9/28/2015phy 711 fall 2015 -- lecture 151 phy 711 classical mechanics and mathematical methods...
TRANSCRIPT
PHY 711 Fall 2015 -- Lecture 15 19/28/2015
PHY 711 Classical Mechanics and Mathematical Methods
10-10:50 AM MWF Olin 103
Plan for Lecture 15:
Continue reading Chapter 4
1. Normal modes for extended one-dimensional systems
2. Normal modes for 2 and 3 dimensional systems
PHY 711 Fall 2015 -- Lecture 15 29/28/2015
PHY 711 Fall 2015 -- Lecture 15 39/28/2015
22323
21212
233
222
211
2
1
2
1
2
1
2
1
2
1
xxkxxk
xmxmxmL
Example – linear molecule
1x2x
3x
m1 m2 m3
PHY 711 Fall 2015 -- Lecture 15 49/28/2015
2333
321231222
1211
2
:motion of equations Coupled
xxkxm
xxxkxxkxxkxm
xxkxm
23332
321222
12112
2
Let
XXkXm
XXXkXm
XXkXm
eX(t)x t-iωii
0 0 01 1 1 2 2 1 12 3 3 1 12 23
2 22 2 21 1 2 2 3 3 2 1 3 2
Let:
1 1 1 1 1
2 2 2 2 2
x x x x x x x x x
L m x m x m x k x x k x x
PHY 711 Fall 2015 -- Lecture 15 59/28/2015
m1 m2 m3
m1 m2 m3
m1 m2 m3
01
Om
k2
CO m
k
m
k 23
PHY 711 Fall 2015 -- Lecture 15 69/28/2015
1ixix 1ix
m m m
Consider an extended system of masses and springs:
0position mequilibriu its to
relative measured is coordinate masseach :Note
ix
.conditionsboundary using treatedbe will
and masses; have wefact,In :Note
2
1
2
1
10
0
2
11
2
N
N
iii
N
ii
xxN
xxkxmVTL
PHY 711 Fall 2015 -- Lecture 15 79/28/2015
0 and 0
2
1
2
1
10
0
2
11
2
N
N
iii
N
ii
xx
xxkxmVTL
2
2
2
2
:equations Lagrange-Euler From
1
11
1232
121
NNN
iiii
xxkxm
xxxkxm
xxxkxm
xxkxm
PHY 711 Fall 2015 -- Lecture 15 89/28/2015
2sin
4
2cos2
2
)( :Try
0ith w2
:equations Lagrange-Euler From
22
2
2
1011
qa
m
k
qam
k
Aeeem
kAe
Aetx
xxxxxkxm
iqajtiiqaiqaiqajti
iqajtij
Njjjj
PHY 711 Fall 2015 -- Lecture 15 99/28/2015
0)(
0)(
:conditionsboundary Impose
)(
:solution General
2sin
4 )( : thatNote
2sin
4 )( :Try
0ith w2
:continued -- equations Lagrange-Euler From
111
0
22
22
1011
NiqatiNiqatiN
titi
iqajtiiqajtij
iqajtij
iqajtij
Njjjj
BeAetx
BeAetx
BeAetx
qa
m
kBetx
qa
m
kAetx
xxxxxkxm
PHY 711 Fall 2015 -- Lecture 15 109/28/2015
1
0,1,2 where1
01sin
0)(
0)(
0)(
:continued -- conditionsboundary Impose
111
111
0
N
νπqa
Nqa
Nqa
eeAetx
-A B
BeAetx
BeAetx
NiqaNiqatiN
NiqatiNiqatiN
titi
PHY 711 Fall 2015 -- Lecture 15 119/28/2015
N
Nm
k
N
jiAetx ti
j
,2,1
are valuesunique trivial,-non that Note
12sin
4
1sin2)(
parameter integer for solution -- Recap
22
PHY 711 Fall 2015 -- Lecture 15 129/28/2015
qa
Example for N=4:
Note that solution form remains correct for N∞
4
sin2 1
k
m N
4 /k m
4 / sinqa k m qa
PHY 711 Fall 2015 -- Lecture 15 13
mm M
9/28/2015
mm M
Consider an infinite system of masses and springs now with two kinds of masses:
,,position mequilibriu its to
relative measured is coordinate masseach :Note00ii yx
0
2
0
2
10
2
0
2
2
1
2
1
2
1
2
1
iii
iii
ii
ii xykyxkyMxm
VTL
ix iy 1iy1ix 2ix
PHY 711 Fall 2015 -- Lecture 15 149/28/2015
qajiti
j
qajitij
jjjj
jjjj
iii
iii
ii
ii
Bety
Aetx
xyxkyM
yxykxm
xykyxkyMxm
VTL
2
2
1
1
0
2
0
2
10
2
0
2
:solution Trial
2
2
:equations Lagrange-Euler
2
1
2
1
2
1
2
1
0
21
1222
22
B
A
kMek
ekkmqai
qai
PHY 711 Fall 2015 -- Lecture 15 159/28/2015
mM
qa
Mmk
M
k
m
k
B
A
kMek
ekkmqai
qai
2cos211
:Solutions
021
12
22
2
22
22
w
qa/p
Mm
Mm
PHY 711 Fall 2015 -- Lecture 15 169/28/2015
Eigenvectors:
1
0
0
1
2
2
:2
For
1
1
1
1
22 0
:0For
NB
AN
B
A
m
k
M
k
qa
NB
AN
B
A
M
k
m
k
qa
PHY 711 Fall 2015 -- Lecture 15 179/28/2015
22
12 2
,
2 22
12 2
, ,
( , ) ( , )
eq eq
eq eq eq eq
eq eq eq
x y
eq eq eq
x y x y
VV x y V x y x x
x
V Vy y x x
xy y
y y
V(x,y)
Potential in 2 and more dimensions
PHY 711 Fall 2015 -- Lecture 15 189/28/2015
Example – normal modes of a system with the symmetry of an equilateral triangle
1
3
2u1
u3
u2
k k
km
m
m
Degrees of freedom for
2-dimensional motion:
2 6N
PHY 711 Fall 2015 -- Lecture 15 199/28/2015
Example – normal modes of a system with the symmetry of an equilateral triangle -- continued
1
3
2u1
u3
u2
13
2
13 13 3 1 13
2
13 3 1
13
2
3 1 3 1
Potential contribution for spring 13:
1
2
1
2
1 1 3
2 2 2x x y y
V k
k
k u u u u
u u
u u
13 13
1 3ˆ ˆ
2 2
x y
PHY 711 Fall 2015 -- Lecture 15 209/28/2015
12 13 23
22
13 3 112 2 1
12 13
2
23 3 2
23
2
2 1
2
3 1 3 1
2 3
Potential contributions:
1 1
2 2
1
2
1
2
1 1 3
2 2 2
1 1 3
2 2
x x
x x y y
x x
V V V V
k k
k
k u u
k u u u u
k u u
u uu u
u u
2
2 32 y yu u
Example – normal modes of a system with the symmetry of an equilateral triangle -- continued
PHY 711 Fall 2015 -- Lecture 15 219/28/2015
Example – normal modes of a system with the symmetry of an equilateral triangle -- continued
m
k
1
2
3
1
2
3
x
x
x
y
y
y
u
u
u
u
u
u
= w2
1
2
3
1
2
3
x
x
x
y
y
y
u
u
u
u
u
u
PHY 711 Fall 2015 -- Lecture 15 229/28/2015
Example – normal modes of a system with the symmetry of an equilateral triangle -- continued
1
3
2u1
u3
u2
k k
km
m
m
m
k2
PHY 711 Fall 2015 -- Lecture 15 239/28/2015
3-dimensional periodic lattices Example – face-centered-cubic unit cell (Al or Ni)
Diagram of atom positions
Diagram of q-space n(q)
PHY 711 Fall 2015 -- Lecture 15 249/28/2015
From: PRB 59 3395 (1999); Mishin et. al. n(q)
PHY 711 Fall 2015 -- Lecture 15 259/28/2015
Lattice vibrations for 3-dimensional lattice
Example: diamond lattice
Ref: http://phycomp.technion.ac.il/~nika/diamond_structure.html
PHY 711 Fall 2015 -- Lecture 15 269/28/2015
bb
baba
aaaa
aaa
a
UUU 0
,
2
00
0
02
1
:equilibriunear function energy Potential
:positions at the located Atoms
RRRR
RRRR
uRR
R
bk
abjk
kjba
aj
ja
aja
aj
aj
bk
abjk
kjba
aj
a
bk
aj
abjk
uDuUumuuL
uDuUU
UD
a
,,,0
,
2
,,,0
2
2
1
2
1,
2
1
thatso
:Define
0
R
RRR
PHY 711 Fall 2015 -- Lecture 15 279/28/2015
replicas denotes
and sites unique
denotes where :Details
1
:formSolution
:motion of Equations
2
1
2
1,
0
,
,,,0
,
2
0
T
τTτR
Rq
aaa
itiaj
a
aj
kb
bk
abjk
aja
bk
abjk
kjba
aj
ja
aja
aj
aj
a
eAm
tu
uDum
uDuUumuuL
PHY 711 Fall 2015 -- Lecture 15 289/28/2015
q
q
q Tq
T
ττq
curves" dispersion" Find
sites. atomic unique
overonly issummation heequation t In this
:equations Eigenvalue
:Define
,
2
kb
bk
abjk
aj
i
ba
iabjkab
jk
AWA
emm
eDW
ba
PHY 711 Fall 2015 -- Lecture 15 299/28/2015
B. P. Pandy and B. Dayal, J. Phys. C. Solid State Phys. 6 2943 (1973)