in the form c(ky) c(k) c(ke) 0 M C(k) 0 c() 0 c(ky) c(k) G) If we approximate and say that only c (k),c(k) and c (k-) are important (others taken as 0). we get M to be a 3 x 3 matrix: (abc) def (c(ko)` c (k,) 0 gbj, (c(k)) What are a, b, c, w i We have an 9-atom crystal with atomic spacing a = 2лÁ = 2x x 10" m So L 9 x (a) 18 x 100m (a small crystal!) A) The general Fourier series for w is: w(x) = c(k,)e** (1) for cyclic boundary conditions, v(x) = (x + L) with L = 9u What are the allowed kr's in the series in equation (1)? --> B) The potential for V (x) is: V(x) = _V₁, c'i‚” 3115 (2) with lattice periodicity: V(x+na) = V(x) (ignoring end effects, which here would be hard to ignore for this small crystal, but will anyway) What are the allowed G.'s? 2x C) Specialize to the simple case where V (x) = V₁cos (GLX) G₁ = a What are the {V} (Only 2 are needed, the rest are zero-use Euler's formula to represent cos in terms of complex exponentials and use cq. (2)) D) Put the special case of A) and C) into the Schrödinger equation: - v°(x) + V(x)¥′(x) = Ev(x) (3) 2m Put (3) into the form: (note: G₁=9+k₁ and G:=2•G]=18•G:) [ ] + [ ] + + [ ] c -ik x + [ + [ ] eks …….. + [ ] + [ ]-n Дл е ... = 0 E) What is What is general expression [ (the term multiplying with k = 2 ) 27 [ 12? the term multiplying e' with ki = •12 - k₁ L 2x [ ]? the term multiplying e^ŕ with k, = •21 k₁ + G₁ 1. 2x [ ] 67 the term multiplying with k = • (-6) = k - G 1. -0) F) Since each Fourier term is independent, we will need = [ ] [ ]=0 [ ]=0 [ ]=0 etc... Use this to write the equations for the set € (ki), c (ki), c(k), c(ks), c (k-), c(k-is), e (k-24) ………..
in the form c(ky) c(k) c(ke) 0 M C(k) 0 c() 0 c(ky) c(k) G) If we approximate and say that only c (k),c(k) and c (k-) are important (others taken as 0). we get M to be a 3 x 3 matrix: (abc) def (c(ko)` c (k,) 0 gbj, (c(k)) What are a, b, c, w i We have an 9-atom crystal with atomic spacing a = 2лÁ = 2x x 10" m So L 9 x (a) 18 x 100m (a small crystal!) A) The general Fourier series for w is: w(x) = c(k,)e** (1) for cyclic boundary conditions, v(x) = (x + L) with L = 9u What are the allowed kr's in the series in equation (1)? --> B) The potential for V (x) is: V(x) = _V₁, c'i‚” 3115 (2) with lattice periodicity: V(x+na) = V(x) (ignoring end effects, which here would be hard to ignore for this small crystal, but will anyway) What are the allowed G.'s? 2x C) Specialize to the simple case where V (x) = V₁cos (GLX) G₁ = a What are the {V} (Only 2 are needed, the rest are zero-use Euler's formula to represent cos in terms of complex exponentials and use cq. (2)) D) Put the special case of A) and C) into the Schrödinger equation: - v°(x) + V(x)¥′(x) = Ev(x) (3) 2m Put (3) into the form: (note: G₁=9+k₁ and G:=2•G]=18•G:) [ ] + [ ] + + [ ] c -ik x + [ + [ ] eks …….. + [ ] + [ ]-n Дл е ... = 0 E) What is What is general expression [ (the term multiplying with k = 2 ) 27 [ 12? the term multiplying e' with ki = •12 - k₁ L 2x [ ]? the term multiplying e^ŕ with k, = •21 k₁ + G₁ 1. 2x [ ] 67 the term multiplying with k = • (-6) = k - G 1. -0) F) Since each Fourier term is independent, we will need = [ ] [ ]=0 [ ]=0 [ ]=0 etc... Use this to write the equations for the set € (ki), c (ki), c(k), c(ks), c (k-), c(k-is), e (k-24) ………..
Introductory Circuit Analysis (13th Edition)
13th Edition
ISBN:9780133923605
Author:Robert L. Boylestad
Publisher:Robert L. Boylestad
Chapter1: Introduction
Section: Chapter Questions
Problem 1P: Visit your local library (at school or home) and describe the extent to which it provides literature...
Related questions
Question
Transcribed Image Text:in the form
c(ky)
c(k)
c(ke)
0
M
C(k)
0
c()
0
c(ky)
c(k)
G) If we approximate and say that only c (k),c(k) and c (k-) are important (others taken as
0). we get M to be a 3 x 3 matrix:
(abc)
def
(c(ko)`
c (k,)
0
gbj,
(c(k))
What are a, b, c,
w
i
![We have an 9-atom crystal with atomic spacing a = 2лÁ = 2x x 10" m
So L 9 x (a) 18 x 100m (a small crystal!)
A) The general Fourier series for w is: w(x) = c(k,)e** (1)
for cyclic boundary conditions, v(x) = (x + L) with L = 9u
What are the allowed kr's in the series in equation (1)?
-->
B) The potential for V (x) is: V(x) = _V₁, c'i‚”
3115
(2)
with lattice periodicity: V(x+na) = V(x) (ignoring end effects, which here would be
hard to ignore for this small crystal, but
will anyway)
What are the allowed G.'s?
2x
C) Specialize to the simple case where V (x) = V₁cos (GLX)
G₁ =
a
What are the {V}
(Only 2 are needed, the rest are zero-use Euler's formula to represent
cos in terms of complex exponentials and use cq. (2))
D) Put the special case of A) and C) into the Schrödinger equation:
- v°(x) + V(x)¥′(x) = Ev(x) (3)
2m
Put (3) into the form: (note: G₁=9+k₁ and G:=2•G]=18•G:)
[ ] + [ ] + + [ ] c
-ik x + [
+ [ ] eks
…….. + [ ] + [ ]-n
Дл
е
...
= 0
E) What is
What is general expression [
(the term multiplying with k = 2
)
27
[
12?
the term multiplying e' with ki
=
•12 - k₁
L
2x
[ ]?
the term multiplying e^ŕ with k,
=
•21 k₁ + G₁
1.
2x
[ ] 67
the term multiplying with k
=
• (-6) = k - G
1.
-0)
F) Since each Fourier term is independent, we will need
=
[ ] [ ]=0 [ ]=0 [ ]=0
etc...
Use this to write the equations for the set
€ (ki), c (ki), c(k), c(ks),
c (k-), c(k-is), e (k-24) ………..](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F093e7116-3e01-49ad-9157-7a22a5caeb16%2Febb9d3d8-056f-4a48-baf5-abe55c72a99d%2Fdgdvmrk_processed.png&w=3840&q=75)
Transcribed Image Text:We have an 9-atom crystal with atomic spacing a = 2лÁ = 2x x 10" m
So L 9 x (a) 18 x 100m (a small crystal!)
A) The general Fourier series for w is: w(x) = c(k,)e** (1)
for cyclic boundary conditions, v(x) = (x + L) with L = 9u
What are the allowed kr's in the series in equation (1)?
-->
B) The potential for V (x) is: V(x) = _V₁, c'i‚”
3115
(2)
with lattice periodicity: V(x+na) = V(x) (ignoring end effects, which here would be
hard to ignore for this small crystal, but
will anyway)
What are the allowed G.'s?
2x
C) Specialize to the simple case where V (x) = V₁cos (GLX)
G₁ =
a
What are the {V}
(Only 2 are needed, the rest are zero-use Euler's formula to represent
cos in terms of complex exponentials and use cq. (2))
D) Put the special case of A) and C) into the Schrödinger equation:
- v°(x) + V(x)¥′(x) = Ev(x) (3)
2m
Put (3) into the form: (note: G₁=9+k₁ and G:=2•G]=18•G:)
[ ] + [ ] + + [ ] c
-ik x + [
+ [ ] eks
…….. + [ ] + [ ]-n
Дл
е
...
= 0
E) What is
What is general expression [
(the term multiplying with k = 2
)
27
[
12?
the term multiplying e' with ki
=
•12 - k₁
L
2x
[ ]?
the term multiplying e^ŕ with k,
=
•21 k₁ + G₁
1.
2x
[ ] 67
the term multiplying with k
=
• (-6) = k - G
1.
-0)
F) Since each Fourier term is independent, we will need
=
[ ] [ ]=0 [ ]=0 [ ]=0
etc...
Use this to write the equations for the set
€ (ki), c (ki), c(k), c(ks),
c (k-), c(k-is), e (k-24) ………..
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