# Blackbody Radiation & Planck's Law > Every warm body glows. Classical physics — Maxwell's waves plus statistical mechanics — makes a > clean, parameter-free prediction for the glow's spectrum, and the prediction is not just wrong > but *infinitely* wrong. Planck's 1900 repair, restricting each radiation mode to energies > $E_n = nh\nu$, is the first appearance of $h$ in physics and the opening shot of the quantum > revolution. A cat on a radiator has always known thermal physics matters; here we learn exactly > how much. ## Learning Objectives After this lesson you will be able to: 1. Explain why the **cavity radiator** realizes an ideal **blackbody**, and state the **Stefan–Boltzmann** and **Wien displacement** laws with correct constants. 2. Derive the **Rayleigh–Jeans law** from standing-wave mode counting plus **equipartition**, and explain the **ultraviolet catastrophe**. 3. Derive **Planck's law** from the hypothesis $E_n = nh\nu$ via Boltzmann-weighted geometric sums, and recover both limits. 4. Obtain the **Stefan–Boltzmann constant** and **Wien's constant** by integrating and maximizing Planck's law. 5. Compute blackbody spectra numerically and apply them to real sources (the Sun, the CMB). --- ## Intuition Heat a poker: first infrared, then dull red, then orange, then white. Two facts are universal — hotter bodies radiate *more* (total power grows steeply with $T$) and radiate *bluer* (the peak shifts up in frequency). For an ideal absorber the entire spectrum depends on **temperature alone**, not on material — so it probes something fundamental about radiation itself. The classical picture is seductive. The field in a hot cavity is a collection of standing-wave modes — exactly the modes of [P.1.1](../c01-theoretical-mechanics/01-waves-and-the-wave-equation.md) — and equipartition assigns each mode average energy $k_BT$. But a cavity supports *ever more* modes at higher frequency, without bound, so classical physics predicts infinite energy density, diverging in the ultraviolet. Planck's fix: a mode of frequency $\nu$ holds energy only in lumps of $h\nu$. High-frequency modes, whose lump dwarfs $k_BT$, freeze out — the spectrum bends over and the total is finite. --- ## Theory ### Light as an electromagnetic wave Maxwell's equations in vacuum make each field component obey the wave equation of [P.1.1](../c01-theoretical-mechanics/01-waves-and-the-wave-equation.md), $\nabla^2E = c^{-2}\,\partial_t^2E$ with $c = 1/\sqrt{\mu_0\varepsilon_0} = 2.998\times10^8\ \mathrm{m\,s^{-1}}$ and $\lambda\nu = c$. Light is **transverse**, with **two independent polarizations** per propagation direction — remember that factor of 2. The spectrum runs from radio ($\nu\sim10^6$ Hz) through visible ($4$–$7.5\times10^{14}$ Hz, 750–400 nm) to gamma rays. Accelerating charges radiate; thermally agitated charges in matter emit a continuous spectrum: **thermal radiation**. ### Blackbodies and the cavity radiator The **spectral radiancy** $R_T(\nu)\,d\nu$ is the power emitted per unit area in $[\nu,\nu+d\nu]$. Real surfaces emit a fraction $\epsilon\le1$ (**emissivity**) of the ideal maximum; a **blackbody** is the ideal $\epsilon=1$ absorber-and-emitter. The lab realization is a cavity with a small hole: entering radiation rattles around and is absorbed before escaping, so the hole absorbs everything — it *is* a blackbody — and its leakage samples the equilibrium **energy density** $u(\nu,T)$ inside. A kinetic-theory angle average relates the two: $R_T(\nu) = \tfrac{c}{4}\,u(\nu,T)$. ### The empirical laws Two facts were nailed down before 1900. **Stefan–Boltzmann** (1879/84): the total radiancy is $$ R_T = \sigma T^4, \qquad \sigma = 5.670\times10^{-8}\ \mathrm{W\,m^{-2}\,K^{-4}} . $$ **Wien displacement** (1893): the peak of the wavelength spectrum obeys $$ \lambda_{max}\,T = 2.898\times10^{-3}\ \mathrm{m\,K} . $$ A 5800 K surface (the Sun) peaks near 500 nm, mid-visible; your 310 K body peaks near 9.4 μm. ### Rayleigh–Jeans: counting modes, feeding them equipartition Take a cubic cavity of side $a$ ($V = a^3$) with conducting walls, so fields vanish at the walls — the 3D standing waves of P.1.1, $E \propto \sin(n_x\pi x/a)\sin(n_y\pi y/a)\sin(n_z\pi z/a)$ with $k_i = n_i\pi/a$, $n_i = 1,2,3,\dots$ The dispersion relation $|\mathbf k| = 2\pi\nu/c$ gives $$ n_x^2+n_y^2+n_z^2 = \left(\frac{2a\nu}{c}\right)^{\!2} \equiv r^2 . $$ Modes are lattice points in the **positive octant** of $\mathbf n$-space (sign flips give the same standing wave). Counting points below frequency $\nu$ as the octant volume of a sphere of radius $r$, times 2 for polarization: $$ N(<\nu) = \underbrace{\tfrac18}_{\text{octant}}\cdot\tfrac43\pi\left(\frac{2a\nu}{c}\right)^{\!3}\cdot\underbrace{2}_{\text{polarizations}} = \frac{8\pi V\nu^3}{3c^3} \quad\Longrightarrow\quad \frac{1}{V}\frac{dN}{d\nu} = \frac{8\pi\nu^2}{c^3} . $$ Classically each mode is a harmonic degree of freedom in equilibrium, and **equipartition** (Boltzmann-weighting over a *continuum* of energies, $\langle E\rangle = \int_0^\infty E e^{-E/k_BT}dE \big/ \int_0^\infty e^{-E/k_BT}dE$) assigns it $\langle E\rangle = k_BT$ independent of $\nu$, with $k_B = 1.381\times10^{-23}\ \mathrm{J\,K^{-1}}$. Modes × energy-per-mode is the **Rayleigh–Jeans law**: $$ u(\nu,T) = \frac{8\pi\nu^2}{c^3}\,k_BT . $$ It fits experiment at low frequency — and the total energy diverges: $\int_0^\infty u\,d\nu \propto \int_0^\infty \nu^2\,d\nu = \infty$. Every oven should contain infinite energy, mostly beyond the ultraviolet: the **ultraviolet catastrophe**. Nothing here is sloppy — the mode counting survives into quantum theory unchanged, and equipartition is a theorem. Classical physics itself is wrong. ### Planck's hypothesis and the quantized average energy Planck (1900) kept the mode counting and changed the energy assignment: a mode of frequency $\nu$ may only hold $$ E_n = n\,h\nu, \qquad n = 0,1,2,\dots, \qquad h = 6.626\times10^{-34}\ \mathrm{J\,s} . $$ Boltzmann factors still weight each allowed energy, but integrals become sums. With $x \equiv h\nu/k_BT$ and $q \equiv e^{-x}$, both sums are geometric: $$ Z = \sum_{n=0}^{\infty} e^{-nx} = \frac{1}{1-e^{-x}}, \qquad \sum_{n=0}^{\infty} n\,e^{-nx} = -\frac{dZ}{dx} = \frac{e^{-x}}{(1-e^{-x})^2}, $$ $$ \langle E\rangle = \frac{\sum_n nh\nu\,e^{-nx}}{\sum_n e^{-nx}} = h\nu\,\frac{e^{-x}}{1-e^{-x}} = \frac{h\nu}{e^{h\nu/k_BT}-1} . $$ Multiplying by the unchanged mode density gives **Planck's law**: $$ \boxed{\,u(\nu,T) = \frac{8\pi\nu^2}{c^3}\,\frac{h\nu}{e^{h\nu/k_BT}-1}\,} $$ **Low frequency** ($h\nu\ll k_BT$): $e^x - 1 \approx x$, so $\langle E\rangle \to k_BT$ — Rayleigh–Jeans recovered exactly where it worked. **High frequency** ($h\nu\gg k_BT$): $\langle E\rangle \approx h\nu\,e^{-h\nu/k_BT}$, the exponential *Wien form* $u \propto \nu^3 e^{-h\nu/k_BT}$. Modes whose quantum exceeds the thermal budget are almost never excited: discreteness starves the ultraviolet. > **Caution.** Planck quantized the **energies of the cavity modes** (equivalently, of the material > oscillators exchanging energy with them) — he did *not* claim free light is made of particles. > Radiation in flight was still, for Planck, a classical wave. The bolder step — the **photon** — > is Einstein's (1905), next course: > [P.3.1 Light as Particles](../c03-wave-particle-duality/01-light-as-particles.md). ### Stefan–Boltzmann from Planck Integrate over all frequencies with $x = h\nu/k_BT$, $d\nu = (k_BT/h)\,dx$: $$ u_{tot}(T) = \frac{8\pi h}{c^3}\left(\frac{k_BT}{h}\right)^{\!4}\int_0^\infty\!\frac{x^3}{e^x-1}\,dx = \frac{8\pi^5k_B^4}{15\,h^3c^3}\,T^4, $$ using $\int_0^\infty x^3/(e^x-1)\,dx = \pi^4/15$. With $R_T = (c/4)\,u_{tot}$: $$ R_T = \sigma T^4, \qquad \sigma = \frac{2\pi^5k_B^4}{15\,h^3c^2} = 5.670\times10^{-8}\ \mathrm{W\,m^{-2}\,K^{-4}} . $$ The empirical $T^4$ law now follows from first principles. Historically the logic ran in reverse: Planck fitted his law to the data and extracted the first accurate values of both $h$ and $k_B$. ### Wien displacement from Planck Maximize $u \propto x^3/(e^x-1)$ over $x = h\nu/k_BT$: setting the derivative to zero, $3x^2(e^x-1) = x^3e^x$, i.e. $$ x = 3\left(1 - e^{-x}\right), $$ a transcendental equation with nonzero root $x \approx 2.821$. Hence $\nu_{max}/T = 2.821\,k_B/h = 5.88\times10^{10}\ \mathrm{Hz\,K^{-1}}$. > **Caution.** The peak of the **wavelength** distribution is *not* at $c/\nu_{max}$. Since > $u(\lambda)|d\lambda| = u(\nu)|d\nu|$ with Jacobian $c/\lambda^2$, the $\lambda$-form is > $u(\lambda,T) = (8\pi hc/\lambda^5)[e^{hc/\lambda k_BT}-1]^{-1}$, whose maximum solves > $x = 5(1-e^{-x})$, root $x \approx 4.965$ — giving the familiar > $\lambda_{max}T = hc/(4.965\,k_B) = 2.898\times10^{-3}\ \mathrm{m\,K}$. Same spectrum, different > binning; always say which variable you maximized over. ### Looking ahead: a cavity mode is a harmonic oscillator The ladder $E_n = nh\nu$ is no accident: each electromagnetic mode is mathematically a **harmonic oscillator**, whose exact quantum levels are $E_n = (n+\tfrac12)h\nu$ — the ladder returns, with a zero-point offset, in [P.5.3 The Harmonic Oscillator](../c05-bound-states/03-harmonic-oscillator.md). The same quantized-mode structure, engineered in microwave resonators coupled to artificial atoms, is circuit quantum electrodynamics — the physics of superconducting quantum processors (Term 4.4). --- ## Worked Examples ### Example 1 — The Sun as a blackbody Solar surface: $T \approx 5800$ K, radius $R_\odot = 6.96\times10^{8}$ m. *Peak:* $\lambda_{max} = 2.898\times10^{-3}/5800 = 5.00\times10^{-7}$ m $= 500$ nm — mid-visible, matching the evolution of eyes. *Flux:* $R_T = \sigma T^4 = 5.670\times10^{-8}\times(5800)^4 = 6.42\times10^{7}\ \mathrm{W\,m^{-2}}$. *Luminosity:* $L = 4\pi R_\odot^2R_T = 4\pi(6.96\times10^8)^2\times6.42\times10^7 = 3.9\times10^{26}$ W — the accepted solar output, from two constants and a temperature. ### Example 2 — Why high-frequency modes freeze out At $T = 300$ K, $k_BT = 4.14\times10^{-21}$ J $= 0.0259$ eV. *Radio mode,* $\nu = 1$ GHz: $h\nu = 6.63\times10^{-25}$ J, so $x = 1.6\times10^{-4}$ and $\langle E\rangle = h\nu/(e^x-1) \approx k_BT(1-x/2) = 0.99992\,k_BT$ — equipartition holds to 0.01%; Rayleigh–Jeans is excellent here. *Visible mode,* $\nu = 5\times10^{14}$ Hz: $h\nu = 3.31\times10^{-19}$ J $= 2.07$ eV, so $x = 80$ and $\langle E\rangle \approx h\nu\,e^{-80} \approx 6\times10^{-54}$ J — below equipartition by a factor $\sim10^{33}$. The mode exists, but the bath almost never scrapes together its first quantum. That suppression *is* the resolution of the catastrophe — and why a warm room is dark. --- ## Hands-on (Python) ```python import numpy as np import matplotlib.pyplot as plt from scipy.integrate import quad from scipy.optimize import brentq h, c, kB = 6.626e-34, 2.998e8, 1.381e-23 # J s, m/s, J/K def planck(nu, T): """Spectral energy density u(nu,T) in J m^-3 Hz^-1.""" return (8 * np.pi * h * nu**3 / c**3) / np.expm1(h * nu / (kB * T)) def rayleigh_jeans(nu, T): return 8 * np.pi * nu**2 * kB * T / c**3 def wien_approx(nu, T): return (8 * np.pi * h * nu**3 / c**3) * np.exp(-h * nu / (kB * T)) # --- 1. Planck vs Rayleigh-Jeans vs Wien at T = 5800 K ------------------- T = 5800.0 nu = np.linspace(1e12, 2.5e15, 2000) plt.plot(nu, planck(nu, T), label="Planck") plt.plot(nu, rayleigh_jeans(nu, T), "--", label="Rayleigh–Jeans (diverges)") plt.plot(nu, wien_approx(nu, T), ":", label="Wien approximation") plt.ylim(0, 1.8 * planck(nu, T).max()); plt.xlabel(r"$\nu$ (Hz)") plt.ylabel(r"$u(\nu,T)$"); plt.legend(); plt.title("T = 5800 K"); plt.show() # Expected: RJ hugs Planck at low nu then blows up; Wien matches the high-nu tail. # --- 2. Verify the T^4 law and extract sigma ------------------------------ for T in [300.0, 1000.0, 3000.0, 5800.0]: nu_max = 100 * kB * T / h # integrand is dead beyond x = 100 u_tot, _ = quad(planck, 1.0, nu_max, args=(T,)) print(f"T = {T:6.0f} K sigma = {(c/4) * u_tot / T**4:.4e}") # Expected: sigma = 5.676e-08 W m^-2 K^-4 for EVERY T -> R_T = sigma T^4. # (CODATA: 5.670e-08; the 0.1% offset is rounding in our 4-digit h, kB.) # --- 3. Wien displacement via brentq, applied to the CMB ------------------ x_nu = brentq(lambda x: x - 3 * (1 - np.exp(-x)), 0.1, 10) x_lam = brentq(lambda x: x - 5 * (1 - np.exp(-x)), 0.1, 10) print(x_nu, x_lam) # 2.8214... 4.9651... print(f"Wien constant = {h*c/(x_lam*kB):.4e} m K") # 2.898e-03 (verified) T_cmb = 2.725 # cosmic microwave background print(f"CMB peak = {x_nu * kB * T_cmb / h / 1e9:.1f} GHz") # ~160.2 GHz ``` The CMB — the 2.725 K afterglow of the Big Bang — is the most perfect blackbody ever measured; COBE found deviations from Planck's law below 50 parts per million. --- ## Exercises **E1 (easy).** Your body: $T = 310$ K, area $A = 1.7\ \mathrm{m^2}$, $\epsilon \approx 1$ in the infrared. Find $\lambda_{max}$ and the total power you radiate.

Solution

$\lambda_{max} = 2.898\times10^{-3}/310 = 9.35\ \mu$m (far infrared — thermal cameras look here). $P = \sigma T^4A = 5.670\times10^{-8}\times(310)^4\times1.7 \approx 890$ W. You also *absorb* $\sigma T_{env}^4A \approx 690$ W from a 293 K room; the net $\sim$150–200 W is your metabolic scale.

**E2 (easy).** Expand Planck's $\langle E\rangle$ for $h\nu \ll k_BT$ one order beyond equipartition and show $\langle E\rangle \approx k_BT - \tfrac12h\nu$.

Solution

With $x \ll 1$: $e^x-1 = x(1 + x/2 + O(x^2))$, so $\langle E\rangle = \frac{h\nu}{x}(1+x/2)^{-1} \approx k_BT(1 - x/2) = k_BT - \tfrac12h\nu$. Quantization always *lowers* the average below equipartition, more at higher frequency.

**E3 (medium).** Reproduce the geometric-sum derivation of $\langle E\rangle = h\nu/(e^{h\nu/k_BT}-1)$ without looking, evaluating both sums explicitly.

Solution

Let $x = h\nu/k_BT$, $q = e^{-x} < 1$. Partition sum $Z = \sum_{n\ge0}q^n = 1/(1-q)$; energy sum $\sum_n nh\nu\,q^n = h\nu\,q\,\frac{d}{dq}\frac{1}{1-q} = h\nu\,\frac{q}{(1-q)^2}$. Ratio: $\langle E\rangle = h\nu\frac{q}{1-q} = \frac{h\nu}{e^x-1}$. Limits: $k_BT$ as $x\to0$; $h\nu e^{-x}$ as $x\to\infty$.

**E4 (medium).** Compute the CMB's total energy density $u_{tot} = \frac{4\sigma}{c}T^4$ at $T = 2.725$ K, in $\mathrm{J\,m^{-3}}$ and $\mathrm{eV\,cm^{-3}}$.

Solution

$a \equiv 4\sigma/c = 7.566\times10^{-16}\ \mathrm{J\,m^{-3}K^{-4}}$ and $T^4 = 55.1$, so $u_{tot} = 4.17\times10^{-14}\ \mathrm{J\,m^{-3}} = 2.60\times10^{5}\ \mathrm{eV\,m^{-3}} = 0.26\ \mathrm{eV\,cm^{-3}}$. Every cubic centimeter of the universe holds about a quarter electron-volt of primordial light.

**E5 (hard).** Derive the $\lambda$-form of Planck's law from the $\nu$-form, maximize it to get $x = 5(1-e^{-x})$, and compute Wien's constant. Then explain in one sentence why $\lambda_{max}\nu_{max} \ne c$.

Solution

Equal energy in corresponding bins: $u(\lambda) = u(c/\lambda)\,|d\nu/d\lambda| = u(c/\lambda)\,c/\lambda^2 = \frac{8\pi hc}{\lambda^5}\frac{1}{e^{hc/\lambda k_BT}-1}$. With $x = hc/\lambda k_BT$, maximizing $x^5/(e^x-1)$ gives $5x^4(e^x-1) = x^5e^x$, i.e. $x = 5(1-e^{-x})$, root $x \approx 4.965$; then $\lambda_{max}T = \frac{hc}{4.965\,k_B} = \frac{(6.626\times10^{-34})(2.998\times10^{8})}{4.965\times1.381\times10^{-23}} = 2.898\times10^{-3}\ \mathrm{m\,K}$. ✓ The two peaks differ because maximizing a *density* depends on the variable it is a density in — the Jacobian $c/\lambda^2$ reweights the spectrum, so $x_\lambda = 4.965 \ne x_\nu = 2.821$.

--- ## Checkpoint 1. Why does a small hole in a cavity behave as a perfect blackbody? 2. Derive the mode density $8\pi\nu^2/c^3$, identifying where the octant factor $\tfrac18$ and the polarization factor 2 enter. 3. Which classical ingredient causes the ultraviolet catastrophe — mode counting or equipartition — and how does Planck's hypothesis repair it? 4. Show in two lines how Planck's law reduces to Rayleigh–Jeans for $h\nu\ll k_BT$ and to Wien's exponential for $h\nu\gg k_BT$. 5. What did Planck quantize — and what did he *not* quantize?

Answers

1. Any ray entering the hole reflects many times and is absorbed before escaping, so the hole absorbs essentially everything ($\epsilon = 1$); in equilibrium it must therefore also emit the ideal blackbody spectrum of the cavity interior. 2. Standing waves require $k_i = n_i\pi/a$ with $n_i > 0$; counting lattice points inside the sphere $|\mathbf n| = 2a\nu/c$ uses the positive octant only (factor $\tfrac18$), and each spatial mode has two transverse polarizations (factor 2): $N = 8\pi V\nu^3/3c^3$, whose $\nu$-derivative per volume is $8\pi\nu^2/c^3$. 3. Equipartition. The mode counting survives into QM; the error is giving every mode $k_BT$. With $E_n = nh\nu$, $\langle E\rangle = h\nu/(e^{h\nu/k_BT}-1) \to 0$ exponentially at high $\nu$, so the total energy integral converges. 4. $h\nu\ll k_BT$: $e^{h\nu/k_BT}-1 \approx h\nu/k_BT$, so $u \to (8\pi\nu^2/c^3)k_BT$. $h\nu\gg k_BT$: drop the $-1$, so $u \to (8\pi h\nu^3/c^3)e^{-h\nu/k_BT}$. 5. He quantized the allowed energies of cavity modes / material oscillators ($E_n = nh\nu$) — energy *exchange* with radiation. He did not quantize free light; the photon is Einstein's step, in [P.3.1](../c03-wave-particle-duality/01-light-as-particles.md).

--- ## Further Reading - **[ER]** Eisberg & Resnick, §1.1–1.6 — thermal radiation, cavity mode counting, and Planck's postulate; the definitive elementary treatment. - **[ER]** Eisberg & Resnick, Appendix to Ch. 1 — the equipartition theorem and its breakdown. - **[Gri]** Griffiths & Schroeter, §5.4 — the photon gas and the blackbody spectrum, revisited with proper quantum statistics. --- ← Prev: [Hamiltonian Mechanics](../c01-theoretical-mechanics/04-hamiltonian-mechanics.md) · Up: [Pre-Term](../README.md) · Next: [Atomic Models & Spectral Series](02-atomic-models-and-spectra.md) →