Physics with Neutrons SS 2015 - sces.ph.tum.de file12/05/2015 E-mail:[email protected] Web:...
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12/05/2015 E-mail:[email protected] Web: www.sces.ph.tum.de Physics with Neutrons SS 2015 Peter Böni Physik-Department E21 Technische Universität München D-85748 Garching VL5, Physics with Neutrons II, May 11, 2015 http://www.ph.tum.de/studium/mh/entry/?mid=PH2054
Transcript of Physics with Neutrons SS 2015 - sces.ph.tum.de file12/05/2015 E-mail:[email protected] Web:...
12/05/2015
E-mail:[email protected]
Web: www.sces.ph.tum.de
Physics with Neutrons
SS 2015
Peter Böni
Physik-Department E21
Technische Universität München
D-85748 Garching
VL5, Physics with Neutrons II, May 11, 2015
http://www.ph.tum.de/studium/mh/entry/?mid=PH2054
He
d
v
VdD
g
3
2
22)(
He
2
Dispersion of Monoatomic Chain
0.0
0.2
0.4
0.6
0.8
1.0
1.23. BZ
4/a3/a/a 2/a0-/a
File Phonons.opj
Ph
on
on
En
erg
y (
un
its
0)
Momentum Q
-2/a
Phonon Dispersion1. BZ 2. BZ2. BZ
a a a Fn
un-2 un-1 un un+1
force constant k
force on atom at position n: Fn = k(un+1 – un) - k(un – un-1)
3
Dispersion May Depend on Propagation Direction
f1
f2
f1
f1
f2 f2 f2 f2
q1
q2
polarization of modes only defined for high symmetry directions
4
Phonon Modes: Longitudinal
q es
5
Phonon Modes: Transverse T1
q
es
6
Phonon Modes: Transverse T2
q
es
7
Dispersion of Phonons in Bravais Crystal
0.0
0.2
0.4
0.6
0.8
1.0
1.2
LA
TA1
TA2
17.01.2011
File Phonons.opj
Ph
on
on
En
erg
y (
un
its o
f
0)
Momentum Q (r.l.u)
/a3/4a/4a /2a0
Acoustic Phonons in Bravais Crystal
simple structures behave along symmetry directions like linear chains
8
The phonon dispersion curves of Al. The experimental points at 80 K are from Stedman and Nilsson115 and those at 300 K from Yarnell et
al.137. The solid lines were obtained by fitting the data to an eight-nearest-neighbor axially symmetric Born-von Kármán model.
Al: n = 8 [100] [110] [111]
T = 80K
L
T L
T2 T1
L
T
TA degenerate:
• [1 0 0]:
- es || [0 1 0]
- es || [0 0 1]
• [1 1 1]:
- es || [2 -1 -1], [-1 -1 2]
L L L
T T
T2
T1
Dispersion of Aluminum (Al)
TA not degenerate:
• [110]:
- es || [1 -1 0]
- es || [0 0 1]
• general directions
T = 300K
9
Linear Mono Atomic Chain: More Neighbors
un-2
a
un-1 un un+1
a a spring constant k
next nearest neighbor interactions:
2sin 2 qa
2a )(sin2 qa
10
lines with red labels:
• first 5 Fourier components
Dispersion of Lead (Pb)
Born – von Kármán model:
Pb:
• crystal structure: fcc
• n = 12 neighbors
• fcc
M²(q) for LA branch in [100] direction
[100]
qa/2 (reduced lattice units)
1
2
4
3 5
M
²(q
) (d
ynes
cm
-1)
[100]
----- 5 planes
12 planes
11
Linear Chain with 2-Atomic Basis
M M M Mm m m m
2 3n -
u2 2n- u2 1n- u2n u2 1n+u2 -3n
2 2n - 2 1n + 2 2n + 2 3n + 2 4n + 2n2 - 1nf
a/2 a/2 a/2 a/2 a/2 a/2 a/2
a
2
2
2
21212 )2(t
umuuuk n
nnn
• equations of motion:
2
12
2
12222 )2(t
uMuuuk n
nnn
• solutions: , )2(
221
)(nqati
n etu
qanti
n etu)12(
1221
)(
• dispersion relation: )(sin4)()(2
22
0
2
0
22
0
2
0
2
0
2
0
2 qa
M
k2
0
m
k2
0
12
2-Atomic Basis: Dispersion Relation
0
2
4
6
8
0 /(2a)-/(2a) /a-/a
21/2
(0
2+
0
2)1/2
21/2
0
21/2
0
acoustic: -
optic: +
Dispersion for frequencies 0 = 3,
0 = 4
File 02_Phonon_n.opj
Fre
qu
en
cy
Wavenumber q
1. BZ
gap
)(sin4)()(2
22
0
2
0
22
0
2
0
2
0
2
0
2 qa
+: optic branch
-: acoustic branch
13
Elongation of Acoustic Modes
acoustic mode, q = 0
M M M Mm m m m
periodicity: a
heavy masses low frequency acoustic
a
acoustic mode, = 2 /(2 ) = /q a a
periodicity: 2a
z z
14
optic mode, = 2 /(2 ) = /q a a
periodicity: 2a
optic mode, q = 0
M M M Mm m m m
periodicity: a
Elongation of Optic Modes
light masses high frequency optic m
k2
0
a
z z
15
Dispersion Relation of Diamond
TA
LA
LO
TO
TA
LA
LO
TO
After: J. L. Warren et al., Phys. Rev. 158, 805 (1967).
diamond: fcc, basis: (000), (¼ ¼ ¼) , T = 296 K, shell model fits
For X-ray data see: E. Burkel, Inelastic Scattering of X-Rays with Very High Energy Resolution, Springer Berlin (1991)
2.0
10
14 r
ad/s
= 1
31
.7 m
eV
16
Dispersion: m M
0
2
4
6
8
0 /a-/a /a-/a
Dispersion M --> m
File UNTITLED.opj
fre
qu
en
cy
wave number q
• lattice constant for crystal with m M: a
• lattice constant for crystal with m = M: ½a (gap vanishes)
intensity vanishes
intensity is large
intensity is large
