X-ray notation
X-ray notation is a standardized system in atomic physics and spectroscopy for designating the principal energy levels (shells) and subshells of atoms, as well as the characteristic spectral lines produced by electron transitions between these levels during X-ray emission or absorption processes.[1] It originated from early 20th-century X-ray studies and is essential for identifying atomic structure and elemental composition in techniques like X-ray fluorescence (XRF) and photoelectron spectroscopy.[2] The notation primarily focuses on inner-shell electrons, using labels such as K for the 1s orbital, L for the 2s and 2p orbitals (subdivided into LI, LII, and LIII), and M for the 3s, 3p, and 3d orbitals (with five subshells), reflecting the quantum mechanical organization of atomic orbitals.[1] Two primary notation systems are employed: the historical Siegbahn notation and the modern IUPAC notation. The Siegbahn system, introduced by Manne Siegbahn in the early 1900s, labels spectral lines by the initial shell vacancy (e.g., K or L) followed by Greek letters indicating the filling shell and relative intensity, such as Kα for the strongest transition from the L shell to the K shell vacancy, and Kβ for transitions from the M shell.[2] This empirical approach, while widely used, lacks explicit reference to subshell quantum states and can be ambiguous for complex spectra. In contrast, the IUPAC notation, recommended since 1991, explicitly denotes transitions by connecting the initial (upper) and final (lower) energy levels with a hyphen, using Arabic numerals for subshells (e.g., K-LIII for the 1s vacancy filled by a 2p3/2 electron, equivalent to Kα1 in Siegbahn notation).[1] This system accommodates satellite lines from multiple ionizations (e.g., labeled as "sat n") and unresolved doublets (e.g., K-L2,3), promoting consistency across emission, absorption, and Auger electron spectroscopies.[1] The adoption of X-ray notation has facilitated precise calibration of spectrometers and analysis of atomic data, with key wavelength standards like the Cu K-LIII line at 1.5405974 Å serving as benchmarks for measurements.[1] Its application extends to fields such as materials science, where it aids in non-destructive elemental analysis, and astrophysics, for interpreting high-resolution spectra from cosmic sources.[2]Principles of notation
Shell and subshell designations
In X-ray notation, the principal electron shells are designated by letters that reflect their increasing energy levels and distance from the nucleus, starting with the innermost shell. The K shell corresponds to the principal quantum number n=1, the L shell to n=2, the M shell to n=3, the N shell to n=4, the O shell to n=5, and the P shell to n=6. These labels originated from early X-ray spectroscopy observations and are used to describe the binding energies of core electrons relevant to X-ray transitions.[3][4] Subshells within these principal shells are further subdivided using numerical indices to distinguish orbitals based on their angular momentum and spin-orbit coupling, particularly important for inner shells where fine structure effects are pronounced. The K shell (n=1) consists of a single subshell with no further division. The L shell (n=2) is divided into three subshells: L1 corresponding to the 2s orbital, L2 to the 2p_{1/2} orbital, and L3 to the 2p_{3/2} orbital. The M shell (n=3) has five subshells: M1 (3s), M2 (3p_{1/2}), M3 (3p_{3/2}), M4 (3d_{3/2}), and M5 (3d_{5/2}). Higher shells like N, O, and P follow similar patterns with increasing numbers of subshells (seven for the N shell), but detailed subshell designations beyond M are less commonly specified in X-ray contexts.[3][5][4] This notation is primarily applied to inner shells up to n=4 (N shell) or occasionally n=5 (O shell), as these have binding energies high enough (typically in the keV range) to produce observable X-ray emissions, whereas outer shells with lower binding energies are more relevant to optical or ultraviolet spectroscopy. For example, K-shell electrons in heavy elements like tungsten exhibit binding energies around 69 keV, enabling characteristic X-ray lines in the hard X-ray regime, while lighter elements like carbon have K-shell binding energies of only about 0.28 keV.[4][3][6] An alternative labeling system, known as spectroscopic notation (e.g., 1s for K, 2s and 2p for L subshells), provides a more detailed description using quantum numbers but is often used alongside X-ray notation for clarity in atomic physics.[5]Transition line notations
In X-ray notation, spectral lines arising from electron transitions between atomic shells are designated using a combination of series identifiers and Greek letters, with the series named after the lower-energy shell that receives the transitioning electron. The K-series encompasses transitions to the K shell (n=1), the L-series to the L shell (n=2, comprising 2s and 2p subshells), and the M-series to the M shell (n=3, including 3s, 3p, and 3d subshells), following the principal shell designations.[7][8] Within each series, Greek letters such as α, β, and γ denote specific transitions ordered by increasing photon energy (decreasing wavelength), with α representing the lowest-energy transition from the adjacent higher shell, β from the next higher shell, and so on. For instance, in the K-series, the Kα lines result from electrons transitioning from the L shell to the K shell, while Kβ lines involve transitions from the M shell to the K shell, producing higher-energy photons due to the greater energy difference. Subscripts further distinguish fine structure from subshell splittings, typically with 1 indicating the transition involving the j=3/2 subshell and 2 the j=1/2 subshell.[7][9] The Kα line, often the most intense in the K-series, exemplifies this notation as the aggregate of Kα₁ (from L₃ or 2p_{3/2} to K or 1s) and Kα₂ (from L₂ or 2p_{1/2} to 1s), where Kα₁ carries slightly higher energy than Kα₂ owing to the subshell binding energy differences. In the L-series, Lα denotes transitions from the M shell to the L₃ subshell, such as Lα₁ (M₅ or 3d_{5/2} to L₃) and Lα₂ (M₄ or 3d_{3/2} to L₃), while Lβ involves higher-energy transitions from N-shell subshells to L subshells. The M-series follows analogously, with Mα for N to M transitions, though these lines are generally weaker and occur at lower energies for heavier elements.[7][8]| Series | Greek Letter | Typical Transition | Example (Mo, energies in eV) |
|---|---|---|---|
| K | α₁ | L₃ → K | 17,479 |
| K | α₂ | L₂ → K | 17,374 |
| K | β₁ | M₃ → K | 19,608 |
| L | α₁ | M₅ → L₃ | 2,293 |
| L | α₂ | M₄ → L₃ | 2,290 |
| L | β₁ | N → L₃ | 2,395 |
Relation to quantum mechanics
Correspondence to quantum numbers
The X-ray notation for atomic shells directly corresponds to the principal quantum number n and azimuthal quantum number l, with subshell splittings arising from the total angular momentum quantum number j due to spin-orbit interactions. The K shell is designated as the n=1 level with l=0, corresponding to the 1s orbital. For the L shell (n=2), the notation distinguishes subshells based on l and j: L1 represents the 2s orbital (l=0, j=1/2); L2 the 2p_{1/2} orbital (l=1, j=1/2); and L3 the 2p_{3/2} orbital (l=1, j=3/2). This splitting of the p subshell into j=1/2 and j=3/2 components is a direct consequence of spin-orbit coupling, where the interaction between the electron's spin and orbital angular momentum lifts the degeneracy of states with the same l but different j.[10] The M shell (n=3) extends this mapping further: M1 is the 3s orbital (l=0, j=1/2); M2 and M3 are the 3p_{1/2} (l=1, j=1/2) and 3p_{3/2} (l=1, j=3/2) orbitals, respectively, again split by spin-orbit coupling; while M4 and M5 correspond to the 3d_{3/2} (l=2, j=3/2) and 3d_{5/2} (l=2, j=5/2) orbitals. Spin-orbit effects become more pronounced in these inner orbitals of heavier elements, where relativistic corrections significantly influence energy levels and transition probabilities.[10][11] This notation primarily addresses inner shells up to n=4 (N shell), as higher principal quantum numbers (n>4) are less relevant for X-ray processes due to their lower binding energies and weaker relativistic influences. The focus on inner orbitals highlights the role of relativistic effects, including spin-orbit coupling, in determining the fine structure observed in X-ray spectra.[10]Conversion from spectroscopic notation
The conversion from spectroscopic notation, which uses the principal quantum number n, azimuthal quantum number l (with letters s for l=0, p for l=1, etc.), and total angular momentum j for fine structure, to X-ray notation involves mapping the core electron shells and subshells based on their quantum mechanical designations. This process is essential for interpreting X-ray spectra, where the notation simplifies labeling of inner-shell transitions while retaining the underlying quantum structure.[12] To perform the conversion step by step, first identify the principal quantum number n from the shell designation in X-ray notation, where K corresponds to n=1, L to n=2, M to n=3, N to n=4, O to n=5, P to n=6, and Q to n=7. Next, determine l from the subshell type: s indicates l=0, p indicates l=1, d indicates l=2, and f indicates l=3. Finally, account for j-splitting due to spin-orbit coupling in subshells with l \geq 1, where X-ray subshell labels (numbered I, II, III, etc., often written as 1, 2, 3) are assigned in order of increasing j within each shell, with ties resolved by increasing l. For example, the 2p subshell splits into $2p_{1/2} (j=1/2) and $2p_{3/2} (j=3/2), mapped to L_{II} and L_{III}, respectively. The following table illustrates key mappings for the K and L shells:| Spectroscopic notation | X-ray notation | n | l | j |
|---|---|---|---|---|
| 1s | K | 1 | 0 | 1/2 |
| 2s | L_I | 2 | 0 | 1/2 |
| 2p_{1/2} | L_{II} | 2 | 1 | 1/2 |
| 2p_{3/2} | L_{III} | 2 | 1 | 3/2 |