Direct Energy

🧭 Overview

🧠 One-sentence thesis

Capacitors, piezoelectric devices, pyroelectric devices, and electro-optic devices all rely on establishing material polarization in a dielectric layer, and understanding how capacitors work through material polarization provides insight into these energy conversion devices.

📌 Key points (3–5)

• Core mechanism: All these devices involve establishing a material polarization (charge separation) throughout a dielectric material.
• What makes piezoelectric devices special: Mechanical strain causes material polarization, enabling two-way energy conversion between mechanical energy and electricity.
• Why start with capacitors: Capacitors are familiar energy storage devices whose operation depends on material polarization; understanding them clarifies piezoelectric, pyroelectric, and electro-optic devices.
• Common confusion: Capacitance C is not just a constant—it depends on the permittivity of the dielectric material, which stores energy by polarizing when the capacitor charges.
• Crystal structure matters: Only 21 of 32 crystal point groups (those without inversion symmetry) can be piezoelectric; the effect depends on crystallographic orientation.

🔋 How capacitors store energy through material polarization

🔋 Basic capacitor relationships

• In introductory circuits, a capacitor is described by i = C dv/dt where C is just a constant.
• The charge-voltage relationship is Q = Cv where Q is charge in coulombs, v is voltage, C is capacitance in farads.
• Energy stored: E = (1/2)Cv² = (1/2)Qv.
• To go beyond this simple model and understand different capacitor types, we need to examine material polarization.

🧲 Material polarization defined

Material polarization: charge separation throughout a material when an external voltage is applied across an insulator.

• More precisely, material polarization P (in C/m²) is defined as the difference between the electric field in the material D and the field that would be present in free space ε₀E:
  • P = D - ε₀E or P = (ε - ε₀)E
• Two electric field parameters exist: displacement flux density D and electric field intensity E, related by D = εE.
• Why two parameters? It separates the description of the electric field inside a material from the field in free space.
• Similarly, magnetic fields use two parameters: magnetic field intensity H and magnetic flux density B.

⚡ How dielectric materials increase energy storage

• For a parallel plate capacitor: C = εA / d_thick where A is plate area, d_thick is distance between plates, ε is permittivity.
• A vacuum-filled capacitor has ε = ε₀ (permittivity of free space).
• Key insight: When you replace vacuum with a dielectric (ε > ε₀), capacitance becomes larger because the dielectric material changes as the capacitor charges.
• What happens during charging:
  • In an insulator, electrons are bound to atoms and cannot flow as current.
  • Instead, electrons move slightly with respect to their nuclei while staying bound.
  • This displacement balances charges on the plates, storing more energy for a given voltage.
  • We say this process induces electric dipoles.
• The larger the permittivity ε, the more energy the material can store by polarizing.
• Example: Tantalum dioxide (Ta₂O₅) has ε = 25ε₀, so it stores 25 times the energy of an air-filled capacitor of the same size at the same voltage.

📊 Permittivity and related measures

All these describe the ability of a material to store energy in the electric field:

Measure	Symbol	Definition	Notes
Permittivity	ε	Material property	Units: F/m
Relative permittivity	εᵣ	εᵣ = ε/ε₀	Unitless
Electric susceptibility	χₑ	χₑ = ε/ε₀ - 1	Unitless
Index of refraction	n	n = c/|v| = speed of light in free space / speed in material	Unitless, n > 1; for good insulators with μ = μ₀: n = √εᵣ

• Material polarization can be written: P = (εᵣ - 1)ε₀E = ε₀χₑE
• Table 2.1 lists relative permittivities: vacuum (1.0), Teflon (2.1), SiO₂ (3.5), Si (11.8), Ta₂O₅ (24), BaSrTiO₃ (300), PbTe (360).

🔷 Anisotropic materials

• In some crystalline materials, permittivity depends on direction.
• A voltage along one crystallographic axis may induce charge separation more easily than along a different axis.
• Such materials are called anisotropic; permittivity is better described by a 3×3 matrix than a scalar.
• Example (Fig. 2.1): atoms with nuclei (black circles) and electron clouds (gray circles); when an electric field is applied in one direction, electrons displace relative to nuclei; the external field needed for the same displacement differs in different crystallographic directions.

🔧 Capacitor properties and types

🔧 Key selection parameters

• Capacitance and maximum voltage: first two measures to consider; exceeding maximum voltage can damage the capacitor.
• Fig. 2.2 shows ranges for different types:
  • Electrolytic: 10⁻⁷ to 1 F, 1 to 1000 V max.
  • Ceramic: 10⁻¹³ to 5·10⁻⁴ F, 1 to 50,000 V max.

🌡️ Other important factors

• Temperature stability: ideally capacitance is independent of temperature; ceramic and electrolytic capacitors are more sensitive than polymer or vacuum capacitors.
• Accuracy/precision: capacitors have tolerances (e.g., ±5%, ±10%), like resistors.
• Equivalent series resistance (ESR): all materials have some resistivity; model a physical capacitor as an ideal capacitor in series with an ideal resistor.
• Leakage: how well a capacitor retains stored charge when disconnected; ideal capacitor has no leakage; electrolytic capacitors have larger leakage.
• Lifetime: ideal capacitor operates for decades; electrolytic capacitors are not designed for long lifetimes.
• Other factors: cost, availability, size, frequency response.

🧱 Capacitor types by dielectric material

Type	Advantages	Disadvantages	Notes
Ceramic	Small, cheap, readily available; tolerate large voltages; low ESR	Small capacitance, poor accuracy, poor temperature stability, moderate leakage; can cause voltage spikes; some are piezoelectric (vibration → noise)
Mica	Good accuracy, small leakage		Mica is a flaky mineral with layered structure; can make very thin dielectric layers; natural forms include biotite and muscovite KAl₂(AlSi₃O₁₀)(OH)₂
Polymer	Good accuracy, temperature stability, leakage		Types: polystyrene, polycarbonate, polyester, polypropylene, Teflon, mylar
Vacuum	Very low leakage		Used in high-voltage or very-low-leakage applications
Oil (liquid dielectric)			Similar applications to vacuum
Electrolytic	Small device can have large capacitance	Poor accuracy, temperature stability, leakage; finite lifetime (liquid degrades); polarized (positive/negative terminals; reversing voltage destroys capacitor)	Dielectric is solid + liquid electrolyte; initial voltage application chemically creates oxide layer (the dielectric)

• Electrolyte: a liquid through which some charges flow more easily than others.
• Don't confuse: "polarized" capacitor (has positive/negative terminals) vs. "material polarization" (charge separation in dielectric).

⚙️ Piezoelectric devices: mechanical stress induces polarization

⚙️ Core principle

• Can we induce material polarization without applying a voltage? Yes—this creates an energy conversion device.
• In piezoelectric devices, mechanical stress causes material polarization.
• When stress is exerted, valence electrons are displaced, but nuclei and other electrons do not move.
• When stress is released, material polarization goes away.
• Don't confuse with permanent crystal structure change (e.g., coal → diamond under high pressure/temperature, or shot peening of steel); piezoelectricity requires little energy and is reversible.

📐 Piezoelectric strain constant

• Material polarization in a piezoelectric insulator:
  • P = D - ε₀E + dς
  • d is the piezoelectric strain constant in m/V.
  • ς is stress in pascals (Pa = J/m³ = N/m²).
• For many materials, d is zero or very small.
• Barium titanate has relatively large d ≈ 3·10⁻¹⁰ m/V.
• Mechanical strain (unitless) vs. stress (units: Pa):
  • Without external electric field: strain = (1 / Young's elastic modulus) · stress.
  • With electric field: strain = (1 / Young's elastic modulus) · stress + E · d.
• Energy stored under stress ς: E = |ς| · A · l · η_eff where A is cross-sectional area (m²), l is deformation (m), η_eff is efficiency.
• Bigger devices, more deformation, or larger piezoelectric constants → more energy stored.

🔬 Nonlinear piezoelectricity

• The linear relationship (P proportional to ς) describes many materials but not all.
• For other piezoelectric crystals, polarization is proportional to the square of stress:
  • |P| = |D| - ε₀|E| + d|ς| + d_quad|ς|²
  • d_quad is another piezoelectric strain constant.
• Some materials need terms with higher powers of stress.

🔮 Crystal structure and piezoelectricity

🔮 Describing crystal structures

• Crystalline materials: atoms arranged periodically; composed of elements (e.g., Si) or compounds (e.g., NaCl).
• Two components describe atom arrangement: lattice and basis.
• Lattice: a periodic array of points in space; specified by n lattice vectors for n-dimensional lattice.
  • For 3D: three vectors a₁, a₂, a₃; travel integer multiples of these vectors to get from one lattice point to any other.
  • Primitive lattice vectors: as short as possible.
• Cell: area (2D) or volume (3D) formed by lattice vectors.
• Primitive cell: smallest possible repeating unit describing a lattice (formed by primitive lattice vectors).
• Crystal basis: arrangement of one or more atoms attached to every lattice point.
• Crystal structure: lattice + crystal basis together.
• Example (Fig. 2.5): 2D lattice with two primitive vectors; basis has two atoms of one type and one of another.

🧊 Bravais lattices and crystal systems

• 14 possible 3D lattice types called Bravais lattices.
• Examples (Fig. 2.6): simple cubic, body centered cubic, face centered cubic, asymmetric triclinic.
  • Simple cubic: all angles between nearest neighbors are right angles; all nearest-neighbor lengths equal.
  • Asymmetric triclinic: no right angles; no equal nearest-neighbor lengths.
• Primitive cell angles labeled α, β, γ; side lengths labeled a, b, c (Fig. 2.9).
• Crystal system: classification based on angles and lengths of primitive cell (e.g., triclinic, monoclinic, orthorhombic, tetragonal, trigonal, hexagonal, cubic/isometric).
• Example crystal structures:
  • Sodium chloride: face centered cubic lattice + basis of one Na and one Cl atom.
  • Silicon: diamond structure = face centered cubic lattice + basis of two Si atoms at (0,0,0) and (l/4, l/4, l/4).
  • Diamond structure: C, Si, Ge, Sn with cell lengths l = 0.356, 0.543, 0.565, 0.646 nm respectively.

🔑 Crystal point groups and symmetry

• Cannot list all possible crystal structures (infinite), but they are classified by symmetries.
• Possible symmetry operations: 2-fold, 3-fold, 4-fold, 6-fold rotations; horizontal/vertical mirror planes; inversion.
• Crystal point groups: 32 possible groups based on symmetry elements.
• Two labeling systems:
  • Hermann-Mauguin notation (1930s): used by chemists, mineralogists, some physicists.
  • Schoenies notation (1891): used by mathematicians, spectroscopists, other physicists.
• Table 2.2 lists all 32 crystal point groups with both notations, crystal system, primitive cell angles/lengths, and whether materials can be piezoelectric, pyroelectric, or Pockels electro-optic.
• Crystal space groups (230 total): based on symmetry transformations including translations + rotations/mirror planes; not discussed further in this text.

🪞 Inversion symmetry

• Inversion operation: in 3D, a shape has inversion symmetry if it is identical when rotated 180° and inverted through the origin.
  • Draw vector V from center to any surface point; if shape has inversion symmetry, point at -V is also on surface.
• Example (Fig. 2.8): left shape has inversion symmetry; right shape does not.
• Center of symmetry: crystal structure with inversion symmetry.
• Noncentrosymmetric: no inversion symmetry.
• 21 of 32 point groups have no center of symmetry (Table 2.2, sixth column).
• 20 of these 21 groups have a polar axis: some axis with different forms on opposite ends.
  • Mechanically stressing along the polar axis → different charge buildup on different sides.
• Dielectric crystalline materials in any of these 21 noncentrosymmetric groups are piezoelectric.

⚠️ Practical considerations

• Even if crystal structure has no inversion symmetry, the piezoelectric effect and strain coefficient d may be too small to measure.
• Effect may only occur for stress along particular axes, not arbitrary orientations.
• Only one crystal point group (asymmetric triclinic) allows random stress to produce material polarization.
• For all other groups, only stresses along certain axes produce polarization.
• In most crystals, stress along one axis produces different polarization than the same stress along a different axis.
• More accurate description: piezoelectric strain coefficient is a 3×6 matrix with elements d_ik = (∂strain along k / ∂electric field along i) at given stress.
  • Electric field has x, y, z components; stress can be along xx, xy, xz, yy, yz, or zz directions.

🌀 Ferroelectricity and poling

🌀 Piezoelectricity in non-crystalline materials

• Previous section discussed crystals; cannot define crystal structure for amorphous materials.
• But piezoelectric devices can be made from polycrystalline and amorphous materials.
• In dielectrics, external electric field induces material polarization (electrons/nuclei displace slightly).
• Charge buildups (electric dipoles) induce additional electric fields (Coulomb's law).
• Secondary effect: once one atom polarizes, nearby atoms polarize.
• Small regions of the same material polarization are called electrical domains.

🌀 Ferroelectric effect

• In certain dielectrics, external mechanical stress induces local material polarization.
• This polarization induces polarization in nearby atoms, forming electrical domains.
• Ferroelectricity: nonlinear process where one atom's polarization induces polarization in nearby atoms, forming domains.
• Can occur in crystalline, amorphous, or polycrystalline materials.
• For non-crystalline materials, the effect is necessarily nonlinear (not well described by linear equations).
• Ferroelectric materials can be ferroelectric piezoelectric, ferroelectric pyroelectric, or ferroelectric electro-optic (next chapter).
• Curie temperature: for many ferroelectric materials, effects occur only below this temperature; heating above it removes the ferroelectric effect.
• Hysteresis: material polarization depends on past history; ferroelectric materials may have polarization even without external stress or field if energy was previously applied.

🧲 Etymology and analogy

• "Ferro-" means iron, but most ferroelectric materials do not contain iron, and most iron-containing materials are not ferroelectric.
• Analogy to ferromagnetic: some iron-containing materials are ferromagnetic.
  • External magnetic field → internal magnetic field in material.
  • Can have permanent magnetic dipole without applied field.
  • Electric dipole modeled as pair of charges; magnetic dipole modeled as small current loop.
  • Ferromagnetic materials exhibit hysteresis and have magnetic domains with aligned dipoles.

🔌 Poling process

• Initially: piezoelectric ferroelectric material has randomly aligned electrical domains and no net polarization; not yet piezoelectric or ferroelectric.
• Poling: process of causing a material to exhibit piezoelectricity and ferroelectricity.
• How to pole: place material in a strong external electric field (e.g., across battery poles—hence the term).
• Poling does not change atomic structure (amorphous material remains amorphous).
• During poling, electrical domains form and remain when external field is removed.
• After poling: material may have net polarization throughout; is now piezoelectric and ferroelectric (external stress induces polarization locally and throughout).
• Electret: material that is piezoelectric due to this type of poling.

🛠️ Piezoelectric materials and applications

🛠️ What makes a good piezoelectric material

1. Electrical insulator: when voltage is applied across a conductor, valence electrons are removed from atoms, so no material polarization accumulates.
2. Large piezoelectric strain constant: d is so small it cannot be detected in many crystals from the 21 known piezoelectric point groups; zero in other groups.
3. Not brittle: should withstand repeated stressing without permanent damage.
4. Thermal properties may also be important.
5. No single material is best for all applications.

🧪 Common piezoelectric materials

Material	Notes
Quartz (crystalline SiO₂)	First material studied (Pierre and Jacques Curie, 1880s); used in crystal oscillators today
Lead zirconium titanate	Relatively high piezoelectric strain constant
Polyvinyldenfluoride (polymer)	Flexibility; withstands repeated stress without damage
Barium titanate (BaTiO₃)	Relatively large d ≈ 3·10⁻¹⁰ m/V
Lithium niobate, tourmaline, Rochelle salt	Also studied

• Manufacturers often do not label whether devices are crystalline, amorphous, or polycrystalline.
• Polycrystalline/amorphous advantages: easier to make into different shapes (cylinders, spheres).
• Polycrystalline/amorphous disadvantages: often lower melting temperatures, higher temperature expansion coefficients, more brittle.
• Crystalline advantages (e.g., quartz): harder, higher melting temperature.

🔌 Electrical component applications

• Oscillators: voltage applied → material bends; voltage released → springs back at natural resonant frequency; integrated with feedback circuit → precise frequency oscillations; often made from crystalline quartz.
• Piezoelectric transformers: used in cold cathode fluorescent lamps (LCD panel backlights); require ~1000 V to turn on, hundreds of volts during use; much smaller than magnetic transformers (small enough for PCB mounting).
  • Traditional transformer: AC electricity → magnetic energy → AC electricity at different voltage (pair of coils).
  • Piezoelectric transformer: AC electricity → mechanical vibrations → AC electricity at different voltage.
  • Energy conserved: high voltage with low current.
  • Example (Fig. 2.10): converts 8-14 V input to up to 2 kV output.
• Small components (Fig. 2.11): microphones, ultrasonic transmitters/receivers, vibration sensors, oscillator crystals.

⚡ Energy harvesting and sensors

• Efficiency: hard to discuss (different assumptions); commercial devices often 6% or less.
• Due to low efficiency, many devices used as sensors.
• Despite low efficiency, some used for energy harvesting:
  • Train station embedded devices in platforms to generate electricity.
  • Convert energy from fluid motion or wind directly to electricity.

🏥 Biomedical applications

• Quartz: piezoelectric, durable, readily available, nontoxic.
• Devices designed for outside and inside the body.
• Sensors: monitor knees/joints; ultrasonic imaging (generate and detect ultrasonic vibrations).
• Power sources: artificial hearts, pacemakers, other devices limited by battery technology; piezoelectric generators have no moving parts, avoid battery changes.
• Physical activity types: continuous (breathing) or discontinuous (walking); both can be energy sources.
• Power requirements vary: artificial heart ~8 W, pacemaker ~few microwatts.
• Examples: device in artificial knee produced 0.85 mW; device in shoe generated 8.4 mW from walking.

🔊 Other applications

• Imaging systems: sonar (WWI military development for detecting boats/submarines; today: detect fish, measure water depth, analyze circuits, detect imperfections/cracks in steel/welds).
• Buttons and keyboards: piezoelectric sensors.
• Accelerometers: measure acceleration.
• Pipe flow measurement.
• Speakers, microphones, buzzers: audio and ultrasonic frequencies.
• Ultrasonic emulsification: dyes, paints, food products (e.g., peanut butter).
• Barbecue grill ignitions: mechanical stress induces electric spark.

🧭 Overview

🧠 One-sentence thesis

Pyroelectric and electro-optic devices convert energy between temperature differences or optical radiation and material polarization in dielectric insulators, expanding the range of energy conversion mechanisms beyond piezoelectricity.

📌 Key points (3–5)

• Pyroelectric devices convert temperature differences to/from electricity through changes in material polarization (not melting or phase changes).
• Electro-optic devices convert optical electromagnetic radiation to/from material polarization through nonlinear dependence on external electric fields.
• Crystal structure determines capability: only 10 of 32 crystal point groups can be pyroelectric; 21 can be piezoelectric or Pockels electro-optic; all pyroelectric materials are also piezoelectric.
• Common confusion: ferroelectric versions of these effects can occur in amorphous/polycrystalline materials with hysteresis, not just crystals; the terminology "ferroelectric" has nothing to do with iron despite the prefix.
• Applications are limited: pyroelectric sensors exist but thermocouples are more common; electro-optic materials are used in controllable optical devices, displays, and frequency conversion.

🌡️ Pyroelectricity fundamentals

🌡️ What pyroelectricity is

Pyroelectric effect: energy conversion between a temperature difference and electricity through changes in material polarization.

• Occurs in insulators, so it is different from the thermoelectric effect (which involves charge flow through junctions, discussed in Chapter 8).
• First studied by Hayashi (1912) and Röntgen (1914).
• The effect involves small energy inputs—enough to change material polarization but not enough to melt the material or change its crystal structure.

Don't confuse:

• Pyroelectricity ≠ melting or phase transitions.
• Pyroelectricity ≠ thermoelectric effect (which occurs in conductors/junctions).

📐 Mathematical description

The material polarization depends on temperature change:

P = D − ε₀E + b·ΔT

Where:

• P = material polarization (C/m²)
• D = displacement flux density (C/m²)
• E = electric field intensity (V/m)
• ε₀ = permittivity of free space (F/m)
• b = pyroelectric coefficient (C/(m²·K))
• ΔT = temperature change

Key features:

• The pyroelectric coefficient b is a vector because polarization may differ along different crystal directions.
• In some materials, the dependence is linear (Eq. 3.1); in others, quadratic and higher terms are needed (Eq. 3.2): P = D − ε₀E + b·ΔT + b_quad·(ΔT)² + ...
• Many materials exhibit pyroelectricity only below a pyroelectric Curie temperature.

Example: A crystal is heated slightly → electrons displace slightly → electric dipoles form → material polarization changes → this can be measured as electricity.

🔷 Which crystals are pyroelectric

To determine if a crystalline material can be pyroelectric:

1. Identify its crystal structure.
2. Determine the corresponding crystal point group (one of 32 classes based on symmetries).
3. Check if it belongs to one of the 10 pyroelectric crystal point groups listed in Table 2.2.

Key relationships:

• All pyroelectric crystals are piezoelectric, but not all piezoelectric crystals are pyroelectric.
• Crystals in 21 of 32 point groups (those without a center of symmetry) can be piezoelectric.
• Only 10 of those 21 can be pyroelectric.

Don't confuse: The number of groups—21 for piezoelectric, 10 for pyroelectric; pyroelectric is a subset.

🧲 Ferroelectric pyroelectricity

Some materials (crystalline, amorphous, or polycrystalline) are ferroelectric pyroelectric:

• Material polarization depends nonlinearly on temperature (not mechanical stress).
• Polarization of one atom induces polarization in nearby atoms → electrical domains with aligned polarization form.
• These materials can have material polarization even when no temperature gradient is applied.
• They exhibit hysteresis: behavior depends on past history.

Don't confuse: "Ferroelectric" does not mean the material contains iron; the term is an analogy to ferromagnetic materials.

🔬 Pyroelectric materials and applications

🔬 Example materials

Pyroelectricity has been studied in:

• Barium titanate (BaTiO₃)
• Lead titanate (PbTiO₃)
• Potassium hydrogen phosphate (KH₂PO₄)
• Chalcogenide glasses (sulfides, selenides, tellurides such as GeTe)

Selection criteria:

• Pyroelectric coefficient magnitude
• Thermal properties: ability to withstand repeated heating/cooling
• Relatively high melting temperature

📡 Applications (limited)

The pyroelectric effect does not have many applications:

Application	Status	Why limited
Infrared optical detectors	Some exist	Most optical detectors are photovoltaic (semiconductor junctions, Chapter 6)
Temperature sensors	Could be used	Thermocouples (thermoelectric effect, Chapter 8) are more convenient to build and operate
General sensing	Limited	Many pyroelectric materials are nonlinear; linear sensors are easier to work with and calibrate

💡 Electro-optic effect fundamentals

💡 What the electro-optic effect is

Electro-optic effect: occurs when an applied external electric field induces a material polarization where the amount depends nonlinearly on the external field.

Key clarifications:

• The name includes "optic" because the external field is often from a visible laser beam, but the field can be from any source at any frequency (even constant DC fields).
• This is not about melting or permanent crystal structure changes—only changes in material polarization that are not permanent.
• Linear materials: polarization proportional to applied field.
• Nonlinear (electro-optic) materials: polarization depends nonlinearly on applied field.

📊 Mathematical description

Material polarization as a power series of the applied external field:

|P| = ε₀χₑ|E| + ε₀χ⁽²⁾|E|² + ε₀χ⁽³⁾|E|³ + ...

Where:

• χₑ = electric susceptibility (unitless)
• χ⁽²⁾ = chi-two coefficient (m/V)
• χ⁽³⁾ = chi-three coefficient (m²/V²)

Two main effects:

Effect	Term	Name	Discoverer	Polarization depends on
Pockels (linear electro-optic)	ε₀χ⁽²⁾|E|²	Chi-two	Friedrich Pockels (1893)	Square of external field
Kerr (quadratic electro-optic)	ε₀χ⁽³⁾|E|³	Chi-three	John Kerr (1875)	Cube of external field

Don't confuse: "Linear" and "quadratic" refer to how the index of refraction changes (see below), not the power of E in the polarization equation.

🌈 Index of refraction formulation

Optics scientists prefer to work with index of refraction (n) rather than polarization. The expansion is rewritten:

1/εᵣ_eo = 1/εᵣ_x + γ|E| + s|E|² + ...

Where:

• γ = Pockels coefficient (m/V)
• s = Kerr coefficient (m²/V²)
• εᵣ_x = relative permittivity without nonlinear contributions
• n_x = index of refraction without nonlinear contributions (n_x² = εᵣ_x)

The overall index of refraction can be approximated as:

n_eo ≈ n_x(1 − ½γn_x²|E| − ½sn_x²|E|²)

Why "linear" and "quadratic"?

• Pockels effect: n_eo depends on the first power of |E| → "linear electro-optic effect"
• Kerr effect: n_eo depends on the second power of |E| → "quadratic electro-optic effect"

Example: A laser beam shines on an electro-optic material → the strong electric field induces material polarization nonlinearly → the index of refraction changes → a second beam passing through experiences altered phase/intensity.

🔷 Electro-optic materials and crystal structure

🔷 Pockels effect in crystals

To determine if a crystalline insulator can exhibit the Pockels effect:

1. Determine the crystal structure.
2. Identify symmetries and determine the crystal point group.
3. Check if it is noncentrosymmetric (no inversion symmetry).

Key result:

• Of 32 crystal point groups, 21 may exhibit the Pockels effect (χ⁽²⁾ and γ are nonzero).
• These are the same 21 groups that are piezoelectric (listed in Table 2.2).
• All piezoelectric materials are also Pockels electro-optic and vice versa.

Important relationship:

• All pyroelectric materials are piezoelectric.
• All piezoelectric materials are Pockels electro-optic.
• Therefore: pyroelectric ⊂ piezoelectric = Pockels electro-optic.

Practical implication: If a device is used as an electro-optic device and is accidentally mechanically stressed or vibrated, material polarization will also be induced by piezoelectricity. Multiple effects can occur simultaneously, making it difficult to identify the primary cause.

🔷 Kerr effect in crystals

The Kerr electro-optic effect can occur in crystals whether or not they have a center of symmetry:

• Some materials exhibit the Kerr effect but not the Pockels effect.
• In many materials, the Kerr effect is quite small.

🧪 Electro-optic effect in amorphous and polycrystalline materials

Table 2.2 (crystal point groups) applies only to crystalline materials. However:

• Crystalline, polycrystalline, and amorphous materials can all be electro-optic.
• In amorphous and polycrystalline materials, the electro-optic effect is necessarily nonlinear.
• Mechanism: external electric field (e.g., from a laser) → material polarization develops → charge separation induces polarization in nearby atoms.
• Ferroelectric electro-optic materials (amorphous/polycrystalline) exhibit hysteresis, similar to ferroelectric piezoelectric and pyroelectric materials.

🛠️ Electro-optic applications

🛠️ Controllable optical devices

Electro-optic materials are used to make:

• Controllable lenses
• Prisms
• Phase modulators
• Switches
• Couplers

Typical operation:

• Two laser beams are used.
• One beam controls the material polarization of the device.
• The intensity, phase, or electromagnetic polarization of the second optical beam is altered as it travels through the device.

Advanced applications:

• Combinations of electro-optic devices → controllable optical logic gates and interconnects for optical computing.

💾 Memory and display applications

Application	Details
Rewritable memory	Some creative designs use electro-optic materials
Hologram storage	Electro-optic materials enable holographic data storage
Liquid crystal displays	Liquid crystals are electro-optic; external voltage alters their material polarization

Note: Most memory devices are not made from electro-optic materials.

🌈 Frequency conversion

Electro-optic materials convert optical beams from one frequency to another:

Second harmonic generation:

• Converts photons of energy E to photons of energy ½E.
• Uses electro-optic materials.

Related processes:

• Third harmonic generation
• Three-wave mixing
• Four-wave mixing
• Optical parametric oscillation
• Stimulated Raman scattering

⚠️ Terminology challenges

⚠️ Inconsistent naming across literature

The excerpt warns that every author seems to have a different name for the same physical phenomena, and the same term may have completely different meanings.

Example confusion:

• Quadratic piezoelectricity (this text): mechanical stress induces polarization proportional to the square of stress.
  • Called "electrostriction" by some references [3] [6].
  • Called "ferroelectricity" by reference [33].
  • Called "photoelasticity" by reference [31].
• Ferroelectricity (reference [26]): assumes crystalline materials, no assumption about linearity.

⚠️ Misleading etymological roots

Looking at Greek/Latin roots does not help and sometimes makes matters worse:

Term	Root meaning	Actual meaning	Why confusing
Ferro-	Iron	Has nothing to do with iron	Analogy to ferromagnetic; ferroelectric materials rarely contain iron
Pyro-	Fire (also in pyrite, pyrrhotite—iron compounds)	Temperature-related	Follows Röntgen's 1914 terminology
Chalcogenide	Copper (chalc- = copper)	Sulfides, selenides, tellurides	Named in analogy to CuS (chalcosulfide)

Additional confusion:

• Iron has periodic table symbol Fe; iridium has symbol Ir.
• "Phase change materials" and "photodarkening" are sometimes applied to electro-optic effects in amorphous chalcogenides (GeAsS, GeInSe, etc.) but not to crystalline materials.

⚠️ Example of jargon translation

Reference [44] describes material polarization in chalcogenide glasses:

"...a material stores energy by a transient exciton which can be visualized as a transient intimate valence alternation defect pair... macroscopic anisotropies result from geminate recombination of electron-hole pairs..."

Translation: The material polarizes. An external optical electric field is applied → electric dipoles form throughout the material. (An exciton is a bound electron-hole pair.)

Key takeaway: When reading literature on piezoelectricity, pyroelectricity, and electro-optics, be aware that there is not much consistency in the terminology used.

🧭 Overview

🧠 One-sentence thesis

Antennas are energy conversion devices that convert between electrical energy and electromagnetic energy, functioning as both transmitters and receivers.

📌 Key points (3–5)

• What antennas do: convert electrical energy (flow of electrons) into electromagnetic energy, and vice versa.
• Dual function: antennas can act as both transmitters and receivers.
• Physical principle: based on Maxwell's equations—time-varying electric fields induce magnetic fields, and time-varying magnetic fields induce electric fields.
• Context: antennas are one type of energy conversion device involving conductors (related to inductors), distinct from motors/generators which use magnets and coils.

🔄 Energy conversion principle

⚡ Maxwell's equations foundation

Maxwell's equations say that time varying electric fields induce magnetic fields and time varying magnetic fields induce electric fields.

• This mutual induction is the core physical mechanism behind many energy conversion devices.
• Example: if a permanent magnet moves near a coil of wire, the time-varying magnetic field induces a current in the coil.
• This principle underlies motors, generators, antennas, and Hall effect devices.

🔌 What antennas convert

• Electrical energy: the flow of electrons (current in conductors).
• Electromagnetic energy: propagating electromagnetic waves (radio waves, microwaves, etc.).
• The conversion works in both directions:
  • Transmitter mode: electrical energy → electromagnetic energy.
  • Receiver mode: electromagnetic energy → electrical energy.

🧲 Relationship to other devices

🧲 Conductors and inductors

• Antennas are energy conversion devices involving conductors and are related to inductors.
• This distinguishes them from the devices discussed in earlier chapters (Chapters 1–3), which involved insulators and were related to capacitors.

🔧 Comparison with motors and generators

Device type	What they convert	Why not covered in this text
Motors & generators	Mechanical ↔ electrical energy	Involve magnets and coils; outside the scope
Antennas	Electrical ↔ electromagnetic energy	Covered in this chapter
Hall effect devices	(To be covered in next chapter)	Also based on time-varying field induction

• Motors and generators are "some of the most common energy conversion devices," but the text focuses on antennas and Hall effect devices instead.
• Don't confuse: all three device types rely on the same Maxwell's equations principle, but antennas do not require permanent magnets or coils.

📡 Transmitter and receiver modes

📡 Dual functionality

• Antennas are not one-way devices; the same antenna structure can perform both functions.
• Transmitter: converts the flow of electrons (electrical current) into outgoing electromagnetic waves.
• Receiver: captures incoming electromagnetic waves and converts them back into electrical current.
• This reversibility is a key feature of antennas as energy conversion devices.

🧭 Overview

🧠 One-sentence thesis

The Hall Effect is a physical phenomenon with applications ranging from magnetohydrodynamics to quantum effects, enabling practical devices for measurement and control.

📌 Key points (3–5)

• What the chapter covers: the physics of the Hall Effect, magnetohydrodynamics, quantum Hall Effect, and practical applications.
• Structure: the chapter begins with fundamental physics, extends to specialized contexts (fluids and quantum regimes), and concludes with device applications.
• Common confusion: the excerpt is a table of contents only—it lists topics but does not explain mechanisms or distinctions between classical and quantum Hall effects.
• Why it matters: Hall Effect devices are used in real-world applications (section 5.5), though specifics are not provided in this excerpt.

📚 Chapter structure and scope

📚 Introduction (5.1)

• The chapter opens with an introduction to the Hall Effect.
• No details are provided in this excerpt about what the Hall Effect is or its historical context.

🔬 Physics of the Hall Effect (5.2)

• This section covers the fundamental physical principles underlying the Hall Effect.
• The excerpt does not describe the mechanism, but the section title suggests it explains the core phenomenon.

🌊 Specialized contexts

🌊 Magnetohydrodynamics (5.3)

• The chapter includes a section on magnetohydrodynamics (MHD).
• This likely connects the Hall Effect to the behavior of electrically conducting fluids in magnetic fields.
• No further details are available in this excerpt.

⚛️ Quantum Hall Effect (5.4)

• A dedicated section addresses the quantum Hall Effect.
• This suggests a distinction from the classical Hall Effect, though the excerpt does not explain the difference.
• Don't confuse: the presence of separate sections (5.2 for general physics, 5.4 for quantum) implies different regimes or scales, but the excerpt provides no clarification.

🛠️ Applications and practice

🛠️ Applications of Hall Effect Devices (5.5)

• The chapter concludes with practical applications of Hall Effect devices.
• The excerpt does not list specific applications or device types.

📝 Problems (5.6)

• A problem set is included for practice.
• No sample problems or topics are shown in this excerpt.

⚠️ Note on excerpt content

This excerpt contains only a table of contents for Chapter 5. It lists section titles and page numbers but provides no substantive explanations, definitions, mechanisms, comparisons, or examples. The actual content—what the Hall Effect is, how it works, how classical and quantum versions differ, what magnetohydrodynamics involves, and what applications exist—is not present in this excerpt.

🧭 Overview

🧠 One-sentence thesis

Photovoltaic devices convert optical electromagnetic energy to electricity by using the energy gap of semiconductors to create electron-hole pairs that are swept apart at a pn junction, enabling solar cells and photodetectors to operate.

📌 Key points (3–5)

• Photovoltaic effect: light shining on a semiconductor creates electron-hole pairs; an external electric field sweeps these charges apart to produce voltage.
• Energy gap determines function: only photons with energy ≥ the semiconductor's energy gap (Eₘ) can excite electrons; this determines which wavelengths a device can absorb.
• Why semiconductors, not insulators or conductors: semiconductors have energy gaps (0.5–3 eV) that match visible/near-infrared photon energies; insulators' gaps are too large, conductors have no gap so electrons decay too quickly.
• Common confusion—direct vs indirect semiconductors: in direct semiconductors (e.g., GaAs), the top of the valence band and bottom of the conduction band occur at the same wave vector |k⃗|; in indirect semiconductors (e.g., Si), they occur at different |k⃗| values, requiring a phonon for efficient absorption.
• Pn junctions enable energy conversion: the built-in electric field at a pn junction sweeps photo-generated electrons and holes apart before they recombine, converting light into usable current.

🌈 Light as wave and particle

🌈 Photon energy and wavelength

Photon: a quantum (small chunk) of light.

• Light behaves both as a wave (wavelength λ) and as particles (photons).
• The energy of a photon is given by E = hf = hc/λ, where h is Planck's constant (6.626·10⁻³⁴ J·s) and c is the speed of light (2.998·10⁸ m/s).
• Optical energy: electromagnetic radiation with wavelengths roughly 400 nm ≤ λ ≤ 650 nm, corresponding to frequencies 4.6·10¹⁴ Hz ≤ f ≤ 7.5·10¹⁴ Hz.
• Example: a red photon (λ = 650 nm) has energy E = 1.908 eV; a blue photon (λ = 400 nm) has energy E = 3.100 eV.

⚡ Energy units: joules, electronvolts, and wave number

• Electronvolt (eV): the energy acquired by a charge of magnitude q = 1.602·10⁻¹⁹ C in the presence of a 1 V potential difference.
• Conversion: E[J] = q · E[eV].
• Convenient relation: 1240 / λ[nm] = E[eV].
• Wave number (cm⁻¹): the reciprocal of wavelength; E[J] = 1.986·10⁻²³ · E[cm⁻¹].
• Example: a radio photon at f = 88 MHz has energy E = 3.640·10⁻⁷ eV; an X-ray photon at λ = 10⁻¹⁰ m has energy E = 1.240·10⁴ eV.

☢️ Ionizing vs non-ionizing radiation

Ionizing radiation: radiation with enough energy per photon to rip an electron from skin or muscle.

• UV and X-ray photons have enough energy per photon to ionize atoms (remove electrons), making them dangerous.
• Example: an X-ray photon has ~1000 times more energy than a green light photon.
• Visible light, radio, and microwave photons do not have enough energy per photon to ionize, though microwaves can still heat tissue if enough photons are present.
• Don't confuse: high energy per photon (ionizing) vs high total power (heating); both can be hazardous but through different mechanisms.

🔋 Semiconductors and energy levels

🔋 Intrinsic vs extrinsic semiconductors

Intrinsic semiconductor: a crystal with no lattice defects or impurities; at T = 0 K, all valence electrons are in chemical bonds, no free carriers.

Extrinsic semiconductor: a semiconductor where the equilibrium concentration of electrons (n) or holes (p) differs from the intrinsic carrier concentration (nᵢ) due to impurities or defects.

• At finite temperature T > 0 K, thermal motion creates some charge carriers even in intrinsic semiconductors.
• Doping: introducing more electrons or holes into a semiconductor.
  • n-type: excess of electrons (e.g., Si doped with phosphorus, a column 15 element with 5 valence electrons).
  • p-type: excess of holes (e.g., Si doped with aluminum, a column 13 element with 3 valence electrons).
• Donor: an impurity atom that donates an electron (e.g., P in Si).
• Acceptor: an impurity atom that accepts an electron, creating a hole (e.g., Al in Si).

🔋 Valence electrons, holes, and charge carriers

Valence electrons: the electrons most easily removed from an atom (outer shell electrons).

Hole: an absence of an electron.

Charge carriers: charged particles (valence electrons and holes) that move through a semiconductor when an external voltage is applied.

• At T = 0 K in an intrinsic semiconductor, no charge carriers are present; at T > 0 K, some electron-hole pairs form due to thermal motion.
• Exciton: an electron-hole pair; these pairs naturally recombine (decay) within a short time.
• Charge carrier concentration is measured in electrons/cm³ or holes/cm³.

🔋 Heterostructures and lattice matching

Heterostructure: a stack of different semiconductor materials grown on top of each other.

• Not all materials can form heterostructures; the lattice constant (atomic spacing) must be nearly the same.
• Example: GaAs (lattice constant 0.545 nm) and AlAs (0.546 nm) can form heterostructures.
• If atomic spacing differs too much, mechanical strain pulls the material apart or introduces defects that trap charge carriers.

📊 Energy level diagrams: isolated atoms vs crystals

System	Temperature	Key features
Isolated Al or P atom	T = 0 K	Discrete energy levels; electrons occupy lowest 13 (Al) or 15 (P) levels; valence electrons easier to remove than inner shell
Isolated atoms	T > 0 K	Energy levels broaden; electrons occasionally excited to higher states, then quickly decay
AlP crystal	T = 0 K	Energy levels form bands; valence band (filled) and conduction band (empty) separated by energy gap Eₘ
AlP crystal	T > 0 K	Bands broaden; some electrons excited to conduction band, then decay back

Valence band: the filled energy level closest to the top of an energy level diagram for a semiconductor.

Conduction band: the energy level above the valence band.

Energy gap (bandgap) Eₘ: the energy difference from the top of the valence band to the bottom of the conduction band.

• At T = 0 K, the valence band is typically filled; the conduction band is empty or partially empty.
• Defects or impurities broaden energy levels and may introduce additional allowed levels within the energy gap.
• Example: in AlP, the energy required to remove a valence electron from the crystal differs slightly from the energy required for an isolated Al or P atom.

🔍 Conductors, insulators, semiconductors, and semimetals

Material type	Conduction band at T = 0 K	Energy gap Eₘ	Behavior
Conductor	Partially filled	None	Valence electrons flow easily; many available energy states
Insulator	Empty	Typically > 3 eV	Electrons do not flow easily; large energy needed to excite electrons
Semiconductor	Empty	0.5 eV ≤ Eₘ ≤ 3 eV	Acts as insulator with small applied voltage; acts as conductor with large voltage or light
Semimetal	Partially filled	None	Low electron concentration (n < 10²² electrons/cm³)

• In a semiconductor, photons with energy E ≥ Eₘ can excite electrons from valence to conduction band.
• Example: AlP has Eₘ = 2.45 eV; blue photons (E = 3.1 eV) are absorbed, red photons (E = 1.9 eV) are not.
• Don't confuse: the electro-optic effect (in insulators, external energy too small to excite electrons, no momentum change) vs photovoltaic effect (in semiconductors, external energy large enough to excite electrons, momentum changes).

🌞 Why semiconductors for solar cells and photodetectors?

Why not insulators?

• Sunlight photons (mostly 1.9 eV < E < 3.1 eV, peaking near green ~2.2 eV) do not have enough energy to excite electrons in insulators (Eₘ > 3 eV).
• Example: most visible photons cannot be absorbed by an insulator.

Why not conductors?

• In conductors, electrons excited by photons quickly decay back to the ground state before they can be swept away.
• The energy converted to electricity per excited electron depends on Eₘ, not the photon energy; conductors have no energy gap, so very little energy is converted.
• Excess photon energy (hf − Eₘ) is lost as heat or phonons (lattice vibrations).

Why semiconductors?

• Semiconductors have energy gaps (0.5–3 eV) that match the energy of visible and near-infrared photons.
• Photons with E ≥ Eₘ are absorbed; electrons are excited to the conduction band.
• A pn junction sweeps electrons and holes apart before they recombine, converting light to electricity.
• The energy gap should be large enough to convert significant energy per photon, but not so large that photons are not absorbed.
• Example: Si (Eₘ = 1.1 eV) absorbs visible and near-infrared light; only energy Eₘ per photon is converted to electricity.

🔢 Fermi energy level and electron distribution

Fermi energy level (Eₘ): the energy level at which the probability of finding an electron is one half.

• At T = 0 K in a pure semiconductor, Eₘ is at the middle of the energy gap; no electrons are found at Eₘ because no electrons can have energy inside the gap.
• At T > 0 K, some electrons are excited to higher energy levels; the probability of finding an electron at energy E varies as e⁻ᴱ/ᵏᴮᵀ, where kᴮ is the Boltzmann constant (8.617·10⁻⁵ eV/K).
• Fermi-Dirac distribution: F(E, T) = 1 / (1 + e⁽ᴱ⁻ᴱᶠ⁾/ᵏᴮᵀ); ranges 0 ≤ F ≤ 1.
  • For E ≫ Eₘ (far above conduction band), F ≈ 0 (electrons unlikely).
  • For E ≪ Eₘ (far below valence band), F ≈ 1 (electrons very likely).
• Impurities shift Eₘ:
  • p-type material (lacks electrons): Eₘ closer to valence band.
  • n-type material (excess electrons): Eₘ closer to conduction band.

🔬 Crystallography and E vs k diagrams

🔬 Real space and reciprocal space

Lattice: the arrangement of points in a crystal.

Basis: how atoms are arranged at each lattice point.

Crystal structure: lattice + basis.

• A 3D lattice is described by three primitive lattice vectors a⃗₁, a⃗₂, a⃗₃ (magnitude ~0.1 nm).
• Reciprocal lattice: a corresponding lattice defined by vectors b⃗₁, b⃗₂, b⃗₃.
  • b⃗₁ = 2π (a⃗₂ × a⃗₃) / (a⃗₁ · (a⃗₂ × a⃗₃)); b⃗₁ is perpendicular to a⃗₂ and a⃗₃, parallel to a⃗₁.
  • |b⃗₁| · |a⃗₁| = 2π; if a⃗₁ is long, b⃗₁ is short.
• Real space lattice vectors have units of length (m); reciprocal space vectors have units m⁻¹ (spatial frequency).
• Brillouin zone: a primitive cell for a reciprocal lattice.
• If planes of atoms are closely spaced in one direction (small |a⃗₁|), the corresponding reciprocal vector |b⃗₁| is large (high spatial frequency).

🔬 E vs k diagrams and crystal momentum

Wave vector k⃗: represents change in spatial frequency (distance in reciprocal space); units m⁻¹.

Crystal momentum: M⃗_crystal = ℏk⃗, where ℏ = h/(2π); represents internal momentum due to lattice vibrations; units kg·m/s.

• Kinetic energy in terms of wave vector: E_kinetic = (ℏ²|k⃗|²) / (2m).
• Energy vs |k⃗| diagrams plot allowed energy levels; the |k⃗| axis represents change in position in reciprocal space.
• Near the top of the valence band and bottom of the conduction band, energy is approximately quadratic in |k⃗|, so plots are parabolic.

🔬 Direct vs indirect semiconductors

Type	Band structure	Examples	Photon absorption
Direct	Top of valence band and bottom of conduction band occur at the same |k⃗|	GaAs, InP, ZnTe	A photon alone can excite an electron from valence to conduction band
Indirect	Top of valence band and bottom of conduction band occur at different |k⃗|	Si, Ge, AlAs, GaP	Excitation requires a photon + a phonon (to conserve momentum), or a higher-energy photon

• In an indirect semiconductor, exciting an electron from the top of the valence band to the bottom of the conduction band requires both energy and a change in crystal momentum.
• Example: in Si, a photon with energy slightly above Eₘ cannot excite an electron alone; a phonon (lattice vibration) is also needed.
• Solar cells and photodetectors can be made from either direct or indirect semiconductors.

⚡ Pn junctions

⚡ What happens at a pn junction

Pn junction: the junction of p-type material (excess holes) and n-type material (excess electrons) in a semiconductor.

• When p-type and n-type materials are joined, charge carriers diffuse:
  • Excess electrons from the n-type region move toward the p-type region.
  • Excess holes from the p-type region move toward the n-type region.
• Depletion layer: a region near the junction lacking charge carriers (electrons and holes have diffused away).
• An electric field forms across the junction due to the net charge distribution:
  • n-type side has excess positive charges (donor atoms that lost their valence electron).
  • p-type side has excess negative charges (acceptor atoms).
• Contact potential V₀: the voltage drop across a pn junction in equilibrium; cannot be measured directly with a voltmeter because additional junctions form at the leads.

⚡ Energy level diagram of a pn junction

• The Fermi level Eₘ is near the valence band for p-type materials (lacking electrons).
• The Fermi level Eₘ is near the conduction band for n-type materials (excess electrons).
• In an energy level diagram vs position, the Fermi levels of the two materials line up.
• The vertical distance qV₀ represents the energy required to move an electron across the junction.

⚡ Forward biased, reverse biased, and unbiased junctions

Bias condition	Voltage across junction	Energy to move charge	Depletion layer	Behavior
Unbiased	V₀	qV₀	Moderate width	Equilibrium; no net current
Forward biased	V₀ − Vₓ	q(V₀ − Vₓ)	Narrower	Current flows easily; acts as a wire
Reverse biased	V₀ + Vₓ	q(V₀ + Vₓ)	Wider	Acts as open circuit; charges do not flow

• Forward biased: current flows from p-type to n-type; holes flow from p-type to n-type, neutralizing excess charges in the depletion layer; the electric field preventing charge flow gets smaller.
• Reverse biased: the voltage across the junction is larger; more energy is needed for charges to flow; charges do not flow.

💡 LEDs, solar cells, and photodetectors as pn junctions

Light emitting diode (LED):

• Converts electricity to optical energy.
• A forward bias is applied; holes flow from p-type to n-type and combine with electrons in the depletion layer.
• Photons are emitted in this process; the energy of the emitted photon corresponds to the energy gap Eₘ.
• Some LEDs have an intrinsic (undoped) layer between p-type and n-type layers to improve efficiency.

Solar cell and photodetector:

• Convert optical energy to electricity.
• When light shines on the device, photons with energy E ≥ Eₘ create electron-hole pairs at the junction.
• The charge distribution across the junction sweeps electrons and holes away from the junction before they recombine.
• This flow of charges is a current; optical energy is converted to electricity.
• Photovoltaic effect: when light shines on a photovoltaic device, a voltage can be measured across the junction.

Thermoelectric effect:

• When a temperature gradient is applied across a pn junction, charges flow (Seebeck effect, discussed in Chapter 8).
• Energetic charges on the hot side diffuse to the cooler side.

🌞 Solar cells

🌞 Solar cell efficiency and losses

Efficiency: η_eff = P_electrical_out / P_optical_in.

Why solar cells are not 100% efficient:

1. Not all sunlight reaches the cell:

   • Atmospheric absorption (wavelength-dependent); e.g., ozone O₃ absorbs UV light.
   • Typical solar cell receives ~0.1 W/cm² in a bright sunny area.

2. Not all light that hits the cell is absorbed:

   • Light may heat the cell instead of exciting electrons.
   • Light may be reflected off the surface (antireflection coatings or rough surfaces reduce this).
   • Photons hitting already-excited electrons are not absorbed.
   • Wires on the surface block some light (some cells use partially transparent conductors like indium tin oxide).

3. After a photon excites an electron:

   • Electron may decay before being swept from the junction.
   • Electron excited above the conduction band quickly decays to the top of the conduction band, losing excess energy (hf − Eₘ) as heat or phonons.
   • Internal resistance in bulk regions or wiring converts electricity to heat.
   • Unmatched loads reduce efficiency.

4. Temperature effects:

   • As temperature increases, efficiency decreases.
   • Energy gap Eₘ decreases (energy levels broaden) → voltage produced decreases roughly linearly.
   • More electron-hole pairs recombine at the junction → current decreases roughly exponentially (main effect).

🌞 Solar cell technologies

Technology	Description	Typical efficiency	Notes
Crystalline	Single crystal or polycrystalline (e.g., Si)	~20%	First generation; ~80% of market; polycrystalline often cheaper, slightly less efficient
Thin film	Thin films (few microns) deposited on substrate (glass, steel)	Lower than crystalline	Often cheaper; materials include amorphous Si, CdTe (Eₘ = 1.45 eV; Cd and Te are toxic)
Multijunction (compound)	Dozen+ layers stacked; multiple pn junctions; larger-gap semiconductors on top	Up to 46% (lab)	Larger-gap layers absorb high-energy photons; smaller-gap layers absorb lower-energy photons
Emerging	Nanotechnology (carbon nanotubes, quantum dots), organic solar cells	Varies	Organic cells: active layer ~100–200 nm; may not require high processing temperatures

• Example multijunction cell: top layer ZnS (Eₘ = 3.6 eV), middle layer ZnSe (Eₘ = 2.7 eV), bottom layer ZnTe (Eₘ = 2.25 eV); photons with E > 3.6 eV absorbed by ZnS, 2.7 eV < E < 3.6 eV by ZnSe, 2.25 eV < E < 2.7 eV by ZnTe.
• Trend: if an element in a semiconductor is replaced with one below it in the periodic table, the energy gap tends to get smaller (e.g., AlP → AlAs → AlSb).

🌞 Solar cell systems and balance of system

Balance of system: components in a solar panel installation other than the solar panel arrays.

Components:

• Solar panel arrays: individual solar cells connected into modules, modules connected into panels.
  • Typical cell produces ~1 W; typical house requires ~4 kW.
  • Typical panel: ~40 cells; typical roof installation: 10–20 panels.
• Mounting or tracking hardware: foundation, supports, brackets, wiring.
  • Fixed position or tracking (single-axis east-west, or two-axis).
• Concentrators: mirrors or lenses to capture more sunlight onto panels (often used with two-axis tracking).
• Batteries: provide power at night or on cloudy days.
  • Typical lifetime: 3–9 years (often first component to need replacement).
  • Solar panels last 30+ years with ~1–2% degradation per year.
• Power conditioner: inverter (DC to AC), phase matching for grid-tied systems, current/voltage limiting to maximize power, safeguards (fuses).
  • Typical lifetime: 10–15 years.

🔍 Photodetectors

🔍 Types of photodetectors

Photodetector: a sensor that converts light (optical or nearby frequencies) to electricity.

Classification by active material:

1. Semiconductor photodetectors: solid semiconductor pn junctions.
   • Choice of semiconductor determines which wavelengths can be absorbed (E ≥ Eₘ).
   • Example: Si (Eₘ = 1.11 eV) absorbs visible (1.9 eV < E < 3.1 eV) and near-infrared (1.1 eV < E < 1.9 eV).
   • Semiconductor p-i-n junction photodetector: thin intrinsic (undoped) layer between p-type and n-type; widens depletion layer, decreases internal capacitance, increases response time.
2. Phototubes: gas-filled vacuum tubes.
   • Voltage across electrodes; light rips electrons off gas atoms; electrons and ions flow to electrodes.
   • Photomultiplier tube: multiple electrodes; each electron hitting an electrode emits additional electrons (cascade → high internal amplification).

Classification by mechanism:

1. Photoelectric detectors (photoelectric emission): incoming photons have energy ≥ energy from valence band to ground level; photons rip electrons off atoms; flow of electrons is a current.
2. Photoconductive (photovoltaic) detectors (photoconductivity): incoming photons have energy = Eₘ (valence to conduction band); photons excite electrons; conductivity increases when light shines on detector.
   • Most solid semiconductor photodetectors operate based on photoconductivity.
   • Phototubes typically operate based on photoelectric emission.

Single element vs arrays:

• Some photodetectors have a single element; others are arrays (e.g., digital cameras with millions of elements).
• Charge-coupled device (CCD): circuitry to sequentially transfer electrical output of each photodetector in an array; invented 1969 by Willard S. Boyle and George E. Smith (2009 Nobel Prize).
• Human eye: retina is an array of ~120 million rod cells and 6–7 million cone cells.

🔍 Measures of photodetectors

Measure	Definition	Units	Notes
Frequency response	Sensitivity vs wavelength (or photon energy)	—	Detector sensitive only within a particular wavelength range; response often not flat
Signal to noise ratio	—	—	Major noise source: thermal noise (random motion of charges); weak-signal detectors are cooled
Noise equivalent power	Optical power that produces signal-to-noise ratio of 1	W	Related to signal-to-noise ratio
Detectivity (D)*	√(Area · Bandwidth) / Noise Equivalent Power	cm·(Hz^(1/2))/W	Measure of output strength assuming 1 W optical input
Response time	Time needed to respond to a step-like optical input	s	Typical range: picoseconds to milliseconds; tradeoff between response time and sensitivity

• Don't confuse: fast operation vs high sensitivity; some detectors are designed for one or the other.

🧭 Overview

🧠 One-sentence thesis

Devices that convert electricity to light operate through three fundamental processes—absorption, spontaneous emission, and stimulated emission—with each process enabling different device types ranging from simple incandescent lamps to highly coherent semiconductor lasers.

📌 Key points (3–5)

• Three energy conversion processes: absorption (light → internal energy), spontaneous emission (excited electron → photon, random direction/phase), and stimulated emission (excited electron + incoming photon → two identical photons).
• Device classification by process: absorption devices (solar cells, photodetectors), spontaneous emission devices (lamps, LEDs), and stimulated emission devices (lasers, optical amplifiers).
• Population inversion requirement: lasers and optical amplifiers only work when more electrons occupy the upper energy level than the lower level, a non-equilibrium condition.
• Common confusion—spontaneous vs. stimulated emission: spontaneous emission produces photons with random direction/phase/polarization and does not require incoming photons; stimulated emission produces photons identical to the triggering photon (same frequency, direction, phase, polarization) and requires both an incoming photon and an excited electron.
• Why it matters: understanding these processes explains why lasers produce narrow-bandwidth coherent light while lamps produce broad-spectrum incoherent light, and why certain materials and designs are chosen for specific applications.

🔬 Three fundamental energy conversion processes

🔬 Absorption

Absorption: the process in which optical energy is converted to internal energy of electrons, atoms, or molecules.

• When a photon is absorbed, an electron moves from a lower to a higher energy level.
• The photon energy must match the energy gap between levels; smaller-energy photons are not absorbed.
• This process occurs in solar cells and photodetectors (covered in Chapter 6).
• Example: neon atoms have an energy gap of 1.96 eV between 2p and 3s levels, corresponding to red photons at 632.8 nm; photons of this energy can be absorbed, exciting electrons to the higher level.

💡 Spontaneous emission

Spontaneous emission: an energy conversion process in which an excited electron or molecule decays to an available lower energy level and in the process gives off a photon.

• Occurs naturally without interaction with other photons.
• The emitted photon may have any direction, phase, and electromagnetic polarization.
• Energy of the emitted photon equals the energy difference between the electron energy levels.
• Average decay time is called the spontaneous emission lifetime (nanoseconds to seconds depending on material).
• Don't confuse: the electron can also decay by emitting heat or mechanical vibrations instead of a photon; only when a photon is produced is it called spontaneous emission.

⚡ Stimulated emission

Stimulated emission: the process in which an excited electron or molecule interacts with a photon, decays to an available lower energy level, and in the process gives off a photon.

• Requires an incoming photon with energy equal to the difference between allowed energy levels.
• The stimulated photon has the same frequency, direction, phase, and electromagnetic polarization as the incoming photon.
• This is the key process enabling lasers and optical amplifiers.
• Example: an incoming photon at the right energy interacts with an excited electron in an atom; the electron decays and produces a second photon that is identical to the first, resulting in two photons where there was one.

🔄 Energy level diagrams

The excerpt illustrates these three processes with energy level diagrams showing two levels (labeled 1 and 2):

• Absorption: electron starts at level 1, absorbs incoming photon (squiggly arrow), moves to level 2.
• Spontaneous emission: electron starts at level 2, spontaneously decays to level 1, emits photon in random direction.
• Stimulated emission: electron starts at level 2, incoming photon triggers decay to level 1, produces second identical photon.

📊 Rate equations and Einstein coefficients

📊 How rates depend on populations and photon density

The excerpt models the rate of each process mathematically:

Process	Rate equation	What it depends on
Absorption	(dn₂/dt)_abs = B₁₂ n₁ u	Population in lower state (n₁) and photon energy density (u)
Spontaneous emission	(dn₂/dt)_spont = −A₂₁ n₂	Only population in upper state (n₂); no photons needed
Stimulated emission	(dn₂/dt)_stim = −B₂₁ n₂ u	Population in upper state (n₂) and photon energy density (u)

• n₁ = number of electrons per unit volume in lower state
• n₂ = number of electrons per unit volume in upper state
• u = spectral energy density per unit bandwidth (J·s/m³)
• A₂₁, B₁₂, B₂₁ = Einstein coefficients (constants of proportionality)

🔗 Relationships between Einstein coefficients

Einstein showed that if one coefficient is known, the others can be calculated:

• A₂₁ / B₂₁ = 8πhf³ / c³
• g₁B₁₂ = g₂B₂₁ (where g₁, g₂ are degeneracy levels)

These relationships come from considering a blackbody radiator at equilibrium.

⚖️ Population inversion condition

Population inversion: the condition n₂ − (g₂/g₁)n₁ > 0, meaning more electrons are in the upper energy level than the lower level (accounting for degeneracy).

• Optical amplification and lasing can only occur when there is a population inversion.
• This is a non-equilibrium condition; energy must be continuously supplied to maintain it.
• At thermal equilibrium, the lower state is always more populated (Boltzmann statistics).
• Don't confuse: "population inversion" has nothing to do with inversion symmetry (a crystallographic concept).

💡 Devices involving spontaneous emission

💡 Incandescent lamps

Incandescent lamp: a device that converts electricity to light by blackbody radiation.

• Construction: solid metal filament inside a glass vacuum tube.
• Current heats the filament to thousands of degrees.
• High temperatures used because visible spectral response of daylight is close to a blackbody radiator at 6500 K.
• Main limitation: low efficiency—much electromagnetic radiation falls outside the visible range.
• Main advantage: simplicity; these were among the earliest lamps developed (practical versions date to the 1850s).

🌡️ Blackbody radiation

Blackbody radiation: spontaneous emission that occurs naturally in any object at a temperature above absolute zero.

• Spectral energy density per unit bandwidth: u = (8πf²/c³) · [hf / (e^(hf/kᵦT) − 1)]
• Higher temperatures produce more photons and a larger fraction in the visible range.
• Photons have a relatively wide range of wavelengths; bandwidth depends on temperature.

🔦 Gas discharge lamps

Gas discharge: when a conducting path forms through a plasma (ionized gas).

• Construction: sealed tube with two electrodes, filled with gas at pressures from 10⁻⁴ Pa to 10⁵ Pa.
• Hundreds to millions of volts applied across electrodes ionize the gas and supply free electrons.
• Optical properties determined by the gas; spontaneous emission occurs between distinct allowed energy levels, producing relatively narrow wavelength ranges.
• Examples: sodium vapor lamps, mercury arc lamps, fluorescent lamps, neon advertising signs.

🌟 Three operating regions

Region	Current level	Characteristics
Dark (Townsend discharge)	Low	Not self-sustaining; no chain reaction
Glow discharge	Medium	Self-sustaining; avalanche ionization; significant spontaneous emission
Arc discharge	High	Self-sustaining; lower voltage needed to maintain than glow discharge

💡 Fluorescent lamps

• A type of gas discharge device using phosphors (chemicals with desired optical properties).
• Gas produces UV photons by arc or glow discharge.
• Phosphors absorb UV and emit visible light at lower frequencies.
• Example phosphors: zinc silicate, calcium tungstate, zinc sulfide.

🔴 LEDs (Light Emitting Diodes)

LEDs: devices that convert electricity to light by spontaneous emission, made from pn junctions in semiconductors.

• Forward bias applied across pn junction injects electrons and holes.
• Electrons and holes near the junction combine and spontaneously emit photons.
• Some LEDs have a thin intrinsic (undoped) layer between p-type and n-type layers to improve efficiency.
• Narrow frequency range: determined by the energy gap of the semiconductor.
• Much narrower than incandescent lamps, but broader than lasers.

🎨 Producing white light from LEDs

• Applications like residential lighting require white light with broader bandwidth.
• Strategy: use phosphors that absorb blue or near-UV light from LED and emit at lower energies across the visible range.
• Blue LEDs were particularly important; developed in 1980s–1990s after decades of effort.
• 2014 Nobel Prize in Physics awarded to Akasaki, Amano, and Nakamura for developing blue LEDs.
• Required development of deposition technology for materials like gallium nitride without mechanical strain.

📱 OLEDs (Organic Light Emitting Diodes)

• Voltage excites electrons in a thin layer (100–200 nm) of organic material.
• Type of organic material determines wavelength emitted.
• Used in flat panel displays; white light achieved by combining red, green, and blue OLEDs.

✅ Advantages of LEDs

• Small size: often fit into a cubic millimeter; easy to integrate into electronics.
• Low voltage: require small amounts of input electrical power.
• Long lifetime: no electrode sputtering like in gas discharge lamps.
• Efficiency: produce less waste heat than incandescent lamps.

🔦 Devices involving stimulated emission

🔦 Laser fundamentals

Laser: Light Amplification by Stimulated Emission of Radiation—devices that produce optical energy through stimulated emission and involve optical feedback.

• Development preceded by theoretical breakthroughs in understanding atomic energy conversion processes (mid-1950s).
• Maser (microwave frequencies) demonstrated ~1955 by Gordon, Zeiger, and Townes.
• First laser (ruby, 694 nm) demonstrated 1960 by Maiman.
• Semiconductor lasers predicted 1961, demonstrated within a year in gallium arsenide.

🧩 Three main laser components

🧩 Pump (power supply)

• Supplies energy to excite electrons or molecules in the active material.
• Electrical pumping: battery or power supply provides DC current (e.g., semiconductor lasers).
• Optical pumping: lamp or another laser provides photons (e.g., early ruby lasers used flashlamps; argon ion lasers pump titanium-doped sapphire lasers).
• Electrically pumped lasers convert electricity → light.
• Optically pumped lasers convert high-energy photons → lower-energy photons.

🎯 Active material

Active material: the material with multiple allowed energy levels where energy conversion occurs as electrons transit between levels.

• Can be solid, liquid, or gas.
• Population inversion required: more electrons must be in the upper energy level than the lower level for lasing to occur.
• Without population inversion, stimulated photons are more likely to be absorbed or decay by other means than to stimulate additional photons.

Two, three, and four level lasers:

• Two level: pump excites electron from level 1 → 2; lasing occurs between the same two levels.
• Three level: pump excites 1 → 3; electron quickly decays 3 → 2 (emitting heat); lasing occurs 2 → 1.
• Four level: pump excites 1 → 4; quick decay 4 → 3 (heat); lasing occurs 3 → 2; quick decay 2 → 1 (heat).
• Four level systems may lase more easily because population inversion can be easier to achieve.

🪞 Cavity

Laser cavity: confines photons to the active material and acts as an optical filter.

• Simplest type: two mirrors forming a Fabry-Perot cavity.
• More complex cavities use multiple mirrors, lenses, prisms to focus desired photons and reject other frequencies.
• Semiconductor lasers often use the semiconductor-air interface as mirrors (no separate mirrors needed).

How cavities filter wavelengths:

• Cavity acts as optical filter even without active material present.
• Light constructively interferes when cavity length equals an integer number of half wavelengths.
• Light destructively interferes at other wavelengths.
• This occurs in both longitudinal direction (along cavity length) and transverse directions.
• Feedback encourages lasing at specific wavelengths corresponding to allowed modes.
• Example: if cavity length = 3λ/2 in longitudinal direction, that wavelength is favored.

📉 Laser efficiency

📉 Overall (wall plug) efficiency

• η_eff = P_optical_out / P_electrical_in (for electrically pumped lasers)
• Overall efficiency is the product: η_eff = η_pump · η_quantum · η_cavity

Three efficiency components:

Component	What it represents	Factors affecting it
η_pump	Pump efficiency	Reflection/transmission losses; conversion to heat; photon energy mismatch; interaction with already-excited electrons
η_quantum (internal quantum efficiency)	Active material efficiency	Ratio of electrons that decay and produce a photon vs. all electrons that decay; depends on temperature, impurities, defects
η_cavity	Cavity efficiency	Mirror reflectivity; absorption in optical components; in semiconductors, index of refraction determines reflection (e.g., GaAs n=3.52 → only 31% reflected)

• Relationship for mirror reflectivity R, absorption coefficient α, and active material length l:
  • η_eff = η_eff-other · [1 / (1 + (1/αl)·ln(1/R))]

📏 Laser bandwidth

📏 Why lasers have narrow bandwidth

• Photons from stimulated emission have the same wavelength as the stimulating photon.
• Cavity only allows constructive interference at specific wavelengths (integer half-multiples of cavity length).
• Result: lasers emit over a much narrower wavelength range than LEDs, gas discharge lamps, or incandescent lamps.

📏 Full width half maximum (FWHM)

FWHM: the wavelength (or frequency) difference between two points at half the maximum intensity.

• Measured from intensity vs. wavelength plot (or intensity vs. frequency plot).
• Can be specified in nm or Hz.
• Quality factor = λ_peak / FWHM[nm] = f_peak / FWHM[Hz]
• Higher quality factor = narrower bandwidth.

Example from dye laser (rhodamine 6G with silver nanoparticles):

• Below lasing threshold (spontaneous emission only): FWHM ≈ 45 nm, quality factor ≈ 13
• Above lasing threshold (stimulated emission dominates): FWHM ≈ 5 nm, quality factor ≈ 115
• Gas lasers can achieve FWHM of 0.01 nm (orders of magnitude narrower).

🎯 Lasing threshold

• Lasing only occurs if the active material is pumped strongly enough.
• Below threshold: spontaneous emission occurs (broad bandwidth).
• Above threshold: stimulated emission dominates (narrow bandwidth).
• Plots of output intensity vs. pump energy show a discontinuity in slope at the lasing threshold.

🔬 Four main types of lasers

🔬 Gas lasers

• Active material: gas (neutral or ionized atoms).
• Construction: gas-filled glass tube with internal electrodes and external mirrors.
• Electrodes supply power to excite gas atoms.
• Examples: helium-neon (632.8 nm, but can operate at other wavelengths like 3.39 μm), argon ion, carbon dioxide.
• Advantages: electrically pumped; can produce very high power (CO₂ lasers: hundreds of kW continuous, terawatts pulsed); used for cutting, welding, weaponry.
• Disadvantages: physically large; high-power versions require water cooling.

🔬 Dye lasers

• Active material: solute in a liquid (often organic molecules).
• Often optically pumped by other lasers.
• Lasing may occur between molecular vibration energy levels.
• Example: rhodamine 6G, tunable from 570 nm < λ < 610 nm.
• Advantages: tunable over wide wavelength range.
• Disadvantages: require regular maintenance because dyes have finite useful lifetime.

🔬 Solid state lasers

• Active material: solid insulating material (often high-purity crystal) doped with an element.
• Lasing occurs between electron energy levels of the dopant.
• External mirrors form cavity.
• Typically optically pumped by lamps or other lasers.

Examples:

• Ruby laser: sapphire (Al₂O₃) doped with ~0.05% chromium (Cr³⁺); three-level laser.
• Nd:YAG laser: yttrium aluminum garnet (Y₃Al₅O₁₂) doped with ~1% neodymium (Nd³⁺); four-level laser; produces infrared at λ = 1.0641 μm.
• Ti:Sapphire laser: sapphire (Al₂O₃) doped with ~1% titanium (Ti³⁺); tunable 700 nm < λ < 1020 nm; tuning achieved through adjustable prism and mirror coatings; used for spectroscopy and materials research.

🔬 Semiconductor lasers

• Active material: solid semiconductor pn junction.
• May include intrinsic (undoped) layer between p-type and n-type to increase depletion region width and improve efficiency.
• Entire device typically fits in a cubic millimeter.
• Wavelength determined by energy gap of semiconductor.
• First semiconductor lasers: gallium arsenide (infrared).
• Blue lasers took 30+ years to develop reliably (1980s–1990s); required GaN and ZnMgSSe technology.

Direct vs. indirect semiconductors:

• Almost all semiconductor lasers use direct semiconductors.
• In direct semiconductors, top of valence band and bottom of conduction band line up in energy vs. wave vector plot.
• Electrons can decay by emitting a photon without changing momentum.
• In indirect semiconductors, momentum change is required; electrons more often decay by producing heat/vibrations instead of photons.
• This is why first semiconductor lasers used GaAs even though silicon processing was more developed.

Two main categories:

• Edge emitters: optical emission from edge; cavity formed by semiconductor edges or by etched grating (distributed feedback lasers).
• VCSELs (Vertical Cavity Surface Emitting Lasers): optical emission from surface; multiple epitaxial layers form mirrors above and below active material.

Advantages:

• Small size; integrate easily into consumer devices (laser pointers, DVD players) and industrial equipment.
• Electrically pumped.
• Often don't need external cooling due to relatively high efficiency.
• Output wavelength can be designed by selecting composition.
• Example: In₁₋ₓGaₓAs₁₋ᵧPᵧ produces infrared 1.1 μm < λ < 1.6 μm, useful for fiber optic communication (silica glass absorption minimum near 1.55 μm, dispersion minimum near 1.3 μm).

Limitations:

• Power output lower than gas lasers (over 1 W for semiconductors vs. orders of magnitude more for gas lasers).

🔊 Optical amplifiers

Optical amplifier: consists of a pump and active material, but does not have a cavity.

• Pump excites electrons to upper energy level.
• Incoming optical signal photons cause additional photons to be generated by stimulated emission.
• Amplification occurs (more photons out than in), but lasing does not occur without optical feedback from a cavity.
• Can be made from all types of active materials: gases, solid state, semiconductors, dyes.

🔊 Erbium doped fiber amplifiers

• Solid state devices where stimulated emission occurs between energy levels of erbium (dopant) in silica glass fibers.
• Semiconductor laser acts as pump.
• Very useful in optical communication networks: amplify optical signals without converting to/from electrical signals.
• Can amplify signals near fiber absorption minimum at 1.55 μm.

🗂️ Classification of devices

🗂️ By energy conversion process

Devices can be classified by whether they primarily involve absorption, spontaneous emission, or stimulated emission:

Process	Device categories	Examples
Absorption	Solar cells, photodetectors	Photodiodes, phototransistors, photomultiplier tubes, solar panels
Spontaneous emission	Lamps, LEDs	Incandescent lamps, gas discharge lamps (glow discharge, arc discharge, neon signs, fluorescent, sodium vapor), LEDs
Stimulated emission	Lasers, optical amplifiers	Gas lasers (argon ion, helium-neon, CO₂), solid state lasers (ruby, Ti:Sapphire, Nd:YAG), semiconductor lasers (edge emitters, VCSELs), dye lasers, erbium doped fiber amplifiers

• Antennas also convert between electricity and electromagnetic radiation (at radio/microwave frequencies rather than optical).
• Receiving antennas involve absorption; transmitting antennas involve spontaneous emission.

🗂️ By active material type

Devices can also be classified by the state of matter of the active material:

Active material	Device examples
Gaseous	Gas discharge lamps (glow/arc discharge, neon signs, fluorescent, sodium vapor), gas lasers (argon ion, helium-neon, CO₂), photomultiplier tubes
Semiconducting	Photodiodes, phototransistors, solar cells, LEDs, semiconductor lasers (edge emitters, VCSELs)
Solid state (insulating)	Solid state lasers (ruby, Ti:Sapphire, Nd:YAG), erbium doped fiber amplifiers
Conductive	Incandescent lamps, antennas

🔄 Multiple processes in one device

• Devices are usually designed for one process, but multiple processes can occur depending on operation.
• Example: a semiconductor laser with current above lasing threshold → stimulated emission (laser).
• Same device with weaker current → spontaneous emission (LED).
• Same device with light shining on it and voltage measured → absorption (photodetector).
• Similarly: photomultiplier tubes, gas discharge lamps, and gas lasers can convert electricity → light or light → electricity depending on how they are operated.

🧭 Overview

🧠 One-sentence thesis

Thermoelectric devices convert temperature differences directly into electricity (or vice versa) through junctions of dissimilar conductors or semiconductors, with efficiency fundamentally limited by the Carnot efficiency and practically determined by the material's figure of merit.

📌 Key points (3–5)

• Core mechanism: Thermoelectric devices exploit three related effects (Seebeck, Peltier, Thomson) that arise when materials have high electrical conductivity but low thermal conductivity.
• Material requirements: Good thermoelectric materials must allow charges to flow easily while blocking heat flow—typically achieved with doped semiconductors or semimetals containing tellurium, bismuth, lead, or antimony.
• Figure of merit: A single measure Z = S²σ/κ summarizes material quality, combining Seebeck coefficient, electrical conductivity, and thermal conductivity.
• Efficiency limits: The Carnot efficiency η = 1 - Tc/Th sets the theoretical maximum for any device converting temperature differences to other energy forms; practical thermoelectric devices typically achieve less than 10% efficiency.
• Common confusion: Don't confuse the three thermoelectric effects—Seebeck generates voltage from temperature difference, Peltier produces heating/cooling from current, and Thomson describes heating/cooling in a single material with both current and temperature gradient.

🌡️ Thermodynamic foundations

🌡️ Four fundamental properties

The excerpt defines four state properties (independent of history):

Property	Symbol	Units	Type	What it measures
Volume	V	m³	Extensive	Size/extent of material
Pressure	P	Pa (N/m²)	Intensive	Force per area
Temperature	T	K	Intensive	Thermal energy level
Entropy	S	J/K	Extensive	Disorder/lack of organization

• Extensive properties depend on system size (volume, entropy).
• Intensive properties are independent of size (pressure, temperature).
• Temperature must use absolute scales (Kelvin) for thermodynamic calculations: T[°C] = T[K] - 273.15.

⚖️ Ideal gas law

The ideal gas law relates pressure, volume, and temperature: PV = NRT

• N = number of moles, R = 8.314 J/(mol·K) = molar gas constant.
• Good approximation for gases; rough approximation for liquids and solids.
• Can incorporate entropy: PV = ST.
• Example: 5 mol argon in 10 L at 15°C → P = NRT/V = 1.20 × 10⁶ Pa ≈ 11.8 atm.

🔋 First law of thermodynamics

Energy conservation: ΔU = Q - W

• ΔU = change in internal energy (stored in the system).
• Q = heat transferred into the system.
• -W = work and other energy forms transferred into the system (minus sign convention).
• In closed systems, energy is conserved—either stored, transferred as heat, or transferred as work/other forms.
• Example: heating gas in a piston with 100 J while piston moves → internal energy increases by 100 J minus mechanical work done by expansion.

🔌 The three thermoelectric effects

⚡ Seebeck effect (1821)

When a junction of two different materials has sides at different temperatures, a voltage develops across the junction.

• Seebeck coefficient S (units: V/K) = voltage difference / temperature difference.
• For a junction: S₁ - S₂ = ΔV₁₂ / ΔT₁₂.
• Most materials have S < 1 μV/K; tellurium compounds have largest values (e.g., PbTe ≈ 400 μV/K).
• Physical mechanism: Charge carriers (electrons in metals, electrons and holes in semiconductors) diffuse from hot to cold side, creating charge separation and electric field.
• Key requirement: Material must have large electrical conductivity but small thermal conductivity—charges move easily, heat doesn't.

🔥 Peltier effect (1834)

When current flows through a junction of two different materials, heat is transferred (heating one side, cooling the other).

• Peltier coefficient Π (units: V) = (rate of heat transfer) / current.
• Π₁₂ = Π₁ - Π₂ = (dQ/dt) / I.
• Relationship to Seebeck: Π₁ - Π₂ = (S₁ - S₂)T.
• Example: PbTe at 300 K has Π = 0.12 V.
• Mechanism: Current carries charges through materials with different thermal conductivities, causing heat accumulation/depletion.
• Works bidirectionally: supply current to create temperature difference, or supply temperature difference to generate current.

🌊 Thomson effect (1860s)

When current passes through a single uniform material with a temperature gradient, heating or cooling occurs.

• Thomson coefficient τ (units: V/K) = (rate of heat generated) / [current × temperature difference].
• τ = (dQ/dt) / [I(Th - Tc)].
• Relationship to Seebeck: Integral of τ/T over temperature equals Seebeck coefficient.
• May be positive, negative, or zero depending on material.
• Don't confuse: Seebeck and Peltier involve junctions of two materials; Thomson involves a single material with both current and temperature gradient.

🧪 Material properties for thermoelectrics

🔋 Electrical conductivity

Electrical conductivity σ (units: 1/(Ω·m)) measures the ability of charges to flow through a material.

• For conductors: σ = nqμₙ where n = electron concentration, q = electron charge, μₙ = electron mobility.
• For semiconductors: σ = q(nμₙ + pμₚ) including both electrons and holes.
• Mobility μ (units: m²/(V·s)) = drift velocity / electric field strength.
• Measured using: ohmmeter to find resistance R, then σ = l/(AR) for known length l and area A.
• Temperature dependent: limited by impurity scattering at low T, phonon scattering at high T; peaks at intermediate temperature.
• Crystals have higher σ than amorphous materials (less scattering).
• Doped semiconductors have higher σ than intrinsic (more charge carriers).

🌡️ Thermal conductivity

Thermal conductivity κ (units: W/(m·K)) measures the ability of heat to flow through a material.

• By definition: κ = -(dQ/dt) / [A(dT/dx)] where dQ/dt = heat flow rate, A = area, dT/dx = temperature gradient.
• Measured using: heater on one end, heat sink on other, thermocouples measure temperature difference at steady state.
• Alternative view: κ = Cv|v|l where Cv = specific heat, |v| = transport velocity, l = scattering length.
• Heat transported by phonons, photons, electrons, or other mechanisms—all temperature dependent.
• Crystals typically have higher κ than glasses (less scattering); radiation can destroy crystallinity and lower κ.
• Temperature dependent: rises to maximum then decreases at higher temperatures.
• For thermoelectrics, want low thermal conductivity (blocks heat flow).

📊 Figure of merit

Figure of merit Z (units: K⁻¹) = S²σ/κ summarizes how good a material is for thermoelectric devices.

• Larger Z = better thermoelectric material.
• Depends on three properties: Seebeck coefficient S, electrical conductivity σ, thermal conductivity κ.
• Often use ZT (dimensionless) to account for temperature dependence.
• Optimization challenge: Want high S and σ but low κ—these properties are coupled and hard to optimize independently.

Factor	Effect on Z	Why
High electrical conductivity	Increases Z	Charges flow easily
Low thermal conductivity	Increases Z	Heat blocked
High Seebeck coefficient	Increases Z	Large voltage per degree
Heavy doping	Usually increases Z	More charge carriers → higher σ
Crystallinity	Complex	Both σ and κ increase in crystals

• Material selection factors beyond Z: Melting temperature limits maximum operating temperature; larger ΔT increases power but material must not melt.
• Common materials: Bi₂Te₃, PbTe, Sb₂Te₃ (semimetals/small-gap semiconductors near Bi, Sb, Pb on periodic table).
• Recent research: Layered materials and superlattices can tailor electrical and thermal properties independently.

📉 Efficiency limits and practical performance

🎯 Carnot efficiency

The Carnot efficiency η = 1 - Tc/Th is the maximum possible efficiency for any device converting a temperature difference to another energy form.

• Applies to: thermoelectric devices, steam turbines, pyroelectric devices, Stirling engines—any heat engine.
• Does NOT apply to: photovoltaic devices, piezoelectric devices (these don't convert temperature differences).
• Derivation logic: Energy in = kB·Th (to maintain hot side), Energy out = kB·Th - kB·Tc (extracted), Efficiency = Eout/Ein = (Th - Tc)/Th.
• Temperatures must be in absolute units (Kelvin or Rankine), not Celsius or Fahrenheit.
• Example: Cold side at 300 K, want 40% efficiency → hot side must be 500 K.
• Example: Body heat (309 K) vs room temp (295 K) → maximum 4.5% efficiency.

Why efficiency is fundamentally limited:

• If Tc = Th (no temperature difference), cannot extract any energy.
• If Tc = room temperature, efficiency cannot reach 100%.
• Larger temperature differences allow higher efficiency, but practical limits exist (melting, material degradation).

🔻 Practical efficiency factors

Real thermoelectric devices achieve much lower efficiency than Carnot limit:

• Resistive heating: All materials have finite resistance; I²R losses convert electrical energy back to heat.
• Heat conduction: Heat flows through device from hot to cold side; not all input heat converts to electricity.
• Typical performance: Most thermoelectric devices achieve less than 10% efficiency.
• Example calculation: 1 mm long, 1 mm² area conductor with ρ = 10⁻⁵ Ω·m, 3 mA current → 9×10⁻⁸ W resistive heating loss.
• Efficiency increases with higher Th, but melting temperature limits maximum (e.g., PbTe melts at 924°C, Bi₂Te₃ at 580°C).

🛠️ Applications and practical considerations

🛠️ Cooling applications

• Electronics cooling: CPUs, graphics cards—small thermoelectric devices increase reliability and lifetime.
• Refrigeration: RVs, submarines, restaurant food dispensers, pharmaceutical transport—less efficient than traditional but small, quiet, low maintenance.
• Air conditioning: Environmentally friendly (no freon) but only a few percent efficient, so rarely used.
• Imaging devices: Cool scanning electron microscopes to reduce atomic vibration that smears images; replaced liquid nitrogen cooling.
• Temperature control: Keep sensors at fixed temperature for accurate, stable measurements.

⚡ Power generation applications

• Space missions: NASA Multi-Mission Radioisotope Thermoelectric Generator powers Mars rover Curiosity—~10 lbs plutonium-238 produces ~2 kW heat → ~120 W electricity (~6% efficiency); no moving parts, no refueling needed for years.
• Temperature sensors: Thermocouples (small thermoelectric junctions) measure temperature accurately and inexpensively by converting small temperature differences to measurable voltages.
• Energy harvesting: Convert waste heat to electricity—car interiors, near light bulbs, industrial processes.

⚠️ Unwanted thermoelectric effects

• Circuit noise: Junctions of different metals (aluminum PCB trace to tin resistor wire, solder to copper wire) create unwanted voltages via Seebeck effect.
• Example: Copper-to-tin-lead-solder junction has S ≈ 2 μV/K—introduces noise/distortion in sensitive circuits.
• Engineers must account for these effects in precision measurement circuits.

🔑 Key advantages

• No moving parts → high reliability, long lifetime.
• No chemical refrigerants → environmentally friendly.
• Compact and quiet.
• Bidirectional operation (heating or cooling; power generation or temperature sensing).

🧭 Overview

🧠 One-sentence thesis

Batteries and fuel cells convert chemical energy stored in bonds into electrical energy through oxidation-reduction reactions, and their performance depends on material choices, charge flow mechanisms, and the accumulation of reaction products.

📌 Key points (3–5)

• What distinguishes batteries from fuel cells: batteries consume or deposit one or both electrodes during operation, while fuel cells consume only the fuel (not the electrodes).
• How charge flows: both electrons (through the external circuit/load) and ions (through the electrolyte) must flow to complete the circuit; electrons flow away from the anode and toward the cathode.
• Multiple measures of energy storage: chemists and engineers use different but related measures—electronegativity, chemical potential, redox potential, cell voltage, specific energy, and capacity—to describe the ability of charges to flow and energy stored.
• Common confusion—theoretical vs practical values: theoretical values calculated from chemical reactions are always higher than practical measured values due to internal resistance, temperature effects, and accumulation of reaction products.
• Why practical voltage drops during use: the Nernst equation shows that as reaction products accumulate in the electrolyte, the practical cell voltage decreases, reducing efficiency and eventually reaching an unusable end voltage.

🔋 Core components and charge flow

⚡ Three main components

All batteries and fuel cells contain:

Component	Role	Material examples
Anode	Electrode electrons flow away from; negative terminal during discharge	Mg, Zn, Li, Pb, carbon
Cathode	Electrode electrons flow toward; positive terminal during discharge	NiO₂, MnO₂, PbO₂, LiCoO₂
Electrolyte	Material through which ions flow more easily than electrons	KOH, H₂SO₄, organic solvents with lithium salts

• The electrodes must be good conductors, often porous to increase surface area.
• The electrolyte has high ionic conductivity but low electrical conductivity, forcing electrons to flow through the external load instead of directly between electrodes.
• A separator (thin polymer membrane) is typically included to prevent electrodes from contacting and to selectively allow certain ions to pass.

🔄 Charge flow during discharge

Oxidation: the process of losing an electron.
Reduction: the process of gaining an electron.

Example: Mg anode and NiO₂ cathode battery

At the anode:

• Mg + 2OH⁻ → Mg(OH)₂ + 2e⁻ (oxidation; electrons released)

At the cathode:

• NiO₂ + 2H₂O + 2e⁻ → Ni(OH)₂ + 2OH⁻ (reduction; electrons consumed)

Complete circuit requires both electron and ion flow:

• Electrons flow from anode → through load → to cathode.
• OH⁻ ions flow from cathode → through electrolyte → to anode.
• Current direction is opposite to electron flow direction.

The battery discharges until one electrode is consumed.

🔌 Charge flow during charging (secondary batteries)

During charging, an external voltage source reverses the reactions:

• The cathode becomes the negative terminal (electrons flow toward it).
• The anode becomes the positive terminal (electrons flow away from it).
• Reactions run in reverse: Mg(OH)₂ + 2e⁻ → Mg + 2OH⁻ at the cathode; Ni(OH)₂ + 2OH⁻ → NiO₂ + 2H₂O + 2e⁻ at the anode.
• Electrical energy is converted back to chemical energy stored in the electrode bonds.

Don't confuse: During discharge, anode = negative terminal and cathode = positive terminal. During charging, these polarities reverse.

⛽ Charge flow in fuel cells

Fuel cells have the same three main components (anode, cathode, electrolyte) plus:

• A membrane (like a separator) that selectively allows ions to pass.
• A catalyst (often platinum) coating the electrodes to speed reactions.
• Continuous supply of fuel (e.g., H₂ gas) at the anode and oxidizer (e.g., O₂ from air) at the cathode.

Example: H₂ fuel and O₂ oxidizer

• At anode: H₂ + 2OH⁻ → 2H₂O + 2e⁻ (fuel oxidized; electrons released)
• At cathode: ½O₂ + 2e⁻ + H₂O → 2OH⁻ (oxidizer reduced; electrons consumed)
• Electrons flow from anode → through load → to cathode; OH⁻ ions flow from cathode → through electrolyte → to anode.
• The fuel and oxidizer are consumed; the electrodes are not.

Fuel cells can operate continuously as long as fuel and oxidizer are supplied and products are removed.

📊 Measures of charge flow and energy storage

🧪 Electronegativity and chemical potential

Chemists and physicists use different terms for closely related concepts:

Semiconductor physics term	Chemistry term	Meaning
Fermi energy level E_f	Chemical potential μ_chem = −χ	Energy required to remove the next electron
Valence band	Highest occupied energy level	Energy level filled with electrons
Conduction band	Lowest unoccupied energy level	Next higher unfilled energy level
Energy gap E_g	Twice the chemical hardness	Minimum energy to excite an electron

Electronegativity χ: the negative of the partial derivative of internal energy U with respect to number of electrons N, at constant electrical potential V and entropy S.

Chemical potential μ_chem: the negative of electronegativity; μ_chem = −χ.

Mulliken electronegativity is an approximate operational definition:

• χ_Mulliken = (I_ioniz + A_aff) / 2
• Where I_ioniz = ionization energy (energy to remove an electron) and A_aff = electron affinity (energy change when adding an electron).
• This equals the magnitude of the Fermi energy at T = 0 K (with opposite sign).

Electronegativity describes the energy in chemical bonds; materials with low electronegativity (weak bonds) have high electrical conductivity because charges flow easily.

⚖️ Redox potential

Redox potential V_rp: a relative measure of the ability of a substance to lose an electron, measured in volts (joules per coulomb).

• It is a macroscopic property describing a larger piece of material, not just an individual atom.
• Measured experimentally by applying an external voltage to balance the internal voltage; often referenced to a platinum electrode at standard conditions (T = 25°C, P = 1 atm).
• Different sign conventions and reference points exist in the literature (American vs European definitions).

Example: The reaction Mg + 2OH⁻ → Mg(OH)₂ + 2e⁻ has V_rp = 2.68 V.

🔢 Cell voltage and capacity measures

Cell voltage V_cell: the voltage between anode and cathode; the sum of redox potentials for the half-reactions at each electrode.

• Theoretical cell voltage is calculated from tabulated redox potentials.
• Practical cell voltage (measured with a voltmeter) is always slightly lower due to internal resistance and other losses.
• Reactions with V_cell > 0 occur spontaneously.

Capacity and related measures:

Measure	Units	Definition
Capacity	mA·h or C	Total charge stored
Specific capacity	mA·h/g or C/kg	Charge stored per unit mass
Charge density	mA·h/L or C/m³	Charge stored per unit volume
Specific energy	J/g or W·h/kg	Energy stored per unit mass = V_cell × specific capacity
Energy density	J/m³ or W·h/L	Energy stored per unit volume = V_cell × charge density

• All measures can be theoretical (calculated from chemical reactions) or practical (measured experimentally).
• Theoretical values are always higher; practical values are typically 25–35% below theoretical.
• Only valence electrons are counted; inner shell electrons do not participate in reactions.

Don't confuse: "Capacity" (charge stored) has nothing to do with "capacitance" (a property of capacitors).

🧮 Example calculation: Mg + NiO₂ battery

Given:

• Mg half-reaction: V_rp = 2.68 V, involves 2 valence electrons, atomic weight = 24.31 g/mol
• NiO₂ half-reaction: V_rp = 0.49 V, involves 2 valence electrons, molecular weight = 90.69 g/mol

Cell voltage: V_cell = 2.68 + 0.49 = 3.17 V

Specific capacity:

• Weight per unit charge for Mg: 24.31 g/mol × (1 mol / 6.022×10²³ atoms) × (1 atom / 2 e⁻) × (1 e⁻ / 1.602×10⁻¹⁹ C) × (1 C / 1 A·s) × (3600 s / 1 h) = 0.454 g/(A·h)
• Weight per unit charge for NiO₂: similarly = 1.69 g/(A·h)
• Total: 0.454 + 1.69 = 2.146 g/(A·h)
• Specific capacity = 1 / 2.146 = 0.466 A·h/g

Specific energy: 3.17 V × 0.466 A·h/g = 1.48 W·h/g = 5.32×10³ J/g

(This calculation ignores the mass of the package, separator, and other components.)

🧪 pH and ionic conductivity

pH: a unitless measure of hydrogen ion concentration in a liquid solution; pH = log₁₀(1 / [H⁺]).

• [H⁺] is the amount concentration of hydrogen ions in mol/L (formerly called molarity).
• pH is defined for liquid solutions at T = 25°C and P = 1 atm.
• pH < 7: acidic (high H⁺ concentration); pH > 7: alkaline or basic (low H⁺ concentration); pH ≈ 7: neutral.

Relation to charge flow:

• Water molecules (H₂O) can ionize into H⁺ and OH⁻ ions.
• Adding solutes can increase ionization, producing more ions.
• More ions → higher electrical conductivity in the liquid.
• Solutions with pH much below or above 7 have many ions present and are good electrical conductors.

Example: A solution with 10²⁰ H⁺ ions in 1 L has [H⁺] = 1.66×10⁻⁴ mol/L, so pH ≈ 4 (acidic).

📉 Practical voltage and efficiency

🔋 Why practical voltage is lower than theoretical

Multiple factors cause practical cell voltage to be less than theoretical:

1. Internal resistance: Electrodes are good but imperfect conductors (e.g., carbon has finite electrical conductivity). Current flow through finite conductivity causes energy loss as heat.
2. Temperature dependence: Ions move faster at higher temperatures, reducing internal resistance but potentially speeding up unwanted reactions and reducing battery life.
3. Accumulation of reaction products: As the battery discharges, reaction products build up in the electrolyte, inhibiting further reactions and reducing voltage.

Voltage measures:

• Nominal voltage: typical voltage during use (often printed on the label).
• Open circuit voltage: voltage under no load (approximately the initial voltage).
• Closed circuit voltage: voltage under load (less than open circuit voltage due to internal resistance).
• End or cutoff voltage: voltage at the end of useful life (battery should be replaced even though some charge remains).

📐 Nernst equation

Nernst equation: V_cell_theor − V_cell_prac = (k_B T) / (N_v q) × ln([products] / [reactants])

Where:

• k_B = Boltzmann constant
• T = temperature
• N_v = number of valence electrons involved in the reaction
• q = magnitude of electron charge
• [products] / [reactants] = activity quotient (ratio of product to reactant concentrations)

Interpretation:

• When the battery is first set up, few products are present, so ln([products]/[reactants]) ≈ 0 and V_cell_prac ≈ V_cell_theor.
• As the battery discharges, products accumulate, the activity quotient grows, and V_cell_prac drops.
• The quantity (k_B T) / (N_v) represents internal energy per valence electron.

📊 Efficiency from the Nernst equation

Efficiency η_eff = E_out / E_in = V_cell_prac / V_cell_theor

Using the Nernst equation: η_eff = 1 − ln([products] / [reactants])

Key insights:

• Efficiency depends on the activity quotient (ratio of products to reactants).
• The activity quotient varies with temperature and changes as the battery discharges.
• This equation is analogous to Carnot efficiency (for thermoelectric and other heat-based devices) and to laser efficiency equations (accounting for mirror reflectivity and absorption).

Don't confuse: Even when the battery reaches its end voltage and is no longer useful, it still contains some stored charge; the voltage is simply too low to be practical.

🔋 Battery types and characteristics

🏷️ Classification schemes

Batteries can be classified in multiple ways:

Primary vs secondary:

• Primary battery: used once, then disposed; simpler construction, no recharging circuitry needed, often high specific energy.
• Secondary battery: rechargeable; can be recharged thousands of times, less waste, may cost more initially but cheaper in the long run, often have flat discharge curves (constant voltage throughout use).

By chemistry (electrode and electrolyte materials): Common families include lead acid, alkaline, nickel metal hydride, and lithium.

🚗 Lead acid batteries

• Anode: Pb; Cathode: PbO₂; Electrolyte: H₂SO₄ (sulfuric acid solution)
• Anode reaction: Pb + SO₄²⁻ → PbSO₄ + 2e⁻ (V_rp = 0.37 V)
• Cathode reaction: PbO₂ + SO₄²⁻ + 4H⁺ + 2e⁻ → PbSO₄ + 2H₂O (V_rp = 1.685 V)
• Overall cell voltage: V_cell = 2.055 V (six cells in series for a 12 V car battery)
• History: dates to ~1795 (Volta); practical devices ~1860 (Planté)
• Applications: car ignitions, backup power systems
• Advantages: handle high current, operate over wide temperature range, flat discharge curve, secondary (rechargeable)
• Limitations: lower specific energy than other types
• Typical practical specific energy: ~35 W·h/kg (theoretical ~252 W·h/kg)

🔋 Alkaline batteries

• Anode: Zn; Cathode: MnO₂; Electrolyte: KOH or NaOH (alkaline solution)
• History: developed for military in WWII, commercially available 1959, popular since 1980s
• Applications: inexpensive electronics, toys, gadgets
• Advantages: handle high current, inexpensive, operate over wide temperature range
• Limitations: sloping discharge curve (voltage decreases noticeably during use)
• Most are primary batteries (some secondary versions exist)
• Typical practical specific energy: ~154 W·h/kg (theoretical ~358 W·h/kg)

🔋 Nickel metal hydride batteries

• Anode: nickel metal alloy saturated with hydrogen (e.g., LaNi₅, TiNi₂, ZrNi₂); Cathode: nickel oxide (NiOOH); Electrolyte: KOH
• Anode reaction: Alloy(H) + OH⁻ → Alloy + H₂O + e⁻
• Cathode reaction: NiOOH + H₂O + e⁻ → Ni(OH)₂ + OH⁻ (V_rp = 0.52 V)
• History: developed 1960s for satellites, research accelerated 1970s–1980s, used in early laptops and cellphones
• Applications: portable tools, cameras, electronics requiring repeated recharging or high current; International Space Station (48 orbital replacement units, each with 38 nickel-hydrogen cells)
• Advantages: flat discharge curve, secondary (rechargeable many times), better for environment than nickel-cadmium
• Limitations: lower energy density than lithium
• Typical practical specific energy: ~100 W·h/kg (theoretical ~240 W·h/kg)

🔋 Lithium batteries

• Anode: Li or carbon; Cathode: MnO₂, LiCoO₂, FeS₂, or others; Electrolyte: organic solvents (e.g., propylene carbonate, dimethoxyethane) mixed with lithium salts (e.g., LiBF₄, LiClO₄); can be liquid or solid
• History: development since 1960s, military use 1970s, widespread consumer use today
• Applications: cellphones, laptops, portable electronics, hearing aids, medical devices
• Advantages: very high specific energy and energy density
• Limitations: often designed for low current output to prevent damage; secondary types require controlled recharging to prevent damage
• Both primary and secondary types available
• Typical practical specific energy: ~200 W·h/kg (theoretical ~448 W·h/kg)
• Over 250 million cells produced per month

🎯 Ideal battery characteristics (tradeoffs required)

An ideal battery would have:

• High specific energy and energy density
• No toxic chemicals (environmentally friendly, easy disposal)
• Safe to use
• Inexpensive
• Rechargeable with no complicated procedure
• Able to output large current
• Able to withstand wide temperature range
• Constant voltage output throughout life (flat discharge curve)
• Long shelf life (remain charged during storage)

Reality: Many of these qualities contradict (e.g., high specific energy inherently requires more safety precautions). Different applications require different tradeoffs, so no single battery type is best for all uses.

⛽ Fuel cells

🏗️ Components and system

Core components (same as batteries):

• Anode, cathode, electrolyte

Additional components:

• Membrane: thin polymer that selectively allows ions (not fuel/oxidizer) to pass; must be chemically stable, electrically insulating, and mechanically stable
• Catalyst: coating on electrodes (often platinum, nickel, or other materials) to speed reactions; must not dissolve or oxidize in the electrolyte and should only catalyze the desired reaction
• Fuel: continuously supplied (gas like H₂ or CO, liquid like methanol or ammonia, or solid like coal)
• Oxidizer: continuously supplied (typically O₂ gas or air)

Fuel cell system components (beyond the core cell):

• Fuel processor: breaks down fuel into usable form, filters impurities (e.g., converts coal to smaller hydrocarbons)
• Flow plates: channel fuel and oxidizer to electrodes, channel away waste and heat
• Heat recovery system: may include thermoelectric devices to convert waste heat back to electricity
• Inverter: converts DC power to AC (for grid connection)
• Control system: regulates fuel/oxidizer flow, monitors temperature, manages overall operation

🏷️ Classification of fuel cells

By operating temperature:

• Low: 25–100°C
• Medium: 100–500°C
• High: 500–1000°C
• Very high: >1000°C
• Higher temperatures speed reactions but require materials that can withstand heat without melting or corroding.

Primary vs secondary:

• Primary (nonregenerative): reactants used once, then discarded
• Secondary (regenerative): reactants reused; external energy source (electrical, thermal, photochemical, or radiochemical) refreshes the fuel

Direct vs indirect:

• Direct: fuel used as is
• Indirect: fuel processed inside the system before reaction (e.g., coal broken down by enzyme into smaller hydrocarbons)

By electrolyte type: Examples include alkaline (KOH solution), phosphoric acid, molten carbonate, solid oxide (ceramic electrolytes).

Common types:

• Proton exchange membrane (PEM): H₂ fuel, O₂ oxidizer, solid electrolyte, platinum catalyst; low temperature; used in buses, aerospace, backup power
• Direct methanol: methanol fuel; low or medium temperature; similar applications

🌍 Practical considerations

Advantages:

• Environmental: some produce no harmful outputs (e.g., H₂ + O₂ → H₂O only)
• Efficiency: high-temperature and high-power units can reach up to 65% efficiency
• Dual output: NASA and military submarines use fuel cells to produce both electricity and pure water since the 1960s
• Continuous operation: can run as long as fuel and oxidizer are supplied

Limitations preventing widespread use:

• Cost: platinum catalysts are expensive
• Fuel sensitivity: some non-platinum catalysts lose efficiency in presence of CO or CO₂ (common in air)
• Fuel storage and delivery: H₂ and methane pose challenges
• Size and infrastructure: most efficient systems are large, require fixed space, cooling, and additional infrastructure (not portable)

History:

• Concept dates to ~1802
• Working demonstrations in 1830s
• First practical device built 1959 (when pure materials became commercially available)
• Used in NASA Gemini and Apollo projects (1960s) and military submarines (1960s–present)

Applications today:

• Large stationary installations generating kilowatts or megawatts
• Buses and aerospace
• Backup power systems
• Specialized applications where water production is valuable (space, submarines)

🔬 Example: H₂ fuel cell in alkaline electrolyte

• Fuel: H₂ gas; Oxidizer: O₂ gas; Electrolyte: KOH solution; Catalyst: platinum; Anode material: carbon cloth
• Anode reaction: H₂ + 2OH⁻ → 2H₂O + 2e⁻ (oxidation of fuel)
• Cathode reaction: ½O₂ + 2e⁻ + H₂O → 2OH⁻ (reduction of oxidizer)
• Electrons flow from anode → through load → to cathode
• OH⁻ ions flow from cathode → through electrolyte → to anode
• Membrane prevents reverse flow and prevents ions from combining directly
• Only byproduct: pure water

🧭 Overview

🧠 One-sentence thesis

A wide variety of energy conversion devices exist beyond magnets and coils, including thermionic devices, radiation detectors, biological systems, resistive sensors, and electrofluidic devices, each converting energy between different forms through distinct physical mechanisms.

📌 Key points (3–5)

• Scope limitation: This chapter covers low-power electrical energy conversion devices that do not involve magnets and coils, representing only a subset of all possible devices.
• Thermionic and radiation devices: Thermionic devices convert heat to electricity through electron evaporation; radiation detectors convert radioactive particles to electrical signals for safety monitoring.
• Biological conversion: The human body and its components (neurons, muscles, photoreceptors) perform multiple energy conversions between chemical, electrical, kinetic, and optical forms.
• Resistive sensors: When resistivity, length, or cross-sectional area changes with an external effect, resistive devices can sense temperature, strain, or pressure while converting electricity to heat.
• Electrofluidics: Electrohydrodynamic devices convert between electrical energy and fluid flow, operating through pressure changes (Bernoulli's equation), Lorentz forces on conductive fluids, or material polarization effects.

🔥 Thermionic Devices

⚡ How thermionic conversion works

Thermionic devices convert thermal energy to electricity using the thermionic effect.

• A vacuum tube contains two electrodes: a heated cathode and a cooler anode.
• The metal cathode is heated (typically to 1500°C) until electrons evaporate off the metal surface.
• These electrons collect at the anode, which is at a colder temperature.
• The distance between electrodes may be as small as 10 micrometers.

🎯 Efficiency and materials

• Efficiency limit: Carnot efficiency limits this effect because a temperature differential is converted to electricity.
• Measured efficiencies reach up to 12%, though other methods of converting temperature differences are often more efficient for the same temperature differential.
• Cathode materials: tungsten, molybdenum, tantalum, and barium oxide (the cathode gets used up and eventually needs replacement).
• Anode materials: copper, cesium, nickel, barium oxide, strontium oxide, and silver.

🔬 Historical note and applications

• First patented by Thomas Edison in 1883.
• Some gas chromatographs use nitrogen phosphorous thermionic detectors.

☢️ Radiation Detectors

🛡️ Purpose and types of radiation

Radiation detectors convert energy from radioactive sources to electricity.

• Why needed: Excessive radiation harms people, but humans cannot sense radioactivity directly—we can only measure it indirectly.
• Used as safety devices.
• Radiation types: alpha particles (positively charged ionized helium nuclei), beta particles (high energy electrons), gamma rays (high energy, short wavelength electromagnetic radiation), and neutrons.

🔌 Ionization chambers and Geiger counters

Both work on the same principle but at different voltages:

Device	Voltage range	Mechanism
Ionization chamber	Few volts to hundreds of volts	Each incoming particle causes a single atom to ionize
Geiger counter	500 V to 2000 V	Incoming particle triggers cascading ionization reaction

How they work:

• A gas is enclosed in a chamber or tube with voltage applied across it.
• Incoming alpha, beta, or gamma rays ionize the gas.
• Positive ions flow to one electrode, negative ions to the other, forming a current.
• Geiger counter advantage: More sensitive due to amplification—the voltage gradient accelerates and separates ion pairs, causing additional atoms to ionize in a cascade.

Example: Many smoke detectors are ionization chambers. When no smoke is present, radiation ionizes air and current is detected. When smoke is present, it scatters the radiation, so no current is detected.

💡 Scintillation counters

A scintillation counter is often made from crystalline material such as sodium iodide; sometimes a phosphor is also used.

• Two-step process: Incoming radioactive particles excite (but do not ionize) atoms of the material. These atoms then decay and emit a photon.
• Semiconductor or other photodetectors convert the photons to electricity.
• Advantages: Can be higher sensitivity than other types; can determine the energy of incoming radiation by spectroscopy.

📸 Photographic film based detectors

• Incoming radioactive particles expose the film, changing its color.
• Materials include aluminum oxide (Al₂O₃) and lithium fluoride.
• Photodetectors convert the information recorded on film to a measurable signal.
• Practical use: Worn as rings or badges by radiology technicians and nuclear power plant employees.
• Must be sent to a lab for analysis; both amount and type of radiation can be determined.

🧬 Biological Energy Conversion

🏃 The human body as an energy converter

The human body takes in chemical energy (food) and converts it to:

• Kinetic energy
• Heat
• Other forms of energy

Energy storage in the body:

• Muscles store energy as they stretch and contract.
• Fat cells store energy in chemical form.
• When walking, the center of mass moves up and down, storing energy in pendulum-like motion.
• Bone, skin, and collagen exhibit piezoelectricity.

👁️ Sensory energy conversion

Different organs convert various energy forms:

Organ/System	Energy conversion
Muscles	Chemical energy → kinetic energy
Photoreceptors (retina)	Optical energy of photons → electrical energy of neurons
Ear	Sound waves → pressure energy in inner ear fluid → kinetic energy of moving hairs in cochlea → electrical energy of neurons

🧠 Neurons

Neurons are nerve cells that convert chemical energy to electrical energy.

Structure: composed of a cell body, an axon, dendrites, and synapses.

• Axon: the fibrous part that transmits information to other neurons.
• Dendrites: the fibrous part that receives information from other neurons.
• Synapse: a gap between neurons.

How they work:

• Ions (such as Na⁺, K⁺, or Cl⁻) build up on the membrane or in the gap between neurons.
• The charge separation of ions causes an electrical potential.
• Ions sometimes cross the gap between neurons.

Neuron types:

1. Sensory afferents: transmit signals from sensory receptors to the nervous system.
2. Interneurons: transmit signals throughout the nervous system.
3. Motoneurons: transmit signals from the nervous system to muscles.

Signal characteristics:

• The human brain has around 10¹¹ neurons.
• Electrical signals involve pulses with a duration of a few milliseconds.
• Information is encoded in the frequency rate of the pulses.

🔧 Resistive Sensors

📐 Fundamental principle

The resistance R of a uniform resistive device is given by R = ρl/A (where ρ is resistivity, l is length, A is cross-sectional area).

If resistivity, length, or cross-sectional area change with respect to any effect, we can make a resistive sensor.

How they work:

• When current is applied through a resistive sensor, energy is converted from electricity to heat.
• The sensor is calibrated so that a given voltage drop corresponds to a known change in some parameter.

🎛️ Types of resistive sensors

Potentiometer:

A potentiometer is a variable resistor.

• As current flows through it, energy converts from electricity to heat.
• When the knob is turned, the length of material through which current flows changes, so the rate of energy conversion changes.

Resistance temperature detector:

• Converts a temperature difference to electricity.
• Works based on the Thomson effect.
• The resistivity varies with temperature.

Strain gauge:

• When strain is applied, both the length and cross-sectional area of the device change.

Pirani hot wire gauges:

• Used to measure pressure in low-pressure environments.
• Current is applied through a metallic filament, heating it up.
• As air molecules hit the filament, heat is transferred away.
• The resistance depends on temperature; the filament cools more quickly in environments with more air molecules than at lower pressure.
• By monitoring the resistance, pressure can be determined.

🔍 Comparison with other sensor types

Sensors may be made from capacitive, inductive, or resistive materials:

Sensor type	Key parameter	What changes
Capacitive	C = εA/d_thick	Permittivity ε, area A, or separation d_thick
Inductive	L = μd_thick/w	Permeability μ, thickness d_thick, or width w
Resistive	R = ρl/A	Resistivity ρ, length l, or area A

💧 Electrofluidics

🌊 Overview of electrohydrodynamic devices

Electrohydrodynamic devices (EHDs) convert between electrical energy and fluid flow; also known as electrokinetic devices.

Microfluidic devices are EHD devices patterned on a single silicon wafer or other substrate, with length scales often less than a millimeter.

Device types: Engineers have built EHD pumps, valves, mixers, separators, and other devices.

Applications: ink jet printers, chemical detectors, machines for DNA sequencing or protein analysis, and insulin pumps.

🌀 Bernoulli's equation mechanism

Some EHD devices operate based on Bernoulli's equation, a direct consequence of energy conservation.

Constricted pipe example:

• A pipe converts energy from a pressure differential to kinetic energy.
• Consider a fluid with zero viscosity and zero thermal conductivity flowing through a horizontal pipe.
• Velocity and pressure differ at locations with different pipe diameter.

Energy conservation derivation:

• For a small mass of water, there are two energy components: kinetic energy and energy due to compressed fluid.
• Change in compression energy: (P₁ - P₂)ΔV
• Change in kinetic energy: (1/2)(ρ_dens ΔV)(|v₁|² - |v₂|²)
• Energy is conserved, leading to Bernoulli's equation:

P + (1/2)ρ_dens|v|² = constant

In EHDs: Electricity induces changes in pressure or volume of a microfluidic channel. By Bernoulli's equation, this change in pressure induces a change in fluid velocity. The fluid may be conductive or insulating.

⚡ Lorentz force mechanism

In other EHDs, applied voltages exert forces on conductive fluids.

The Lorentz force equation: F = Q(E + v × B)

Two scenarios:

1. With electric field, no magnetic field (this chapter's focus):

   • The liquid must be conductive.
   • When voltage is applied, an electric field is induced, causing the liquid to flow.
   • This effect is due to a streaming potential.

2. Related effect—electrophoresis:

   • Occurs in liquids containing charged particles.
   • If an electric field is applied, these particles move.
   • Demonstrated with charged DNA molecules and charged protein molecules in solutions.

Don't confuse: Streaming potential (conductive liquid flows) vs. electrophoresis (charged particles in liquid move).

🔋 Dielectrophoresis

Dielectrophoresis operates by changing material polarization of an insulating liquid.

How it differs:

• If we apply an electric field across a conductor (solid or liquid), charges flow.
• If we apply an electric field across a dielectric, the material may polarize—net charge displacement occurs even if all electrons remain bound to atoms.
• The external electric field causes both the atoms of the liquid to polarize and these polarized atoms to flow.

🌊 Electroosmosis

Electroosmosis can occur in fluids with a surface charge.

• In some liquids, ions build up on the surface due to unpaired chemical bonds, ions adsorbed onto the surface, or other reasons.
• (The excerpt ends here without further detail on this effect.)

🧭 Overview

🧠 One-sentence thesis

The calculus of variations provides a general mathematical framework for describing any energy conversion process by finding the path that minimizes the action (integral of the Lagrangian), which reveals how systems evolve as energy transforms from one form to another.

📌 Key points (3–5)

• Core principle: The Principle of Least Action states that physical systems follow the path that minimizes the action, defined as the integral of the Lagrangian over time.
• Key mathematical tools: The Lagrangian (difference between two energy forms) and the Euler-Lagrange equation (a differential equation derived from minimizing action) describe how energy conversion processes evolve.
• Advantage over other methods: Working with energy (a scalar quantity) is more manageable than working with potentials (which may be vectors).
• Common confusion: Lagrangian vs Hamiltonian—the Lagrangian is the difference between energy forms, while the Hamiltonian is the sum (total energy).
• Broad applicability: The same framework applies to mechanical systems (mass-spring), electrical systems (capacitor-inductor), and quantum mechanics (Schrödinger's equation).

🎯 Foundational concepts

🎯 The generalized path

Generalized path y(t): the quantity that evolves during an energy conversion process, with one independent variable t and one dependent variable y.

• The units depend on the specific system being studied.
• In a mass-spring system, it represents the position of the mass (meters).
• In a capacitor-inductor circuit, it represents the charge on the capacitor (coulombs).
• The framework assumes only one energy conversion process occurs (an idealization).

⚡ Lagrangian and Hamiltonian

Lagrangian definition:

L(t, y, dy/dt) = (First form of energy) − (Second form of energy)

Hamiltonian definition:

H(t, y, dy/dt) = (First form of energy) + (Second form of energy)

• Both have units of joules.
• The Lagrangian captures the difference between energy forms.
• The Hamiltonian represents the total energy of the system.
• Energy conservation can be expressed as dH/dt = 0.

Don't confuse: The Lagrangian is not the total energy; it's the difference. The Hamiltonian is the total.

📐 Generalized quantities

Generalized potential:

• Defined as the partial derivative of the Lagrangian with respect to the path: ∂L/∂y.
• Units are joules divided by the units of the path.
• Example: In a mass-spring system, this becomes force (newtons); in a circuit, it becomes voltage (volts).

Generalized momentum:

M = ∂L/∂(dy/dt)

• Different from ordinary momentum but related in mechanical systems.
• Units depend on the specific system.

Generalized capacity:

• Ratio of generalized path to generalized potential.
• In electrical systems, this is capacitance; in mechanical systems, it's the inverse of the spring constant.

🔬 The Principle of Least Action

🔬 What is action?

Action S: the magnitude of the integral of the Lagrangian along the path.

Formula (in words): Action equals the absolute value of the integral from initial time to final time of the Lagrangian with respect to time.

• Units: joule-seconds (when t represents time).
• The action quantifies how much the energy difference accumulates over the entire path.

🎲 Why "least" action?

• Among all possible paths between two points, nature follows the path that minimizes the action.
• This is an experimental observation, not a derived result.
• The idea contains the principle of energy conservation.
• Also known as Hamilton's principle.

Example: A mass attached to a compressed spring could theoretically follow many paths when released, but it follows only the one with minimum action.

🧮 Functions vs functionals

Key distinction:

• A function takes a scalar as input and returns a scalar.

• A functional takes a function as input and returns a scalar.

• Both L and H are functionals: they take the function y(t) as input and return a scalar (energy in joules).

• To find the minimum of a functional, we use calculus of variations instead of ordinary calculus.

• The process involves taking derivatives with respect to the function y(t), not just a variable.

📏 The Euler-Lagrange equation

📏 The equation itself

Euler-Lagrange equation: ∂L/∂y − (d/dt)(∂L/∂(dy/dt)) = 0

• This is a second-order differential equation.
• It expresses the Principle of Least Action as a differential relationship.
• If the Lagrangian L is known, this equation can be simplified to find the equation of motion.
• The equation of motion describes the path y(t) that minimizes action.

🔧 How to use it

Step-by-step process:

1. Identify the two forms of energy in the conversion process.
2. Write the Lagrangian as their difference.
3. Calculate ∂L/∂y (the generalized potential).
4. Calculate ∂L/∂(dy/dt) (related to generalized momentum).
5. Take the time derivative of the result from step 4.
6. Set the difference (step 3 minus step 5) equal to zero.
7. Solve the resulting differential equation for y(t).

🔄 Alternative formulations

Hamilton's equations (pair of first-order equations):

• dM/dt = −∂H/∂y
• dy/dt = ∂H/∂M

These contain the same information as the Euler-Lagrange equation but expressed differently.

Conservation law interpretation:

• The Euler-Lagrange equation is a conservation law for the generalized potential.
• Each term has units of joules per unit of path.
• In mechanical systems, this becomes conservation of force (Newton's second law).
• In electrical systems, this becomes Kirchhoff's voltage law.

🔩 Mass-spring example

🔩 System description

• A mass m (kg) attached to a spring with spring constant K (J/m²).
• Energy converts between spring potential energy and kinetic energy.
• Position of mass: x(t) in meters.
• Assume no friction or other energy losses.

Energy forms:

• Spring potential energy: (1/2)Kx²
• Kinetic energy: (1/2)m(dx/dt)²

⚙️ Applying the framework

Lagrangian: L(t, x, dx/dt) = (1/2)Kx² − (1/2)m(dx/dt)²

Hamiltonian: H = (1/2)Kx² + (1/2)m(dx/dt)²

Generalized potential: ∂L/∂x = Kx (in newtons, which equals the negative of the spring force)

Generalized momentum: M = ∂L/∂(dx/dt) = −m(dx/dt) (units of momentum: kg·m/s)

🎯 Equation of motion

Applying the Euler-Lagrange equation: Kx + m(d²x/dt²) = 0

Interpretation:

• This is Newton's second law: force equals mass times acceleration.
• It's also the wave equation, a second-order linear differential equation.
• Solution: x(t) = c₀ cos(√(K/m) t) + c₁ sin(√(K/m) t)
• The mass oscillates, converting energy back and forth.

🔁 Energy conservation

• Both H and L do not explicitly depend on time: ∂H/∂t = 0 and ∂L/∂t = 0.
• This is called time translation symmetry or time invariance.
• The system behaves the same at any point in time.
• Can verify dH/dt = 0 algebraically using the equation of motion.

⚡ Capacitor-inductor example

⚡ System description

• A circuit with a capacitor and an inductor.
• Energy converts between electrical energy in the capacitor and magnetic energy in the inductor.
• Generalized path: charge Q(t) on the capacitor (coulombs).
• Assume no resistance (idealization).

Energy forms:

• Capacitor energy: (1/2)Cv² = (1/2)Q²/C
• Inductor energy: (1/2)L(iₗ)²

🔌 Key relationships

Capacitor:

• Q = Cv (charge equals capacitance times voltage)
• Current: dQ/dt = C(dv/dt)

Inductor:

• Ψ = Liₗ (magnetic flux equals inductance times current)
• Voltage: v = L(diₗ/dt)

⚙️ Applying the framework

Lagrangian: L(t, Q, dQ/dt) = (1/2)Q²/C − (1/2)L(dQ/dt)²

Hamiltonian: H = (1/2)Q²/C + (1/2)L(dQ/dt)²

Generalized potential: ∂L/∂Q = Q/C = v (voltage in volts)

Generalized momentum: M = ∂L/∂(dQ/dt) = −L(dQ/dt)

🎯 Equation of motion

Applying the Euler-Lagrange equation: Q/C + L(d²Q/dt²) = 0

Interpretation:

• This is Kirchhoff's voltage law.
• In terms of voltage and current: v − L(diₗ/dt) = 0
• Again, this is the wave equation.
• The circuit oscillates, with energy converting back and forth.

🔄 Parallel with mechanical systems

Property	Capacitor-Inductor	Mass-Spring
Generalized path	Charge Q (C)	Displacement x (m)
Generalized potential	Voltage v (V)	Force F (N)
Generalized capacity	Capacitance C (F)	1/K (m²/J)
Conservation law	Kirchhoff's voltage law	Newton's second law

🌊 Connection to quantum mechanics

🌊 Wave function and probability

Wave function ψ: describes the probability of finding a particle (like an electron) in a particular energy state.

• The probability is given by |ψ|², which must be between 0 and 1.
• Quantum mechanics fundamentally uses Hamiltonian and calculus of variations concepts.

⚛️ Quantum mechanical Hamiltonian

Structure: H_QM = E_kinetic + E_potential

Kinetic energy:

• E_kinetic = (1/2m)(M_QM)²
• M_QM is the quantum mechanical momentum operator: M_QM = jℏ∇
• ℏ is Planck's constant divided by 2π.
• ∇ is the del operator (spatial derivative).

Rewritten form: H_QM = −(ℏ²/2m)∇² + E_potential

🔬 Schrödinger's equation

Fundamental relationship: H_QM ψ = E_total ψ

Time-independent Schrödinger equation: ∇²ψ + (2m/ℏ²)(E_total − E_potential)ψ = 0

• One of the most fundamental equations in quantum mechanics.
• Energy level diagrams (allowed energies for electrons in atoms) can be derived by solving this equation.
• Shows how calculus of variations extends from classical to quantum systems.

🔗 Broader implications

🔗 Symmetries and conservation laws

Noether's theorem:

• There is a close relationship between symmetries and conservation laws.
• Time translation symmetry (system behaves the same at all times) implies energy conservation.
• Symmetries of the equation of motion lead to additional conservation laws and invariants.

🔗 First variation concept

Definition: The first variation δS represents the linear change in action when the path is slightly perturbed.

• For the path y(t) to minimize action, the first variation must equal zero: δS = 0.
• This is analogous to setting a derivative equal to zero to find a minimum in ordinary calculus.
• The derivation involves considering nearby paths and expanding in a small parameter ε.

Don't confuse: Setting the first variation to zero is a necessary condition for a minimum, but not always sufficient—the action could be at a maximum or saddle point.

🔗 Integration by parts technique

The derivation of the Euler-Lagrange equation uses integration by parts to transform the integral condition into a differential equation. Key assumption: the endpoints of all paths being compared are the same (boundary conditions).

🧭 Overview

🧠 One-sentence thesis

Calculus of variations provides a unifying mathematical language to describe energy conversion processes across electrical, mechanical, thermodynamic, and chemical systems, revealing deep structural similarities between seemingly disparate physical phenomena.

📌 Key points (3–5)

• Unifying framework: Calculus of variations lets scientists describe energy conversion in circuits, springs, thermodynamic systems, and chemical bonds using the same mathematical structure (generalized path, generalized potential, Lagrangian).
• Fundamental laws emerge: Applying the Euler-Lagrange equation to different systems recovers Newton's second law, Kirchhoff's voltage/current laws, Gauss's laws, conservation of momentum, and the second law of thermodynamics.
• Analogies across disciplines: By comparing tables of parameters, one can draw analogies—e.g., capacitance in circuits corresponds to spring constant in mechanics and bulk modulus in thermodynamics.
• Common confusion: The same physical system can be described by choosing different variables as the generalized path (e.g., charge vs. voltage for a capacitor), leading to different but equivalent formulations.
• Interdisciplinary insight: Understanding these parallels helps researchers transfer ideas between fields and identify open questions at disciplinary boundaries.

⚡ Electrical energy conversion

🔌 Circuit-level description

Electrical circuits are described by four fundamental parameters:

• Charge Q (coulombs)
• Voltage v (volts)
• Magnetic flux Ψ (webers)
• Current i (amperes)

Devices like resistors, capacitors, and inductors convert or store electrical energy. The excerpt treats these as point-like elements with no spatial extent.

🧲 Capacitors and inductors

Capacitor: stores energy in charge separation between plates.

• If charge Q is the generalized path, voltage v becomes the generalized potential.
• If voltage v is the generalized path, charge Q becomes the generalized potential.
• Constitutive relationship: Q = Cv, where C is capacitance (farads).
• Energy stored: ½Cv² or ½Q²/C.
• Equation of motion: Kirchhoff's Voltage Law (KVL) or conservation of charge, depending on the choice.

Inductor: stores energy in a magnetic field.

• If magnetic flux Ψ is the generalized path, current i is the generalized potential.
• If current i is the generalized path, magnetic flux Ψ is the generalized potential.
• Constitutive relationship: Ψ = Li, where L is inductance (henries).
• Energy stored: ½Li² or ½Ψ²/L.
• Equation of motion: Kirchhoff's Current Law (KCL) or conservation of magnetic flux.

Don't confuse: The choice of generalized path is flexible; swapping path and potential gives an equivalent but complementary description.

🌊 Electromagnetic field description

Circuits can also be described using four vector fields (functions of position and time):

• Displacement flux density D (C/m²) generalizes charge Q
• Electric field intensity E (V/m) generalizes voltage v
• Magnetic flux density B (Wb/m²) generalizes magnetic flux Ψ
• Magnetic field intensity H (A/m) generalizes current i

Materials with permittivity ε > ε₀ store energy in distributed charge separation (like a capacitor). Materials with permeability μ > μ₀ store energy in magnetic fields (like an inductor).

Applying calculus of variations to these fields yields Gauss's laws:

• For electric fields: ∇·D = ρ_ch (charge density)
• For magnetic fields: ∇·B = 0

Example: Piezoelectric, pyroelectric, Hall effect, and magnetohydrodynamic devices all involve converting electrical energy to/from energy stored in material polarization or magnetic fields, and can be described using these electromagnetic field variables.

🔑 Generalized capacity

By analogy to capacitance C (ability to store charge), the excerpt introduces generalized capacity: a parameter representing the ability of any device to store energy.

• For a capacitor: capacitance C (farads)
• For an inductor: inductance L (henries)
• For a dielectric material: permittivity ε (F/m)
• For a magnetic material: permeability μ (H/m)

🔧 Mechanical energy conversion

🏋️ Point-like mechanical devices

Mechanical systems are described by eight possible vector fields (force, momentum, velocity, displacement, angular momentum, angular displacement, torque, angular velocity). The excerpt focuses on point-like devices of mass m.

Linear spring: stores energy in compression/extension.

• If force F is the generalized path, displacement x is the generalized potential.
• If displacement x is the generalized path, force F is the generalized potential.
• Constitutive relationship (Hooke's law): F = Kx, where K is spring constant (J/m²).
• Energy stored: ½K|x|² or ½(1/K)|F|².
• Equation of motion: Newton's second law F = ma.

Moving mass: stores kinetic energy.

• If momentum M is the generalized path, velocity v is the generalized potential.
• If velocity v is the generalized path, momentum M is the generalized potential.
• Constitutive relationship: M = mv, where m is mass (kg).
• Energy stored: ½m|v|² or ½|M|²/m.
• Equation of motion: conservation of momentum.

🌀 Rotational devices

Flywheel: stores energy in rotational motion.

• Generalized path can be angular velocity ω_ang or angular momentum L_am.
• Constitutive relationship: L_am = Iω_ang, where I is moment of inertia (kg·m²).
• For a point mass: I = m|r|² (distance from origin).
• For a solid shape: I = ∫|r|² dm.
• Energy stored: ½I|ω_ang|² or ½(1/I)|L_am|².
• Equation of motion: conservation of angular momentum.

Torsion spring: stores energy in twisting.

• Generalized path can be torque τ or angular displacement θ.
• Constitutive relationship: τ = Kθ, where K is torsion spring constant (J/rad²).
• Energy stored: ½K|θ|² or ½K|τ|².
• Equation of motion: conservation of torque.

Example: Flywheels are used in some electric/hybrid vehicles because storing rotational kinetic energy requires fewer energy conversion steps than storing energy in a battery.

🔗 Connection to electromagnetism

There is a remarkable mathematical parallel between Maxwell's equations (for electromagnetic fields) and equations relating mechanical fields.

The transformation:

• D → M (displacement flux → momentum)
• E → v (electric field → velocity)
• B → τ (magnetic flux → torque)
• H → θ (magnetic field → angular displacement)

...converts source-free Maxwell's equations into valid equations relating mechanical quantities.

Don't confuse: This is a mathematical analogy, not a physical equivalence—the systems are different, but their governing equations have the same structure.

🌡️ Thermodynamic energy conversion

🎈 Four thermodynamic parameters

Thermodynamic systems are described by:

• Volume V (m³)
• Pressure P (pascals)
• Temperature T (kelvin)
• Entropy S (J/K)

Many devices convert between some form of energy and energy stored in confined volume, pressure, temperature difference, or disorder.

🔄 Volume and pressure

Confined gas (e.g., balloon): stores energy in confined volume.

• If volume V is the generalized path, pressure P is the generalized potential.
• If pressure P is the generalized path, volume V is the generalized potential.
• Generalized capacity: V/B (where B is bulk modulus, Pa) or B/V.
• Constitutive relationship: ΔV = -(V/B)ΔP or ΔP = -(B/V)ΔV.
• Energy: ∫VdP or ∫PdV; for small changes: VΔP or PΔV.
• Equation of motion: Bernoulli's equation.
• Assumptions: Temperature and entropy held constant (unrealistic but simplifies analysis).

Example: A balloon tied to a toy car—releasing air converts energy stored in confined volume (and stretched rubber) to kinetic energy of the car.

Example: A piston, airfoil, or constricted pipe converts pressure difference to kinetic energy.

🔥 Temperature and entropy

Hot liquid or ordered system (e.g., two separated gases): stores energy in temperature difference or order.

• If temperature T is the generalized path, entropy S is the generalized potential.
• If entropy S is the generalized path, temperature T is the generalized potential.
• Generalized capacity: T/C_v or C_v/T, where C_v is specific heat at constant volume (J/(g·K)).
• Constitutive relationship: ΔT = (T/C_v)ΔS or ΔS = (C_v/T)ΔT.
• Energy: ∫TdS or ∫SdT; for small changes: TΔS or SΔT.
• Equation of motion: second law of thermodynamics (conservation of entropy).
• Assumptions: Pressure and volume held constant.

Second law of thermodynamics: ΔS = ∫(δQ/T) + S_produced

In words: change in entropy = entropy out due to heat transfer + entropy produced by the system.

Example: A Sterling engine converts a temperature difference to kinetic energy.

Don't confuse: Entropy can increase or decrease within a system (ΔS can be positive or negative), but the total entropy (including surroundings) cannot decrease.

🔋 Chemical energy conversion

⚛️ Macroscopic vs. microscopic descriptions

Macroscopic: uses charge density ρ_ch (C/m³) and redox potential V_rp (volts).

• If ρ_ch is the generalized path, V_rp is the generalized potential (or vice versa).
• Constitutive relationship: ∫ρ_ch dV = CV_rp (analogous to Q = Cv for a capacitor).
• Energy: ∫ρ_ch V_rp dV.
• Equation of motion: Nernst equation (analogous to KVL) or conservation of charge.

Microscopic: uses number of electrons N and chemical potential μ_chem (J/atom).

• If N is the generalized path, μ_chem is the generalized potential (or vice versa).
• Generalized capacity: 1/E_g or E_g, where E_g is the energy gap (J).
• Constitutive relationship: ΔN = (1/E_g)Δμ_chem or Δμ_chem = E_g ΔN.
• Energy: Nμ_chem.
• Equation of motion: Nernst equation or conservation of charge.

Chemical potential (μ_chem): also known as Fermi energy at T = 0 K; represents the average between the highest occupied and lowest unoccupied energy levels.

Example: Batteries and fuel cells store energy in chemical bonds; calculus of variations can describe electron distribution around atoms and energy conversion in electrochemical cells.

Don't confuse: Charge Q (second column of circuit table) is the integral of charge density ρ_ch over volume; displacement flux density D (electromagnetic table) is related but a vector field in C/m².

🔗 Cross-disciplinary patterns

📊 Structural similarities

All the tables in the excerpt share the same structure:

• Generalized path: the variable whose evolution we track
• Generalized potential: the "force" driving changes in the path
• Generalized capacity: ability to store energy (capacitance, inductance, mass, spring constant, bulk modulus, specific heat, energy gap)
• Constitutive relationship: equation relating path and potential
• Energy: expression for stored energy
• Law for potential: fundamental physical law that emerges as the equation of motion

System	Example capacity	Example law
Electrical	Capacitance C (farads)	Kirchhoff's Voltage Law
Mechanical	Mass m (kg)	Newton's second law
Thermodynamic	Bulk modulus B (Pa)	Bernoulli's equation
Chemical	Energy gap E_g (J)	Nernst equation

🌉 Why analogies matter

The excerpt emphasizes that:

• Specialists in each discipline use their own favorite quantities, but energy conversion is a common topic.
• Calculus of variations provides a unifying language across disciplines.
• Open research questions are often found at boundaries between disciplines, where there is less expertise.
• Ideas from one field may answer questions in another; challenges in one field may pose interesting research questions in another.

Don't confuse: These are mathematical analogies revealing structural similarities, not claims that (for example) a capacitor "is" a spring—the physical mechanisms differ, but the mathematical descriptions are isomorphic.

⚠️ Simplifying assumptions

The excerpt repeatedly notes drastic assumptions:

• Only one energy conversion process occurs at a time (e.g., ignoring friction, gravity, heating).
• Devices are treated as point-like (no spatial variation), though the ideas can be generalized.
• External reservoirs or constraints hold some variables constant (e.g., temperature and entropy constant for volume/pressure analysis).
• These assumptions are unrealistic but pedagogically useful for isolating core mechanisms.

Example: A real mass-spring system involves friction (kinetic → thermal energy), gravity (kinetic ↔ gravitational potential), and possibly other effects—but the excerpt focuses only on kinetic ↔ spring potential energy conversion.

🧭 Overview

🧠 One-sentence thesis

The Thomas Fermi equation, derived using calculus of variations with voltage as the generalized path, provides the simplest ab initio method to calculate where electrons are statistically most likely to be found around an atom, forming the foundation for modern density functional theory.

📌 Key points (3–5)

• Core goal: Find the charge density ρ_ch(r) and voltage V(r) around an atom by minimizing the action integral over position (not time).
• Energy conversion mechanism: As an electron moves radially, energy converts between Coulomb interaction energy (nucleus-electron plus electron-electron) and kinetic energy of electrons.
• Reciprocal space insight: Though we cannot describe how energy varies with real-space position r directly, we know how it varies with wave vector k (spatial frequency), enabling the derivation.
• Severe assumptions required: The method assumes spherical symmetry, temperature near absolute zero, uniform electron cloud, no spin, and many electrons—making results inaccurate but foundational.
• Common confusion: The independent variable is position r (in spherical coordinates), not time t; the final equation uses a scaled position variable also called t, which is not time.

🎯 The fundamental question and approach

🎯 What we're trying to find

Charge density ρ_ch(r): tells us, statistically on average, where electrons are most likely to be found around an atom.

• At temperatures above absolute zero, electrons are in continual motion.
• The Heisenberg uncertainty principle prevents knowing position and momentum simultaneously with complete accuracy.
• Despite these challenges, we can find the statistical distribution.
• Why it matters: Understanding electron distribution is vital for chemical properties (bond strength) and electrical properties (energy required to remove electrons).

🛠️ The calculus of variations setup

The method uses calculus of variations where:

• Generalized path: voltage V(r)
• Generalized potential: charge density ρ_ch(r)
• Independent variable: radial position r (not time)
• Lagrangian form: L = L(r, V, dV/dr)
• Principle: The path found in nature minimizes the action δ∫L dr = 0

Don't confuse: This is the Principle of Least Action applied to spatial position; the integral is over position r, not over time.

🔋 Energy components and the Lagrangian

🔋 Total energy of the atom

The atom's energy consists of four terms:

• E_Coulomb_e_nucl: Coulomb interaction between nucleus protons and the electron under consideration
• E_e_e_interact: Coulomb interaction between the electron under consideration and all other electrons (N-1 electrons)
• E_kinetic_nucl: kinetic energy of nucleus ≈ 0 at T ≈ 0 K (ignored)
• E_kinetic_e: kinetic energy of all electrons (cannot be zero even at T = 0 K)

⚡ Coulomb energy in terms of voltage

The first energy term (per unit volume):

E_Coulomb_e_nucl / V + E_e_e_interact / V = (ε/2)|∇V|²

Where:

• ε represents permittivity
• V represents volume (not voltage here)
• The electric field E⃗ = -∇V (negative gradient of voltage)
• Energy density due to electric field = (ε/2)|E⃗|²

Why this form: The N protons attract the electron while the N-1 other electrons repel it; these interactions somewhat cancel but don't disappear completely. The net effect is captured by the electric field energy density.

🌀 Kinetic energy via reciprocal space

The kinetic energy term (per unit volume) requires reciprocal space:

E_kinetic_e / V = c₀V^(5/2)

Where:

• c₀ = [(5mq)/(3ℏ²)]^(3/2) · [q/(3π²)] is a constant
• m is electron mass
• q is magnitude of electron charge
• ℏ is Planck constant

Key insight: We cannot describe how kinetic energy varies with real-space position r, but we can describe it in reciprocal space (wave vector k space) because at T = 0 K, electrons fill the lowest energy states up to the Fermi wave vector k_f.

📐 The Hamiltonian and Lagrangian

Hamiltonian (sum of energy forms per unit volume): H = (ε/2)|∇V|² + c₀V^(5/2)

Lagrangian (difference of energy forms per unit volume): L = (ε/2)|∇V|² - c₀V^(5/2)

Both have units J/m³ (energy per unit volume).

🧮 Reciprocal space concepts

🧮 Why reciprocal space is needed

• Real space: position r⃗ measured in meters; describes where atoms/electrons are located
• Reciprocal space: wave vector k⃗ measured in inverse meters (m⁻¹); describes spatial frequency
• The problem: We don't know how charge density ρ_ch varies with position r, but we do know something about how it varies with wave vector k.

🎚️ Fermi energy and Fermi wave vector

At T = 0 K:

• Electrons occupy the lowest allowed energy states up to the Fermi energy E_f (also called chemical potential μ_chem by chemists)
• Correspondingly, wave vectors are occupied up to the Fermi wave vector k_f
• Relationship: E_f = (ℏ²k_f²)/(2m)

Filled states: |k⃗| = r̃ < k_f (filled), |k⃗| = r̃ > k_f (empty)

🔗 Connecting charge density to wave vector

Key relationships derived:

• ρ_ch = -(q/3π²)k_f³
• k_f = [(-3π²/q)ρ_ch]^(1/3)
• Voltage and charge density relation: V = -(3ℏ²)/(5mq) · (-3π²/q)^(2/3) · ρ_ch^(2/3)

Example: For a large atom with many electrons, the Coulomb interaction between any one electron and the nucleus is shielded by all other electrons. By assuming uniform electron cloud distribution in reciprocal space up to k_f, we can integrate to find total kinetic energy.

🎓 Deriving the equation of motion

🎓 The Euler-Lagrange equation

For spherical symmetry (no θ or φ dependence):

∂L/∂V - ∇·[∂L/∂(dV/dr)]â_r = 0

Evaluating each term:

• ∂L/∂V = (5/2)c₀V^(3/2) (this is essentially the generalized potential)
• ∂L/∂(dV/dr) = ε∇V (this is the generalized momentum M)

🔄 Simplifying to the equation of motion

Substituting into Euler-Lagrange:

(5/2)c₀V^(3/2) - ε∇²V = 0

Or: ∇²V = (5/2)c₀V^(3/2)

Where ∇²V is the Laplacian (qualitatively, the second derivative with respect to spatial position).

🔀 Variable transformation

To clean up the equation, define: V = -y/r

Where y has units V·m (volt-meters).

After substituting and simplifying using the Laplacian in spherical coordinates: d²y/dr² = c₁r^(-1/2)y^(3/2)

Where c₁ = (5/2)[(5mq)/(3ℏ²)]^(3/2) · q/(3π²)

⭐ The Thomas Fermi equation

Final variable change: t = c₁^(-2/3)r (scaled position, not time)

Thomas Fermi equation: d²y/dt² = t^(-1/2)y^(3/2)

• This is a nonlinear differential equation requiring numerical solution.
• Once y(t) is found, voltage V(r) and charge density ρ_ch(r) can be calculated using V = -y/r and the relation between V and ρ_ch.
• Don't confuse: The variable t is a scaled version of position r, not time; the independent variable throughout is position.

🔬 Assumptions and limitations

🔬 Severe assumptions made

To make the problem manageable, the derivation assumes:

Assumption	Implication
Spherical symmetry	No angular dependence (θ or φ) in energy, charge density, voltage
T ≈ 0 K	All electrons occupy lowest allowed energy states
Isolated atom	No other charged particles nearby; not ionized; not part of molecule
Many electrons	One electron feels uniform cloud effect from others
No spin	Electrons have no spin or internal angular momentum

📊 Accuracy and improvements

• Ab initio calculation: From first principles, no experimental data used to find parameters
• Simplest method: Thomas Fermi is the simplest ab initio solution for electron charge density
• Not very accurate: Due to extreme assumptions, results are not very accurate
• Historical context: Developed 1926-1928 when the atom concept itself was still quite new
• Modern refinement: Density functional theory (DFT) is the improved version, making fewer and less severe assumptions

🚀 From Thomas Fermi to density functional theory

Functional: takes in a function and returns a scalar value (unlike a function, which takes in a scalar and returns a scalar).

Modern density functional theory:

• Uses charge density or voltage as the generalized path
• Calculates angular dependence of charge density
• Handles molecules, not just isolated atoms
• Works at temperatures above absolute zero
• Accounts for electron spin
• Improves treatment of E_e_e_interact term
• Requires numerical simulation with parallel processing (supercomputers, graphics cards)
• Active area of research with specialized software packages

Example: While Thomas Fermi assumes uniform electron cloud, DFT can calculate the actual non-uniform distribution and predict allowed energy levels for electrons in complex molecules.

🔗 Connection to other concepts

🔗 Fermi energy and chemical potential

Derived relationship: E_f = -(5q/6)V

• Fermi energy is just a scaled version of voltage with respect to ground at r = ∞
• Electrical engineers: use term "Fermi energy" and "voltage"
• Chemists: use term "chemical potential" μ_chem
• Same concept: Both represent how difficult it is to move electrons around

Don't confuse: Voltage (electrical engineers) and chemical potential (chemists) refer to the same fundamental quantity, just scaled by a constant.

📍 Coordinate systems

The derivation uses spherical coordinates (r, θ, φ):

• Gradient: ∇V = (∂V/∂r)â_r + (1/r)(∂V/∂θ)â_θ + (1/(r sin θ))(∂V/∂φ)â_φ
• Laplacian: ∇²V = (1/r²)(∂/∂r)(r²∂V/∂r) + angular terms
• Simplified for spherical symmetry: ∇V = (∂V/∂r)â_r and ∇²V = (1/r²)(∂/∂r)(r²∂V/∂r)
• Volume element: dV = r² sin θ dr dθ dφ
• Sphere volume integral: For V(r) only, ∫∫∫ V(r)r² sin θ dr dθ dφ = 4π∫V(r)r² dr

🧭 Overview

🧠 One-sentence thesis

Lie analysis provides a systematic procedure to identify continuous symmetries of equations of motion—even nonlinear ones—and use those symmetries to derive conserved quantities (invariants) without solving the equation.

📌 Key points (3–5)

• What Lie analysis does: finds continuous symmetries of differential equations that describe energy conversion processes, expressed as infinitesimal generators.
• Why it matters: symmetries reveal invariants (conserved quantities like energy or momentum) and families of related solutions, even when the equation is nonlinear or has no closed-form solution.
• Noether's theorem: connects continuous symmetries to conservation laws—if a symmetry exists, a corresponding invariant can often be derived.
• Common confusion: discrete vs continuous symmetries—only continuous symmetries (those built from infinitesimal transformations) are analyzed here; discrete symmetries (like time reversal or parity) are not.
• Practical use: knowing one solution and a symmetry lets you generate a whole family of solutions without re-solving the equation.

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🔍 Core concepts

🔍 What Lie analysis is

• The excerpt applies Lie analysis to equations of motion derived from calculus of variations (Chapter 11).
• These equations describe energy conversion processes and may be nonlinear.
• Lie analysis is a systematic technique for extracting information from differential equations by identifying their continuous symmetries.
• It was developed by Sophus Lie in the late 19th century.

🧩 Continuous symmetries

Continuous symmetry: a transformation of the independent and dependent variables that can be expressed as a sum of infinitesimally small symmetries related by a continuous parameter.

• A symmetry means the solutions of the equation remain the same before and after the transformation.
• Example: the wave equation is invariant under the transformation ( t \to t + \varepsilon ) (time shift by a small constant).
• Don't confuse: continuous symmetries (built from infinitesimal steps) vs discrete symmetries (like flipping sign, ( y \to -y ), which cannot be built from infinitesimal steps).

🔧 Infinitesimal generators

Infinitesimal generator: an operator ( U = \xi \partial_t + \eta \partial_y ) that describes how the independent variable ( t ) and dependent variable ( y ) transform under a continuous symmetry.

• ( \xi(t, y) ) describes infinitesimal variation in the independent variable.
• ( \eta(t, y) ) describes infinitesimal variation in the dependent variable.
• The finite transformation is related by ( t \to e^{\varepsilon U} t ) and ( y \to e^{\varepsilon U} y ).
• Example: for the wave equation, ( U = \partial_t ) corresponds to the symmetry ( t \to t + \varepsilon ), ( y \to y ).

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🛠️ Finding symmetries: the procedure

🛠️ The symmetry condition

• Start with a differential equation ( F(t, y, \dot{y}, \ddot{y}, \ldots) = 0 ).
• For a transformation to be a symmetry, not only the solutions but also all their derivatives must remain unchanged.
• The symmetry condition is: [ \text{pr}^{(n)} U , F = 0 ] where ( \text{pr}^{(n)} U ) is the prolongation of the infinitesimal generator (it describes how ( t, y, \dot{y}, \ddot{y}, \ldots ) all transform).

🔨 Prolongation

Prolongation: a generalization of the infinitesimal generator that includes transformations of derivatives of the dependent variable.

• For a second-order equation: [ \text{pr}^{(n)} U = \xi \partial_t + \eta \partial_y + \eta_t \partial_{\dot{y}} + \eta_{tt} \partial_{\ddot{y}} + \cdots ]
• The functions ( \eta_t ) and ( \eta_{tt} ) are defined in terms of ( \xi ) and ( \eta ) (see Eqs. 14.43, 14.44 in the excerpt).

📝 Solving for ( \xi ) and ( \eta )

• Substitute the prolongation into the symmetry condition.
• Use the original equation to eliminate higher derivatives (e.g., replace ( \ddot{y} ) using the equation of motion).
• Regroup terms by powers of ( \dot{y} ).
• Each coefficient must be zero, yielding a system of equations for ( \xi ) and ( \eta ).
• Solve this system to find all allowed infinitesimal generators.

Example: for the Thomas–Fermi equation ( \ddot{y} = y^{3/2} t^{-1/2} ), the only continuous symmetry is ( U = t \partial_t - 3y \partial_y ), corresponding to the transformation ( t \to c_6 t ), ( y \to c_6^{-3} y ).

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🔗 Symmetries and solutions

🔗 Generating families of solutions

• If you know one solution ( y(t) ) and a symmetry ( U ), you can generate a family of related solutions without re-solving the equation.
• Example: the wave equation ( \ddot{y} + \omega_0^2 y = 0 ) has the symmetry ( U = \partial_t ). If ( y(t) ) is a solution, then ( y(t + \varepsilon) ) is also a solution for any constant ( \varepsilon ).
• Another symmetry: ( U = y \partial_y ) (scaling). If ( y(t) ) is a solution, so is ( (1 + \varepsilon) y(t) ).

🧮 Group theory

• Continuous symmetries form a Lie group.
• A group has four properties: identity, inverse, associativity, and closure.
• The commutator ([U_1, U_2] = U_1 U_2 - U_2 U_1) of two generators is also a generator (closure).
• A complete set of infinitesimal generators describes all possible continuous symmetries of the equation.

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🎯 Noether's theorem: symmetries → invariants

🎯 What Noether's theorem says

Noether's theorem: there is a relationship between continuous symmetries of an equation of motion (derived from a Lagrangian) and conserved quantities (invariants).

• If you know a symmetry ( U = \xi \partial_t + \eta \partial_y ) and the Lagrangian ( L(t, y, \dot{y}) ), you can derive an invariant ( \Upsilon ).
• The invariant is a quantity that does not change during the energy conversion process.

🧪 Finding the invariant

1. Find a quantity ( G ) that satisfies: [ \frac{dG}{dt} = \text{pr}^{(n)} U , L + L \frac{d\xi}{dt} ]
2. Then the invariant is: [ \Upsilon = \eta \frac{\partial L}{\partial \dot{y}} + \xi L - \xi \dot{y} \frac{\partial L}{\partial \dot{y}} - G ]
3. This quantity ( \Upsilon ) is conserved: ( \frac{d\Upsilon}{dt} = 0 ).

🔬 Physical meaning

• Invariants often correspond to physical quantities like energy, momentum, or angular momentum.
• Example: for the pendulum equation with Lagrangian ( L = \frac{1}{2} m \dot{y}^2 - mg \cos y ), the symmetry ( U = \partial_t ) (time invariance) yields the invariant ( \Upsilon = -\frac{1}{2} m \dot{y}^2 - mg \cos y ), which is the total energy (Hamiltonian).
• General rule: if the Lagrangian does not explicitly depend on ( t ), the system has the symmetry ( U = \partial_t ), and energy is conserved.

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📚 Examples

📚 Wave equation

• Equation: ( \ddot{y} + \omega_0^2 y = 0 ).
• The excerpt lists three of eight continuous symmetries:
  1. ( U_1 = \partial_t ): time shift ( t \to t + \varepsilon ).
  2. ( U_2 = y \partial_y ): scaling ( y \to (1 + \varepsilon) y ).
  3. ( U_3 = \sin(\omega_0 t) \partial_y ): ( y \to y + \varepsilon \sin(\omega_0 t) ).
• Each symmetry generates a family of solutions.

📚 Line equation

• Equation: ( \ddot{y} = 0 ) (straight line).
• The excerpt derives eight infinitesimal generators by solving the symmetry condition.
• Two examples:
  • ( U_1 = \partial_t ): time shift.
  • ( U_6 = t \partial_y ): ( y \to y + \varepsilon t ).
• Lagrangian: ( L = \frac{1}{2} \dot{y}^2 ).
• Invariants:
  • From ( U_1 = \partial_y ): ( \Upsilon = \dot{y} ) (slope is constant).
  • From ( U_6 = t \partial_y ): ( \Upsilon = y - t \dot{y} ) (y-intercept is constant).

📚 Thomas–Fermi equation

• Equation: ( \ddot{y} = y^{3/2} t^{-1/2} ) (describes electron charge density around an isolated atom).
• Only one continuous symmetry: ( U = t \partial_t - 3y \partial_y ).
• Finite transformation: ( t \to c_6 t ), ( y \to c_6^{-3} y ).
• If ( y(t) ) is a solution, so is ( c_6^{-3} y(c_6 t) ) for any constant ( c_6 ).

📚 Pendulum equation

• Equation: ( \ddot{y} = g \sin y ).
• Lagrangian: ( L = \frac{1}{2} m \dot{y}^2 - mg \cos y ).
• Symmetry: ( U = \partial_t ) (time invariance).
• Invariant: ( \Upsilon = -\frac{1}{2} m \dot{y}^2 - mg \cos y ) (total energy).

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🧠 Key distinctions

🧠 Types of symmetries

Type	Description	Example
Continuous	Built from infinitesimal transformations	( t \to t + \varepsilon )
Discrete	Cannot be built from infinitesimal steps	( y \to -y ) (parity)
Regular	Transform ( t ) and ( y ) only	( U = \xi \partial_t + \eta \partial_y )
Dynamical	Also transform derivatives	Beyond scope here
Geometrical	Direct transformations	Covered here
Nongeometrical	Involve Fourier transforms	Beyond scope here

🧠 Don't confuse

• Symmetry vs invariant: a symmetry is a transformation that leaves the equation unchanged; an invariant is a quantity that does not change during the process.
• Infinitesimal generator vs finite transformation: the generator ( U ) describes infinitesimal changes; the finite transformation is ( e^{\varepsilon U} ).
• Lagrangian symmetry vs equation symmetry: all symmetries of the variational problem ( \delta \int L , dt = 0 ) are symmetries of the Euler–Lagrange equation, but not vice versa.

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🎓 Summary

• Lie analysis identifies continuous symmetries of differential equations by solving a symmetry condition for infinitesimal generators ( U = \xi \partial_t + \eta \partial_y ).
• Symmetries reveal families of related solutions and simplify analysis of nonlinear equations.
• Noether's theorem connects symmetries to conserved quantities (invariants), which often have physical meaning (energy, momentum, etc.).
• The procedure applies to equations of motion from calculus of variations, including nonlinear and unsolvable equations.
• Example: if the Lagrangian does not depend on time, the system has time-translation symmetry, and energy is conserved.