The Atom

Benjamin Crowell · Simple Nature Ch. 12 · CC BY-SA 3.0

Quantum mechanics explains the atom. The Bohr model gets the energy levels right. Full quantum mechanics adds angular momentum quantum numbers, spin, and the exclusion principle. Together they build the periodic table from first principles.

The Bohr model

Bohr postulated that electrons orbit the nucleus in quantized orbits where the angular momentum is n * h_bar. This gives energy levels E_n = -13.6 eV / n^2 for hydrogen. The model is wrong in detail (electrons are not point particles on orbits) but gets the energy levels right, which is why spectral lines match.

Scheme

; Hydrogen energy levels: E_n = -13.6 eV / n^2
; Photon energy for transition: E = E_upper - E_lower
; Wavelength: lambda = h*c / E

(define h 6.626e-34)
(define c 3e8)
(define eV 1.6e-19)

(define (hydrogen-energy n)
  (/ -13.6 (* n n)))

(define (transition-energy n-upper n-lower)
  (- (hydrogen-energy n-upper) (hydrogen-energy n-lower)))

(define (transition-wavelength n-upper n-lower)
  (let ((E-eV (transition-energy n-upper n-lower)))
    (/ (* h c) (* (abs E-eV) eV))))

(display "Hydrogen energy levels:") (newline)
(define (show-E n)
  (display "  n=") (display n)
  (display ": E = ") (display (hydrogen-energy n))
  (display " eV") (newline))
(for-each show-E (list 1 2 3 4 5))

(display "Transitions (emission):") (newline)
; Lyman series (to n=1)
(display "  Lyman-alpha (2->1): ")
(display (* (transition-wavelength 2 1) 1e9))
(display " nm") (newline)

; Balmer series (to n=2) -- visible light!
(display "  H-alpha (3->2): ")
(display (* (transition-wavelength 3 2) 1e9))
(display " nm (red)") (newline)
(display "  H-beta  (4->2): ")
(display (* (transition-wavelength 4 2) 1e9))
(display " nm (blue-green)")

; Hydrogen energy levels: E_n = -13.6 eV / n^2
; Photon energy for transition: E = E_upper - E_lower
; Wavelength: lambda = h*c / E

(define h 6.626e-34)
(define c 3e8)
(define eV 1.6e-19)

(define (hydrogen-energy n)
  (/ -13.6 (* n n)))

(define (transition-energy n-upper n-lower)
  (- (hydrogen-energy n-upper) (hydrogen-energy n-lower)))

(define (transition-wavelength n-upper n-lower)
  (let ((E-eV (transition-energy n-upper n-lower)))
    (/ (* h c) (* (abs E-eV) eV))))

(display "Hydrogen energy levels:") (newline)
(define (show-E n)
  (display "  n=") (display n)
  (display ": E = ") (display (hydrogen-energy n))
  (display " eV") (newline))
(for-each show-E (list 1 2 3 4 5))

(display "Transitions (emission):") (newline)
; Lyman series (to n=1)
(display "  Lyman-alpha (2->1): ")
(display (* (transition-wavelength 2 1) 1e9))
(display " nm") (newline)

; Balmer series (to n=2) -- visible light!
(display "  H-alpha (3->2): ")
(display (* (transition-wavelength 3 2) 1e9))
(display " nm (red)") (newline)
(display "  H-beta  (4->2): ")
(display (* (transition-wavelength 4 2) 1e9))
(display " nm (blue-green)")

Quantum numbers

The full quantum description of an electron in an atom requires four quantum numbers. n (principal): energy level, 1, 2, 3, ... l (angular momentum): 0 to n-1. m_l (magnetic): -l to +l. m_s (spin): +1/2 or -1/2. The Pauli exclusion principle says no two electrons can share all four numbers.

Scheme

; Count states for each shell (principal quantum number n)
; l ranges from 0 to n-1
; m_l ranges from -l to +l (2l+1 values)
; m_s = +1/2 or -1/2 (2 values)
; Total states in shell n = 2 * n^2

(define (states-in-subshell l)
  (* 2 (+ (* 2 l) 1)))

(define (states-in-shell n)
  (let loop ((l 0) (total 0))
    (if (>= l n)
        total
        (loop (+ l 1) (+ total (states-in-subshell l))))))

(define subshell-names (list "s" "p" "d" "f"))

(define (show-shell n)
  (display "n=") (display n) (display ": ")
  (let loop ((l 0))
    (if (< l n)
        (begin
          (display (list-ref subshell-names l))
          (display "(") (display (states-in-subshell l)) (display ") ")
          (loop (+ l 1)))
        (begin
          (display "= ") (display (states-in-shell n))
          (display " total") (newline)))))

(for-each show-shell (list 1 2 3 4))

; This explains the periodic table:
; Period 1: 2 elements (1s)
; Period 2: 8 elements (2s + 2p)
; Period 3: 8 elements (3s + 3p)
; Period 4: 18 elements (4s + 3d + 4p)
(display "Formula: 2n^2 states per shell")

; Count states for each shell (principal quantum number n)
; l ranges from 0 to n-1
; m_l ranges from -l to +l (2l+1 values)
; m_s = +1/2 or -1/2 (2 values)
; Total states in shell n = 2 * n^2

(define (states-in-subshell l)
  (* 2 (+ (* 2 l) 1)))

(define (states-in-shell n)
  (let loop ((l 0) (total 0))
    (if (>= l n)
        total
        (loop (+ l 1) (+ total (states-in-subshell l))))))

(define subshell-names (list "s" "p" "d" "f"))

(define (show-shell n)
  (display "n=") (display n) (display ": ")
  (let loop ((l 0))
    (if (< l n)
        (begin
          (display (list-ref subshell-names l))
          (display "(") (display (states-in-subshell l)) (display ") ")
          (loop (+ l 1)))
        (begin
          (display "= ") (display (states-in-shell n))
          (display " total") (newline)))))

(for-each show-shell (list 1 2 3 4))

; This explains the periodic table:
; Period 1: 2 elements (1s)
; Period 2: 8 elements (2s + 2p)
; Period 3: 8 elements (3s + 3p)
; Period 4: 18 elements (4s + 3d + 4p)
(display "Formula: 2n^2 states per shell")

The periodic table

Electrons fill orbitals from lowest energy up (Aufbau principle). Each orbital holds at most 2 electrons (Pauli exclusion). Electrons in the same subshell spread out before pairing (Hund's rule). These three rules, plus the quantum numbers, generate the entire periodic table. Chemistry is applied quantum mechanics.

Scheme

; Electron configurations
; Fill order: 1s, 2s, 2p, 3s, 3p, 4s, 3d, 4p, ...
; Each subshell holds 2*(2l+1) electrons

(define fill-order
  (list (list 1 0) (list 2 0) (list 2 1) (list 3 0) (list 3 1)
        (list 4 0) (list 3 2) (list 4 1) (list 5 0) (list 4 2)))

(define subshell-names (list "s" "p" "d" "f"))

(define (max-electrons l) (* 2 (+ (* 2 l) 1)))

(define (electron-config Z)
  (let loop ((remaining Z) (orbitals fill-order) (config ""))
    (if (or (<= remaining 0) (null? orbitals))
        config
        (let* ((nl (car orbitals))
               (n (car nl))
               (l (cadr nl))
               (maxe (max-electrons l))
               (e (min remaining maxe))
               (name (list-ref subshell-names l))
               (label (string-append (number->string n) name (number->string e) " ")))
          (loop (- remaining e) (cdr orbitals) (string-append config label))))))

; Show configurations for familiar elements
(define elements (list (list 1 "H") (list 2 "He") (list 6 "C")
                       (list 8 "O") (list 11 "Na") (list 26 "Fe")))

(for-each (lambda (pair)
  (display (cadr pair)) (display " (Z=") (display (car pair))
  (display "): ") (display (electron-config (car pair))) (newline))
  elements)

; Electron configurations
; Fill order: 1s, 2s, 2p, 3s, 3p, 4s, 3d, 4p, ...
; Each subshell holds 2*(2l+1) electrons

(define fill-order
  (list (list 1 0) (list 2 0) (list 2 1) (list 3 0) (list 3 1)
        (list 4 0) (list 3 2) (list 4 1) (list 5 0) (list 4 2)))

(define subshell-names (list "s" "p" "d" "f"))

(define (max-electrons l) (* 2 (+ (* 2 l) 1)))

(define (electron-config Z)
  (let loop ((remaining Z) (orbitals fill-order) (config ""))
    (if (or (<= remaining 0) (null? orbitals))
        config
        (let* ((nl (car orbitals))
               (n (car nl))
               (l (cadr nl))
               (maxe (max-electrons l))
               (e (min remaining maxe))
               (name (list-ref subshell-names l))
               (label (string-append (number->string n) name (number->string e) " ")))
          (loop (- remaining e) (cdr orbitals) (string-append config label))))))

; Show configurations for familiar elements
(define elements (list (list 1 "H") (list 2 "He") (list 6 "C")
                       (list 8 "O") (list 11 "Na") (list 26 "Fe")))

(for-each (lambda (pair)
  (display (cadr pair)) (display " (Z=") (display (car pair))
  (display "): ") (display (electron-config (car pair))) (newline))
  elements)

Nuclear physics and radioactive decay

The nucleus holds protons and neutrons together with the strong force. Unstable nuclei decay: alpha decay emits a helium-4 nucleus, beta decay converts a neutron to a proton (or vice versa), and gamma decay emits a photon. Radioactive decay is exponential: N(t) = N_0 * e^(-t/tau), where tau is the mean lifetime.

Scheme

; Radioactive decay: N(t) = N0 * exp(-t / tau)
; Half-life: t_1/2 = tau * ln(2)
; tau = mean lifetime

(define (decay N0 tau t)
  (* N0 (exp (/ (- t) tau))))

(define (half-life tau)
  (* tau (log 2)))

; Carbon-14: half-life = 5730 years
(define t-half-C14 5730)
(define tau-C14 (/ t-half-C14 (log 2)))

(display "Carbon-14:") (newline)
(display "  Half-life: ") (display t-half-C14) (display " years") (newline)
(display "  Mean lifetime: ") (display (inexact->exact (round tau-C14)))
(display " years") (newline)

; How much remains after various times?
(define N0 1000000)
(define (show-remaining years)
  (let ((N (decay N0 tau-C14 years)))
    (display "  After ") (display years) (display " years: ")
    (display (inexact->exact (round N)))
    (display " remain (")
    (display (inexact->exact (round (* 100 (/ N N0)))))
    (display "%)") (newline)))

(for-each show-remaining (list 1000 5730 11460 20000 50000))
; This is how carbon dating works:
; measure the ratio of C-14 to C-12, solve for t.

; Radioactive decay: N(t) = N0 * exp(-t / tau)
; Half-life: t_1/2 = tau * ln(2)
; tau = mean lifetime

(define (decay N0 tau t)
  (* N0 (exp (/ (- t) tau))))

(define (half-life tau)
  (* tau (log 2)))

; Carbon-14: half-life = 5730 years
(define t-half-C14 5730)
(define tau-C14 (/ t-half-C14 (log 2)))

(display "Carbon-14:") (newline)
(display "  Half-life: ") (display t-half-C14) (display " years") (newline)
(display "  Mean lifetime: ") (display (inexact->exact (round tau-C14)))
(display " years") (newline)

; How much remains after various times?
(define N0 1000000)
(define (show-remaining years)
  (let ((N (decay N0 tau-C14 years)))
    (display "  After ") (display years) (display " years: ")
    (display (inexact->exact (round N)))
    (display " remain (")
    (display (inexact->exact (round (* 100 (/ N N0)))))
    (display "%)") (newline)))

(for-each show-remaining (list 1000 5730 11460 20000 50000))
; This is how carbon dating works:
; measure the ratio of C-14 to C-12, solve for t.

Neighbors

Foundations (Wikipedia)

Translation notes

The Bohr model is historically important but physically wrong: electrons do not orbit like planets. The Schrodinger equation gives probability clouds (orbitals), not orbits. However, the energy levels E_n = -13.6/n^2 eV are exact for hydrogen. Our electron configuration code follows the Aufbau filling order, which has exceptions for chromium (Z=24) and copper (Z=29) due to half-filled subshell stability. The radioactive decay law is exact for large numbers of atoms; individual decays are random.

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