Quang phổ nguyên tử và quy tắc chọn lọc
Đã giới thiệu| English | Ý nghĩa |
|---|---|
| selection rules/sɪˈlekʃn ruːlz/ | Conditions on quantum-number changes that a given transition mechanism allows. |
| metastable/ˌmetəˈsteɪbl/ | Long-lived because its decay route is forbidden at leading order. |
| fine structure/faɪn ˈstrʌktʃə/ | Small splittings of atomic levels from spin-orbit and relativistic effects. |
Same energy, no line
- In hydrogen, $2p \to 1s$ and $2s \to 1s$ release the same 10.2 eV. Only the first gives a strong spectral line.
- GRE atomic physics tests hydrogen-like energies, quantum numbers, selection rules 选择定则 and the limits of the simple model.
Hydrogen-like energies
- $E_n = -\dfrac{13.6\,Z^2}{n^2}$ eV. Emitted photon: $E_\gamma = E_i - E_f > 0$, and $\lambda = \dfrac{hc}{E_\gamma}$ with $hc \approx 1240$ eV·nm.
- $2 \to 1$ in H: $10.2$ eV, $\lambda \approx 122$ nm. In He⁺ ($Z = 2$) every level is 4 times deeper: $2 \to 1$ gives 40.8 eV.
Photon energy (eV) for hydrogen $n = 3 \to 2$, to two decimal places?
Độ khó 2/5 · 140 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
$13.6(\tfrac14 - \tfrac19)$.
Ground-state energy of He⁺ in eV?
Độ khó 2/5 · 140 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
Hydrogen-like energies scale as $Z^2$: $-13.6 \times 2^2 = -54.4$ eV for He⁺.
Quantum numbers
- $l = 0, \dots, n - 1$; $m_l = -l, \dots, l$; spin $m_s = \pm\tfrac12$. Level $n$ holds $2n^2$ states.
- Total angular momentum $\mathbf J = \mathbf L + \mathbf S$, with $j = l \pm \tfrac12$ for one electron ($l > 0$).
How many electron states (including spin) does the $n = 3$ level of hydrogen hold?
Độ khó 2/5 · 140 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
Level $n$ has $n^2$ orbital states, each with two spin states: $2n^2 = 18$ for $n = 3$.
Electric-dipole selection rules
- $\Delta l = \pm1$, $\Delta m_l = 0, \pm1$, spin unchanged. The photon carries one unit of angular momentum.
$2p \to 1s$: $\Delta l = -1$, allowed, fast. $2s \to 1s$: $\Delta l = 0$, electric-dipole forbidden, so $2s$ is metastable 亚稳态的 and decays slowly by other routes (two-photon emission). "Forbidden" means forbidden for this mechanism, not impossible.
Allowed or forbidden?
Apply the Δl = ±1 rule to hydrogen transitions.
Which transition is electric-dipole forbidden?
Độ khó 1/5 · 100 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
Electric-dipole transitions need $\Delta l = \pm1$. A $3d \to 1s$ jump changes $l$ from 2 to 0, so it is forbidden.
Beyond the simple model
- Spin–orbit coupling gives fine structure 精细结构; a magnetic field splits $m$ levels (Zeeman effect).
- In many-electron atoms shielding breaks the $l$-degeneracy: you cannot plug $Z$ into $-13.6Z^2/n^2$ for an outer electron of sodium.
Inner electrons shield most of the nuclear charge from sodium's outer electron.
Độ khó 1/5 · 100 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
True. The 10 inner electrons screen the nucleus, so the 3s electron feels an effective charge near 1–2, not 11, and its energy is about −5 eV.
Recap
- $E_n = -13.6Z^2/n^2$ eV; photon energy is the positive level difference; $\lambda = 1240/E$ nm.
- $2n^2$ states per level.
- E1 lines need $\Delta l = \pm1$.
- selection rules: Conditions on quantum-number changes that a given transition mechanism allows.
- metastable: Long-lived because its decay route is forbidden at leading order.
- fine structure: Small splittings of atomic levels from spin-orbit and relativistic effects.