# Notations

Ryan James York · https://ryanjamesyork.com/notations

> Symbols, units and conventions used throughout the library — cryptographic, spectroscopic and electronic-structure notation.

The notation below is used throughout the chapters and findings. Where a finding departs from it, the departure is stated in the finding.

## General conventions

| Convention | Meaning |
|---|---|
| §*c*.*s* | Section *s* of Chapter *c*; §2026.1(ii) is the second section of Finding 2026.1 |
| *c*.*s*.*n* | Equation *n* of §*c*.*s*; 2026.1.*n* is equation *n* of Finding 2026.1 |
| Figure *c*.*n* | Figure *n* of Chapter *c* (or of a finding, by its number) |
| YYYY-MM-DD | All dates are written in ISO 8601 form |
| [FR00] | A citation key, resolved in the [Bibliography](https://ryanjamesyork.com/bibliography) |
| SI | SI units throughout; Raman shifts in cm⁻¹, energies in eV, lengths in nm or Å as customary |

## Cryptographic notation

| Symbol | Meaning |
|---|---|
| $m$ | A message: the exact bytes of a file or record |
| $H(m)$ | The SHA-256 digest of $m$, written as 64 lowercase hexadecimal characters |
| $a \Vert b$ | Concatenation of byte strings $a$ and $b$ |
| $(\mathit{sk}, \mathit{pk})$ | An Ed25519 secret and public key pair |
| $\mathrm{Sign}_{\mathit{sk}}(m)$ | An Ed25519 signature over $m$; verified by $\mathrm{Verify}_{\mathit{pk}}(m, \sigma)$ |
| $r_i$ | The $i$-th receipt (record) appended to a ledger |
| $h_i$ | The chain head after the $i$-th receipt; $h_0$ is a fixed genesis value |
| $n_A$ | The per-actor sequence number of actor $A$ |

The chain head is defined recursively:

$$
h_i = H\big(h_{i-1} \Vert r_i\big)
$$ {$i \ge 1$}

so that any change to a receipt $r_j$ changes every head $h_i$ with $i \ge j$. A signature $\sigma_i = \mathrm{Sign}_{\mathit{sk}}(h_i)$ over a head therefore commits to the whole prefix of the ledger.

## Raman spectroscopy of carbon

| Symbol | Meaning |
|---|---|
| $\omega$ | Raman shift, in cm⁻¹ |
| $\lambda_L$, $E_L$ | Excitation wavelength and photon energy; 514.5 nm corresponds to 2.41 eV |
| D | The disorder-activated band near 1350 cm⁻¹ at 2.41 eV excitation |
| G | The in-plane sp² stretching band near 1580 cm⁻¹ in graphite |
| D′ | The weaker disorder band near 1620 cm⁻¹ |
| 2D (G′) | The second-order band near 2700 cm⁻¹ at 2.41 eV excitation |
| $I(\mathrm{D})/I(\mathrm{G})$ | The ratio of D to G intensity, the principal disorder measure |
| $\Gamma$ | Full width at half maximum of a fitted band |
| $L_a$ | In-plane crystallite size |
| sp², sp³ | Carbon bonding fractions |

Band positions depend on excitation energy. The D band disperses by about 50 cm⁻¹ per eV of excitation energy and the 2D band by roughly twice that [FR01], [MPDD09], so positions are always quoted with the excitation wavelength. For crystallite sizes above a few nanometres the Tuinstra–Koenig relation holds,

$$
\frac{I(\mathrm{D})}{I(\mathrm{G})} \propto \frac{1}{L_a}
$$ {$L_a \gtrsim 2$ nm}

while below that size the relation reverses and $I(\mathrm{D})/I(\mathrm{G}) \propto L_a^{2}$ [TK70], [FR00]. The three-stage amorphisation trajectory of Ferrari and Robertson, which organises these regimes, is described in [§3.2](https://ryanjamesyork.com/3#s2).

## Electronic structure of graphene

| Symbol | Meaning |
|---|---|
| $a$ | Carbon–carbon distance, 1.42 Å |
| $a_0 = \sqrt{3}\,a$ | Lattice constant, 2.46 Å |
| $t$ | Nearest-neighbour hopping energy, about 2.7 eV |
| $\mathbf{K}$, $\mathbf{K}'$ | The two inequivalent corners of the Brillouin zone |
| $E_F$ | Fermi energy |
| $v_F$ | Fermi velocity, about $10^{6}$ m s⁻¹ |

In the nearest-neighbour tight-binding model the π bands of a single graphene layer are [W47], [CGPNG09]

$$
E_{\pm}(\mathbf{k}) = \pm\, t\,\sqrt{3 + f(\mathbf{k})}, \qquad
f(\mathbf{k}) = 2\cos\!\big(\sqrt{3}\,k_y a\big) + 4\cos\!\Big(\tfrac{\sqrt{3}}{2}\,k_y a\Big)\cos\!\Big(\tfrac{3}{2}\,k_x a\Big)
$$

and near the $\mathbf{K}$ points the dispersion is linear,

$$
E_{\pm}(\mathbf{q}) \approx \pm\, \hbar v_F\,|\mathbf{q}|, \qquad v_F = \frac{3 t a}{2\hbar}
$$ {$\mathbf{q} = \mathbf{k} - \mathbf{K}$, $|\mathbf{q}| \ll |\mathbf{K}|$}

Wave vectors are measured from the zone centre unless written as $\mathbf{q}$.