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Electrons as Waves
1897
m=
J.J. Thomson determines mass-to-charge ratio
for negatively charged “corpuscles”
0.4 – 0.5 x 10-7 g/C (modern value is 0.569 x 10-7 g/C)
mo
1-v2/c2
1905
H.A. Lorentz determines that mass varies
with velocity:
1905
A. Einstein publishes his famous equation: E = mc2 = ρ2c2 + m2c4
0
Reduces to E = ρc for photons (m0 = 0)
Energy of a photon can also be defined by the photoelectric effect as:
E = hν or E = hc/λ
Combining equations we obtain: E = ρc = hc/λ → λ = h/ρ = h/mv
1924
L. deBroglie postulated that particles (electrons) may also behave like waves…
MSE 321 Structural Characterization
E
rest
energy
}
}
Electrons as Waves
Part 1
KE = eV = mc2 – m0c2 → m = m0 +
mo
eV
=
c2
1 - v 2/c 2
m = m0 +
eV
c2
(
[m
0
2
)
+
m
[
]
eV
c2
{
ρ = mv =
m02c 2
eV
m0 + 2
c
→ v = c2 -
c2 -
1
Now, taking the inverse:
Multiplying both sides by m0 and squaring:
∴ v2 = c2 -
m02c 2
eV
m0 + 2
c
(
2
)
m02c 2
eV
m0 + 2
c
(
mo
and we already know that m =
]
2
)
v
MSE 321 Structural Characterization
m02
eV
m0 + 2
c
(
1 - v 2/c 2
mo
=
eV
m0 + 2
c
2
)
1 - v 2/c 2
Now solve for v 2…
= 1 - v 2/c 2
Now we can calculate momentum, ρ:
And squaring both sides:
ρ2
=
[
eV
m0 + 2
c
[
2
]
c2 -
m02c 2
(m
0
+
eV
c2
]
2
)
Electrons as Waves
Part 2
Multiplying
out…
[m
ρ2 = c 2
0 +
eV
c2
2
]c - (m
2
0 +
eV
c2
[ 2mc eV + e cV ] = c1
2
0
2
2
4
2
[
2
) (
m02c 2
eV
m0 + 2
c
]
2
)
(2m0c 2eV + e 2V 2)
[
= c 2 m0 +
eV
c2
2
]–m c
2 2
=
0
Therefore: ρ =
[
c 2 m02+
2m0eV e 2V 2
+ 4 - m02
c
c2
1
(2m0c 2eV + e 2V 2)
c
Now, substituting into deBroglie’s relation:
Factor out
1
c4
h
h
λ = mv = ρ = hc(2m0c 2eV + e 2V 2)-½
MSE 321 Structural Characterization
Electrons as Waves
The wave nature of electrons makes possible interference and diffraction.
It was discovered jointly with Ni crystals in 1927 by…
Clinton Joseph Davisson and
Lester Halbert Germer
Bell Labs, USA
George Paget Thomson and
Alexander Reid†
Aberdeen, UK
C.J. Davisson and G.P. Thomson shared the 1937 Nobel Prize for Physics.
But Joseph John Thomson (G.P.’s father) won the Nobel Prize in 1906 for
his work on negatively-charged “corpuscles”, which showed that electrons
were particles! They were both RIGHT.
Electrons behave as both waves and particles!
MSE 321 Structural Characterization
]
Electrons
ee+
mv 2/r
KE
}
∞
∴v =
e2
4πε0rm
∴ ½mv 2 =
e2
8πε0r
PE
e2
W = ½mv 2 4πε0r
∴W =
e2
=
4πε0r 2
r
}
lim = 0
mv 2
}
e2/(4πε0r2)
r
r→
e = 1.6022x10-19 C
ε0 = 8.854x10-12 F/m
m0 = 9.1094x10-31 kg
(PE= F d )
The electron is bound to the nucleus
by the positive energy –W.
( )
e2
-e 2
e2
(always negative) But this model would continuously
=
8πε0r
4πε0r
8πε0 r
radiate energy and so be unstable!
Bohr suggested that electrons can only lose energy of a certain size – only orbits where L = nh/2π are stable.
mvr =
∴W =
(
nh
2π
-e 2
8πε0
n = 1, 2, 3…
)(π
∴v =
nh
e2
=
2πrm
4πε0rm
)
-13.6
me 2
-me 4 1
= 2 2 2 ≈
eV
n2
ε0h 2n 2
8ε0h n
∴ rn =
ε0 h 2 2
n ≈ 0.529n2 Å
πme 2
h = 6.6261x10-34 Js
(Planck’s constant)
n = 1 13.6 eV binding energy
n = 2 3.4 eV binding energy
(n determines energy)
(1 eV = 1.6022x10-19 J)
Electronic transitions are quantised: hν
ν = |Wn2 – Wn1|
Niels Henrik David Bohr (1885 – 1962)
MSE 321 Structural Characterization
Quantum Numbers
n=2
Principal quantum number, n
n = 1, 2, 3… (K, L, M…)†
energy “shell”
n=1
Orbital quantum number, l
AKA azimuthal, orbital, or angular-momentum quantum number
l = 0, 1, 2… (n – 1)
l = 0: sharp
l = 1: principal
l = 2: diffuse
l = 3: fundamental
orbital angular momentum = {l (l + 1)}½h
Magnetic quantum number, ml
-l ≤ ml ≤ +l
Spin quantum number, s
s = ±½
intrinsic angular momentum
MSE 321 Structural Characterization
Mg
†Charles
Barkla
1877-1944
Total Angular Momentum
Total angular momentum quantum number, j
j =l ±s
|l–s|≤j≤l+s
Total angular momentum = { j ( j + 1)}½h
Aufbau Principle
Orbitals fill in order of
increasing n + l
Where two orbitals have the
same value of n + l, they are
filled in order of increasing n
MSE 321 Structural Characterization
Notation
spdf
IUPAC
n
l
ml
s
j
Siegbahn
1s
2s
2p
2p
3s
3p
3p
3d
3d
4s
4p
4p
4d
4d
4f
4f
K
L1
L2
L3
M1
M2
M3
M4
M5
N1
N2
N3
N4
N5
N6
N7
1
2
2
2
3
3
3
3
3
4
4
4
4
4
4
4
0
0
1
1
0
1
1
2
2
0
1
1
2
2
3
3
0
0
0
±1
0
0
±1
±1
0, ±2
0
0
±1
±1
0, ±2
0, ±2
±1, ±3
±½
±½
±½
±½
±½
±½
±½
±½
±½
±½
±½
±½
±½
±½
±½
±½
½
½
½
3
/2
½
½
3
/2
3
/2
5/
2
½
½
3/
2
3
/2
5
/2
5
/2
7
/2
1s
2s
2p1/2
2p3/2
3s
3p1/2
3p3/2
3d3/2
3d5/2
4s
4p1/2
4p3/2
4d3/2
4d5/2
4f5/2
4f7/2
=ml (max)+½
MSE 421/521 Introduction to Electron Microscopy
Electron Transitions
Rules:
∆n < 0
∆l = ±1
∆ml = 0 or ±1
1.
2.
3.
Karl Manne Georg Siegbahn
1886 - 1978
MSE 321 Structural Characterization
Electron Transitions
Rules:
1.
2.
3.
∆n < 0
∆l = ±1
∆ml = 0 or ±1
Karl Manne Georg Siegbahn
1886 - 1978
MSE 321 Structural Characterization
Electron Transitions
MSE 321 Structural Characterization