Phase contrast physics ====================== Understanding electron wave phase ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ Electron microscopy faces a fundamental challenge: **we cannot directly measure phase**, yet phase often carries the most important information about our samples. This is known as the phase problem. How phase is lost ^^^^^^^^^^^^^^^^^ Here is what actually happens in the microscope: 1. **Incident wave enters sample**: .. math:: \psi_{\text{in}}(x,y) = e^{ik_0 z} 2. **Sample modifies the wave** (object transmission function): .. math:: \psi_{\text{exit}}(x,y) = \psi_{\text{in}}(x,y) \cdot O(x,y) = \psi_{\text{in}}(x,y) \cdot A(x,y) e^{i\phi(x,y)} The object :math:`O(x,y) = A(x,y) e^{i\phi(x,y)}` encodes both amplitude changes :math:`A` (scattering, absorption) and phase shifts :math:`\phi` (potential). 3. **Exit wave propagates to detector**: .. math:: I(k_x, k_y) = |\mathcal{F}\{\psi_{\text{exit}}(x,y)\}|^2 We measure the diffraction pattern intensity in reciprocal space (k-space). 4. **Phase is lost**: The detector measures :math:`|\cdot|^2`, which destroys the complex phase :math:`e^{i\phi}`. We lose the information connecting :math:`\phi(x,y)` to the potential :math:`V(x,y)`. **Phase retrieval** is the inverse problem: Given measured intensities :math:`I(k_x,k_y)`, reconstruct the phase :math:`\phi(x,y)` (and thus the potential :math:`V` through :math:`\phi = \sigma V`). Why we need phase ^^^^^^^^^^^^^^^^^ Phase and intensity provide **complementary information**: - **Intensity** (what we measure): Shows where electrons scatter strongly. Scales as ~:math:`Z^2` (atomic number squared). Heavy elements (high Z) show strong contrast. - **Phase** (what we want): Shows electromagnetic field distributions. Scales as :math:`\int V \, dz` (integrated potential). Even light elements create substantial phase shifts. This distinction matters for **light-element imaging**. Consider a biological sample made of carbon, nitrogen, and oxygen (:math:`Z = 6, 7, 8`): - **Weak scattering**: Low Z → low intensity contrast (~:math:`Z^2`) → barely visible in conventional TEM - **Strong potentials**: Atoms still create electrostatic potentials → substantial phase accumulation (:math:`\phi \propto \int V \, dz`) A thin biological sample may be nearly invisible in intensity images but creates large, measurable phase shifts. **Phase retrieval is essential for imaging soft matter and light-element structures.** .. plot:: :include-source: False import numpy as np import matplotlib.pyplot as plt # Create 1D position array x = np.linspace(0, 10, 500) # Atom positions (in nm) atom_positions = [2, 4, 6, 8] # Build electrostatic potential as sum of Gaussian peaks V = np.zeros_like(x) for x0 in atom_positions: V += 3 * np.exp(-((x - x0)**2) / 0.1) # Create figure fig, ax = plt.subplots(figsize=(10, 4)) # Plot potential ax.plot(x, V, 'b-', linewidth=2.5, label='Electrostatic potential V(x)') ax.fill_between(x, 0, V, alpha=0.2, color='blue') # Mark atom positions for x0 in atom_positions: ax.axvline(x0, color='red', linestyle='--', alpha=0.5, linewidth=1.5) ax.plot(x0, 3, 'ro', markersize=10, zorder=5) # Labels and formatting ax.set_xlabel('Position x (nm)', fontsize=14, fontweight='bold') ax.set_ylabel('Potential V (arbitrary units)', fontsize=14, fontweight='bold') ax.set_title('Electrostatic potential from atomic nuclei\nPositive charges create potential "hills"', fontsize=16, fontweight='bold', pad=15) ax.legend(['V(x) from nuclei', 'Atom positions'], fontsize=12, loc='upper right') ax.grid(True, alpha=0.3, linestyle='--') ax.set_ylim(-0.2, 3.5) ax.tick_params(labelsize=12) plt.tight_layout() plt.show() The electrostatic potential forms "hills" at atomic positions. Negative electrons are attracted toward these peaks (positive nuclei), causing deflection and phase accumulation. Physics of phase shift ---------------------- When an electron wave interacts with a sample, it undergoes two distinct changes: 1. **Amplitude changes** :math:`A(x,y)`: Electrons are absorbed or scattered away - Easy to measure directly - Shows where electrons interact strongly - Scales as ~:math:`Z^2` (atomic number squared) - Heavy elements give strong contrast 2. **Phase shifts** :math:`\phi(x,y)`: The wave's oscillation is delayed by the sample's potential - Cannot measure directly - Contains crucial information about fields and potentials - Scales linearly with potential - Even light elements create substantial shifts .. important:: The phase problem arises because our detectors can only measure intensity :math:`|\psi|^2 = |A|^2`, which tells us about amplitude but completely erases phase information. Mathematical foundations ^^^^^^^^^^^^^^^^^^^^^^^^ The phase retrieval problem can be expressed mathematically as: .. math:: I(x,y) = |\psi(x,y)|^2 = |A(x,y)|^2 where: - :math:`I(x,y)` is the measured intensity - :math:`\psi(x,y)` is the complex wave function - :math:`A(x,y)` is the amplitude - We want to recover :math:`\phi(x,y)` where :math:`\psi(x,y) = A(x,y)e^{i\phi(x,y)}` The measurement challenge ^^^^^^^^^^^^^^^^^^^^^^^^^^^ When an electron wave passes through a sample, it undergoes two types of changes: 1. **Amplitude changes** :math:`A(x,y)`: electrons are absorbed or scattered away 2. **Phase shifts** :math:`\phi(x,y)`: the wave's oscillation is delayed by the sample's electrostatic potential The **exit wave** (the wave leaving the sample) carries both amplitude and phase information. However, detectors can only measure **intensity** = :math:`|\psi_{\text{exit}}|^2`, which contains amplitude information but completely erases the phase. This is problematic because **the exit wave's phase** :math:`\phi(x,y)` directly encodes the sample's electrostatic potential :math:`V(x,y)`. .. _phase-shift-physics: How phase shifts occur ^^^^^^^^^^^^^^^^^^^^^^ .. note:: **Key assumptions in this phase shift model:** The following summarizes the main approximations used to derive the phase object approximation :math:`\phi = \sigma V`. **Non-relativistic quantum mechanics** - *What it means*: Use Schrödinger equation (not Dirac equation) to describe electron wavefunction - *Valid when*: Works well for typical TEM energies (80-300 keV). We still account for relativistic effects on electron mass and wavelength - *Note*: See separate quantum mechanics page for detailed discussion of relativistic corrections **Elastic scattering** - *What it means*: No energy transfer to sample - *Valid when*: Dominant at small angles. Inelastic scattering increases with thickness and Z **Single scattering** - *What it means*: Electron interacts with potential only once - *Valid when*: Thin samples (< 10-50 nm). Multiple scattering increases with thickness and Z **Paraxial approximation** - *What it means*: Electron travels primarily along beam direction; transverse momentum changes small - *Valid when*: :math:`|\partial A/\partial z| \ll k_0 |A|`. Valid for small angles (< few mrad) **Projection approximation** - *What it means*: Neglect transverse diffraction: :math:`\nabla_{\perp}^2 A \approx 0` - *Valid when*: Sample thickness ≪ transverse feature size. Valid for t < 20-50 nm **Weak-phase object** - *What it means*: Phase shift small: :math:`|\sigma V| \ll 1`, allowing :math:`e^{i\sigma V} \approx 1 + i\sigma V` - *Valid when*: Thin samples with low Z. Breaks down when :math:`\sigma V > 0.5` radians **Electrostatic approximation** - *What it means*: Magnetic interactions negligible - *Valid when*: Valid except for ferromagnets. Magnetic phase :math:`\sim 10^{-3}` of electrostatic **When these assumptions break down:** - **Thick samples**: Multiple scattering, breakdown of projection approximation, strong phase shifts - **Heavy elements**: Large phase shifts violate weak-phase approximation, increased inelastic scattering - **Large scattering angles**: Paraxial approximation fails, need full dynamical diffraction theory - **Crystalline samples at zone axis**: Dynamical diffraction (Bragg scattering) dominates over kinematic scattering .. note:: **Notation used in this section:** - :math:`V(x,y,z)`: Electrostatic potential (volts) at position (x,y,z) - :math:`V(x,y) = \int V(x,y,z) dz`: Projected potential (integrated through thickness) - :math:`\phi(x,y)`: Phase shift (radians) accumulated by electron wave - :math:`\phi = \sigma V`: Phase object approximation (see :ref:`phase-object-approximation`) - :math:`\sigma = \pi/(E\lambda)`: Interaction constant (see :ref:`interaction-constant-derivation`) - :math:`E`: Electron kinetic energy (keV) - :math:`\lambda = h/p`: Electron de Broglie wavelength - :math:`q = 1.602 \times 10^{-19}` C: Elementary charge magnitude (positive) - Electron charge: :math:`-q` (negative) - :math:`k_0 = \sqrt{2mE}/\hbar = 2\pi/\lambda`: Incident wave vector - :math:`\nabla_{\perp} = (\partial/\partial x, \partial/\partial y)`: Transverse gradient operator - :math:`\nabla_{\perp} \phi = \sigma \nabla_{\perp} V`: Phase gradient relationship (see :ref:`phase-gradient-relationship`) - :math:`E_{\text{total}} = E_{\text{kinetic}} - qV`: Energy conservation (see :ref:`energy-conservation-electrons`) - :math:`\hbar = h/(2\pi)`: Reduced Planck constant - :math:`m`: Electron mass **Electrostatic potential in the sample** The electrostatic potential :math:`V(x,y,z)` describes the influence that charged particles (nuclei and electrons) in the sample exert on other charged particles passing through. It represents the potential energy per unit charge at each point in space. In a material, positively charged atomic nuclei create regions of elevated (more positive) potential. We can think of the potential landscape as having "peaks" or "hills" centered at each atomic position, with the potential decreasing as you move away from the nuclei. **How electrons respond to the potential landscape** A positive test charge placed on top of a potential hill would naturally "roll down" toward lower potential (away from the positively charged nuclei), moving from high potential to low potential to minimize its potential energy. Conversely, a **negative charge** (like an electron) experiences the opposite behavior—it is attracted **up** the potential hills, toward the positively charged nuclei at the peaks. When an electron beam passes through this potential landscape, each electron experiences a force because it carries a negative charge. Electrons are attracted toward the positively charged nuclei (the potential peaks), causing the electron wave to be deflected and to accumulate phase as it propagates through the sample. **The projected potential** The projected potential :math:`V(x,y)` is the integral through the sample thickness: .. math:: V(x,y) = \int V(x,y,z)\,dz This projection represents the total potential "seen" by an electron traveling along the beam direction (z-axis). When an electron wave passes through a sample, it accumulates a phase shift proportional to this projected potential: .. math:: \phi(x,y) = \sigma V(x,y) = \frac{\pi}{E\lambda} \int V(x,y,z) \, dz where :math:`\sigma = \pi/(E\lambda)` is the **interaction constant**, :math:`E` is electron energy, and :math:`\lambda` is wavelength. **The phase object approximation** .. _phase-object-approximation: In the **phase object approximation** (also called the **projection approximation**, valid for thin samples), the transmitted electron wave can be written as: .. math:: \psi_{\text{exit}}(x,y) = \psi_{\text{incident}}(x,y) \cdot e^{i\sigma V(x,y)} where :math:`\sigma = \pi/(E\lambda)` and :math:`V(x,y) = \int V(x,y,z) dz` is the projected potential. .. _interaction-constant-derivation: Derivation of interaction constant ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ .. _interaction-constant: To derive the interaction constant :math:`\sigma`, we solve the **Schrödinger equation** for a fast electron passing through a thin sample. Starting point: Time-independent Schrödinger equation """"""""""""""""""""""""""""""""""""""""""""""""" For an electron in a potential :math:`V(\mathbf{r})`: .. math:: -\frac{\hbar^2}{2m} \nabla^2 \psi + V(\mathbf{r}) \psi = E \psi Step 1: Separate longitudinal and transverse coordinates """""""""""""""""""""""""""""""""""""""""""""""""""" Write :math:`\psi(\mathbf{r}) = \psi(x,y,z)` and assume the electron propagates primarily in the z-direction with high energy E. Use the ansatz: .. math:: \psi(x,y,z) = A(x,y,z) e^{ik_0 z} where :math:`k_0 = \sqrt{2mE}/\hbar` is the incident wave vector and :math:`A(x,y,z)` is a slowly varying envelope function. .. _incident-wave-vector: Understanding k₀ """""""""""""" The incident wave vector :math:`k_0 = \sqrt{2mE}/\hbar` comes from the relationship between energy and momentum in quantum mechanics. For a free electron: For a free electron (V = 0) traveling in the z-direction, the wavefunction is a plane wave: .. math:: \psi = e^{ik_0 z} The energy eigenvalue equation gives: .. math:: -\frac{\hbar^2}{2m} \frac{d^2 \psi}{dz^2} = E \psi Substituting :math:`\psi = e^{ik_0 z}`: .. math:: -\frac{\hbar^2}{2m}(ik_0)^2 e^{ik_0 z} = E e^{ik_0 z} .. math:: \frac{\hbar^2 k_0^2}{2m} = E Therefore: .. math:: k_0 = \sqrt{\frac{2mE}{\hbar^2}} = \frac{\sqrt{2mE}}{\hbar} This relates the wave vector to the electron's kinetic energy. Using :math:`p = \hbar k_0`, this is equivalent to the classical relation :math:`E = p^2/(2m)`. **Step 2: Apply paraxial approximation** Since the electron has high energy and the sample is thin, :math:`A(x,y,z)` varies slowly with z compared to :math:`e^{ik_0 z}`. We can approximate :math:`\partial^2 A/\partial z^2 \ll k_0 \partial A/\partial z`. Full derivation ^^^^^^^^^^^^^ Computing the Laplacian """""""""""""""""""""" Start with :math:`\psi = A(x,y,z) e^{ik_0 z}`. We need to compute :math:`\nabla^2 \psi`: 1. Longitudinal derivatives ''''''''''''''''''''''''' First derivative with respect to z: .. math:: \frac{\partial \psi}{\partial z} = \left(\frac{\partial A}{\partial z} + ik_0 A\right) e^{ik_0 z} Second derivative with respect to z: .. math:: \frac{\partial^2 \psi}{\partial z^2} &= \frac{\partial}{\partial z}\left[\left(\frac{\partial A}{\partial z} + ik_0 A\right) e^{ik_0 z}\right] \\ &= \left(\frac{\partial^2 A}{\partial z^2} + ik_0 \frac{\partial A}{\partial z}\right) e^{ik_0 z} + \left(\frac{\partial A}{\partial z} + ik_0 A\right) ik_0 e^{ik_0 z} \\ &= \left[\frac{\partial^2 A}{\partial z^2} + 2ik_0 \frac{\partial A}{\partial z} - k_0^2 A\right] e^{ik_0 z} 2. Transverse derivatives '''''''''''''''''''''''' Since :math:`e^{ik_0 z}` is constant with respect to x and y: .. math:: \frac{\partial^2 \psi}{\partial x^2} &= \frac{\partial^2 A}{\partial x^2} e^{ik_0 z} \\ \frac{\partial^2 \psi}{\partial y^2} &= \frac{\partial^2 A}{\partial y^2} e^{ik_0 z} 3. Complete Laplacian '''''''''''''''''''' The full Laplacian is the sum of all second derivatives: .. math:: \nabla^2 \psi &= \frac{\partial^2 \psi}{\partial x^2} + \frac{\partial^2 \psi}{\partial y^2} + \frac{\partial^2 \psi}{\partial z^2} \\ &= \left[\nabla_{\perp}^2 A + \frac{\partial^2 A}{\partial z^2} + 2ik_0 \frac{\partial A}{\partial z} - k_0^2 A\right] e^{ik_0 z} where :math:`\nabla_{\perp}^2 = \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2}` is the transverse Laplacian. Paraxial approximation """"""""""""""""""""" For fast electrons (high :math:`k_0`), the envelope function :math:`A(x,y,z)` varies slowly with z: .. math:: \left|\frac{\partial^2 A}{\partial z^2}\right| \ll \left|k_0 \frac{\partial A}{\partial z}\right| This leads to the paraxial wave equation: .. math:: \nabla_{\perp}^2 A + 2ik_0 \frac{\partial A}{\partial z} = -\frac{2m}{\hbar^2}V(x,y,z)A This equation describes how the electron wave evolves as it propagates through the sample's potential. .. math:: \nabla^2 \psi = \left[\nabla_{\perp}^2 A + \frac{\partial^2 A}{\partial z^2} + 2ik_0 \frac{\partial A}{\partial z} - k_0^2 A\right] e^{ik_0 z} where :math:`\nabla_{\perp}^2 = \partial^2/\partial x^2 + \partial^2/\partial y^2`. **Substitute into Schrödinger equation** :math:`-\frac{\hbar^2}{2m}\nabla^2\psi + V\psi = E\psi`: .. math:: -\frac{\hbar^2}{2m}\left[\nabla_{\perp}^2 A + \frac{\partial^2 A}{\partial z^2} + 2ik_0 \frac{\partial A}{\partial z} - k_0^2 A\right] e^{ik_0 z} + V A e^{ik_0 z} = E A e^{ik_0 z} Divide by :math:`e^{ik_0 z}`: .. math:: -\frac{\hbar^2}{2m}\left[\nabla_{\perp}^2 A + \frac{\partial^2 A}{\partial z^2} + 2ik_0 \frac{\partial A}{\partial z} - k_0^2 A\right] + VA = EA **Use** :math:`k_0^2 = 2mE/\hbar^2` **to cancel terms:** The :math:`-k_0^2 A` term becomes: .. math:: -\frac{\hbar^2}{2m}(-k_0^2 A) = \frac{\hbar^2 k_0^2}{2m} A = \frac{\hbar^2}{2m} \cdot \frac{2mE}{\hbar^2} A = EA This exactly cancels the :math:`EA` on the right side: .. math:: -\frac{\hbar^2}{2m}\left[\nabla_{\perp}^2 A + \frac{\partial^2 A}{\partial z^2} + 2ik_0 \frac{\partial A}{\partial z}\right] + VA = 0 **Neglect** :math:`\partial^2 A/\partial z^2` (paraxial approximation): .. math:: -\frac{\hbar^2}{2m}\left[\nabla_{\perp}^2 A + 2ik_0 \frac{\partial A}{\partial z}\right] + VA = 0 Rearrange: .. math:: 2ik_0 \frac{\partial A}{\partial z} + \nabla_{\perp}^2 A + \frac{2m}{\hbar^2} VA = 0 This gives the **paraxial wave equation**: .. math:: 2ik_0 \frac{\partial A}{\partial z} + \nabla_{\perp}^2 A + \frac{2m}{\hbar^2} V(x,y,z) A = 0 **Step 3: Projection approximation (thin sample)** For a thin sample, we can neglect transverse diffraction during propagation (:math:`\nabla_{\perp}^2 A \approx 0`). This gives: .. math:: \frac{\partial A}{\partial z} = -\frac{im}{\hbar^2 k_0} V(x,y,z) A **Step 4: Integrate through sample** Integrating from z = 0 (entrance) to z = t (exit): .. math:: A(x,y,t) = A(x,y,0) \exp\left[-\frac{im}{\hbar^2 k_0} \int_0^t V(x,y,z) dz\right] **Recall the full wavefunction:** :math:`\psi = A(x,y,z) e^{ik_0 z}`, so at the exit: .. math:: \psi_{\text{exit}} = A(x,y,0) \exp\left[-\frac{im}{\hbar^2 k_0} \int_0^t V(x,y,z) dz\right] e^{ik_0 t} Combine the exponentials: .. math:: \psi_{\text{exit}} = A(x,y,0) \exp\left[ik_0 t - \frac{im}{\hbar^2 k_0} \int_0^t V(x,y,z) dz\right] The **phase shift** :math:`\phi(x,y)` is the change from the incident phase :math:`k_0 t`: .. math:: \phi(x,y) = -\frac{m}{\hbar^2 k_0} \int_0^t V(x,y,z) dz = -\frac{m}{\hbar^2 k_0} V(x,y) **Step 5: Express in terms of electron wavelength** Using :math:`k_0 = 2\pi/\lambda` and :math:`E = \hbar^2 k_0^2/(2m)`, we can rewrite: .. math:: \boxed{\phi(x,y) = -\frac{m}{\hbar^2 k_0} V(x,y) = \frac{\pi}{E\lambda} V(x,y) = \sigma V(x,y)} where :math:`\sigma = \pi/(E\lambda)` is the interaction constant and :math:`V(x,y) = \int V(x,y,z) dz` is the projected potential. **Key insights**: 1. Phase shift proportional to projected potential 2. Higher potential → larger phase accumulation 3. Lower energy electrons (larger λ) → more sensitive to potential 4. Linear relationship enables quantitative analysis The electron wave accumulates phase as it travels through the sample. Regions with higher potential (near atoms) cause a **larger phase shift** compared to regions with lower potential (vacuum or between atoms). This phase modulation encodes information about the atomic structure. For very thin samples where :math:`\sigma V(x,y) \ll 1`, we can use a first-order Taylor expansion (the **weak-phase object approximation**): .. math:: \psi_{\text{exit}}(x,y) \approx \psi_{\text{incident}}(x,y) \cdot [1 + i\sigma V(x,y)] This linearizes the relationship between the potential and the exit wave, simplifying phase retrieval algorithms. Most DPC and ptychography reconstructions assume this weak-phase regime. .. _phase-gradient-relationship: Phase gradient derivation ^^^^^^^^^^^^^^^^^^^^^^ .. _phase-gradient-derivation: From the phase object approximation, :math:`\phi(x,y) = \sigma V(x,y)`. Taking the gradient in the transverse plane: .. math:: \nabla_{\perp} \phi(x,y) = \sigma \nabla_{\perp} V(x,y) where :math:`\nabla_{\perp} = (\partial/\partial x, \partial/\partial y)`. Electric field integration """""""""""""""""""""""" The transverse electric field integrated along the beam path: .. math:: \int (E_x, E_y) dz = -\nabla_{\perp} \int V(x,y,z) dz = -\nabla_{\perp} V(x,y) Momentum transfer """"""""""""""" The momentum transfer to the electron: .. math:: \Delta \mathbf{p}_{\perp} = -q \int (E_x, E_y) dz = q \nabla_{\perp} V(x,y) Final relationship """""""""""""""" Therefore: .. math:: \nabla_{\perp} \phi = \sigma \nabla_{\perp} V = \frac{\sigma}{q} \Delta \mathbf{p}_{\perp} .. note:: The deflection angle :math:`\theta \approx \Delta p_{\perp}/p_z` is directly related to the phase gradient. **Connection to electron deflection** The gradient of the phase tells us the deflection angle: .. math:: \nabla \phi(x,y) = \sigma \nabla V(x,y) Since the electric field is :math:`\mathbf{E} = -\nabla V`, electrons (being negatively charged) are deflected in the direction of :math:`+\nabla V`, which is **opposite** to the electric field direction. They are attracted toward the positively charged nuclei (the potential peaks). By measuring the **center of mass (CoM)** shift of the diffraction pattern at each probe position, we can determine the local deflection angle, and from that, reconstruct the phase gradient :math:`\nabla \phi`. Integrating this gradient gives us the phase :math:`\phi(x,y)`, and thus the projected potential :math:`V(x,y)`. This is the principle behind **Differential Phase Contrast (DPC)** imaging. **The physics of phase accumulation** In quantum mechanics, an electron wavefunction :math:`\psi = A e^{i\phi(t)}` evolves according to its total energy: .. math:: \frac{d\phi}{dt} = -\frac{E_{\text{total}}}{\hbar} Time evolution derivation ^^^^^^^^^^^^^^^^^^^^^ .. _time-evolution-derivation: This fundamental relationship comes from the **time-dependent Schrödinger equation**. Here's the derivation: Starting point """"""""""""" The time-dependent Schrödinger equation: .. math:: i\hbar \frac{\partial \psi}{\partial t} = \hat{H} \psi where :math:`\hat{H}` is the Hamiltonian operator (total energy operator) and :math:`\hbar` is the reduced Planck constant. Step 1: Assume wavefunction form """""""""""""""""""""""""""""" Write the wavefunction as: .. math:: \psi(t) = A e^{i\phi(t)} where :math:`A` is the amplitude (constant) and :math:`\phi(t)` is the time-dependent phase. Step 2: Take time derivative """"""""""""""""""""""""" .. math:: \frac{\partial \psi}{\partial t} = A e^{i\phi(t)} \cdot i \frac{d\phi}{dt} = i \frac{d\phi}{dt} \psi Step 3: Substitute into Schrödinger equation .. math:: i\hbar \left( i \frac{d\phi}{dt} \psi \right) = \hat{H} \psi .. math:: i^2 \hbar \frac{d\phi}{dt} \psi = \hat{H} \psi .. math:: -\hbar \frac{d\phi}{dt} \psi = \hat{H} \psi **Step 4: For energy eigenstate** For a state with definite energy :math:`E_{\text{total}}`, the Hamiltonian acts as: .. math:: \hat{H} \psi = E_{\text{total}} \psi **Step 5: Final result** Substituting and dividing both sides by :math:`\psi`: .. math:: -\hbar \frac{d\phi}{dt} = E_{\text{total}} .. math:: \frac{d\phi}{dt} = -\frac{E_{\text{total}}}{\hbar} **Physical interpretation**: Higher energy → faster phase oscillation. The negative sign means the phase **decreases** with time (rotates clockwise in the complex plane). .. _energy-conservation-electrons: When the electron passes through a region with electrostatic potential :math:`V(z)`, its total energy becomes: .. math:: E_{\text{total}} = E_{\text{kinetic}} + E_{\text{potential}} = E_{\text{kinetic}} + (-q)V(z) = E_{\text{kinetic}} - qV(z) where :math:`q = 1.602 \times 10^{-19}` C is the elementary charge magnitude (positive). The electron carries charge :math:`-q` (negative). **Important**: The total energy :math:`E_{\text{total}}` is conserved—it remains constant throughout the sample. However, the electron's **kinetic energy** temporarily changes as it interacts with the potential. Near a positively charged nucleus (where :math:`V > 0`, a potential "hill"), the negatively charged electron is attracted and **speeds up**, gaining kinetic energy (the :math:`-qV` term becomes more negative, so :math:`E_{\text{kinetic}}` increases). Far from nuclei (between atoms), it **slows down**. Think of it like a ball rolling through a landscape of hills and valleys—the ball speeds up going downhill and slows down going uphill, but its total energy stays constant. This change in kinetic energy corresponds to a change in the electron's wavelength (via de Broglie relation :math:`\lambda = h/p`). Since the wavefunction's phase advance depends on how many wavelengths fit into a distance, different regions accumulate phase at different rates. As the electron propagates through thickness :math:`dz` in time :math:`dt = dz/v`, the accumulated phase shift is: .. math:: d\phi = -\frac{E_{\text{total}}}{\hbar} dt = -\frac{E - qV}{\hbar} \frac{dz}{v} Integrating through the sample thickness and using the de Broglie relation :math:`\lambda = h/p` with relativistic expressions for electron energy and momentum yields the interaction constant :math:`\sigma = \pi/(E\lambda)`. **Understanding the interaction constant** The factor :math:`\sigma = \pi/(E\lambda)` converts electrostatic potential (volts) to phase shift (radians). It depends on the microscope's operating parameters: - **E** is the accelerating voltage: 80 kV, 200 kV, or 300 kV → electrons gain 80 keV, 200 keV, or 300 keV of kinetic energy - :math:`\lambda` is the electron wavelength: higher energy → shorter wavelength (300 keV → :math:`\lambda \approx 0.002` nm, comparable to atomic spacings) - :math:`\sigma` has units (keV·nm)⁻¹, making :math:`\phi = \sigma V` dimensionless (radians) **Example**: For 300 keV electrons, :math:`\sigma \approx 0.729` rad/(V·nm). A single carbon atom (creating ~1 V·nm of integrated potential) shifts the phase by ~0.7 radians. **Key insight**: Lower-energy electrons (larger :math:`\lambda`, smaller E) have larger :math:`\sigma` → more sensitive to potentials → accumulate more phase for the same sample. This is why **low-voltage electron microscopy enhances phase contrast for light elements**.