Download Chap 13 Phonons • classical theory of vibration • 1-dim, 3

Chap 13 Phonons
• classical theory of vibration
• 1-dim, 3-dim
• quantum theory of vibration
• phonon specific heat
• Einstein model, Debye model
• thermal expansion
• neutron scattering
Dept of Phys
M.C. Chang
One dimensional vibration
• consider only longitudinal motion
• consider only NN coupling
d 2un
M
= α (un +1 − un ) − α (un − un −1 )
2
dt
PBC:
Assume un = Aei ( kX n −ωt ) , where X n = na ,
then we'll get
M ( −ω 2 )eikna = −α 2eikna − eik ( n +1) a − eik ( n −1) a ,
which leads to
ω (k )=ωM sin(ka / 2) , ωM = 2 α / M
ω (k )=ωM sin(ka / 2)
Dispersion curve
ω
λ < 2a
λ > 2a λ > 2a
λ < 2a (redundant)
k
-π/a
π/a
The waves with wave numbers k and k+2π/a describe the same
atomic displacement
Therefore, we can restrict k to within the first BZ [-π/a, π/a]
Displacement of the n-th atom
un (t ) = Aei ( kX n −ωt ) , X n = na
Pattern of vibration:
• k ～ 0, exp(ikXn) ～ 1.
Every atom move in unison. Little restoring force.
• k ～ π/a,exp(ikXn) ～ (-1)n.
Adjacent atoms move in opposite directions. Maximum restoring force.
Velocity of wave:
• k ～ 0, ω = (ωMa/2)k
Linear dispersion, phase velocity = group velocity
• k ～ π/a, group velocity ～ 0
PBC
Number of “normal modes”:
u N = u0 ⇒ exp(ikNa) = 1
m 2π
△k=2π/Na
∴k =
N a
, m∈Z
Within the 1st BZ, there are N k-points.
Each k describes a normal mode of vibration
(i.e. a vibration with a specific frequency)
Vibration of a crystal with 2 atoms in a unit cell
a
d 2u2 n +1
= −α (2u2 n +1 − u2 n − u2 n + 2 ),
M2
2
dt
d 2u2 n + 2
= −α (2u2 n + 2 − u2 n +1 − u2 n +3 ).
M1
dt 2
⎛ u2 n +1 ⎞ ⎛ A1eikX 2 n+1 ⎞ − iωt X 2 n +1 = (2n + 1)a / 2
Assume ⎜
e ,
⎟=⎜
ikX 2 n+ 2 ⎟
X 2 n + 2 = (n + 1)a
⎝ u2 n + 2 ⎠ ⎝ A2 e
⎠
⇒
⎛ 2α − M 2ω 2
−2α cos(ka / 2) ⎞ ⎛ A1 ⎞
⎜
⎟ ⎜ ⎟ = 0,
2
2α − M 1ω ⎠ ⎝ A2 ⎠
⎝ −2α cos(ka / 2)
⇒
⎛ 2α − M 2ω 2
−2α cos(ka / 2) ⎞
det ⎜
⎟ = 0.
2
2
α
cos(
ka
/
2)
2
α
M
ω
−
−
1
⎝
⎠
2
⎛ 1
⎛ 1
1 ⎞
1 ⎞ 4sin 2 (ka / 2)
2
⇒ ω± = α ⎜
.
+
+
⎟ ±α ⎜
⎟ −
M
M
M
M
M
M
⎝ 1
⎝ 1
2 ⎠
2 ⎠
1
2
Two branches of dispersion curves (assume M2 > M1)
a
d
c
b
Patterns of vibration
similar
See a nice demo at http://dept.kent.edu/projects/ksuviz/leeviz/phonon/phonon.html
Three dimensional vibration
Along a given direction of propagation, there are
1 longitudinal wave and 2 transverse waves,
each may have different velocities
Sodium
(bcc)
3D crystal with atom basis
FCC lattice with 2-atom basis
cm-1
Rules of thumb
• For a 3-dim crystal, if each unit cell
has p atoms, then there are
3 acoustic branches,
3(p-1) optical branches
• If a crystal has N unit cells, then each branch
has N normal modes。
• As a result, the total number of normal modes
are 3pN (= total DOF of this system)
Quantum theory of vibration
Review: 1D simple harmonic oscillator (DOF=1)
p2 α 2
H=
+ x
2m 2
• Classically, it oscillates with a single freq ω=(α/m)1/2
• The energy ε can be continuously changed.
Quantization:
[ x, p ] = i
1 ⎛
i
⎞
ω
+
m
x
p
⎜
⎟
2 ⎝
mω ⎠
define
a=
then
⎡⎣ a, a ⎤⎦ = 1
1⎞
⎛
H = ⎜ a†a + ⎟ ω
2⎠
⎝
1⎞
⎛
H n = ⎜n+ ⎟ ω n
2⎠
⎝
†
Note:
a n = n n −1
a† n = n + 1 n + 1
a†a n = n n
1⎞
⎛
• After quantization, the energy becomes discrete ε n = ⎜ n + ⎟ ω
2⎠
⎝
• The number n of energy quanta depends on the amplitude of the oscillator.
Quantization of a 1-dim vibrating lattice (DOF=N)
⎡ p2 α
H = ∑⎢
+ ( u +1 − u
2
=1 ⎣ 2m
N
)
2
⎤
⎥
⎦
• Classically, for a given k, it vibrates with a single frequency ω(k).
The amplitude ( and hence energy ε) can be continuously changed.
Quantization:
[u , p ' ] = i
Fourier transf.
u =
then
δ
'
1
eik a uk
∑
N k
1
p =
eik a pk
∑
N k
⎡⎣uk , pk† ' ⎤⎦ = i δ kk '
⎛ 1 †
mωk2 † ⎞
⇒ H = ∑⎜
pk pk +
uk uk ⎟
2
2
m
k ⎝
⎠
Note:
u† = u ;
→ uk† = u− k ;
p† = p
pk† = p− k
k=2πm/L, L=Na.
A collection of N
independent oscillators !
Similarly, define
1
ak =
2
⎛
⎞
i
pk ⎟
⎜ mωk uk +
⎜
⎟
mωk
⎝
⎠
→ ⎡⎣ ak , ak†' ⎤⎦ = δ kk '
1⎞
⎛
→ H = ∑ ⎜ ak† ak + ⎟ ωk
2⎠
k ⎝
eigenstate
n1 , n2 ...nk ,...
1⎞
⎛
H n1 , n2 ...nk ,... = ∑ ⎜ nk + ⎟ ωk n1 , n2 ...nk ,...
2⎠
k ⎝
• The number of energy quanta (called phonons) for the k-mode is nk.
• There are no interaction between phonons → “free” phonon gas.
• If there are p-atoms in a unit cell (p branches), then the total
vibrational energy of the lattice is
1⎞
⎛
U = ∑∑ ⎜ nk , s + ⎟ ωk , s
2⎠
k s =1 ⎝
p
Specific heat: experimental fact
Specific heat
approaches 3R
(per mole) at
high temperature
(Dulong-Petit law)
Specific heat
drops to zero at
low temperature
After rescaling the temperature by θ (Debye temperature), which
differs from material to material, a universal behavior emerges:
Debye temperature
In general, a harder material has a higher Debye temperature
Specific heat: theoretical framework
• Internal energy U of a crystal is the summation of vibrational energies
(consider an insulator so there’s no electronic energies)
U (T ) = ∑ (nk , s + 1/ 2) ωk , s , s =L/T, A/O...
k ,s
• For a crystal in thermal equilibrium, the average phonon number is
nk , s =
• Therefore,
• Specific heat
1
e
ωk ,s / kT
⎛
U (T )= ∑ ⎜
k ,s ⎝ e
−1
1
ωk ,s / kT
Bose-Einstein
distribution
1⎞
+ ⎟ ωk ,s
−1 2 ⎠
CV = ( ∂U / ∂T )V
• Density of states (similar to electron energy band)
∑f
k
k
=V ∫
d 3k
( 2π )
⎛ L ⎞
D (ω ) = ⎜
⎟
⎝ 2π ⎠
3
∫
3
f k = ∫ dω D(ω ) fω
dSω
∇k ω
Ex: In 3D
if ω = vk , then D(ω ) = V ω 2 / 2π 2 v 3
Einstein model (1907)
Assume that each atom vibrates independently of each other,
and every atom has the same vibration frequency ω0
DOS
D(ω ) = 3N δ (ω − ω0 )
3 dim × number of atoms
ω0
ω0
1⎞
⎛
+ 3N
U = 3N ⎜ n + ⎟ ω0 = 3 N
2⎠
exp( ω0 / kT ) − 1
2
⎝
e ω0 / kT
⎛ ω0 ⎞
CV = (∂U / ∂T )V = 3Nk ⎜
⎟
⎝ kT ⎠ e ω0 / kT − 1
2
(
≈ e−
ω0 / kT
as T → 0 K
(Activation behavior)
)
2
Debye model (1912)
Atoms vibrate collectively in a wave-like fashion.
U (T ) = ∑ nk , s
ωk , s ( ωk ,s /2 neglected)
k ,s
= ∑ ∫ d ω Ds (ω )
s
ω
e
ω / kT
−1
• Debye assumed a simple dispersion relation:
ω = vsk. Therefore, Ds (ω ) = V ω 2 / 2π 2 vs3
Also, the 1st BZ is approximated by
a sphere with the same volume
3
∑ ∫ dω D (ω ) = 3N
s =1
s
V ωD3
→ ∑ 2 3 = 3N
s =1 6π vs
3
3
3
1
≡
∑
v3 s =1 vs3
→ ωD = v(6π n) , n = N / V
2
1/3
If vg = ∇ k ω = 0, then there is
"van Hove singularity".
3
U (T ) = ∑
s =1
=
V
2π 2 vs3
ωD
∫
d ωω 2
0
3V
⎛ kT ⎞
⎜
⎟
2π 2 v3 ⎝
⎠
⎛T ⎞
=9 NkT ⎜ ⎟
⎝θ ⎠
4 xD
ω
e
∫ dx
0
ω / kT
−1
Debye temperature
ωD θ
x
,
x
=
= , kθ ≡ ωD
D
x
e −1
kT
T
3
3 xD
x3
∫0 dx e x − 1 = π4/15 as T→ 0
12π 4
⎛T ⎞
Nk ⎜ ⎟ ∝ T 3 as T → 0
∴ CV =
5
⎝θ ⎠
3
solid Argon
(θ=92 K)
At low T, Debye’s curve drops
slowly because long wavelength
vibration can still be excited.
A simple explanation of the T3 behavior:
Suppose that
1. All the phonons with wave vector k<kT are excited
with thermal energy kT.
2. All the modes between kT and kD are not excited.
kD
kT
Then the fraction of excited modes
= (kT/kD)3 = (T/θ)3.
energy U ～ kT⋅3N(T/θ)3, and the heat capacity
C ～ 12Nk(T/θ)3
Thermal expansion
Coeff. Of volume
expansion:
Bulk modulus:
1 ⎛ ∂V ⎞
⎜
⎟ ,
V ⎝ ∂T ⎠ P
⎛ ∂P ⎞
B = −V ⎜
⎟
⎝ ∂V ⎠T
β=
1 ⎛ ∂P ⎞
⎜
⎟
B ⎝ ∂T ⎠V
⎛ ∂F ⎞
P = −⎜
⎟
⎝ ∂V ⎠T
→β =
F = Eelastic + Fphonon , Eelastic
Free energy:
Next page we’ll show that,
U (T )
⎛ ∂E ⎞
P = − ⎜ els ⎟ + γ
V
⎝ ∂V ⎠T
Indep of T
⇒
1 ⎛ ∂P ⎞
γ
=
=
β
cV
⎜
⎟
B ⎝ ∂T ⎠V
B
(cV = CV / V )
∝ T 3 at low T
use
⎛ ∂x ⎞
1
and
⎜ ⎟ = ∂y
y
∂
⎝ ⎠z ⎛ ⎞
⎜ ⎟
⎝ ∂x ⎠ z
⎛ ∂x ⎞ ⎛ ∂y ⎞ ⎛ ∂z ⎞
⎜ ⎟ ⎜ ⎟ ⎜ ⎟ = −1
⎝ ∂y ⎠ z ⎝ ∂z ⎠ x ⎝ ∂x ⎠ y
B ⎛ ΔV ⎞
= ⎜
⎟
2⎝ V ⎠
2
Partition function:
Z ph = ∑ e − Ei / kT
(Ei are the macroscopic eigen-energies)
i
⎡ 1
= ∑ exp ⎢ −
⎣ kT
{nks }
⎤
1⎞
⎛
ω
+
n
∑
⎜ ks
⎟ ks ⎥
2⎠
ks ⎝
⎦
⎡ 1
=∏ ∑ exp ⎢ −
ks nks = 0
⎣ kT
∞
⎤
1⎞
⎛
+
n
ω
⎜ ks
⎟ ks ⎥
2⎠
⎝
⎦
Grüneisen parameter
d ωks
ωks
Fph = − kT ln Z ph
(
= kT ∑ ln e
ωks /2 kT
− e−
ωks /2 kT
⇒β =
)
ks
⎛ ∂Fph ⎞
1 ⎛ ∂ ωks ⎞ e
P' ≡ −⎜
⎟ = − ∑⎜
⎟
2 ks ⎝ ∂V ⎠T e
⎝ ∂V ⎠T
∴P' =
1
γ ksU ks (T ),
∑
V ks
⎛
where U ks (T ) ≡ ωks ⎜
⎝e
ωks / kT
ωks / kT
+1
−1
ωk s d ωk s V
ωk s
≡ −γ k s
V ωk s dV
V
=
= −γ ks
dV
V
1 ⎛ ∂P ' ⎞
⎜
⎟
B ⎝ ∂T ⎠V
γ CV
1 1
γ
C
=
∑ ks V ,ks B V
B V ks
∑γ C
where γ ≡
∑C
ks
V , ks
ks
V , ks
ks
CV = ∑ CV ,ks
1
ωks / kT
1⎞
+ ⎟
−1 2 ⎠
ks
(γ～1-2 for most materials)
Neutron scattering
En
( k)
=
Why neutron?
2
2mn
= 2.07 k 2 meVA 2
• Neutron has no charge (can probe bulk properties)
• Neutron wavelength comparable to interatomic spacings (1-5 Å)
• Neutron energy comparable to phonon’s (5-100 meV)
• Neutron has spin (can probe magnetic structure and magnetic excitations)
Measure phonon dispersions by neutron scattering
More than one phonon mode may be excited
Δnq.s = n 'q.s − nq.s
out E’,p’
in E, p
Conservation of energy
En '+ ∑ ωq , s Δnq , s = En
q,s
Conservation of crystal momentum (for a proof. see App. M of A+M)
p '+ ∑ q Δnq , s = p + G
q,s
(momentum of phonon with λ< a must be shifted by G)
One phonon scattering
En ' = En ± ωs (q )
p' = p ± q + G
Neutron energy:
p '2
p2
=
± ωs (q ),
2mn 2mn
En=p2/2mn
Consider phonon absorption
(
p '2
p2
−
= ωs k '− k
2mn 2mn
⇒ f k (q ) ≡
2
( p' ≡
k '; p ≡ k
q = ± (k '− k )
k=0
)
2 2
| q + k |2
k
−
= ωs ( q )
2mn
2mn
q and ωs can be determined from
the intersections in the figure.
)
2
q2
2M n
ω (q )
q
k≠0
several solutions at a given direction.
→ a series of peaks in the data
Phonon dispersion curve for Si,
comparing experimental data and
ab initio calculation (Wei and Chou 1994)
• Width of one-phonon peaks due to anharmonic effect
• Multi-phonon scatterings give a continuous background
Neutron scattering: formal theory
2π
∑
Ψ f V Ψ i δ ( E f − Ei )
2
• Fermi golden rule
(transition rate)
Ri →[ f ] =
• Neutron states scattered to
d3k’ (per unit time)
mn k ' d ε d Ω
d 3k '
Rk →k ' ⋅ V
R
V
=
⋅
k →k '
(2π )3
(2π )3
f
j = ρv
• Differential cross-section
Rk →[d Ω ,d ε ]
⎛ d 2σ ⎞
⎜
⎟ d Ωd ε ≡
ε
d
d
incident particle flux
Ω
⎝
⎠
⎛ d 2σ ⎞ k ' (Vmn ) 2
Rk →k '
⇒⎜
⎟=
3
ε
π
(2
)
Ω
d
d
k
⎝
⎠
• Quantum state of “neutron + crystal”
Ψ = ψ k (r )Φ crystal , ψ k (r ) =
initial:
final:
Ψ i = ψ k ( r )Φ i ,
Ψ f = ψ k ' ( r )Φ f ,
1 ik ⋅r
e
V
2
k2
Ei =
+ Ei
2mn
Ef =
2
2
k'
+ Ef
2mn
Neutron Crystal
energy energy
= ψk
2
k 1 k
=
mn V mn
• The energy gained by a neutron
due to a phonon in crystal
2
2 2
k '2
k
−
= Ei − E f = ω
2mn 2mn
Momentum transfer
q = k '− k
Neutron-ion interaction potential
(
)
V ( r ) = ∑ v r − r ( R) , where r ( R ) = R + u ( R)
v(r) is the atomic
potential
R
Matrix element
(
)
let v r − r ( R ) =
1
vq eiq ⋅r e − iq⋅r ( R )
∑
V q
Ψ f V Ψ i = ∑ ψ k 'Φ f v ψ k Φ i
R
1
vq ψ k ' eiq⋅r ψ k Φ f e − iq ⋅r ( R ) Φ i
∑
V R ,q
δ k + q ,k '
vk ' − k ∼ v0
1
since Δk for phonon ∼ 108 cm −1
= ∑ vk '− k Φ f e − i ( k '− k )⋅r ( R ) Φ i
V R
the range of vq ∼ 1013 cm −1
=
⎛ dσ
∴⎜
⎝ d Ωd ε
⎞ k ' ⎛ v0 mn ⎞
⎟= ⎜
2 ⎟
⎠ k ⎝ 2π ⎠
2
∑δ (E
f
f
− E i ) ∑ Φ f eiq ⋅r ( R ) Φ i
Sum over a complete set
of phonon states
R
2
⎛ dσ ⎞
One can re-write ⎜ d Ωd ε ⎟ using dynamical structure factor
⎝
⎠
1 ⎛ E f − Ei
δ ( E f − Ei ) = δ ⎜
⎝
use e
i ( E f − Ei ) t /
⎛ dσ
then ⎜
⎝ d Ωd ε
⎞ 1
+ω⎟=
⎠ 2π
iω t
dt
e
e
∫
i ( E f − Ei ) t /
for phonon absorption
(always from neutron’s viewpoint)
Φ f A Φ i = Φ f A(t ) Φ i
⎞ k ' 1 ⎛ mn v0 ⎞
⎟=
⎜
2 ⎟
⎠ k h ⎝ 2π ⎠
2
ω
∫ dt e ∑
i t
Φ i eiq ⋅r ( R ') e − iq ⋅r ( R ,t ) Φ i
R,R '
2
k ' 1 ⎛ mn v0 ⎞
=
2π NSi (q , ω )
⎜
2 ⎟
k h ⎝ 2π ⎠
Dynamical structure factor
Si ( q , ω ) ≡
Density operator (for ions)
1 dt iωt
iq ⋅r ( R ') − iq ⋅r ( R ,t )
e
e
e
Φ
Φi
∑
i
∫
N 2π
R,R '
ρ (r ) = ∑ δ (r − r ( R))
R
ρ q = ∫ d 3r eiq⋅r ρ (r ) = ∑ eiq⋅r ( R )
dt iωt 1
=∫
e
Φ i ρ q (0) ρ − q (t ) Φ i
2π
N
R
Density correlation function
For a crystal at finite temperature
Φ i ρ q (0) ρ − q (t ) Φ i
→
ρ q (0) ρ − q (t )
T
=
∑e
i
− Ei / kT
Φ i ρ q ρ − q (t ) Φ i
∑e
i
− Ei / kT
Evaluation of the dynamical structure factor
eiq ⋅r ( R ') e − iq⋅r ( R ,t )
T
= eiq⋅( R '− R ) eiq ⋅u ( R ') e − iq ⋅u ( R ,t )
It can be shown that, for A, B linear in a , a
e Ae B
T
=e
1 2
A + 2 AB + B 2
2
use ⎡⎣ q ⋅ u ( R ') ⎤⎦
2
T
T
†
D. Mermin, J Math.Phys. 7,1038 (1966)
= ⎡⎣ q ⋅ u ( R, t ) ⎤⎦
and ⎡⎣ q ⋅ u ( R ') ⎤⎦ ⎡⎣ q ⋅ u ( R, t ) ⎤⎦
r ( R) ≡ R + u ( R)
T
T
= [ q ⋅ u (0)]
2
≡ 2W
2
T
T
= [ q ⋅ u (0) ] ⎡⎣ q ⋅ u ( R − R ', t ) ⎤⎦
T
Translation symmetry
of the system
Debye-Waller factor
⇒ S (q , ω ) ≡ e
e
T
−2W
=1 +
dt iωt
− iq ⋅ R [ q ⋅u (0)]⎡⎣ q ⋅u ( R ,t ) ⎤⎦ T
∫ 2π e ∑R e e
1 2
+
+ ...
T
T
2
0-phonon 1-phonon process …
Exact so far
(for a harmonic crystal)
• Rough estimate (Kittel. App.A)
Zero-phonon process
2W =
S0 (q , ω ) = e
−2W
δ (ω ) ∑ e
R
DebyeWaller
factor
Elastic
scattering
− iq ⋅ R
N ∑ δ q ,G
G
Laue’s diffraction
condition
T
T
cos 2 θ
T
1
= G2 u2
T
3
m
3
use ion ω 2 u 2 = kT
T
2
2
−2W
=e
−
G2
mionω 2
kT
2
⎞ k ' 1 ⎛ mn v0 ⎞
2π NS (q , ω )
⎟=
⎜
2 ⎟
⎠ k h ⎝ 2π ⎠
k' N
or = a 2 S (q , ω )
k
⎛ dσ
⎜
⎝ d Ωd ε
2
Intensity I=I0 e-2W (I0 for a rigid lattice)
Differential cross-section
Cf:
)
= G2 u2
→ e
⎛ dσ
⎜
⎝ d Ωd ε
(
G ⋅u
⎞ N
⎟ = S (q , ω )
⎠
2π 2 a
v0 ≡
mn
For X-ray scattering
(the same S)
Scattering length
For more discussion,
see A+M, App. N
use aks† aks
• A more accurate evaluation
1
aks =
2
a−† ks =
1
2
uks =
pks = −i
⎛
⎞
i
pks ⎟ ⋅ eˆks
⎜ mωks uks +
⎜
⎟
mωks
⎝
⎠
⎛
⎞
i
ˆ
m
u
p
ω
−
⎜
ks ks
ks ⎟ ⋅ eks
⎜
⎟
mωks
⎝
⎠
(eˆ*− ks =eˆks )
a
(
ω (k )
2mion
ks
+a
†
− ks
) eˆ
ks
s
2mionωs (k )
(a
ks
− a−† ks ) eˆks
⇒
(
G ⋅u
)
= nks
T
2
=
T
1
N
G ⋅ eˆks
∑ 2m
ks
2
ωs ( k )
(2nks + 1)
ion
In 3D Debye model (at T =0!)
G2
3
2W =
4 mion v k D
(Prob. 7)
In the calculation, one has
kD
1 kD D − 2
D −1
k
k
d
= k dk
∫0
k ∫0
• In 3D, W weakens the diffraction peaks.
• In 2D, W is finite at T=0 but infinite at finite-T.
1
uks eik ⋅R
∑
N k ,s
1
p( R) =
pks eik ⋅R
∑
N k ,s
u ( R) =
• In 1D, no long-range order even at T=0
Mermin-Wagner theo (Mermin PR1968)
There is no long-range crystalline order at
finite-T in 2D.
One-phonon process
S1 (q , ω ) = e −2W ∫
dt iωt
e ∑ e − iq⋅R [ q ⋅ u (0) ] ⎡⎣ q ⋅ u ( R, t ) ⎤⎦
2π
R
T
1/2
⎛
⎞ ik ⋅R
1
†
q ⋅ u ( R, t ) =
⎜
⎟ e ( aks (t ) + a− ks (t ) ) q ⋅ eˆks
∑
N ks ⎝ 2mionωks ⎠
⇒ S1 (q , ω ) = e−2W ∑
s
2mionωqs
( q ⋅ eˆks )
2
aks (t ) = e − iωks t aks
⎣⎡(1 + nqs )δ (ω + ωqs ) + nqsδ (ω − ωqs ) ⎦⎤
Phonon emission
absorption
Delta peaks are broadened only
if anharmonic effect (phononphonon interaction) is included.
One-phonon cross-section
⎛ dσ
⎜
⎝ d Ωd ε
⎞ k' 2 N
S1 (q , ω )
⎟= a
k
⎠