Quantum Mechanics

Light

Energy Levels and Electronic Transitions

Atoms and molecules possess discrete energy levels, which are a direct consequence of the wave-like nature of electrons and the quantized angular momentum they carry. When a photon of light encounters a molecule, its energy can be absorbed by an electron which then transitions to a higher energy level, entering an excited state. The energy difference between the initial and final states of the electron corresponds to the energy of the photon, as dictated by the equation: $$\Delta E=E_f-E_i=h\nu$$ where $\Delta E$ is the energy difference, $E_f$ and $E_i$ are the final and initial energy levels respectively, h is Planck's constant, and nu is the frequency.

Absorption and Emission

The process described above is known as absorption. Following absorption, the electron occupies an excited state, but this state is often unstable, leading the electron to seek a return to a lower energy level. This transition back to a lower energy level can result in the emission of a photon, a process that can manifest as either fluorescence or phosphorescence depending on the nature of the electronic transition.

Types of Transitions

Fluorescence

The downward transition from the excited state to the ground state with the emission of a photon. This process occurs relatively quickly, typically in nanoseconds.

Intersystem Crossing (ISC)

A spin-flip transition from a singlet excited state to a triplet excited state, which is a precursor to phosphorescence.

Phosphorescence

The delayed emission of a photon as the molecule transitions from the triplet excited state to the ground state. This process can take place over a much longer timescale, ranging from microseconds to seconds.

Electron Microscopy

Introduction

Why do we need to use electrons to view cellular substructure? How does the energy of a photon relate?

Optics and Wavelength

Diffraction causes light waves emanating from a point source to spread out, forming a pattern called an Airy disk. When two objects are too close together, their Airy disks overlap, making them indistinguishable.

Abbe Diffraction Limit

The Abbe diffraction limit states that light has a minimum resolvable distance given by $$d=\frac{\lambda}{2n\sin(\theta)}$$ where d is the minimum resolvable distance, $\lambda$ is the wavelength of light, and $n\sin(\theta)$ is the "numerical aperture."

As humans are limited to the visible portion of the electromagnetic spectrum, this also limits the ability of traditional light microscopes. For a numerical aperture of 1.5 and a wavelength of 300nm, we would be limited to a minimum resolvable distance of 100nm. However, this is not sufficient for biologists interested in cellular substructure.

Wave-Particle Duality and De Broglie Wavelength

One of the important fundamental concepts in quantum mechanics is wave-particle duality. The De Broglie wavelength associated with matter has a wavelength given by $$\lambda=\frac{h}{p}$$ where $\lambda$ is the wavelength, p is the momentum, and h is Planck's constant.

Moving to Electrons

Mathematically, when we use the values for an electron, we discover that we get a smaller wavelength than we have for light. Electrons in an electron microscope are accelerated to high velocities, giving them significant momentum and very short de Broglie wavelengths. This means that electron microscopes can achieve much higher resolution than optical microscopes because they can resolve smaller features due to their shorter wavelengths.

Electron Scattering

Introduction to Electron Scattering

Electron scattering occurs when electrons interact with atoms in the specimen being imaged. As the electrons pass through or near the specimen, their trajectories are deflected due to electromagnetic interactions with atomic nuclei and electrons. The scattering of electrons is what enables the formation of images in electron microscopy.

Types of Scattering

In elastic scattering, the kinetic energy of the electrons is conserved, although their direction may change. Elastic scattering is primarily responsible for image contrast in transmission electron microscopy (TEM).

In inelastic scattering, the kinetic energy of the electrons is not conserved as they can lose or gain energy during interaction with the specimen. Inelastic scattering contributes to image contrast but can also cause radiation damage to the specimen.

Interactions

The angle at which electrons are scattered is crucial for image formation. High-angle scattering often corresponds to interactions with atomic nuclei, while low-angle scattering typically arises from interactions with electrons. The volume of the specimen with which the electrons interact, known as the interaction volume, also plays a significant role in determining the resolution and image contrast.

Electron scattering leads to diffraction patterns which are captured on a detector. The diffraction patterns are transformed into real-space images through a process called Fourier transformation. The quality and clarity of the images depend significantly on the scattering process and the subsequent diffraction.

Result

Contrast

Electron scattering creates variations in intensity across the image, which results in contrast. The degree of scattering is influenced by the atomic number of the elements in the specimen, the thickness of the specimen, and the electron beam energy.

Resolution

The resolution of electron microscopy images is often limited by the scattering of electrons. By minimizing inelastic scattering and optimizing the imaging conditions, higher resolution images can be obtained.

Radiation Damage

Inelastic scattering can cause radiation damage to the specimen, which is a significant concern, especially in biological electron microscopy.

Types of Electron Microscopy

Transmission Electron Microscopy

In TEM, a beam of electrons is transmitted through an ultra-thin specimen, interacting with the specimen as it passes through. The transmitted electrons are then focused to form an image on a phosphorescent screen or detector.

Scanning Electron Microscopy

Unlike TEM, SEM creates images by detecting secondary electrons which are emitted from the surface of the specimen when it is scanned by a focused beam of electrons.

Scanning Transmission Electron Microscopy

STEM combines principles of both TEM and SEM. It scans a focused electron beam across the specimen and detects both transmitted and scattered electrons to form images.

Cryo-Electron Microscopy

Cryo-EM involves flash freezing specimens in liquid ethane to preserve their native structures, followed by imaging with TEM or STEM under cryogenic conditions.

Background Math

In Quantum Mechanics, we state that a system has a defined property if the wavefunction is in an eigenstate of that property's associated operator. One example would be for a particle having a defined position. The position operator is $$\hat{x}\psi(x)=x\cdot \psi(x)$$ So the wavefunction that would correspond to a particle having a defined position would satisfy $$x\psi(x) = k\psi(x)$$ where k is a scalar representing the x position of the particle. The solution to the equation would be $$\psi(x)=\delta(x-k)$$ where $\delta$ is the Dirac delta "function." So does this particle with a defined position also have a defined momentum? The momentum operator, p, is defined as $$\hat{p}=-i\hbar\nabla$$ Taking the derivative of the Dirac delta function does not yield a scalar multiple of it (if we attempt to define it), so we can conclude that a particle with a defined position does not have a defined momentum. $$\frac{d}{dx}\delta(x)=-\frac{\delta(x)}{x}\not\propto\delta(x)$$