AcademyAtomic Quantum Structure
Academy
X-Ray Spectra
Level 1 - Physics topic page in Atomic Quantum Structure.
Principle
Atomic x-ray spectra come from transitions involving inner electron shells. If an inner-shell vacancy is created, an outer electron can drop to fill it and emit a photon with energy equal to the level difference.
Because inner-shell energies depend strongly on nuclear charge, characteristic x-ray lines identify elements.
Notation
\(E_i,E_f\)
initial and final atomic energy levels
\(\mathrm{eV}\)
\(E_\gamma\)
emitted photon energy
\(\mathrm{eV}\)
\(\lambda\)
photon wavelength
m, nm
\(Z\)
atomic number
1
\(K_\alpha\)
transition from L shell to K shell
1
\(K_\beta\)
transition from M shell to K shell
1
Method
Derivation 1: Use energy differences
The photon energy equals the loss of atomic energy in the transition.
Photon energy
\[E_\gamma=E_i-E_f\]
Wavelength
\[\lambda=\frac{hc}{E_\gamma}\]
Derivation 2: Name common lines
The shell that receives the electron names the series. The shell the electron comes from identifies the line.
K alpha
\[L\to K\]
K beta
\[M\to K\]
Derivation 3: Estimate nuclear-charge scaling
For inner shells, shielding can be approximated with an effective nuclear charge.
Moseley scaling
\[f\propto (Z-b)^2\]
Energy scaling
\[E_\gamma=hf\]
Rules
Characteristic photon
\[E_\gamma=E_i-E_f\]
Photon wavelength
\[\lambda=\frac{hc}{E_\gamma}\]
Moseley scaling
\[f\propto(Z-b)^2\]
K alpha transition
\[K_\alpha:L\to K\]
Examples
Question
A characteristic x-ray has energy
\[8.0\,\mathrm{keV}\]
Find its wavelength using \[hc=1240\,\mathrm{eV\,nm}\]
Answer
\[\lambda=\frac{1240}{8000}=0.155\,\mathrm{nm}\]
Checks
- Characteristic x-rays depend on target element.
- The continuous x-ray spectrum comes from electron deceleration, not shell transitions.
- K-series lines end in the K shell.
- Higher photon energy means shorter wavelength.