Notations

Contents

	General conventions
	Cryptographic notation
	Raman spectroscopy of carbon
	Electronic structure of graphene

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
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
$a ∥ b$	Concatenation of byte strings $a$ and $b$
$(sk, pk)$	An Ed25519 secret and public key pair
$Sign_{sk} (m)$	An Ed25519 signature over $m$ ; verified by $Verify_{pk} (m, σ)$
$r_{i}$	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:

N.1

h_{i} = H (h_{i - 1} ∥ r_{i})

i \geq 1

so that any change to a receipt $r_{j}$ changes every head $h_{i}$ with $i \geq j$ . A signature $σ_{i} = Sign_{sk} (h_{i})$ over a head therefore commits to the whole prefix of the ledger.

Raman spectroscopy of carbon

Symbol	Meaning
$ω$	Raman shift, in cm⁻¹
$λ_{L}$ , $E_{L}$	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
$I (D) / I (G)$	The ratio of D to G intensity, the principal disorder measure
$Γ$	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,

N.2

\frac{I ( D )}{I ( G )} \propto \frac{1}{L _{a}}

L_{a} ≳ 2

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

Electronic structure of graphene

Symbol	Meaning
$a$	Carbon–carbon distance, 1.42 Å
$a_{0} = 3 a$	Lattice constant, 2.46 Å
$t$	Nearest-neighbour hopping energy, about 2.7 eV
$K$ , $K^{'}$	The two inequivalent corners of the Brillouin zone
$E_{F}$	Fermi energy
$v_{F}$	Fermi velocity, about $1 0^{6}$ m s⁻¹

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

N.3

E_{\pm} (k) = \pm t 3 + f (k), f (k) = 2 cos (3 k_{y} a) + 4 cos (\frac{3}{2} k_{y} a) cos (\frac{3}{2} k_{x} a)

and near the $K$ points the dispersion is linear,

N.4

E_{\pm} (q) \approx \pm ℏ v_{F} ∣ q ∣, v_{F} = \frac{3 t a}{2ℏ}

q = k - K

∣ q ∣ ≪ ∣ K ∣

Wave vectors are measured from the zone centre unless written as $q$ .