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jupytext:
text_representation:
extension: .md
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jupytext_version: 1.10.3
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(ch:electron-microscopybasics)=
# Electron Microscopy basics
(sec:introduction-microscopybasics)=
## Introduction
In this chapter, we will deal with basic knowledge about microscopes in general and electron microscopes specifically. From the basic, general layout and components of a microscope, we will explain the two main modes of image acquisition: wide field recording versus scanning. We will show why the use of electrons lowers the diffraction limit compared to the use of light (or photons) and could theoretically allow for resolution far below a nanometer. In a later chapter, we will see that there are other, practical limitations to the resolution that can be obtained with electron microscopy.
The second part of this chapter, paragraphs 7 and 8, explain the basic components that we encounter in an electron microscope with their relevant properties: electron sources including the concepts of brightness and coherence, and both electrostatic and magnetic electron lenses. The chapter will be concluded with some practical considerations related to the use of electron microscopes.
(sec:basic-microscope-layout)=
## Basic microscope layout
In all microscopes, we can identify some key elements: (i) a source for radiation, e.g. light, (ii) a system with lenses and other components to bring this radiation on (iii) the sample, and (iv) another system with lenses and other components to bring signal from the sample to (v) a detector.
---
---
(sec:wide-field-versus-scanning)=
## Wide field versus scanning
In light microscopy as well as in electron microscopy, we have systems that operate with wide-field illumination (for instance regular, wide-field light microscopes and transmission electron microscopes) and systems that rely on scanning a focused or shaped beam (confocal laser scanning microscopy or light-sheet microscopy, and scanning electron microscopy).
---
---
(sec:the-diffration-limit)=
## The diffraction limit
The main reason for developing microscopy based on electrons, was to find a way to circumvent the diffraction limit of light microscopy. The diffraction limit relates to the wave character of light. Diffraction ({numref}`subsec:diffraction-limit`) occurs when a wave is only partly captured by an aperture (or lens). This leads to a blurring in the image produced by a lens. This blurring is described by the Point Spread Function (PSF). The PSF is the mathematical function for the intensity profile generated by a lens or microscope in case of an infinitely small point source. Due to the blurring, two objects can only be distinguished if they are sufficiently separated. The diffraction limit gives the smallest separation d for two partially overlapping PSFs can still be visibly distinguished ({numref}`fig:AiryDisks`). Thus, it gives a measure for the minimal distance at which the images of two point objects can still be resolved and thereby for the resolution of the lens or microscope. An expression for the diffraction limit was first derived by [Ernst Abbe](https://en.wikipedia.org/wiki/Ernst_Abbe) and is also often referred to as the Abbe limit:
$$
d = \frac{\lambda}{2n \: sin(\alpha)}
$$ (diffraction-limit)
where $\lambda$ is the wavelength of light used for imaging, $n$ is the refractive index of the medium between lens and object and α is the opening angle of the beam. The product $n \: sin(α)$ is also know as the numerical aperture (NA) of the lens. We note that for small angles, $sin(α)$ can be approximated as $\alpha$, if we express the angle in radians. We will later see that we can use this approximation for optics and microscopy with electrons. Note also that for visible light, the diffraction limit is approximately $250-300 \: nm$.
```{code-cell} ipython3
:tags: [remove-input]
import numpy as np
from scipy.special import j1
import plotly.graph_objects as go
from plotly.subplots import make_subplots
from myst_nb import glue
wavelength = 0.625 # Microns (example value)
aperture_diameter = 1 # Arbitrary unit
k = 2 * np.pi / wavelength # Wave number
NA = 0.38
maxseperation = 0.61*wavelength/NA
# Define the Airy function
def airy(rho):
with np.errstate(divide='ignore', invalid='ignore'):
intensity = (2 * j1(k * aperture_diameter * rho ) / (k * aperture_diameter * rho))**2
intensity[np.isnan(intensity)] = 1
return intensity
def rho(seperation):
return np.sqrt((X+seperation/2)**2 + Y**2)
def rho2(seperation):
return np.sqrt((X-seperation/2)**2 + Y**2)
# Create a grid of values
grid_size = 100 # Resolution of the image
xylim = 1.25
x = np.linspace(-xylim, xylim, grid_size)
y = np.linspace(-xylim, xylim, grid_size)
X, Y = np.meshgrid(x, y)
# create figure
fig = make_subplots(rows = 1, cols = 2,
specs=[[{'is_3d': True} , {'is_3d': False}]],
column_widths=[0.2, 0.2],
horizontal_spacing = 0.02)
#slider traces
for step in np.arange(0, maxseperation, 0.05):
Z = (airy(rho(step)) + airy(rho2(step)))
fig.add_trace(
go.Surface(
visible = False,
name = "d = " + str(step*0.05),
x = X,
y = Y,
z = Z,
colorscale = "Turbo"
),row = 1, col = 1)
for step in np.arange(0,maxseperation, 0.05):
Z = (airy(rho(step)) + airy(rho2(step)))
fig.add_trace(
go.Heatmap(
visible = False,
z = Z,
colorscale = "Turbo"
),row = 1, col = 2)
#initial
fig.data[9].visible = True
fig.data[9+ round(len(fig.data)/2)].visible = True
#slider
steps = []
for ii in range(round(len(fig.data)/2)):
step = dict(
label = '',
method="update",
args=[{"visible": [False] * len(fig.data)}], # layout attribute
)
step["args"][0]["visible"][ii] = True # Toggle i'th trace to "visible"
step["args"][0]["visible"][ii + round(len(fig.data)/2)] = True # Toggle i'th trace to "visible"
steps.append(step)
sliders = [dict(
active=10,
currentvalue = {"prefix": "Separation between disks" },
pad={"t": 10},
steps=steps
)]
fig.update_layout(
scene = dict(
xaxis = dict(nticks=1, range=[-xylim,xylim],),
yaxis = dict(nticks=1, range=[-xylim,xylim],),
zaxis = dict(nticks = 3, range = [-0.1,2]),
zaxis_title = "Intensity",
aspectmode = "manual",
aspectratio = dict(x=1,y=1,z=0.5),
camera = dict(
eye = dict(x=3.0,y=3.0, z=5.5)
),
dragmode = False
),
scene_dragmode = False,
autosize = False,
width= 600,
height= 400,
title={
'text' : " ",
'x':0.5,
'xanchor': 'center',
"font" : dict(size =25),
},
margin=dict(r=30, l=30, b=30, t=80),
scene_aspectmode="manual",
scene_aspectratio = dict(x = 2, y = 2, z = 1),
sliders = sliders,
xaxis = dict(
tickmode = 'array',
tickvals = [],
ticktext = []),
yaxis = dict(
tickmode = 'array',
tickvals = [],
ticktext = []),
)
fig.update_traces(contours_z=dict(show=True, usecolormap=True,
highlightcolor="limegreen", project_z=True),
row = 1, col = 1)
#fig.show()
#save the graph and load it later)
glue("AiryDisks",fig, display=False)
```
```{glue:figure} AiryDisks
:align: center
:name: fig:AiryDisks
Illustration of the diffraction limit: two ideal infinitely small point sources will appear in the microscope image with two finite, "blurred", intensity profiles that are described by the microscope's Point Spread Function. When the two sources are well separated, these two intensity profiles can be clearly distinguished. The slider allows to change the distance between the two point sources. When this distance decreases, both profiles start to overlap until at some distance, the two peaks cannot be discriminated anymore. The distance at which this happens is called the diffraction limit.
```
Based on equation {eq}`diffraction-limit`, we can see that the resolution in light microscopes can be improved by maximizing the opening angle and/or the refractive index. This is typically done in high-resolution light microscopy by using water- or oil-immersion lenses with $NA = 1 – 1.5$, or air-immersion lenses with an NA close to $1$. Another option is to decrease the wavelength. Blue light gives a lower value for the diffraction limit compared to red light and this can be pushed further using UV or X-rays. However, using shorter wavelength and thereby higher energy radiation poses additional challenges such as sample damage and different interaction and thereby contrast mechanisms.
Alternatively, one can make sure that every image is composed of point sources with separation larger than the diffraction limit. In other words, there should be sufficient sparsity in the distribution of sources to allow recording well separated intensity profiles for each source. For small point-like emitters, this profile will be the PSF. For each source, the source location can then be retrieved by fitting the profile to find the center position. Thus all point sources contributing to the image can be localized with a resolution below the diffraction limit. Techniques that use or induce object sparsity or use other approaches such as non-linear excitation schemes to image below the diffraction limit are jointly denoted as super resolution light microscopy. Different forms of super resolution light microscopy have been developed roughly since the start of the 21st century and were awarded a [Nobel prize in chemistry in 2014](https://www.nobelprize.org/prizes/chemistry/2014/summary/). Throughout most of the 20th century, however, the Abbe limit was considered a fundamental resolution limit for visible light microscopy and researchers sought for approaches with lower wavelength radiation to achieve higher resolution microscopy. A potential solution came from the emerging field of quantum mechanics.
(sec:quantum-mechanics)=
## Quantum mechanics and the wave character of electrons
In the early 20th century, the debate whether light was behaving as particles, as postulated by Isaac Newton, or as waves as followed from the work of Christiaan Huygens, reached a conclusion with Max Planck’s work on black-body radiation and Albert Einstein’s description of the photoelectric effect. It appeared light had both particle- and wave-like properties, i.e. it came in discretized energies like particles do but also displayed wave phenomena like diffraction and interference. The particles displaying wave-like properties were termed photons. In 1924, PhD student Louis de Broglie took a bold next step, hypothesizing that where photons are waves with particle-like discrete energy and momentum, particles like electrons could also exhibit wave-like properties. In other words, electrons are wave packets that have discrete energy and mass but that can also interfere like photons do. This is now known as wave-particle duality and is a general property of all matter, albeit only observable for materials with extremely small momentum (and thus mass) like electrons or small atoms and molecules.
The de Broglie wavelength associates a wavelength $\lambda$ to a particle with mass $m$ and velocity $v$ (thus momentum $p = mv$):
$$
\lambda = \frac{h}{mv}
$$
(de-broglie-wavelength)
where $h = 6.6 \cdot 10^{-34} \: m^2kg/s$ is a fundamental constant known as Planck’s constant. We can easily see that for macroscopic objects having large momentum, the resulting wavelength becomes indiscernibly small. However, for an electron, the rest mass is very small, $m_e = 9.1 \cdot 10^{-31} \: kg$ and an electron accelerated by a few kilovolt potential difference already travels at a fraction of the speed of light. So by doing an order of magnitude estimation based on equation {eq}`de-broglie-wavelength`, we can see that the wavelength of an accelerated electron can become on the order of picometers ($1 pm = 10-3 nm$), i.e. smaller than atomic distances! In fact, doing all the proper calculations, including physics that we do not deal with here such as relativistic corrections, we can find that an electron travelling with 1kev energy has $\lambda = 39 \: pm$, and an electron travelling at 300 keV, $\lambda = 2 \: pm$. Clearly, imaging with electrons could thus theoretically give even better resolution than working with X-rays or extreme UV radiation.
Since these early days of quantum mechanics and the experimental demonstration of the wave behaviour of electrons, researchers have worked on using electrons to circumvent the light microscopy diffraction limit. Pioneering work started in the 1930s and Ernst Ruska was the most prominent figure pushing the development of the electron microscope. [In 1986, he was awarded half the Nobel prize](https://www.nobelprize.org/prizes/physics/1986/summary/) {cite}`NobelPhysics1986` for *“his fundamental work in electron optics, and for the design of the first electron microscope”* (the other half of the prize went to Gerd Binnig and Heinrich Rohrer, the inventors of the scanning tunnelling microscope).
(sec:photons-vs-electrons)=
## Photons vs electrons
Wave-particle duality and the electron wavelengths calculated in the preceding paragraph make a clear case for development of electron microscopes. Nevertheless, there are profound differences in working with electrons compared to photons. While both share their combination of observable wave- and particle-like properties, electrons have a mass ($m_e = 9.1 \cdot 10^{-31} \: kg$) and a charge ($q_e = -1.6 \cdot 10^{-19} \: C$) where photons have zero mass and zero charge. This makes for big differences in the way we handle the particles in the microscope (for instance for focusing) as well as in the way the particles interact with a sample. First and foremost, electrons interact much stronger with materials than photons do, which means that they easily scatter. Thus, just a tiny amount of material in the beam path of an electron microscope will distort the trajectories of electrons and thus limit our focusing capabilities and control over the electron beam path and profile. This is why an electron microscope operates in vacuum. Next, for manipulation of the electron beam, we need to exert a force on the electrons in vacuum, which can be done with either electric or magnetic fields. Recalling the field-force relationships for both (see also {numref}`subsec:electric-fields` and {numref}`subsec:current-ampere-magnetism`), it may be apparent that it is not trivial to create a focusing field with either of the two. We will touch upon this later in this chapter.
The sample that we are going to inspect with electron microscopy thus also needs to be mounted in the vacuum environment of the microscope. This is a harsh and unnatural environment for almost all biological materials, which thus need substantial treatment before inspection with an electron microscope can be performed. Proper sample preparation which allows mounting in vacuum but at the same time maintains as close as possible the native, hydrated state of the sample is paramount for all biological electron microscopy. We will discuss sample preparation in detail in {numref}`ch:sample-preparation`.
Finally, we have a beam of electrons penetrating the sample, thus effectively injecting electrons in the sample that interact and scatter in the sample material. Thus, we effectively have a current running through our sample, which, without mitigation, may lead to charging. Also the (compared to photons) strong interaction of electrons with the sample because of their mass and charge gives rise to very different contract mechanisms and signals in electron microscopy compared to light microscopy. Further, while radiation damage may also be an issue in light microscopy, the stronger interaction substantially aggravate this problem in electron microscopy.
(sec:electron-sources)=
## Electron sources
(subsec:the-source-unit-and-emission-of-electrons)=
### The source unit and emission of electrons
To build a microscope operating with electrons instead of light, we need to be able to construct the basic constituents of a microscope identified in {numref}`sec:basic-microscope-layout` for electrons instead of light. This means we need a source with which we can create an electron beam and then we need lenses to manipulate the beam and bring it to the sample focused for either wide-field or scanning probe illumination. In this and the next sections, we will look into these two basic components, sources and lenses. Together with the other components in the beam path, they constitute what we refer to as the electron column. As can be seen in {numref}`fig:Electron-Microscope`, the electron source is typically located at the top of the column.
```{figure} images/Images03/Electron_Microscope.png
:name: fig:Electron-Microscope
Example of a scanning electron microscope with the electron column with the source unit on top and a larger chamber where the sample can be placed below. All other components that can be seen attached to the chamber add functionality or means to manipulate the sample while in the microscope.
```
Of course, to do microscopy, we also need to detect a signal coming from the sample after irradiation with the electron beam, but this in part relates to the same principles we encounter in understanding electron lenses. Further, understanding the detector part of the electron microscope needs a basic knowledge of electron-matter interactions, the various signals these interactions can give rise to and what information about the sample these signals carry. We will deal with all these aspects in {numref}`ch:signals-electron-microscopy`.
The basic function of an electron source is to extract electrons from the source material into vacuum and project them into a beam. Obviously, this needs the source to be a conducting material that is connected to an electric circuit that can replenish the electrons lost into vacuum. In order to extract electrons, we need to supply sufficient energy to the source. In order to guide the electrons after extraction into the column that contains the lenses and other microscope components, we need to apply an electric field that accelerates electrons in the right direction. The extraction of electrons from the source can be done in three ways:
1. Application of a strong electric field. The electric field now serves two purposes: it pulls electrons out of the source and then accelerates them towards the microscope column.
2. Heating the source. Heating provides the electrons in the source material with more energy, which makes it easier for the electrons to escape into vacuum.
3. Photo-emission. Absorption of light with sufficient energy, typically in the UV range, can also provide electrons with the energy required to leave the source material.
Note that in the latter two cases, there always also needs to be an electric field present to guide the electrons after escape in the direction of the column. So in practise an electron source always utilizes a strong electric field, possibly in combination with heating or photo-emission. It should also be noted that photo-emission is only used in very specific circumstances, mostly when the aim is to create a pulsed electron beam. In this case, irradiation of the source with laser pulses leads to the emission of an electron pulse and the time between the electron pulses can be controlled by adjusting the laser pulse frequency. This situation is not relevant for biological electron microscopy and what we always have is the strong electric field, either with additional heating (a heated source) or not (a cold source). The material and shape of the actual electron emitter in the source unit can vary, with the most common situations shown in {numref}`fig: illustration-of-different-electron-sources`.
```{figure} images/Images03/illustration-of-different-electron-sources.svg
:name: fig: illustration-of-different-electron-sources
Schematic illustration of the general layout of an electron source unit. The actual electron emitter is connected via metallic rods (red) to a current supply (blue arrows) encaged by insulator material (light grey). The actual emitter can be of different materials and shaped with three examples given.
[Figure adapted from myscope.training](https://myscope.training/SEM_How_the_gun_works), CC BY-SA.
```
(subsec:the-Field-Emission-Gun)=
### The Field Emission Gun
A special situation that should be mentioned here, is the exploitation of field enhancement. A sharp tip can be used to locally enhance the electric field (see {numref}`fig:field-enhancement`) and thus facilitate the emission of electrons. Moreover, since in this case the electrons originate from a relatively small area (the sharp tip of the source), we naturally have a small electron source which, as we will see later, can be an advantage if we want to demagnify the source into a small focused probe for scanning microscopy. A source exploiting electric field enhancement is generally known as a Field Emission Gun or FEG. FEGs are very often used in high resolution electron microscopes. A FEG that operates without further heating, thus where the local electric field alone provides the force driving electrons out of the source material is known as a ColdFEG. The tip of a ColdFEG can be extremely sharp, down to a few atoms in diameter. Where every electron source is susceptible to contamination (molecules absorbing to the emission surface thereby lowering the electron emission efficiency), this is particularly true for a ColdFEG. The source unit of an electron microscope is therefore at a very high vacuum level, typically around $10^{-9} \: mbar$, where at other parts of the microscope, specifically in the sample chamber of a scanning electron microscope, the vacuum may be lower, e.g. $~5 \cdot 10^{-5} \: mbar$.
```{figure} images/Images03/field-enhancement.png
:name: fig:field-enhancement
:width: 600px
Illustration of the effect of electric field enhancement around a sharp tip. In grey is a metallic object, e.g. our electron source, kept at constant electric potential. In white is the surrounding vacuum with the drawn lines denoting equipotential lines. Electric field lines are everywhere perpendicular to the equipotential lines, so we can see that field lines concentrate near the sharp tip of the structure. As the electric field is the derivative of the potential, the closer the equipotential lines, the higher the electric field. Thus, we can also see that the electric field is highest at the tip. This electric field enhancement and guiding of electric field lines is used in lightning rods and often referred to as the lightning rod effect.
```
(subsec:the-Schottky-FEG)=
### The Schottky FEG
A special type of source is the heated FEG. This is commonly denoted as Schottky FEG or Schottky source. The Schottky FEG combines the enhanced local field of the sharp tip with heating to increase the amount of electrons that can escape the (tungsten) material into vacuum. The heating is done via the current fed to the tip, which is typically in the order of a few Ampères. The temperature at which the tip is operated is around 1800 K (see {numref}`fig:Schottky-source`). The tip of a Schottky FEG is not as sharp as that of a ColdFEG. Because of the heating, less field enhancement is needed to emit sufficient electrons, and in addition a tip as sharp as a ColdFEG tip would not be stable at these high operating temperatures. The emission face of a Schottky FEG is a single crystal plane of a few hundred nanometer in diameter. The Schottky source is the most common type of electron source because it can deliver a high current in a relatively narrow beam.
```{figure} images/Images03/schottky-source-noitalics.png
:name: fig:Schottky-source
The Schottky Field Emission Gun at different levels of magnification. [Figure adapted from the thesis of M. Bronsgeest](https://repository.tudelft.nl/record/uuid:7975ef5e-c2ea-4056-be43-6bb6c062c884) Used with permission; no further use is allowed.
```
(subsec:brightness-and-coherence)=
### Brightness and coherence
The electron source already determines several key properties of the resulting electron beam. We already mentioned above that the size of the emission area may matter as the (de)magnification of the lens system may be limited. Also whether a source is heated or not will influence the energy spread of electrons in the beam, which can be substantially reduced using a cold source. We will deal with these aspects later when we discuss the parameters that influence the beam focus on the sample. Clearly also the electron current that we can extract from the source is important as the amount of electrons that hit the sample determines the amount of signal we generate in a certain time and thereby our measurement time and signal to noise level. However, it is not so much directly the current that plays a role here but rather the amount of current that is emitted into a certain beam angle. The property that is related to this is the brightness. In the video we will explain the concept of brightness and especially the reduced brightness.
---
---
```{table} Typical characteristic parameters from the different source types
:name: table:characteristic-parameters
| | $Tungsten \: hairpin$ $(heated)$ | $LaB_6 $ $ (heated)$ | $Schottkey FEG$ | $ColdFEG$ |
| :--: | :--: | :--: | :--: | :--: |
| $Reduced \: brightness \: (A/m^2srV)$ | $10^3-10^4$ | $10^4-10^5$ | $10^7-10^8$ | $10^8$ |
| $Energy \: spread \: (eV)$ | $0.6 - 1.5$ | $10$ | $0.02 - 0.05$ | $0.005 - 0.01$ |
| $Source \: size \: (\mu m)$ | $20$ | $10$ | $0.02 - 0.05$ | $0.005 - 0.01$ |
```
Another important property in an electron beam is coherence. Coherence in the beam is needed for phase contrast imaging (e.g. in transmission electron microscopy) and for electron diffraction experiments. From a physics perspective, obtaining coherence in an electron beam is not trivial: as stated by the [Pauli exclusion principle](https://en.wikipedia.org/wiki/Pauli_exclusion_principle), no two electrons can occupy the same quantum mechanical state. Thus, two electrons can never be coherent with respect to each other, in stark contrast to light where for instance in a laser many photons are in the same coherent state. Thus, in electron microscopy, when we refer to coherence in the electron beam, we always refer to each electron being coherent with respect to itself. This means, if we know the electron phase at a particular position in the beam trajectory, do we then also know the phase at a later position? If so, the beam is coherent and the electron can interfere with itself. As all electrons will then interfere in the same way, we can build phase images or diffractograms. In the below video, we explain coherence in electron beams in a bit more detail and also demonstrate an intricate relation between coherence and brightness.
---
---
(sec:electron-lenses)=
## Electron lenses
(subsec:electrostatic)=
### Electrostatic
Once we have extracted electrons from the source, we need lenses for focusing the electron beam. The first type of electron lens we will look it is the electrostatic electron lens, a lens that is encountered already in the source unit of the microscope
---
---
(subsec:magnetic)=
### Magnetic
Besides electric fields, also magnetic fields can be used to create an electron lens. In practise, mostly magnetic lenses are used in high-resolution electron microscopes.
---
---
(sec:practical-considerations)=
## Practical considerations
In this chapter we have explored some basic knowledge behind the use of electrons for microscopy instead of light. We have seen that there are many conceptual similarities and similar types of components in electron microscopy as there are in light microscopy. However, the physics for using electrons instead of photons is very different as electrons carry a mass and a charge, leading to some profound differences between the two types of microscopes. Inside an electron microscope, we work in a vacuum, limiting the type of samples we can put in the microscope and putting a lot of emphasis on proper sample preparation procedures for biological materials. Also, inside the microscope there may be strong electric and magnetic fields for guiding and focusing the electrons. This means we have to take care not to put magnetic or magnetizable materials inside an electron microscope as they may be subjected to relatively strong forces. Any material or device that we put in the microscope that carries itself an electric or magnetic field could in turn affect the electron beam and lead to aberrations and imaging artefacts. External fields, i.e. in the surroundings of the electron microscope, will also distort the electron trajectories resulting in loss of resolution or distorted images. Highest resolution electron microscopes are therefore placed in an environment well shielded from external electromagnetic field or with active measurement and cancellation of the external fields, e.g. with [Helmholt coils](https://en.wikipedia.org/wiki/Helmholtz_coil).
In {numref}`sec:quantum-mechanics`, we have seen that theoretically the diffraction limit for energetic electrons can be extremely small. In practise, lenses never work perfectly and deviations of the actual lens action from theoretical perfection lead to aberrations in the images and a loss of resolution. For light microscopy, glass lenses can be shaped in different forms, precisely and finely polished, and coated with thin layers in order to optimize the performance and reduce aberrations. In high-resolution light microscopes a single lens often consists of a stack of multiple individual lenses all precisely shaped and arranged to jointly give a near perfect behaviour. For electron microscopy, lenses are macroscopic and bulky, which easily leads to imperfections. Electric and magnetic lens fields are difficult to manipulate and always have substantial, positive aberration coefficients. This makes it much harder to correct for aberrations compared to light optics. Thus, as we will see in more detail in the next chapter, electron microscopes are almost always limited by aberrations and not by diffraction. In fact, while the first electron lenses and electron microscopes were developed in the 1930’s, it took over 60 years for aberration correctors to be developed to take the step towards actual atomic resolution imaging.
## References
```{bibliography}
:filter: docname in docnames
```