AcademyMatter Waves
Academy
Continuous Spectra
Level 1 - Physics topic page in Matter Waves.
Principle
Continuous spectra arise when emitted photon energies can vary over a range rather than fixed level gaps.
Notation
\(E_gamma\)
photon energy
\(\mathrm{J}\)
\(f\)
frequency
\(\mathrm{Hz}\)
\(\lambda\)
wavelength
\(\mathrm{m}\)
\(T\)
absolute temperature
\(\mathrm{K}\)
\(\lambda_{\max}\)
peak thermal wavelength
\(\mathrm{m}\)
\(b\)
Wien displacement constant
\(\mathrm{m\,K}\)
Method
Derivation 1: Contrast line and continuous spectra
Line spectra come from fixed atomic energy gaps. Continuous spectra come from many available energies or thermal motion.
Photon energy
\[E_\gamma=hf=\frac{hc}{\lambda}\]
Line spectrum
\[E_\gamma=E_i-E_f\]
Derivation 2: Use thermal radiation scale
A hot dense object emits a broad spectrum whose peak shifts with temperature.
Wien law
\[\lambda_{\max}T=b\]
Hotter source
\[T\uparrow\Rightarrow \lambda_{\max}\downarrow\]
Derivation 3: Identify bremsstrahlung continua
Accelerated charged particles can emit photons with a continuous range of energies, up to the available kinetic energy.
Rules
Photon energy
\[E_\gamma=hf=\frac{hc}{\lambda}\]
Wien law
\[\lambda_{\max}T=b\]
Endpoint limit
\[E_{\gamma,\max}=K_{\mathrm{available}}\]
Examples
Question
A thermal spectrum peaks at
\[500\,\mathrm{nm}\]
Estimate the temperature using \[b=2.90\times10^{-3}\,\mathrm{m\,K}\]
Answer
\[T=\frac{b}{\lambda_{\max}}=\frac{2.90\times10^{-3}}{500\times10^{-9}}=5.80\times10^3\,\mathrm K\]
Checks
- A continuous spectrum is not evidence for one fixed transition.
- Hotter thermal sources peak at shorter wavelengths.
- Line spectra can sit on top of a continuum.
- A high-energy cutoff reveals the maximum available particle energy.