomes steeper. 187 The larger the background, the smaller the temporal change of (𝑔𝑡𝑤𝑜(𝑡) − 1)/𝛽.  188 Next, we evaluated the accuracy of fitting using equation (6) with various values of the 189 observation time window, 𝜏, 𝛾, and 𝑏𝑔. The absolute values of the relaxation and observation 190 times are not critical; instead, the ratio between them is crucial. Therefore, we utilize the ratio 191 between them (observation time/relaxation time (R)). (If the number of time steps for 192 measurement is the same, fitting under the condition of relaxation time 1000 s and observation 193 time 500 s is identical to fitting under the condition of relaxation time 10 s and observation 194 time 5 s). It is worth noting that in this simulation, since the observation time is assumed to be 195 constant, the meaning of measuring or selecting R and 𝜏 is nearly equivalent. 196 [Figure 3 here] 197 The fitting ranges for R (𝜏), 𝛾, and 𝑏𝑔 were set to 0–1000, 0.01–3, and 0–1, respectively. 198 Accuracy and precision were evaluated through four steps: Step 1 involved selecting values of 199 R, 𝛾 , and 𝑏𝑔,   and calculating 𝑔𝑡𝑤𝑜(𝑡)  using equation (6) for 𝑡 consisting of 500 evenly 200 spaced values on a linear scale from 0 to R. Step 2 introduced noise following a normal 201 distribution with a mean of 0 and a standard deviation of 0.02 to 𝑔𝑡𝑤𝑜(𝑡) at each 𝑡. This 202 normal distribution is identical to the distribution observed in experiments with a dose of 11.5 203 12  pA. It is worth noting that the noise becomes larger at larger 𝑡 in the experiment because the 204 number of 𝐺(𝑘,  𝑡𝑖,  𝑡𝑖+𝑗) used for calculating 𝑔𝑡𝑤𝑜(𝑡) become fewer at larger 𝑡. Such effect 205 is not included in this noise. Thus, added noise is a simplified model. Step 3 entailed fitting 206 𝑔𝑡𝑤𝑜(𝑡) with equation (6) to obtain R, 𝛾 and 𝑏𝑔. Steps 1–3 were repeated 50 times, and the 207 averages and standard deviations of the measured values were calculated. The averages of 208 obtained R, 𝛾 , and 𝑏𝑔  were used to calculate the differences from the selected values, 209 indicating accuracy, while the standard deviation reflected precision. Step 4 encompassed 210 repeating steps 1–3 for R ranging from 0.01 to 100, 𝛾 ranging from 0.2 to 3, and 𝑏𝑔 ranging 211 from 0 to 1. In the Supplementary Fig. S6, the effect of noise amplitude and the number of 212 steps is also examined using the same procedure. 213 The results are depicted in Figure 4. Generally, as R decreases and the background value 214 increases, both the accuracy and precision of 𝜏, 𝛾, and 𝑏𝑔 diminish. A smaller R means that 215 only the early stage of the temporal decay of 𝑔𝑡𝑤𝑜(𝑡) is observed, resulting in lower accuracy 216 and precision. A larger background leads to a smaller temporal decay of 𝑔𝑡𝑤𝑜(𝑡), reducing 217 robustness against noise and resulting in low accuracy and precision. With a background value 218 of 1, no temporal decay occurs, so it is natural that the value cannot be accurately estimated. 219 [Figure 4 here] 220 We then focused on the difference caused by the 𝛾  value. There is a common trend in 221 accuracy and precision for R, 𝛾, and 𝑏𝑔, except for R with 𝛾 less than 0.5. As 𝛾 decreases, 222 13  even with a small R, the accuracy and precision improve. This is because, as shown in Figure 223 3(b), when 𝛾 is small, a temporal decay of 𝑔𝑡𝑤𝑜(𝑡) can be observed even at small 𝑡, leading 224 to better accuracy and precision even with a short measurement time (small R). However, for 225 R with 𝛾 less than 0.5, as 𝛾 decreases, the accuracy and precision of R worsen. This could 226 be because when 𝛾 becomes extremely small, as shown in Figure 3(b), the temporal decay of 227 𝑔𝑡𝑤𝑜(𝑡)  progresses rapidly within a few time steps, making it challenging to accurately 228 measure the relaxation time via fitting. Roughly speaking, if the ratio between observation and 229 relaxation times is above 0.3 and the background is below 0.9, measurements can be made with 230 an error of less than 10%. The required observation time for accurate measurement using TTCF 231 is at least 60 times smaller compared to OTCF. This advantage is well-suited for studying the 232 glass transition phenomenon where relaxation times vary rapidly near the glass transition 233 temperature, allowing for measurement across wider temperature ranges. 234 Here, we outline the advantages and disadvantages of OTCF and TTCF. OTCF can measure 235 relaxation time without averaging over the azimuthal direction in the diffraction pattern, 236 providing resolution in the azimuthal direction 9. Therefore, OTCF has the potential to uncover 237 azimuthal dependencies of relaxation time. Indeed, radius dependence on relaxation time has 238 already been reported 15,16, and a possibility exists that relaxation time also depends on the 239 azimuthal angle. On the other hand, although the TTCF method loses resolution along the 240 azimuthal direction, it can measure longer relaxation times relative to the observation time. 241 14  This advantage is suitable for measuring slow dynamics observed in glass at low temperature. 242  243 Sensitivity analysis 244 In the previous section, we explored how the length of observation time affects measurement 245 accuracy by assessing the error caused by noise. Here, we verify the extent of reliability of the 246 obtained results through sensitivity analysis. Sensitivity analysis is a method for quantifying 247 the extent to which variations in input parameters influence the corresponding changes in 248 output parameters. In this study, we introduce noise into the experimental 5D-STEM data and 249 examine the robustness of the fitting results against this noise to evaluate the reliability of 250 extracted quantities, such as the relaxation time. This analysis used the experimental 5D-STEM 251 data with an 11.5 pA dose. The measured distributions of relaxation time, 𝛾, and background 252 from the data before adding noise are displayed in Figure 5 (a)–(c). When comparing the 253 distribution of relaxation times and background, we observed mainly two types of 254 combinations: short relaxation time with high background and long relaxation time with low 255 background. Large 𝛾 are observed with small background and relaxation time of 103~104 (pair 256 plot shown in Figure S2).  257 [Figure 5 here] 258 (𝑔𝑡𝑤𝑜(𝑡) − 1)/𝛽  calculated from the experimental 5D-STEM data contains noise of a 259 normal distribution with a mean of 0 and a standard deviation of 0.02. We generated noise from 260 15  the same normal distribution, readded it to the experimental 5D-STEM data, and fitted it with 261 the modified KWW function with background (equation (6)) 50 times, measuring averages and 262 standard deviation. To quantify the spread in the measured values against the average value, 263 we evaluate the dispersion of the relaxation time—whose range is relatively large—using its 264 coefficient of variation (C. V.; standard deviation/average), whereas the smaller-range 265 parameters 𝛾 and background are assessed by their respective standard deviations. Large C. 266 V. or standard deviation values suggest that the measured results are significantly affected by 267 noise, indicating lower reliability of those measured values. 268 The distributions of the C. V. and standard deviations are shown in Figure 5(d)–(f), with 269 relative frequencies and cumulative relative frequencies displayed in Figure 5(g)–(i). Over 70% 270 of points have a C. V. of relaxation times smaller than 0.1. Regarding 𝛾 and background, over 271 80% of points have standard deviations for 𝛾  and background smaller than 0.05. This 272 indicates that measurements for over 70% of points have errors with a C. V. of less than 10% 273 for the relaxation time and errors with a standard deviation of less than 0.05 for 𝛾  and 274 background, for over 80% of points. Thus, most points are robust against noise.  275 To validate points with low precision, we created pair plots of measured values against C. V. 276 or standard deviations (Detailed pair plots are shown in Fig. S7). The results are displayed in 277 Figure 6. As shown in the upper right panel in Fig. 6, the precision of combinations involving 278 short relaxation time with high background and long relaxation time with low background are 279 16  high. These combinations of values are situated in the small standard deviation (high precision) 280 areas in Figure 4 (upper left and bottom right areas of each figure in Figure 4). 281 [Figure 6 here] 282 As illustrated in the top row in Fig. 6, C. V. increases with long relaxation time, small 𝛾, and 283 medium backgrounds. This occurs because, when 𝛾 is small, the change in 𝑔𝑡𝑤𝑜(𝑡) towards 284 the end of relaxation becomes more gradual, making it challenging to distinguish background 285 noise from the late stage of prolonged and gradual relaxation. For similar reasons, the standard 286 deviations in background values are also large, particularly when relaxation times are long and 287 𝛾 is small (shown in the right bottom panel in Fig. 6). However, the standard deviations of 𝛾 288 do not exhibit any clear correlation between relaxation time, 𝛾, and background. If anything, 289 the standard deviation of 𝛾 increases when the background has larger values. These points 290 where the measurement precision deteriorates correspond to the upper right area of each panel 291 in Figure 4, aligning with the results in the previous section. 292 Additionally, as depicted in the top right panel in Fig. 6, the length of relaxation time where 293 the assumed precision of relaxation time begins to degrade is also around 3000 s. This value is 294 approximately three times the observation time(1115.6 s) and corresponds to R = 0.3, in 295 agreement with the results presented in the previous section. 296 Based on these results, it is feasible to estimate the measurable relaxation time from the size 297 of the noise and the observation time. Furthermore, it is advisable to conduct a sensitivity 298 17  analysis to assess actual reliability. 299  300 Beam irradiation effect 301 Finally, we investigated the effect of beam irradiation on TTCF relaxation measurements, an 302 aspect that has not been thoroughly evaluated to date. Electron beam irradiation may potentially 303 reduce the relaxation time by modifying the sample structure 11,17,18, a phenomenon known as 304 beam damage. Beam damage encompasses thermal effects, ionization, charge up, and knock-305 on damage 19. It is already known that irradiation increases the sample temperature by only a 306 few millikelvins and does not impact the relaxation time of metallic glass 4,10. Ionization and 307 charge up primarily affect insulating samples; metallic glass, being a conductor, is less affected. 308 Knock-on is displacement caused by the electron irradiation, could be the most significant in 309 our experiment. This type of damage occurs when the acceleration voltage surpasses a 310 threshold voltage, depending on atomic structures. The threshold voltage varies with atomic 311 species, surrounding atoms, irradiation direction, and temperature. Estimating knock-on 312 damage, especially in glass materials lacking periodic atomic structures, proves challenging. 313 [Figure 7 here] 314 To gauge the irradiation effect, we replicated the experiment with varying doses. Figure 7 315 displays the positional average of the 𝑔𝑡𝑤𝑜(𝑡) function measured by the TTCF at different 316 doses of 11.5, 5.8, and 2.8 pA. The time dependencies of 𝑔𝑡𝑤𝑜(𝑡) were consistent across all 317 18  dose levels, with no noticeable acceleration of relaxation with increasing dose. This could be 318 due to the limited effective dose amount. In this method, the electron beam scans, and the 319 irradiation time per position is quite brief. The electron counts per scan are 9500, 4800, and 320 2300 e- at 11.5, 5.8, and 2.8 pA electron beams, respectively. Scanning occurs approximately 321 every 1.9 s, resulting in a net irradiation rate of 5000, 2500, 1200 e-/s at 11.5, 5.8, and 2.8 pA 322 electron beams, respectively. Figure 8 showcases the spatial distributions and histograms of 323 relaxation time, 𝛾, and background at each dose rate. Although the observation areas were not 324 perfectly identical, we compared the histograms of relaxation time, 𝛾, and background. The 325 fitting ranges for relaxation time, 𝛾 , and background were 0–5 × 109, 0–3, and 0–1, 326 respectively. As a reference, supplemental Fig. S8 shows the results obtained from the same 327 dataset using OTCF. 328 [Figure 8 here] 329 It is worth noting that the measured relaxation time is longer than previous research20. This 330 discrepancy is attributed to aging12. Indeed, when the relaxation time was measured promptly 331 after heating using TTCF, it showed good agreement with results obtained by other 332 measurement methods8. As evident in the histograms of relaxation time, 𝛾, and background, 333 their shapes remained consistent across different doses. Since no significant differences were 334 observed among the various doses, we conclude that dose rate does not impact relaxation time 335 measurements via TTCF. This result is different from previous result which indicates that the 336 19  dose rate affects relaxation time measurements by ECM21. This discrepancy derives from the 337 difference in irradiation mode and dose rate. The previous results used TEM mode and 338 continuously irradiated electron beam. On the other hand, we used STEM mode and 339 intermittently irradiated electron beam. This difference makes the dose rate different. The dose 340 rate in previous research ranged from 1 × 105 to 4 × 105 𝑒−/nm2/s, and our dose rate ranged 341 from 2.5 ×  103 to 1 ×  104 𝑒− /nm2/s. Our dose rate is extremely low compared with the 342 previous research. Thus, we conclude that the dependence of relaxation time on dose rate was 343 not observed in our experiment.  344  345 Conclusion 346 Although the measurement of relaxation times in amorphous materials using 5D-STEM still 347 faces limitations—such as the lack of depth resolution and the inability to resolve detailed 348 atomic-scale structural changes—it nonetheless provides a powerful approach, enabling 349 measurements on the nanometer scale. In this study, our focus was on comparing OTCF and 350 the TTCF for measuring relaxation time. While TTCF sacrifices azimuthal direction resolution 351 in the diffraction pattern, it remains unaffected by the length of observation time when 352 measuring temporal decay. Hence, the TTCF proves more adept at measuring long relaxation 353 times. We assessed the errors in measured relaxation time, 𝛾, and backgrounds stemming from 354 insufficient measurement duration. 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Average differences (accuracy) and standard deviations (precision) are shown in the 440 1st and 2nd rows for the ratio (Observation time/Relaxation time), in the 3rd and 4th rows for 𝛾, 441 26  and in the 5th and 6th rows for background. These values are presented at each point along the 442 vertical axis (Observation time/Relaxation time), horizontal axis (background), and 𝛾 443 (column). We show enlarged view of vertical axis (Observation time/Relaxation time) and 444 horizontal axis (background) at the left end and bottom of the figures. In all the figures, the 445 values on the vertical and horizontal axes are consistent.  446 27   447 Figure 5. Spatial distributions of (a) relaxation time, (b) 𝛾 , (c) background, (d) C.V. of 448 relaxation time, (e) standard deviation of 𝛾, and (f) standard deviation of background. Relative 449 frequency and cumulative relative frequency of (g) C. V. of relaxation time, (h) standard 450 deviation of 𝛾 , and (i) standard deviation of background. These data were measured 451 experimentally from Zr50Cu40Al10 metallic glass.  452 28   453 Figure 6. The relationships among the relaxation time (τ), 𝛾 , the background (bg), the 454 coefficient of variation (C.V.) of relaxation time, and the standard deviations of 𝛾  and 455 background. Here, τ denotes the relaxation time and bg represents the background. The top 456 panels show how the C.V. (τ) varies with relaxation time, γ, and background. The middle panels 457 depict the corresponding relationship for the standard deviation of 𝛾, while the bottom panel 458 presents it for the standard deviation of background.   459 29   460 Figure 7. Positional average of (𝑔(𝑘,  𝑡) − 1)/𝛽 for each dose amount (11.5, 5.8, and 2.8 pA).  461 30   462 Figure 8. Spatial distributions and histograms of relaxation time, 𝛾, and background for each 463 dose amount. These data were measured experimentally from Zr50Cu40Al10 metallic glass. 464  465