Integrated intensity 1
Sunday, October 07, 2007
11:55 AM
Integrated intensity 2
Sunday, October 07, 2007
11:57 AM
Integrated intensity 3
Sunday, October 07, 2007
12:10 PM
Integrated intensity 5
Sunday, October 07, 2007
12:15 PM
Powder diffraction 1
Sunday, October 07, 2007
12:17 PM
Powder diffraction 2
Sunday, October 07, 2007
12:20 PM
Powder diffraction 3
Sunday, October 07, 2007
12:21 PM
extinction
Sunday, October 07, 2007
4:49 PM
Summary of elastic scattering
Sunday, October 07, 2007
5:38 PM
The laws of momentum and energy conservation
governing all diffraction and scattering
experiments are well known.
QF Ki Ks momentum
conservation
1012=12? d KI -2kikf costs
hw: Ei-Ef energy
conservation
Energy conservation law can be written
as
throw Ii-ka) tri since E=tric
2mn
for elastic scattering
Kit = Heft = 181
Let's consider a 2-D representation of reciprocal
space which shows an Ewald circle with radius k
Ki . A Q
320,
the
The circle has radius K and passes
through 2 points on the
diagram. The condition for Bragg
scattering is satisfied when
one-Paine f
181=121 Prissiness
Bragg's law
_
incise motion
obiter Instances.
pacing
ol
In diffraction experiment, the magnitude of Q is
controlled by adjusting the angle 2Os between ki
and kf. The orientation of Q is set by rotating the
sample.
IV
Any point in reciprocal space can be measured
by choosing the right ki, 2Os and the orientation
of the sample relative to ki.
which is also written as
ion
Ewa ld. construction
-given the incident wave vector Ta, a circle (or sphere)
is constructed
around i with radius Kl.
-Bragg peaks corresponding to reciprocal lattice vector
be will be
observed only if I gives a rec. lattice point on the
circle.
Methods for measuring B.
Sunday, October 07, 2007
5:39 PM
There are 3 methods for measuring Bragg
Scattering
1.Laue -using a single crystal and a white beam ;
crystal orientation is fixed
2.free rotation of a crystal -monochromatic beam
of neutrons on a crystal that can rotate
3.Powder method -beam of neutrons on a
powder sample
Debye-Scherrer cone for Bragg scattering
. no m
>r
E
K
Brillouin zone
Sunday, October 07, 2007
5:42 PM
INELASTIC scattering
In this case Kitties). The difference is the
energy transferred to the sample
In an experiment, one typically holds either ki or kf
constant.
.Q
K: i \
"t | "t
g,
For a single crystal, energies depend only on
relative momentum hq
defined within a Brillouin zone. It is convenient to
reference the momentum transfer to the nearest
reciprocal lattice vector
Q= Ergs
It Kinks,
hw = this 42-14) 70 Neutron
energy loss
2m I
Ena
z
k Kilkt,
hw = tri 4-4) to Neutron
energy gain
2m (
tilts office.
'
'2
(Stokes scattering)
excitation created
Ki . 1
@
(anti. Stokes scattering)
excitation annihilated.
Kr w
Brillouin zone
Sunday, October 07, 2007
5:44 PM
The first Brillouin zone is defined as the WignerSeitz primitive cell of the reciprocal lattice, or it
could be defined as the set of points in k space
that can be reached from the origin without
crossing any Bragg plane.
The second Brillouin zone is the set of points that
can be reached from the first zone by crossing only
one Bragg plane.
First zone
Third zone
sc
fcc
Second zone
bcc
Inelastic phonon scattering
Sunday, October 07, 2007
5:57 PM
Phonon 2
Sunday, October 07, 2007
6:01 PM
Phonon 3
Sunday, October 07, 2007
6:03 PM
Phonon 4
Sunday, October 07, 2007
6:05 PM
Phonon 5
Sunday, October 07, 2007
6:08 PM
Phonon 6
Sunday, October 07, 2007
6:09 PM
Phonon 7
Sunday, October 07, 2007
6:11 PM
Phonon 8
Sunday, October 07, 2007
6:13 PM
Phonon 9
Sunday, October 07, 2007
6:14 PM
Inelastic phonon scat.
Thursday, February 01, 2007
10:56 AM
Phonons in crystals
1.For a harmonic lattice (phonons have infinite
lifetime) the neutron energy has to match the
lattice dispersion for one-phonon scattering to
occur
2.Anharmonic effects modify the cross-sections
in a non-trivial manner
3.Multiphonon process are significant at
intermediate and large scattering vectors and
contribute to the structure of the one-phonon
processes
Assume crystal has N-atoms with 3N
modes.
Omro mind each atomis
a
}
ohm omit
Initial state, D. described by a set of
quantum I's (nynz,ns.. ..n3N)
final state, I, described lay n! nr... nor
The scattering is classified according to the change
in quantum numbers:
Elastic, Ni In: where i is oscillator index
One-phonon, Ni = Ni or all i except a
no In # f
Two-phonon, nil In: except for 2, B
oscillators
no Inx II n'p=np=c
where phonons can be created or absorbed
if we consider that the atoms are moving in a
coherent way producing normal modes, then
possible to get inelastic phonon scattering
The displacement of an atom is
given by will) l-lattice site
Position vector of atom is
harmonic oscillator
t-=-kx.
Ck-spring constant)
Reft Ute)
One would need to then solve the
Hamiltonian for the equation
of motion
H = PI r KK } very
simplified!
2m
From this obtain lattice dispersion
frequencies
wigs where j has 3 values
(b/c of what?)
2B.
long.
Iran.
Example of an fcc lattice:
W ^ went. : I I
: as
II!
_ 2th a To
This is along (100)
W ^ ; reclined | w ^ \
long transverse
I|
'g
This is along (101)
$
extend to the next
Britain gone
of
Phonon 11
Sunday, October 07, 2007
10:43 PM
Phonon 12
Sunday, October 07, 2007
10:59 PM
Detailed balance
Sunday, October 07, 2007
11:00 PM
Let's consider the case where neutron loses energy in
the scattering process
trio = alias this contributes to SCtr,w)
E
lattice
os is @
neutron
Let's consider the case where neutron gains energy in
the scattering process
of at
Thw
thew
this contributes to S (-15>-20)
neutron lat
sc-ixia) = eon
tire
w
SCia,w)
There is less chance for the system to be in the higher
energy state
initially and then undergo a transition to the lower state.
This is
why Sinew) is multiplied by E Btw term.
If Detailed balance correction
Phonon 14
Sunday, October 07, 2007
11:02 PM
Phonon 15
Sunday, October 07, 2007
11:03 PM
Phonon 16
Sunday, October 07, 2007
11:05 PM
Phonon 17
Sunday, October 07, 2007
11:07 PM
Phonon 18
Sunday, October 07, 2007
11:09 PM
Phonon 19
Sunday, October 07, 2007
11:11 PM
Phonon 20
Sunday, October 07, 2007
11:19 PM
Phonon 21
Sunday, October 07, 2007
11:20 PM
Phonon 22
Sunday, October 07, 2007
11:23 PM