Vibrational spectroscopy, plasmonics and SERS, from first principles

Every molecule carries a signature written in the way its atoms move. Shine light on it and that signature leaks out — as missing colours in transmitted infrared light, or as a faint shifted glow in scattered laser light. Reading the signature is vibrational spectroscopy, and it is one of the few ways to identify a molecule without touching it, labelling it or knowing in advance what to look for. This article builds the subject from the ground up: first the physics of a single vibrating bond, then the two great techniques that read vibrations (infrared absorption and Raman scattering), then the practical art of reading spectra of biological material, and finally the plasmonic physics that boosts Raman signals by factors of a million or more in surface-enhanced Raman scattering.

It follows Light from first principles, where Maxwell's equations, refraction, absorption and the Drude model of metals are derived. Numbers in square brackets point to the references at the end.

01Light, energy and the units of spectroscopy

Light is an electromagnetic wave: an electric field and a magnetic field oscillating at right angles to each other and to the direction of travel. Quantum mechanically it comes in packets, photons, each with an energy fixed by its frequency \(\nu\):

$$E = h\nu = \frac{hc}{\lambda} = hc\,\tilde\nu, \qquad \tilde\nu = \frac{1}{\lambda}$$

where \(E\) is the photon energy, \(\nu\) its frequency (oscillations per second, Hz), \(\lambda\) its wavelength, \(h = 6.626\times10^{-34}\) J s Planck's constant and \(c = 2.998\times10^{8}\) m s⁻¹ the speed of light. (Elsewhere in this article \(E\) also denotes the electric field of the light; the context makes clear which.) The last form introduces the spectroscopist's favourite unit, the wavenumber \(\tilde\nu\) in cm⁻¹: the number of wavelengths per centimetre. It is proportional to energy, so energy differences are simple subtractions — which is exactly what Raman spectroscopy measures.

Figure 1

02What a molecule can do with a photon

Because electrons are thousands of times lighter than nuclei, they rearrange almost instantly as the nuclei move (the Born–Oppenheimer approximation). The energy of a molecule then separates, to a good approximation, into an electronic part, a vibrational part and a rotational part:

$$E \approx E_\text{el} + E_\text{vib} + E_\text{rot}, \qquad \text{typical spacings: } 10^4\text{–}10^5,\;\; 10^2\text{–}10^3,\;\; 1\text{–}10\ \mathrm{cm^{-1}}$$

Here \(E\) is the total energy of the molecule and \(E_\text{el}\), \(E_\text{vib}\), \(E_\text{rot}\) its electronic, vibrational and rotational parts; the spacings are the typical gaps between neighbouring levels of each kind, quoted in wavenumbers.

A photon can interact with these levels in three fundamentally different ways:

• Absorption: the photon disappears and its energy lifts the molecule to a higher level. It only happens when the photon energy matches a gap. Infrared absorption excites vibrations; UV–visible absorption excites electrons.
• Emission: an excited molecule gives the energy back as a photon. Fluorescence is emission from an excited electronic state after it has first relaxed to its lowest vibrational level.
• Scattering: the photon is redirected without being absorbed. The light's field pushes the electron cloud, which re-radiates. Almost always the scattered photon has the same energy (Rayleigh scattering); about one in ten million times it has given some energy to a vibration (Stokes Raman) or taken some from one (anti-Stokes Raman). Scattering does not need a resonance — any photon will do.

Figure 2

03A chemical bond is a spring

The classical oscillator

Near its equilibrium length \(r_e\), the energy of a bond rises roughly quadratically when it is stretched or compressed — exactly like a spring obeying Hooke's law, \(F = -k\,x\), where \(F\) is the restoring force, \(x = r - r_e\) the stretch of the bond length \(r\) away from equilibrium, and \(k\) the spring (force) constant. Two masses \(m_1, m_2\) joined by such a spring oscillate at

$$\omega = \sqrt{\frac{k}{\mu}}, \qquad \mu = \frac{m_1 m_2}{m_1 + m_2}, \qquad \boxed{\;\tilde\nu = \frac{1}{2\pi c}\sqrt{\frac{k}{\mu}}\;}$$

where \(\omega\) is the angular frequency of the vibration (rad s⁻¹; \(\omega = 2\pi\nu\)), \(\tilde\nu\) the same frequency as a wavenumber (\(c\) in cm s⁻¹ gives cm⁻¹), \(m_1\) and \(m_2\) the masses of the two atoms, \(\mu\) the reduced mass and \(k\) the force constant (bond stiffness, in N/m). Two rules follow that organise every vibrational spectrum:

• Stiffer bonds vibrate faster. A triple bond (k ≈ 1600 N/m) sits higher than a double bond (≈ 1000 N/m), which sits higher than a single bond (≈ 450 N/m): C≡C ≈ 2150, C=C ≈ 1650, C–C ≈ 1100 cm⁻¹.
• Light atoms vibrate faster. Any bond to hydrogen has μ ≈ 1 u, so X–H stretches lie far above everything else (C–H ≈ 2900, O–H ≈ 3400 cm⁻¹). Replacing H by deuterium doubles the mass and divides the frequency by almost √2: C–D ≈ 2200 cm⁻¹.

Figure 3 · Interactive

The quantum oscillator and anharmonicity

Quantum mechanics allows only discrete energies. For the harmonic potential \(V = \tfrac12 k (r - r_e)^2\) the Schrödinger equation gives

$$E_v = h c\,\tilde\nu\left(v + \tfrac12\right), \qquad v = 0, 1, 2, \dots$$

where \(V\) is the potential energy, \(E_v\) the energy of vibrational level \(v\), and \(v\) the vibrational quantum number (0 for the ground state). The levels have equal spacings and a zero-point energy \(\tfrac12 hc\tilde\nu\): a bond never stops vibrating. Real bonds are not harmonic — stretch one far enough and it breaks. The Morse potential \(V = D_e\,[1 - e^{-a(r - r_e)}]^2\) captures this, and its levels crowd together near the dissociation energy:

$$E_v = hc\left[\tilde\nu_e\left(v + \tfrac12\right) - \tilde\nu_e x_e\left(v + \tfrac12\right)^2\right]$$

In the Morse potential \(D_e\) is the depth of the potential well (the dissociation energy measured from its bottom) and \(a\) sets its width; in the level formula \(\tilde\nu_e\) is the harmonic wavenumber and \(x_e\) the dimensionless anharmonicity constant (typically 0.005–0.02), which makes successive levels close up. Anharmonicity has three visible consequences: overtones (v = 0 → 2, weak, at slightly less than twice the fundamental — the basis of near-infrared spectroscopy), combination bands (two modes excited at once), and Fermi resonance, where an overtone and a fundamental of the same symmetry and similar energy mix and share intensity (the famous CO₂ Raman doublet at 1285 and 1388 cm⁻¹ is one) [1, 2].

Figure 4 · Interactive

04Normal modes: stretching, bending, wagging…

A molecule of \(N\) atoms has \(3N\) ways to move its atoms (x, y, z for each). Three of these move the whole molecule (translation) and three rotate it (two for a linear molecule), leaving

$$3N - 6 \text{ vibrations (non-linear)}, \qquad 3N - 5 \text{ (linear)}.$$

These are not the motions of individual bonds but normal modes: collective motions in which every atom moves in phase, at the same frequency, along its own direction. Mathematically they are the eigenvectors of the mass-weighted matrix of force constants [3]; physically they are the independent “notes” the molecule can play. Each has a name describing what it does:

Figure 5 · Interactive animation

Some vibrations stay close to a single bond and appear at nearly the same wavenumber in any molecule containing it — group frequencies such as C=O at 1650–1750 cm⁻¹ or C–H near 2900 cm⁻¹. Others involve the whole skeleton and are unique to each molecule; they crowd the region below about 1800 cm⁻¹, which is therefore called the fingerprint region.

05Infrared absorption and FTIR

The selection rule

An oscillating electric field can only drive a vibration that makes the molecule's electric dipole moment \(\boldsymbol\mu\) oscillate. Expanding the dipole moment in the normal coordinate \(Q\),

$$\boldsymbol\mu(Q) = \boldsymbol\mu_0 + \left(\frac{\partial\boldsymbol\mu}{\partial Q}\right)_0 Q + \dots \qquad\Rightarrow\qquad \text{IR intensity} \;\propto\; \left|\frac{\partial\boldsymbol\mu}{\partial Q}\right|^2,$$

where \(\boldsymbol\mu_0\) is the permanent dipole moment at the equilibrium geometry and \((\partial\boldsymbol\mu/\partial Q)_0\) how much the dipole changes as the molecule moves along the normal coordinate \(Q\) (the collective displacement of the atoms in that vibration, section 4). A vibration is infrared-active only if this derivative is non-zero; in the harmonic approximation only \(\Delta v = \pm1\) transitions (one quantum gained or lost) are allowed. Polar bonds that stretch (C=O, O–H, N–H, P=O) absorb strongly; symmetric motions of non-polar groups (C=C in a symmetric environment, S–S) are weak or invisible. Absorption in a sample follows the Beer–Lambert law:

$$A = -\log_{10}\frac{I}{I_0} = \varepsilon\, c\, \ell$$

where \(A\) is the absorbance, \(I_0\) and \(I\) the intensities entering and leaving the sample, ε the molar absorption coefficient (L mol⁻¹ cm⁻¹), \(c\) the concentration (mol L⁻¹) and ℓ the path length (cm). (This ε is unrelated to the permittivity ε used later for metals.)

The water problem and ATR

Liquid water absorbs intensely in the mid-infrared (O–H stretch near 3400 cm⁻¹ and H–O–H bend at 1640 cm⁻¹ — right on top of the protein amide I band). Path lengths in water must be a few micrometres, or the measurement done by attenuated total reflection (ATR): the beam is totally reflected inside a diamond or germanium crystal and samples only the evanescent field, which decays into the sample over

$$d_p = \frac{\lambda}{2\pi n_1\sqrt{\sin^2\theta - (n_2/n_1)^2}}$$

where \(d_p\) is the penetration depth (the distance over which the evanescent field falls to 1/e of its value at the surface), \(\lambda\) the vacuum wavelength of the infrared light, \(n_1\) the refractive index of the ATR crystal, \(n_2\) that of the sample, and \(\theta\) the angle of incidence inside the crystal — about 1–2 µm for diamond at 45°. The same evanescent-wave physics reappears in surface plasmon resonance (section 11).

Why every modern IR spectrometer uses a Michelson interferometer

A Fourier-transform infrared (FTIR) spectrometer sends the broadband IR light through a Michelson interferometer. One mirror moves, creating a path difference δ; the detector records the interferogram

$$I(\delta) = \int_0^\infty B(\tilde\nu)\,\big[1 + \cos(2\pi\tilde\nu\delta)\big]\,d\tilde\nu, \qquad B(\tilde\nu) \propto \int I(\delta)\cos(2\pi\tilde\nu\delta)\,d\delta,$$

where \(I(\delta)\) is the detector signal at optical path difference \(\delta\) between the two arms, and \(B(\tilde\nu)\) the spectrum — the intensity reaching the detector at each wavenumber \(\tilde\nu\) (source spectrum times sample transmission). The spectrum is the cosine (Fourier) transform of the interferogram. All wavelengths are measured at once (the multiplex or Fellgett advantage), without a slit (the throughput or Jacquinot advantage), and calibrated by a reference laser (the Connes advantage) [11]. The resolution is set by how far the mirror travels: \(\Delta\tilde\nu \approx 1/\delta_\text{max}\), where \(\Delta\tilde\nu\) is the smallest separation of two lines that can be resolved and \(\delta_\text{max}\) the largest path difference (a path difference of 1 cm gives about 1 cm⁻¹).

Figure 6 · Interactive

06Raman scattering

In 1928 C. V. Raman and K. S. Krishnan reported that a small fraction of light scattered by liquids changes colour [4]. The explanation is simplest classically.

Classical picture: an oscillating polarisability mixes frequencies

The light's electric field \(E = E_0\cos\omega_0 t\) — amplitude \(E_0\), angular frequency \(\omega_0\) of the laser, time \(t\) — induces a dipole in the electron cloud, \(\mu_\text{ind} = \alpha E\), where \(\alpha\) is the polarisability — how easily the cloud is distorted. If the molecule vibrates along its normal coordinate \(Q\) with amplitude \(Q_0\) and vibrational angular frequency \(\omega_v\), \(Q = Q_0\cos\omega_v t\), the polarisability changes with it (\(\alpha_0\) is its value at equilibrium):

$$\alpha = \alpha_0 + \left(\frac{\partial\alpha}{\partial Q}\right)_0 Q_0\cos\omega_v t$$

and the product of two cosines is a sum of two others:

$$\mu_\text{ind} = \underbrace{\alpha_0E_0\cos\omega_0 t}_{\text{Rayleigh}} \;+\; \frac{\alpha' Q_0E_0}{2}\Big[\underbrace{\cos(\omega_0-\omega_v)t}_{\text{Stokes}} + \underbrace{\cos(\omega_0+\omega_v)t}_{\text{anti-Stokes}}\Big]$$

Here \(\alpha'\) is shorthand for \((\partial\alpha/\partial Q)_0\), the rate at which the polarisability changes during the vibration. An oscillating dipole radiates at its own frequencies, so the scattered light contains the laser frequency and two sidebands displaced by exactly the vibrational frequency. Hence the Raman selection rule: a vibration is Raman active only if it changes the polarisability, \(\partial\alpha/\partial Q \ne 0\) (α is really a 3×3 tensor; it is enough that one component changes) [5, 6].

Figure 7 · Interactive

Intensity, the ν⁴ law and the Stokes/anti-Stokes ratio

A dipole radiates power \(\propto \omega^4|\mu|^2\), so Raman intensity rises steeply with the excitation frequency:

$$I_\text{Raman} \;\propto\; (\tilde\nu_0 \mp \tilde\nu_v)^4\,\left(\frac{\partial\alpha}{\partial Q}\right)^2 N\,I_0$$

Here \(\tilde\nu_0\) is the laser wavenumber, \(\tilde\nu_v\) the vibrational wavenumber (the Raman shift), the minus sign applies to Stokes and the plus to anti-Stokes lines, \(N\) is the number of molecules probed and \(I_0\) the laser intensity. Going from a 785 nm to a 532 nm laser gives (785/532)⁴ ≈ 4.7 times more signal — but also more fluorescence from biological samples, which is why 785 nm is the usual compromise for tissue. The quantum treatment adds one more ingredient: anti-Stokes scattering starts from an excited vibrational level, whose population is set by the Boltzmann distribution, so

$$\frac{I_\text{aS}}{I_\text{S}} = \left(\frac{\tilde\nu_0 + \tilde\nu_v}{\tilde\nu_0 - \tilde\nu_v}\right)^4 \exp\!\left(-\frac{hc\,\tilde\nu_v}{k_BT}\right)$$

where \(I_\text{aS}\) and \(I_\text{S}\) are the anti-Stokes and Stokes intensities of the same band, \(k_B = 1.381\times10^{-23}\) J K⁻¹ is Boltzmann's constant and \(T\) the absolute temperature; the exponential is the fraction of molecules already in the first excited vibrational level — a contact-free thermometer.

Figure 8 · Interactive

How weak Raman is

Typical Raman cross-sections are around 10⁻²⁹ cm² per molecule — some 10¹³ times smaller than the absorption cross-section of a good dye. Only about one photon in 10⁶–10⁸ is Raman scattered, and any fluorescence from the sample, even a weak one, can bury it. Two routes make Raman stronger by many orders of magnitude: tuning the laser to an electronic absorption (resonance Raman, 10³–10⁶×, the reason carotenoids dominate many biological spectra) and placing the molecule in the enhanced field near a metal nanostructure (SERS, section 13).

Figure 9

Polarisation and the depolarisation ratio

Measuring the scattered light polarised parallel and perpendicular to the laser gives the depolarisation ratio \(\rho = I_\perp/I_\parallel\). For totally symmetric vibrations \(\rho < 3/4\) (often close to 0); for all others \(\rho = 3/4\). It is a quick experimental way to tell symmetric from antisymmetric modes [5].

07IR, Raman and their relatives compared

IR and Raman see the same vibrations through different windows: IR asks whether the dipole moment changes, Raman whether the polarisability changes. They are complementary: polar, asymmetric motions are strong in IR; symmetric motions of polarisable groups (C=C, S–S, aromatic rings) are strong in Raman. In a molecule with a centre of symmetry the two sets do not overlap at all (mutual exclusion).

Figure 10

08Reading a biological spectrum

A cell, a tissue section or a drop of serum is a mixture of lipids, proteins, nucleic acids, carbohydrates and water. Its Raman spectrum is, to a good approximation, the weighted sum of theirs — which is why multivariate and machine-learning methods work so well on biological spectra. The spectrum divides naturally into three regions:

• Fingerprint region, ≈ 400–1800 cm⁻¹: skeletal C–C, C–O, C–N stretches, ring modes, CH bends, amide bands, phosphate stretches. Dense, specific, and where most diagnostic information lies.
• Silent region, ≈ 1800–2800 cm⁻¹: essentially empty for biomolecules (section 9).
• High-wavenumber region, ≈ 2800–3700 cm⁻¹: C–H stretches (strong, informative about the lipid / protein ratio) and the broad O–H stretch of water.

Figure 11 · Interactive

Raman bands of biomolecules

Assignments are from the compilations of Movasaghi et al. [16] and Butler et al. [17]; exact positions shift by a few cm⁻¹ with environment, instrument calibration and conformation.

Infrared bands of biomolecules

Amide I splits by secondary structure because the C=O oscillators of neighbouring peptide groups couple through hydrogen bonds — α-helix near 1650–1658 cm⁻¹, β-sheet near 1620–1640 cm⁻¹ (with a weak component near 1690 for antiparallel sheets), random coil near 1640–1648 cm⁻¹ in IR [18, 19]. Fitting the band with these components is a standard way to follow protein folding and aggregation.

The fingerprint region

The name is literal. Below about 1800 cm⁻¹ most vibrations are not localised on one bond but spread over the skeleton (C–C, C–O, C–N stretches and bends couple strongly because their frequencies are similar). The result is a dense pattern of bands that depends on the whole molecular structure, so two different molecules almost never share it — like fingerprints. It is also the region where most biological disease signatures appear, because it reports conformation (amide III, C–C backbone), composition (phosphate, phenylalanine) and saturation (C=C) at once.

09The silent region, and why it is silent

Look again at Figure 3 and the formula \(\tilde\nu = \frac{1}{2\pi c}\sqrt{k/\mu}\). To vibrate between about 1800 and 2800 cm⁻¹ a bond must be either very stiff and between heavy atoms (a triple bond: C≡C, C≡N), or involve a mass between hydrogen and carbon (deuterium: C–D), or involve less common atoms (Si–H, metal carbonyls). Biomolecules have none of these in meaningful quantity:

• single bonds between C, N, O: k ≈ 400–500 N/m, μ ≈ 6–7 u → below ≈ 1300 cm⁻¹;
• double bonds C=C, C=O, C=N: k ≈ 900–1200 N/m → 1500–1800 cm⁻¹;
• bonds to hydrogen: μ ≈ 1 u → 2800–3700 cm⁻¹;
• the overtones and combinations that could fall in between are weak (section 3).

So for cells and tissues the window 1800–2800 cm⁻¹ is a flat, empty baseline (apart from a weak, broad water combination band near 2100 cm⁻¹). That emptiness is valuable. Anything that does vibrate there stands out against zero background, so small molecules can be tagged with a single alkyne, nitrile or deuterium — tags far smaller than a fluorophore that barely perturb the molecule. Alkyne-tag Raman imaging (Yamakoshi et al., 2012) [20] and the multicolour “Raman dyes” of Wei et al. (2017) [21] are built on exactly this, and deuterated nutrients (C–D at ≈ 2100–2300 cm⁻¹) report metabolic activity of single cells.

10Plasmonics: the free electrons of a metal

To understand SERS we need to understand why gold and silver nanoparticles are so good at concentrating light. The answer is their free electrons.

The Drude model

Treat the conduction electrons of a metal (density \(n\), mass \(m\), charge \(-e\)) as a gas driven by the light field and damped by collisions at rate \(\gamma\):

$$m\ddot{\mathbf x} + m\gamma\dot{\mathbf x} = -e\mathbf E_0 e^{-i\omega t} \quad\Rightarrow\quad \mathbf P = -ne\mathbf x = -\frac{ne^2}{m(\omega^2 + i\gamma\omega)}\mathbf E$$

Here \(\mathbf x\) is the displacement of an electron, the dots denote first and second time derivatives (velocity and acceleration), \(\mathbf E_0 e^{-i\omega t}\) is the light field of angular frequency \(\omega\) written in complex form (\(i = \sqrt{-1}\); the physical field is the real part), and \(\mathbf P\) the polarisation, the induced dipole moment per unit volume. Since the relative permittivity (dielectric function) is \(\varepsilon = 1 + P/(\varepsilon_0E)\), with \(\varepsilon_0 = 8.854\times10^{-12}\) F m⁻¹ the permittivity of vacuum,

$$\boxed{\;\varepsilon(\omega) = \varepsilon_\infty - \frac{\omega_p^2}{\omega^2 + i\gamma\omega}, \qquad \omega_p^2 = \frac{ne^2}{\varepsilon_0 m}\;}$$

\(\omega_p\) is the plasma frequency (ħω_p ≈ 9 eV for Au and Ag), and \(\varepsilon_\infty\) accounts for the bound d-electrons. Below \(\omega_p\) the real part of ε is negative: the electrons move out of phase with the field and screen it, which is why metals reflect. A negative permittivity is the ingredient every plasmonic effect needs [22, 23].

The Drude model misses one thing: interband transitions, where light lifts d-band electrons into the conduction band. In gold they start below about 520 nm and add absorption — the reason gold looks yellow and its plasmons are damped in the blue; silver's interband edge is near 320 nm, so its plasmons are sharper [24].

Figure 12 · Interactive

Three kinds of plasmon

• Bulk (volume) plasmon: the whole electron gas oscillating against the ions at ω_p, in the ultraviolet; not excited by light at normal incidence [25].
• Surface plasmon polariton (SPP): a charge-density wave travelling along a flat metal surface, bound to it by an evanescent field (section 11).
• Localized surface plasmon (LSP): the electrons of a nanoparticle sloshing back and forth; excited directly by light (section 12).

11Surface plasmon polaritons

At a flat interface between a metal (ε_m) and a dielectric (ε_d), Maxwell's equations allow a transverse-magnetic wave bound to the surface, decaying exponentially into both media. Matching the fields at the boundary gives its dispersion relation [23]:

$$k_\text{SPP} = \frac{\omega}{c}\sqrt{\frac{\varepsilon_m\,\varepsilon_d}{\varepsilon_m + \varepsilon_d}}$$

Here \(k_\text{SPP}\) is the wavevector of the surface plasmon along the surface (\(2\pi\) divided by its wavelength), \(\omega\) the angular frequency, \(c\) the speed of light, and \(\varepsilon_m\), \(\varepsilon_d\) the permittivities of the metal and the dielectric. Because \(\varepsilon_m < -\varepsilon_d\), the square root is larger than \(\sqrt{\varepsilon_d}\): the SPP always has more momentum than light of the same frequency in the dielectric. Light hitting a smooth metal surface from the dielectric side can therefore never excite it. Two tricks supply the missing momentum: a grating (\(k_\parallel + m\,2\pi/\Lambda\), where \(k_\parallel\) is the in-plane wavevector of the incident light, \(\Lambda\) the grating period and \(m\) an integer diffraction order), or a prism of higher index under a thin metal film — the Kretschmann [26] and Otto [27] configurations. Light totally reflected in the prism has \(k_\parallel = n_p\,(\omega/c)\sin\theta\), with \(n_p\) the prism's refractive index and \(\theta\) the angle of incidence; at the angle where this equals \(k_\text{SPP}\) the energy is transferred to the plasmon and the reflectance dips sharply. Because \(k_\text{SPP}\) depends on \(\varepsilon_d\), that angle reports the refractive index of whatever is on the surface — SPR biosensing.

Figure 13 · Interactive

The SPP field decays into the dielectric over \(\delta_d = \frac{1}{k_0}\left|\frac{\varepsilon_m + \varepsilon_d}{\varepsilon_d^2}\right|^{1/2}\), where \(k_0 = \omega/c = 2\pi/\lambda\) is the vacuum wavevector — typically 100–300 nm in the visible and near-infrared, and it propagates micrometres to tens of micrometres along the surface before absorption in the metal stops it. That short decay is why SPR measures binding at the surface, not the bulk solution.

12Localized plasmons in nanoparticles

The quasi-static sphere

A sphere of radius \(R \ll \lambda\) sees a field that is uniform across it at any instant. Solving Laplace's equation with the boundary conditions gives a uniform field inside and, outside, the incident field plus that of a point dipole with polarisability [22, 23]

$$\alpha = 4\pi R^3\,\frac{\varepsilon - \varepsilon_m}{\varepsilon + 2\varepsilon_m}$$

Here \(\alpha\) is the particle's polarisability (in units of volume: the induced dipole is \(\varepsilon_0\varepsilon_m\alpha E_0\)), \(R\) its radius, \(\lambda\) the wavelength, \(\varepsilon(\omega)\) the metal's dielectric function and — note the change of meaning from section 11 — \(\varepsilon_m\) now the permittivity of the surrounding medium (\(\varepsilon_m = n_m^2\), 1.78 for water). The denominator vanishes — the response becomes resonant — when

$$\boxed{\;\mathrm{Re}\,\varepsilon(\omega) = -2\varepsilon_m\;}\qquad\text{(the Fröhlich condition)}$$

which for gold in water happens near 520–530 nm and for silver near 400 nm. Absorption and scattering follow from the dipole:

$$C_\text{ext} = k\,\mathrm{Im}\,\alpha, \qquad C_\text{sca} = \frac{k^4}{6\pi}|\alpha|^2, \qquad C_\text{abs} = C_\text{ext} - C_\text{sca}$$

where \(C_\text{ext}\), \(C_\text{sca}\) and \(C_\text{abs}\) are the extinction, scattering and absorption cross-sections (the area of the beam the particle removes, scatters or absorbs) and \(k = 2\pi n_m/\lambda\) the wavevector in the medium. Scattering grows as R⁶ and absorption as R³: small particles absorb (and heat), large ones scatter (and glow in dark-field microscopes). Because the resonance condition contains ε_m, the peak shifts to the red as the surrounding refractive index increases — the basis of LSPR sensing.

Beyond quasi-statics: Mie theory

For larger particles the field varies across the particle; retardation shifts the resonance to the red, broadens it, and allows quadrupole and higher modes. Gustav Mie solved the sphere exactly in 1908 by expanding the fields in vector spherical harmonics [28, 29]:

$$Q_\text{ext} = \frac{2}{x^2}\sum_{n=1}^{\infty}(2n+1)\,\mathrm{Re}(a_n + b_n), \qquad Q_\text{sca} = \frac{2}{x^2}\sum_{n=1}^{\infty}(2n+1)\left(|a_n|^2 + |b_n|^2\right), \qquad x = \frac{2\pi n_m R}{\lambda}$$

Here \(Q_\text{ext}\) and \(Q_\text{sca}\) are efficiencies — cross-sections divided by the geometric cross-section \(\pi R^2\); \(x\) is the size parameter, with \(n_m\) the refractive index of the medium; the sum runs over multipole orders \(n\) (1 = dipole, 2 = quadrupole, …); and \(a_n\), \(b_n\) are the electric and magnetic scattering coefficients of order \(n\), built from Riccati–Bessel functions of \(x\) and of \(m x\), where \(m\) is the particle's refractive index relative to the medium. The figure below evaluates them exactly, with the measured optical constants of gold and silver.

Figure 14 · Interactive (exact Mie theory)

The near field and hot spots

Just outside the sphere, on the axis of polarisation, the field is \(E = E_0(1 + 2g)\) with \(g = (\varepsilon - \varepsilon_m)/(\varepsilon + 2\varepsilon_m)\). At resonance |g| is large (limited only by Im ε), so the intensity is enhanced tens of times for gold and hundreds of times for silver — and the enhancement falls off with distance as \((R/(R+d))^3\) in field.

Figure 15 · Interactive

Shape and proximity concentrate the field far more than a single sphere can. Sharp tips (nanorods, nanostars, triangles) act like lightning rods; and in the narrow gap between two particles the plasmons of both couple, the resonance red-shifts, and the field can be enhanced 10³–10⁵ times in intensity — a hot spot. These configurations have no simple formula; they are solved numerically (section 14). Our own RNBeamX Layer Studio ▸ Particles tab computes spheres, core–shells, dimers and nanorods, including the gap field.

13Surface-enhanced Raman scattering (SERS)

A short history

In 1974 Fleischmann, Hendra and McQuillan observed unexpectedly strong Raman spectra of pyridine on a roughened silver electrode [30]. In 1977 Jeanmaire and Van Duyne [31] and, independently, Albrecht and Creighton [32] showed that the signal was enhanced by about a million per molecule, far more than the extra surface area could explain. The electromagnetic theory was developed in the following years [33, 34], and in 1997 two groups detected Raman spectra of single molecules [9, 10].

The electromagnetic mechanism: why |E|⁴

A molecule near a plasmonic particle is enhanced twice. First, the local field that drives its induced dipole is larger than the incident field, by \(|E_\text{loc}(\omega_0)|/|E_0|\). Second, the Raman-shifted dipole does not radiate alone: it polarises the particle at \(\omega_R\), which radiates far more efficiently — by the reciprocity theorem, by the same factor that the particle would enhance a field at \(\omega_R\) arriving from the detector's direction. Multiplying the two intensity factors gives [7, 35]

$$\boxed{\;G_\text{SERS} \approx \frac{|E_\text{loc}(\omega_0)|^2}{|E_0|^2}\cdot\frac{|E_\text{loc}(\omega_R)|^2}{|E_0|^2} \approx \frac{|E_\text{loc}|^4}{|E_0|^4}\;}$$

Here \(E_0\) is the amplitude of the incident field, \(E_\text{loc}\) the local field at the molecule, \(\omega_0\) the laser frequency, \(\omega_R\) the Raman-scattered frequency, and \(G_\text{SERS}\) the enhancement of the Raman intensity. This is the |E|⁴ approximation, accurate when the Raman shift is small compared with the plasmon width. A field enhancement of 10 gives 10⁴; of 100, 10⁸; and in the best gaps 10¹⁰ and more, enough for single molecules.

Figure 16 · Interactive

The chemical mechanism

Molecules that bind to the metal can gain a further factor of about 10–100 through charge transfer: new electronic states formed between the molecule and the metal's Fermi level make the scattering partly resonant, and binding changes the molecule's polarisability. This chemical enhancement is specific to the molecule and its binding, explains differences between molecules on the same substrate, and can change relative band intensities [7, 36].

Enhancement factors: being careful with numbers

“Enhancement factor” is used for several different quantities [37]. The analytical enhancement factor compares signals at equal concentrations, \(\mathrm{AEF} = (I_\text{SERS}/c_\text{SERS}) / (I_\text{RS}/c_\text{RS})\); the substrate enhancement factor divides by the number of molecules actually probed, \(\mathrm{SSEF} = (I_\text{SERS}/N_\text{surf}) / (I_\text{RS}/N_\text{vol})\). Averaged over a whole substrate, typical values are 10⁵–10⁸; the 10¹⁰ of single-molecule work belongs to the rare molecules in the best hot spots. A spectrum is dominated by those few — one reason SERS is powerful for detection and challenging for quantification.

Surface selection rules and what changes in a SERS spectrum

Near a metal surface the field is mostly normal to it, so vibrations whose polarisability changes along the surface normal are enhanced most. Together with chemical binding, this is why a SERS spectrum can differ from the normal Raman spectrum of the same molecule in relative intensities and by small shifts — something to remember when assigning bands [7].

From effect to measurement

• Substrates: colloidal Au / Ag nanoparticles (often aggregated to make gaps), nanostructured films made by lithography or deposition, particle-on-mirror gaps, and shell-isolated particles (SHINERS) [15] that keep the metal away from the sample.
• Label-free versus label-based: label-free SERS reads the molecules near the surface directly; label-based SERS uses nanoparticles coated with a strong reporter (often with a silent-region band) as bright, multiplexable tags. Both are developed in detail in Label-free and labelled SERS.
• Reproducibility: because hot spots dominate, spectra fluctuate; averaging over many spots, internal standards and multivariate or machine-learning analysis turn SERS into a quantitative tool — which is the direction of receptor-free diagnostic platforms.

14Simulating spectra and plasmonic fields

A sensible workflow: Mie or quasi-static models for intuition and checks; BEM, FDTD or FEM for the real geometry; density-functional calculations of the molecule to assign its bands; and always a comparison with a measured spectrum.

15From the instrument to a clean spectrum

• Choose the laser: 532 nm for strong signal and resonance with many SERS substrates; 633 nm as a compromise; 785 nm for tissue and gold nanostructures (less fluorescence, deeper penetration); 1064 nm when fluorescence is overwhelming.
• Calibrate the Raman shift axis with silicon (520.7 cm⁻¹) or a standard such as 4-acetamidophenol or polystyrene, and the intensity axis with a white-light or NIST standard if spectra from different instruments are compared [17].
• Watch the power density: a microscope focuses milliwatts into a micrometre — enough to damage cells and to reshape metal nanostructures.
• Pre-process: remove cosmic-ray spikes, subtract the fluorescence background (polynomial or asymmetric least squares), smooth gently (Savitzky–Golay), normalise (vector, area, or to a band such as 1003 cm⁻¹), then compare groups or train classifiers. Every step changes the numbers, so report them. This is the pipeline built into RNSpectraX.

16Glossary

17References

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The interactive figures are small simulations written for this article. The Mie calculation is exact; the near-field and SERS figures use the quasi-static sphere; the biological spectrum is a sum of Lorentzian bands at literature positions with illustrative heights. Optical constants of gold and silver are from Johnson & Christy [24]. Corrections and questions are welcome at rakesh_n@cpqct.iitr.ac.in.

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