Corrigendum to: “Relative merits and limiting factors for x-ray and electron microscopy of thick, hydrated organic materials” [Ultramicroscopy 184 (2018) 293–309]

Abstract

During follow-on calculations, we found a few minor errors in our previous publication [Ultramicroscopy 184, 293–309 (2018); doi:10.1016/j.ultramic.2017.10.003]. We present here the necessary corrections. The full revised manuscript, with the corrected parts indicated in blue color text, is available at https://arxiv.org/abs/2004.10069.

The authors regret that a few minor mistakes were found in our previous publication, [M. Du and C. Jacobsen, “Relative merits and limiting factors for x-ray and electron microscopy of thick, hydrated organic materials,” Ultramicroscopy 184, 293–309 (2018); doi:10.1016/j.ultramic.2017.10.003]. Although the overall effects of the errors do not alter our major conclusions, we would like to make corrections to several equations 5 and figures presented in the paper. This Corrigendum shows only the changes. One can also see the full manuscript, with the changes indicated in blue color text, at https://arxiv.org/abs/2004.10069.

Eqs. 80 and 81 in the original paper are intended to describe the intensity loss due to out-of-aperture scattering and absorption from the entire feature-containing column. However, the form they are presented in the paper would account for only the intensity loss within the feature slice in the middle when the entrance intensity is I₀. The correct equation to calculate the signal loss that only occurs in the feature slice should be

$$I_{\mathrm{out}/\text{f},f}=\frac{\left[I_0-I_{\text{out},b}\left(t_b/2\right)-I_{\text{abs},b}\left(t_b/2\right)\right]K_{\text{out},f}}{K_{\text{out},f}+K_{\text{abs},f}}\left(1-\exp [-(K_{\text{out},f}+K_{\text{abs},f})t_f]\right)$$ (80)

$$I_{\text{abs}/\text{f},f}=\frac{\left[I_0-I_{\text{out},b}\left(t_b/2\right)-I_{\text{abs},b}\left(t_b/2\right)\right]K_{\text{abs},f}}{K_{\text{out},f}+K_{\text{abs},f}}\left(1-\exp [-(K_{\text{out},f}+K_{\text{abs},f})t_f]\right).$$ (81)

Based on these, the original Eq. 80 should be written as

$$I_{\text{out},f}=I_{\text{out},b}\left(t_b/2\right)+I_{\text{out}/\text{f},f}+\frac{I_0-I_{\text{out},b}\left(t_b/2\right)-I_{\text{abs},b}\left(t_b/2\right)-I_{\text{out}/\text{f},f}-I_{\text{abs}/\text{f},f}}{I_0}{I}_{\text{out},b}\left(t_b/2\right)$$ (82a)

The equation consists of three terms, describing the amount of photons scattered out in the overlaying background material, out of the feature slice in the middle, and out of the underlying background material. Following the definition in original Eq. 72, I_out,b (t_b/2) gives the amount of photons scattered out in the background material of thickness t_b/2, given an incident intensity of I₀, which is the thickness of the overlying slab. Within the feature slice at the middle, the amount of photons scattered out is I_out/f,f. The third term is the amount of out-of-aperture photons contributed by the underlying slab, but the incident intensity I₀ in I_out,b(t_b/2) has to be replaced by the beam intensity after being attenuated by the overlying and middle slab, which is accounted for by the prefactor of I_out,b(t_b/2). For t_f ≪ t_b, the first and third terms in the equation can be collectively replaced by I_out,b(t_b), resulting in

$$I_{\text{out},f}\simeq I_{\text{out},b}\left(t_b\right)+I_{\text{out}/\text{f},f}.$$ (82b)

This approximation is based on the assumption that the attenuation caused in the middle slice does not significantly alter the beam intensity at the upper boundary of the underlying material. Along with similar considerations for photoelectric absorption, original Eq. 81 should be

$$I_{\text{abs},f}=I_{\text{abs},b}\left(t_b/2\right)+I_{\text{abs}/\text{f},f}+\frac{I_0-I_{\text{out},b}\left(t_b/2\right)-I_{\text{abs},b}\left(t_b/2\right)-I_{\text{out}/\text{f},f}-I_{\text{abs}/\text{f},f}}{I_0}{I}_{\text{abs},b}\left(t_b/2\right)$$ (82c)

where the approximation at t_f ≪ t_b similarly applies to Eq. 82c. Finally, this set of equations are completed by the expression

$$I_{\text{inel}}=I_0-I_{\text{out},f}-I_{\text{abs},f}-I_{\text{in},\text{noinel},f}.$$ (82d)

Additionally, we realized that the electron dose plot in Fig. 11 was created using a slightly erroneous equation. The constrast parameter Θ for electron microscopy, without and with energy filter, were correctly listed in the original manuscript as

$${\Theta }_{\text{e},\text{unfiltered}}=\frac{\left|I_{\text{noscat},f}-I_{\text{noscat},b}+I_{1\text{el},f}-I_{1\text{el},b}\right|+2\sqrt{I_{\text{noscat},f}{I}_{1\text{el}/\text{f},f}}}{\sqrt{I_{\text{in},f}+I_{\text{in},b}}}.$$ (129)

when a zero-loss energy filter is absent, and

$${\Theta }_{\text{e},\text{filtered}}=\frac{\left|I_{\text{noscat},f}-I_{\text{noscat},b}+I_{1\text{el},f}-I_{1\text{el},b}\right|+2\sqrt{I_{\text{noscat},f}{I}_{1\text{el}/\text{f},f}}}{\sqrt{I_{\text{in},\text{noinel},f}+I_{\text{in},\text{noinel},b}}}.$$ (130)

The “signal intensity” in the numerators should count only unscattered or single elastically scattered electrons. However, the Θ values used for electron microscopy in Fig. 11 were calculated with plural scattered electrons mistakenly included in the numerators. After fixing the issue in our simulation code, we see a minor yet observable change in the electron curves.

Figure 11:

Estimated radiation dose associated with 10 nm resolution imaging of protein features in amorphous ice. This shows the case for soft x-ray microscopy at 0.5 keV and hard x-ray microscopy at 10 keV, both for absorption and Zernike phase contrast. In the case of electron microscopy, accelerating voltages of 100 and 300 kV are shown for phase contrast imaging with and without the use of a zero-loss energy filter. In all cases, the imaging system is assumed to have 100% efficiency. The slight reduction in electron microscopy dose in ice thicknesses in the 100 nm range comes about due to an improved balance between unscattered and single-scattered electrons, leading to a slight improvement in contrast.

Another minor revision is made to the left hand side of Eq. 126, which should be I_inel,f instead of I_inel.

With the above stated changes, we have regenerated the plotted shown in Fig. 5–7 and 11–14 in the original manuscript.

Figure 5:

Contrast parameter Θ for soft x-ray (0.5 keV) imaging of t_f = 20 nm protein features as a function of amorphous ice thicknesses t_b. At left is shown the case for Zernike phase contrast using the pure-phase thin sample approximation of Eq. 36, the conventional model of Eq. 35, and the complete model of Eq. 88 with phase contrast (φ = π/2) . The discrepancy between the pure-phase thin sample approximation and the conventional model is due to the fact that there is significant absorption at the soft x-ray energy of 0.5 keV, even though this is within the “water window” spectral region between the carbon (0.290 keV) and oxygen (0.540 keV) x-ray absorption edges.

Figure 7:

Contrast parameter Θ for Zernike phase contrast imaging with hard x-rays (15 keV) as a function of feature thickness t_f for protein in amorphous ice. In this case no overlying or underlying thickness was assumed (that is, t_b,o = t_b,u = 0 in Fig. 1), so this is just for a protein feature of the indicated thickness in an equal thickness slab of ice. As can be seen, the pure-phase thin sample expression of Eq. 36 gives inaccurate predictions for ice thicknesses of even a few tens of micrometers; the conventional model of Eq. 35 works well for thicknesses up to several hundreds of micrometers at which point the more complete expression of Eq. 88 with φ = π/2 gives correct results.

Figure 14:

Combined contour and brightness map of the required x-ray radiation dose in Gray for imaging 100 nm features in amorphous ice as a function of both x-ray photon energy and overall ice thickness. This figure shows the lower of absorption or phase contrast imaging at each point; in nearly all cases phase contrast provides the lowest dose. The grayscale image shows log₁₀(Gray), with the overlaying contour line labeled 6 representing a dose of 10⁶ Gray and so on. Of course it would be very challenging to obtain amorphous ice over these organ-scale thicknesses, but on the other hand ice crystal artifacts that obscure features in few-nanometer-resolution cryo electron microscopy studies might be unnoticeable in 100 nm resolution imaging.

Figure 6:

Contrast parameter Θ for Zernike phase contrast imaging with hard x-rays (15 keV) as a function of overall amorphous ice thickness. The case for a small protein feature (t_f = 20 nm) is shown at left, while the case for a larger protein feature (t_f = 1000 nm) is shown at right. The conventional Zernike phase contrast model of Eq. 35 works well for describing fine features in ice layers up to tens of micrometers thick, but the more complete model of Eq. 88 with phase contrast (φ = π/2) becomes necessary with thicker features and ice layers. Absorption contrast is not shown because it is quite weak for hard x-ray imaging of organic materials in ice.

Figure 12:

Combined contour and brightness map of the required x-ray radiation dose in Gray for imaging 10 nm features in amorphous ice as a function of both x-ray photon energy and overall ice thickness. This figure shows the lower of absorption or phase contrast imaging at each point; in nearly all cases phase contrast provides the lowest dose. The grayscale image shows log₁₀(Gray), with the overlaying contour line labeled 6 representing a dose of 10⁶ Gray and so on. The soft x-ray “water window” energy range [1] ([84] in original manuscript) between the carbon K edge at 0.29 keV and the oxygen K edge at 0.54 eV provides minimum dose imaging for specimens in ice layers up to about 10–20 μm. thick, while phase contrast requires a slightly higher dose at multi-keV energies while accommodating thicker specimens overall. Note that the presence of sulfur in our model protein leads to the contour feature at the S K edge at 2.47 keV.

Figure 13:

Required radiation dose as a function of resolution. In this case of imaging protein in 10 μm. of amorophous ice, the dose for SNR=5 imaging was calculated as t_F was varied, for both soft x-rays in the water window (0.5 keV) or for hard X rays (10 keV), for the better of absorption or Zernike phase contrast at each thickness. The trend of required dose increasing as the fourth power of improvements in spatial resolution (decreases in t_f) as expected from Eqs. 94 and 95 is clearly seen. Also shown are the radiation doses associated with various detrimental effects in biological specimens, as discussed in Sec. 7.3.