Experimental Physics EP3
Atoms and Molecules
– Atomic model –
Thomson’s model, Rutherford scattering,
spectral lines
http://uni-leipzig.de/~valiu/
Experimental Physics III - Atomic structure
1
Thomson’s plum padding model of atom
2R
b
b
The plum pudding model of the atom by J. J.
Thomson, who discovered the electron in
1897, was proposed in 1904 before the
discovery of the atomic nucleus in order to
add the electron to the atomic model.
E=
Q
4πε 0 r 2
Zer 3 1
= 3
R 4πε 0 r 2
b= R
2πb
zZe2
θ = ∫ θ (b) 2 db =
2
π
R
16
ε
Rm
υ
0
0
b =0
Experimental Physics III - Atomic structure
 θ2 
n (θ ) ∝ exp −
2
 Nθ 
2
Rutherford scattering
υ ∞ cos θ
υ∞
y
υ ∞ sin θ
υ∞
F cos ϕ
F sin ϕ
F
υ0
ϕ
b
r
θ
x
Particle charge
q = + ze
qQ
F=
4πε 0 r 2
Nucleus charge
Q = + Ze
r
dL r r
= r ×F =0
dt
2 dϕ
L = mbυ0 = mr
dt
Alpha particle in a helium chamber
r r
mr × υ = const
dt
r 2 = bυ0
dϕ
Experimental Physics III - Atomic structure
r
r
r
v
θ
r⊥
3
Rutherford scattering
υ ∞ cos θ
υ∞
y
υ ∞ sin θ
υ∞
r 2 = bυ0
F cos ϕ
F sin ϕ
F
υ0
b
ϕ
r
θ
x
q' sin ϕdϕ = mbυ0dυ y
υ y (ϕ = π − θ ) = υ0 sin θ
θ
q'
(1 − cosϕ )
⇒υy =
mbυ0
sin θ
=
mυ02b 1 + cos θ
q'
zZe2
tan =
2 4πε 0mυ02b
dt
dϕ
dυ x
F cos ϕ = m
dt
dυ y
F sin ϕ = m
dt
υ∞ = υ0
Energy conservation
zZe2
q' ≡
4πε 0
Experimental Physics III - Atomic structure
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Rutherford scattering
I
∆A
Number of particles passing
through Unit area per unit time
dθ
R
θ
dσ
∆b
b
dN = Idσ = I 2πb∆b
Number of particles passing through
an area element dσ
σ
 2 q'  cos(θ / 2 )

dNθ = ndI 2π 
dθ
3
 E0  sin (θ / 2 )
2
Number of particles per unit length nd
dNθ = ndI 2πb∆b
db 2q'
1
=
dθ E0 sin 2 (θ / 2 )
∆A = 2πR sin θRdθ
b=
2
ndI  zZe 
1


dN A =
∆A
2 
4
4 R  8πε 0 E0  sin (θ / 2 )
2
2 q'
θ
cot
E0
2
∆b =
db
∆θ
dθ
Experimental Physics III - Atomic structure
5
Rutherford scattering
Experimental Physics III - Atomic structure
6
Emission spectrum of atomic hydrogen gas
λ=
c
ν
B ≈ 364 nm
m = 3,4..
n=2
Bm 2
λ= 2 2
m −n
n
λ (nm)
3
656
4
486
5
434
6
410
7
397
∞
365
The Balmer formula is an empirical equation discovered by Johann Balmer in 1885.
Experimental Physics III - Atomic structure
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The Rydberg equation
In 1888 the physicist Johannes Rydberg generalized the Balmer
equation for all transitions of hydrogen.
1

1
= RH  2 − 2 
 n1 n 
2 

1
λvac
n1
n2
Name
n1 < n2
RH ≈ 1.097 × 107 m −1
Converges toward
1916
1
2→∞
Lyman series
91.13 nm (UV)
2
3→∞
Balmer series
364.51 nm (Visible)
3
4→∞
Paschen series
820.14 nm (IR)
4
5→∞
Brackett series
1458.03 nm (Far IR)
1922
5
6→∞
Pfund series
2278.17 nm (Far IR)
1924
6
7→∞
Humphreys series
3280.56 nm (Far IR)
v12 = λ−1 = Tn1 − Tn 2
v13 = Tn1 − Tn 3
Ritz, 1908
v12 > v13
⇒ n2 > n3
v12 − v13 = Tn 3 − Tn 2
The Rydberg-Ritz Combination Principle is the theory proposed by Walter Ritz in 1908 to explain relationship of
the spectral lines for all atoms. The principle states that the spectral lines of any element include frequencies that are
either the sum or the difference of the frequencies of two other lines.
Experimental Physics III - Atomic structure
8
To remember!
Thomson anticipated that atom is a positively charged sphere with
homogeneously distributed point-like electrons.
Thomson’s atoms may scatter charged particles only to small
angles (Gaussian distribution for a collection of atoms).
Rutherford scattering equation is derived under assumption of
positive charge condensed in a center of atom.
The main deficiency of the Rutherford model
is its inability to explain the stability of atoms.
It also predicts well-defined emission
frequencies, which are not consistent with
the Ritz-Rydberg combinatory principle.
Experimental Physics III - Atomic structure
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