Unformatted text preview: MatE 153 Spring 2008
Midterm 2
SOLUTIONS
Class average was 79%
1.) An electron from the 2s shell in B (Z=5) is excited to the 5th n shell. What wavelength
of light is given off when the electron returns to the 2s shell? (10 points) EI = mq 4
= 13.6eV
2
8ε 0 h 2 En = − Z2E I
n2 1⎞
⎛1
ΔE = −5 2 ⋅ 13.6eV⎜ 2 − 2 ⎟ = −71.4eV
5⎠
⎝2
The negative sign represents the light being given off.
ΔE = λ= hc
λ ΔE
=
hc 4.136 × 10 −15 eV ⋅ s ⋅ 3 × 10 8
71.4eV m
s = 1.7 × 10 −8 m = 17 nm The selection rules are n must change, Δl = ±1, Δm l = 0,±1 . Briefly state why
there are selection rules. (5 points)
l and ml are related to the angular momentum of the electron. The selection rules are
needed because only certain transitions of l and ml are possible to ensure that momentum is
conserved. 2.) 3.) Using the selection rules, give an allowed set of 4 quantum numbers for an electron in
the 2s of B and an allowed set of 4 quantum numbers for the electron that is excited to
the 5th n shell (problem 1). (5 points)
For your 2s electron, you could have given either of these:
n=2, l=0, ml= 0, ms = + ½
n=2, l=0, ml= 0, ms =  ½
For the 5th shell electron, you could have given any of these:
n=5, l=1, ml= 0, ms = + ½
n=5, l=1, ml= 0, ms =  ½
n=5, l=1, ml= 1, ms = + ½
n=5, l=1, ml= 1, ms =  ½
n=5, l=1, ml= 1, ms = + ½
n=5, l=1, ml= 1, ms =  ½ 1 4.) Write the electron configuration for Co (Z=27). (5 points)
Co has Z=27 (27 electrons). Using the given chart showing the electron energies, you need
to fill in the orbitals with the lowest energies first. This gives: 1s22s22p63s23p64s23d7.
5.) Assuming Co has 2 valence electrons, calculate the Fermi energy at 0K for Co. (10
points)
electrons
g
atoms
2
⋅ 8.9 3 ⋅ 6.02 × 10 23
# valence ⋅ d ⋅ N A
atom
mol = 1.818 × 10 23 electrons
cm
n=
=
g
M at
cm 3
58.93
mol
2 ( )(
) ) 2 6.626 × 10 −34 J s ⎡ 3 1.818 × 10 29 m 3 ⎤ 3
h ⎛ 3n ⎞ 3
−18
EF =
=
⎜⎟
⎢
⎥ = 1.87 × 10 J = 11.7eV
−31
π
8m e ⎝ π ⎠
8 9.1 × 10 kg ⎣
⎦
Note: To get the units to work out you needed h in Joules and n in m3
2 2 ( 6.) Does the mean speed of a conduction electron in a metal have a weak or strong
dependence on temperature? Briefly explain why. (10 points)
This problem was directly testing the concepts you learned in HW 4 Question 1. The
mean speed of the electron relates to the average energy of the electron at that temperature.
The temperature dependence of the average energy is the temperature dependence of the
Fermi energy (equation 4.26). As we saw in our homework, this is a very weak
dependence on temperature. That is to say, the Fermi energy, and in return, the average
energy and the mean speed of the electron is only weakly dependent on temperature.
7.) Describe what Ψhyb represents in this figure. Explain in words how LCAO resulted in
ΨA and ΨB having different energy. (10 points)
The valence electrons in the 3s and 3p shells of Si interact to form a new orbital that is an
overlap of the sand p orbitals. This new orbital has 4 lobes in the shape of a tetrahedral
and is referred to as sp3 hybridization. Ψhyb is the wavefunction of an electron in this new
orbital. When the Si atoms come together in a solid, Si covalently bonds with it’s
neighbors. This means that the wavefunction from one atom now combines with the
wavefunction from the other atom. This combination is known as LCAO (linear
combination of atomic orbitals). The Ψhyb wavefunctions of two atoms can combine
constructively to form a new wavefunction, ΨB. The Ψhyb wavefunctions of two atoms can
combine destructively to form a new wavefunction, ΨA. ΨA is higher in energy because it
has nodes (and results in antibonding).
Extra: Not required for the answer: The figure also shows the valence band and the
conduction band. When more than two atoms come together, all the ΨB from all the pairs
of atoms covalently sharing electrons shift energy with respect to each other in order not to
violate Pauli’s exclusion principle. This results in the broad valence band rather than a
discrete state. 2 8.) Calculate the intrinsic carrier concentration in Ge at 150K. (10 points)
⎛ − EG ⎞
n i = N C N V exp⎜
⎟
⎝ 2kT ⎠
⎛ 2πm * kT ⎞
e
⎟
N C = 2⎜
⎜ h2 ⎟
⎝
⎠ 3/ 2 ⎛ 2πm * kT ⎞
h
⎟
N V = 2⎜
⎜ h2
⎟
⎝
⎠ ⎛
⎞
J
⎜ 2π ⋅ 0.56 ⋅ 9.1 × 10 −31 kg ⋅ 1.38 × 10 − 23 150K ⎟
K
⎟
= 2⎜
3
⎜
⎟
2 kg ⋅ m
6.626 × 10 −34 Js
⎜
⎟
2
Js
⎝
⎠ ( 3/ 2 3/ 2 ) ⎛
⎞
J
⎜ 2π ⋅ 0.4 ⋅ 9.1 × 10 −31 kg ⋅ 1.38 × 10 − 23 150K ⎟
K
⎟
= 2⎜
3
⎜
⎟
2 kg ⋅ m
6.626 × 10 −34 Js
⎜
⎟
Js 2
⎝
⎠ ( ) = 3.71 × 10 24 m −3 3/ 2 = 2.24 × 10 24 m −3 Note: You needed to use h and k in Joules to get the units to work out.
As in the homework problems and example problem 5.1, you need to use the (b) value for
effective mass for the density of states calculation.
⎛
⎞
⎜
⎟
− 0.66eV
⎛ − EG ⎞
24
−3
24
−3
⎜
⎟
n i = N C N V exp⎜
⎟ = 3.71 × 10 m ⋅ 2.24 × 10 m exp
⎜
⎟
− 5 eV
⎝ 2kT ⎠
150K ⎟
⎜ 2 ⋅ 8.62 × 10
K
⎝
⎠
13
−3
= 2.38 × 10 m 9.) If Ge is doped with Ga, is it ntype or ptype? (3 points)
Ga has 3 valence electrons compared to Ge’s 4 so Ge doped with Ga is ptype.
10.) Using a band diagram, compare the likely relative magnitudes of the band gap
energy of Ge with the ionization energy of Ga as a dopant in Ge. (7 points) Conduction Band EG (band gap)
ΔEA
Ionization energy
Valence Band 3 ...
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 Fall '08
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 Electron, Atomic orbital, Condensed matter physics

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