Electron Microscope - Science & References

🌊 De Broglie Wavelength

Louis de Broglie proposed in 1924 that all matter exhibits wave-like behavior. The wavelength of a particle is inversely proportional to its momentum. For electrons accelerated through a potential difference $V$, the de Broglie wavelength is:

De Broglie Wavelength (non-relativistic)

$$\lambda = \frac{h}{\sqrt{2 m_e e V}}$$

Where: $h$ = Planck's constant (6.626 × 10⁻³⁴ J·s), $m_e$ = electron mass (9.109 × 10⁻³¹ kg), $e$ = electron charge (1.602 × 10⁻¹⁹ C), $V$ = accelerating voltage (V).

Relativistic Correction (for V > 100 kV)

$$\lambda = \frac{h}{\sqrt{2 m_e e V \left(1 + \frac{eV}{2 m_e c^2}\right)}}$$

At high voltages, the electron velocity approaches a significant fraction of the speed of light. The relativistic correction factor $\left(1 + \frac{eV}{2m_ec^2}\right)$ accounts for the increase in effective mass. At 200 kV, the correction reduces the wavelength by ~10%.

For comparison: visible light has wavelengths of 400–700 nm. A 100 kV electron has a wavelength of ~0.0037 nm — about 100,000 times shorter than visible light.

🔍 Resolution & Resolving Power

The resolving power of any microscope is fundamentally limited by the wavelength of the radiation used. The Rayleigh criterion defines the minimum resolvable distance:

Rayleigh Criterion

$$d = \frac{0.61 \, \lambda}{n \sin \alpha} = \frac{0.61 \, \lambda}{\text{NA}}$$

Where: $d$ = minimum resolvable distance, $\lambda$ = wavelength, $n$ = refractive index of medium, $\alpha$ = half-angle of the maximum cone of light entering the objective, NA = numerical aperture.

Abbe Diffraction Limit

$$d = \frac{\lambda}{2 \, \text{NA}}$$

Ernst Abbe showed that the finest detail resolvable is approximately half the wavelength divided by the numerical aperture. For an optical microscope (λ ≈ 550 nm, NA ≈ 1.4), d ≈ 200 nm. For a TEM at 200 kV (λ ≈ 0.0025 nm), the theoretical limit is sub-angstrom.

🧲 Electromagnetic Lenses

Electron microscopes use electromagnetic lenses instead of glass lenses. A coil carrying current creates a magnetic field that exerts a Lorentz force on moving electrons, bending their trajectories.

Lorentz Force on Electron

$$\vec{F} = -e(\vec{v} \times \vec{B})$$

The force is always perpendicular to both the velocity $\vec{v}$ and the magnetic field $\vec{B}$. This causes electrons to spiral along helical paths, creating a focusing effect analogous to a glass lens.

Focal Length of Magnetic Lens

$$\frac{1}{f} = \frac{e}{8 m_e V} \int_{-\infty}^{\infty} B_z^2(z) \, dz$$

The focal length depends on the axial magnetic field distribution $B_z(z)$ and the accelerating voltage $V$. Stronger fields or lower voltages produce shorter focal lengths (stronger focusing).

Key difference from glass lenses: Electromagnetic lenses always converge — there are no diverging electron lenses. Aberration correction requires complex multipole elements.

⚛ Electron-Specimen Interactions

When the electron beam hits the specimen, multiple signals are generated simultaneously:

Secondary Electrons (SE): Low-energy electrons (< 50 eV) ejected from the specimen surface. Provide topographic information. Escape depth: ~5–50 nm.
Backscattered Electrons (BSE): High-energy electrons reflected by elastic scattering from atomic nuclei. Intensity depends on atomic number $Z$ (heavier elements scatter more). Provide compositional contrast.
Characteristic X-rays: Emitted when inner-shell electrons are ejected and outer-shell electrons fill the vacancy. Each element produces unique X-ray energies, enabling elemental analysis (EDS/EDX).
Auger Electrons: Alternative to X-ray emission — energy is transferred to another electron which is ejected. Surface-sensitive technique.
Cathodoluminescence: Visible light emitted by certain materials when excited by the electron beam.
Transmitted Electrons (TEM): Electrons that pass through thin specimens. Elastic scattering provides diffraction contrast; inelastic scattering provides chemical information (EELS).

Rutherford Scattering Cross-Section (BSE)

$$\sigma = \frac{Z^2 e^4}{16 E^2} \cot^2\left(\frac{\theta_{\min}}{2}\right)$$

The probability of backscattering increases with the square of the atomic number $Z$, making BSE imaging sensitive to composition. $E$ is the beam energy and $\theta_{\min}$ is the minimum scattering angle detected.

⚡ Electron Beam Energy & Penetration

The kinetic energy of accelerated electrons determines their penetration depth into the specimen:

Kanaya-Okayama Penetration Depth

$$R = \frac{0.0276 \, A \, E_0^{1.67}}{Z^{0.89} \, \rho}$$

Where: $R$ = penetration depth (μm), $A$ = atomic weight (g/mol), $E_0$ = beam energy (keV), $Z$ = atomic number, $\rho$ = density (g/cm³). Higher beam energy means deeper penetration but lower surface sensitivity.

🧮 De Broglie Wavelength Calculator

Calculate the electron wavelength for a given accelerating voltage. Both non-relativistic and relativistic results are shown.

Interactive Calculator

Accelerating Voltage: 100 kV

Non-Relativistic Wavelength

0.00386 nm

About 142,000× shorter than green light (550 nm)

Relativistic Wavelength

0.00370 nm

Relativistic correction: 4.1% difference at 100 kV

Electron Velocity

1.88 × 10⁸ m/s

62.6% of the speed of light

🔍 Resolution Calculator

Estimate the theoretical resolution limit for different microscope types.

Compare Resolution Limits

Light Microscope NA: 1.40

Light Wavelength (nm): 550 nm

EM Voltage (kV): 200 kV

Light Microscope Resolution

~240 nm

Electron Microscope Theoretical Resolution

~0.0015 nm

~160,000× better than the light microscope (limited by lens aberrations to ~0.05–0.1 nm in practice)

📐 Magnification & Field of View

Understand the relationship between magnification and the area you can observe.

Field of View Calculator

Magnification: 10,000×

Screen Width (mm): 150 mm

Field of View

15.0 μm

At 10,000× magnification, you can see an area about 15.0 μm wide — roughly the size of a red blood cell.

📊 Light Microscope vs Electron Microscope

Feature	Light Microscope	SEM	TEM
Radiation Source	Visible light (400–700 nm)	Electrons	Electrons
Wavelength	400–700 nm	~0.01–0.001 nm	~0.005–0.001 nm
Max Magnification	~1,500×	~500,000×	~50,000,000×
Resolution	~200 nm	~1–5 nm	~0.05–0.1 nm
Image Type	2D, color	3D surface, grayscale	2D projection, grayscale
Specimen Prep	Minimal	Conductive coating	Ultra-thin sectioning (~50–100 nm)
Environment	Air / liquid	High vacuum	High vacuum
Living Specimens	Yes	No (except ESEM)	No (cryo-EM possible)
Depth of Field	Very shallow	Very large	Moderate
Cost	$100 – $50,000	$100,000 – $1,000,000	$500,000 – $10,000,000
Lenses	Glass	Electromagnetic	Electromagnetic

⚖ SEM vs TEM — Detailed Comparison

Aspect	SEM	TEM
Beam Interaction	Scans surface; detects secondary & backscattered electrons	Transmits through specimen; detects transmitted electrons
Specimen Thickness	Bulk (any thickness)	Ultra-thin (< 100 nm)
Voltage Range	1–30 kV	60–300 kV
Information	Surface morphology, topography	Internal structure, crystal lattice
Elemental Analysis	EDS, WDS	EDS, EELS
3D Imaging	Yes (natural 3D appearance)	Tomography possible
Sample Prep Difficulty	Low–Medium	High (FIB, ultramicrotomy)

📏 Scale of Observable Structures

Structure	Size	Visible With
Human hair	~80 μm	Naked eye
Red blood cell	~7 μm	Light microscope
Bacterium (E. coli)	~2 μm	Light microscope
Mitochondrion	~500 nm	Light / SEM
Virus (HIV)	~120 nm	SEM / TEM
Ribosome	~25 nm	TEM
DNA double helix width	~2 nm	TEM
Single atom	~0.1–0.3 nm	Aberration-corrected TEM

📜 History of Electron Microscopy

1924

Louis de Broglie proposes wave-particle duality — electrons have wavelengths inversely proportional to their momentum. This theoretical foundation makes electron microscopy conceivable.

1926

Hans Busch demonstrates that magnetic fields can focus electron beams, analogous to glass lenses focusing light — the birth of electron optics.

1931

Ernst Ruska & Max Knoll build the first electron microscope prototype in Berlin. It achieves only 17× magnification but proves the concept.

1933

Ruska builds an electron microscope that exceeds the resolution of optical microscopes for the first time — a watershed moment in microscopy.

1935

Max Knoll produces the first scanning electron microscope (SEM) image by scanning an electron beam across a surface.

1938

First commercial TEM produced by Siemens, designed by Ruska and Bodo von Borries. Resolution: ~10 nm.

1942

Zworykin, Hillier & Snyder at RCA develop the first practical SEM with secondary electron detection.

1965

Cambridge Instrument Company produces the first commercial SEM (Stereoscan), designed by Charles Oatley's group. Revolutionizes surface imaging.

1975

Development of STEM (Scanning Transmission Electron Microscope) by Albert Crewe, achieving atomic resolution with heavy atoms on thin carbon films.

1986

Ernst Ruska receives the Nobel Prize in Physics for his fundamental work in electron optics and the design of the first electron microscope.

1990s

Development of Environmental SEM (ESEM) allowing imaging of wet, non-conductive specimens without coating.

1998

Aberration-corrected electron microscopes become practical, pushing TEM resolution below 0.1 nm (sub-angstrom).

2017

Jacques Dubochet, Joachim Frank, Richard Henderson receive the Nobel Prize in Chemistry for developing cryo-electron microscopy (cryo-EM) for high-resolution structure determination of biomolecules.

2020s

Modern TEMs achieve resolutions of ~0.04 nm. AI-assisted image processing and automated data collection transform structural biology and materials science.

💡 Key Facts & Numbers

0.04 nm

Best TEM Resolution

Modern aberration-corrected TEMs can resolve individual atoms, seeing details 10,000× smaller than a virus.

50,000,000×

Maximum Magnification

TEMs can magnify up to 50 million times — enough to see individual atoms in a crystal lattice.

10⁻⁷ Pa

Operating Vacuum

The column operates at ultra-high vacuum, about one-trillionth of atmospheric pressure, to prevent electron scattering.

300 kV

Max Accelerating Voltage

High-voltage TEMs accelerate electrons to ~78% of the speed of light at 300 kV.

2,700°C

Tungsten Filament Temp

The tungsten filament in a thermionic gun is heated to extreme temperatures to emit electrons via thermionic emission.

~50 nm

TEM Specimen Thickness

TEM specimens must be ultra-thin (50–100 nm) so electrons can pass through. That's about 500 atoms thick.

1931

Year of Invention

Ernst Ruska and Max Knoll built the first electron microscope in Berlin, Germany.

3 Nobel Prizes

Nobel Recognition

Electron microscopy has contributed to at least 3 Nobel Prizes: Ruska (1986), cryo-EM (2017), and AlphaFold-related structural biology.

⚡ Electron Speed at Different Voltages

Voltage (kV)	Wavelength (nm)	Velocity (m/s)	% Speed of Light

❓ Frequently Asked Questions

Electron microscopes require high vacuum inside the column (10⁻⁴ to 10⁻⁷ Pa). Living cells contain water, which would instantly evaporate in vacuum, destroying the cell. Additionally, the high-energy electron beam causes radiation damage to biological specimens. However, cryo-electron microscopy (cryo-EM) freezes specimens so rapidly that water forms vitreous ice (non-crystalline), preserving near-native structure for imaging.

Color is a property of visible light (electromagnetic radiation with wavelengths 400–700 nm). Electrons are particles, not photons, so they don't have "color." The detectors measure electron intensity (number of electrons hitting each point), which is displayed as brightness levels — hence grayscale. Colored EM images you see in textbooks are artificially colorized (false color) for clarity.

Magnification is how much larger the image appears compared to the actual specimen. Resolution is the ability to distinguish two closely spaced points as separate. You can magnify an image infinitely (like zooming into a digital photo), but beyond the resolution limit, you only see blur — no new detail. An electron microscope's power comes from its high resolution (due to short electron wavelength), not just high magnification.

Non-conductive specimens (biological tissue, polymers, ceramics) accumulate electric charge when hit by the electron beam. This charge buildup deflects the beam and causes bright spots and image distortion (charging artifacts). A thin conductive coating (gold, platinum, carbon — typically 5–20 nm thick) provides a path for charge to dissipate. Environmental SEM (ESEM) and low-voltage SEM can image uncoated specimens by using gas molecules to neutralize charge.

TEM specimens must be electron-transparent, typically 50–100 nm thick (about 500 atoms). Preparation methods include:

Ultramicrotomy: Cutting ultra-thin sections with a diamond knife
Focused Ion Beam (FIB): Milling with gallium ions for site-specific preparation
Electropolishing: Electrolytic thinning for metals
Ion milling: Argon ion bombardment for ceramics and semiconductors
Crushing/dispersion: For nanoparticles and powders

Cryo-EM involves rapidly freezing biological specimens in liquid ethane (~-180°C) so that water forms vitreous (amorphous) ice instead of crystals. This preserves the native 3D structure of proteins and other biomolecules. Combined with single-particle analysis (averaging thousands of images of identical molecules in random orientations), cryo-EM can determine protein structures at near-atomic resolution (2–3 Å) without crystallization. This earned the 2017 Nobel Prize in Chemistry.

Electromagnetic lenses suffer from aberrations that limit practical resolution far below the theoretical diffraction limit:

Spherical aberration (Cs): Electrons farther from the optical axis are focused more strongly, causing a disc of confusion instead of a point. This is the dominant aberration in TEM.
Chromatic aberration (Cc): Electrons with slightly different energies are focused at different points. Reduced by using monochromators.
Astigmatism: Non-circular lens fields cause different focal lengths in different directions. Corrected with stigmator coils.

Modern aberration correctors (multipole elements) can reduce Cs to near zero, enabling sub-angstrom resolution.

📚 Bibliography & References

A curated list of foundational textbooks, landmark papers, and authoritative resources on electron microscopy.

📖 Foundational Textbooks

[1] Williams, D.B. & Carter, C.B. (2009). Transmission Electron Microscopy: A Textbook for Materials Science (2nd ed.). Springer. ISBN: 978-0-387-76500-6.
DOI: 10.1007/978-0-387-76501-3

[2] Goldstein, J.I., Newbury, D.E., Michael, J.R., Ritchie, N.W.M., Scott, J.H.J. & Joy, D.C. (2018). Scanning Electron Microscopy and X-Ray Microanalysis (4th ed.). Springer. ISBN: 978-1-4939-6674-5.
DOI: 10.1007/978-1-4939-6676-9

[3] Reimer, L. & Kohl, H. (2008). Transmission Electron Microscopy: Physics of Image Formation (5th ed.). Springer Series in Optical Sciences, Vol. 36. ISBN: 978-0-387-40093-8.
DOI: 10.1007/978-0-387-40093-8

[4] Egerton, R.F. (2016). Physical Principles of Electron Microscopy: An Introduction to TEM, SEM, and AEM (2nd ed.). Springer. ISBN: 978-3-319-39876-7.
DOI: 10.1007/978-3-319-39877-4

[5] Hawkes, P.W. & Spence, J.C.H. (Eds.) (2019). Springer Handbook of Microscopy. Springer Handbooks. ISBN: 978-3-030-00068-4.
DOI: 10.1007/978-3-030-00069-1

📄 Landmark Papers

[6] de Broglie, L. (1925). Recherches sur la théorie des quanta. Annales de Physique, 10(3), 22–128.
DOI: 10.1051/anphys/192510030022

[7] Knoll, M. & Ruska, E. (1932). Das Elektronenmikroskop. Zeitschrift für Physik, 78(5–6), 318–339.
DOI: 10.1007/BF01342199

[8] Ruska, E. (1987). The development of the electron microscope and of electron microscopy (Nobel Lecture). Reviews of Modern Physics, 59(3), 627–638.
DOI: 10.1103/RevModPhys.59.627

[9] Crewe, A.V., Wall, J. & Langmore, J. (1970). Visibility of single atoms. Science, 168(3937), 1338–1340.
DOI: 10.1126/science.168.3937.1338

[10] Haider, M., Uhlemann, S., Schwan, E., Rose, H., Kabius, B. & Urban, K. (1998). Electron microscopy image enhanced. Nature, 392(6678), 768–769.
DOI: 10.1038/33823

🧬 Cryo-Electron Microscopy

[11] Dubochet, J., Adrian, M., Chang, J.J., Homo, J.C., Lepault, J., McDowall, A.W. & Schultz, P. (1988). Cryo-electron microscopy of vitrified specimens. Quarterly Reviews of Biophysics, 21(2), 129–228.
DOI: 10.1017/S0033583500004297

[12] Frank, J. (2006). Three-Dimensional Electron Microscopy of Macromolecular Assemblies: Visualization of Biological Molecules in Their Native State. Oxford University Press. ISBN: 978-0-19-518218-7.
DOI: 10.1093/acprof:oso/9780195182187.001.0001

[13] Kühlbrandt, W. (2014). The resolution revolution. Science, 343(6178), 1443–1444.
DOI: 10.1126/science.1251652

🌐 Online Resources

[14] Nobel Prize Committee. (1986). The Nobel Prize in Physics 1986 — Ernst Ruska. nobelprize.org/prizes/physics/1986

[15] Nobel Prize Committee. (2017). The Nobel Prize in Chemistry 2017 — Cryo-EM. nobelprize.org/prizes/chemistry/2017

[16] NIST. Scanning Electron Microscopy — National Institute of Standards and Technology. nist.gov/programs-projects/scanning-electron-microscopy

[17] MyScope — Microscopy Training. Online training for electron microscopy techniques. myscope.training

[18] Microscopy Society of America (MSA). Educational resources and publications. microscopy.org

📘 Additional Reading

[19] Pennycook, S.J. & Nellist, P.D. (Eds.) (2011). Scanning Transmission Electron Microscopy: Imaging and Analysis. Springer. ISBN: 978-1-4419-7199-9.
DOI: 10.1007/978-1-4419-7200-2

[20] Fultz, B. & Howe, J.M. (2013). Transmission Electron Microscopy and Diffractometry of Materials (4th ed.). Springer. ISBN: 978-3-642-29760-1.
DOI: 10.1007/978-3-642-29761-8

⚛ Electron Microscope

The Science of Electron Microscopy