Embed Size (px)
Citation preview
'
)0(~
)(~
'41'*'*
)0(~
')(
~
''2
121
2
2
22
'
'
))1((1
2
1),(
)0()()()(1
2
1),(
),('
),()ˆˆ('
'
jjT
itijjjj
jN
jN
jjtinuc
Ti
jti
jjj
jjti
mag
nucmag
jj
jj
eeIIbbbbN
dteS
eJetJgFgFN
dteS
Sk
kNS
mc
e
k
kN
dEd
d
RκRκ
RκRκ
κ
κ
κκκκ
Unpolarisierte Neutronen- Van Hove Streufunktion S(κ,ω)
'
'41'*'*
'' ))1((1
)(jj
iijjjj
jN
jN
WWjjelnuc
jjjj eIIbbebbN
S RκRκ
)()(~
ttjjj
uRR
''
''21
'21 )()(
1)( jjjj WWii
TjjTjjj
j
elmag eeJgFJgF
NS RκRκ
Aufteilung von S: elastische – inelastische Streufunktion
inelmag
elmagmag
inelnuc
elnucnuc
SSS
SSS
)(2
.../2
)()(
1)'(
')'(1
)(
...
')'(......)(
0
0
/2)'(
2/
2/
/'2
0
/2
2/
2/
/'2
0
/2
qa
eLxqa
c
xcx
eL
xx
dxexfeL
xf
dxexffwithefxf
n
iqna
n
Lxxin
L
L
Lnxi
n
Linx
L
L
Lnxin
n
Linxn
A shortExcursionto FourierAnd DeltaFunctions ....
it follows by extending the range of x to more than –L/2 ...L/2 andgoing to 3 dimensions (v0 the unit cell volume)
τ
GκGκ τκ
..0
3
'
)()2(
'
lattrezG
kk
ii
vNe kk
ddd
dN
B
d
Wdd
B
dd
WWidd
Blattrez
elnuc
IIbN
ebbN
eebbNv
S
d
dddd
)1(1
)(
1)(
1)(
)2()(
41
2
222
'
)('
*
..0
3''
BBκ
τ
τκ
Neutronen – Diffraktion
'412'*
)1(||11
)( ''
jj jjj
jN
WWiijjelnuc IIb
Neebb
NS jjjj RκRκ
Gitter G mit Basis B: dkjkdj BGR
)........(
„Isotopen-inkoherente-Streuung“
„Spin-inkoherente-Streuung“
unabhängig von κ:
Gitterfaktor Strukturfaktor
ein Element(NB=1): )1()(44 41222
dd
dN
incel
nucinc
elnuc IIbbbN
d
d
i 2||4 bc
ddd
dN
B
W
ddd
B
dd
WWidd
Blattrez
elnuc
IIbN
ebbN
eebbNv
S
d
dddd
)1(1
)(
1)(
1)(
)2()(
41
2
222
'
)('
*
..0
3''
BBκ
τ
τκ
Gitterfaktor Strukturfaktor
The Nobel Prize in Physics 1994
"In simple terms, Clifford G. Shull has helped answer the question of where atoms are, ...“, (Nobel citation)
Inco
min
g
Neutr
on
τ
Oa*
c*
Bragg’s Law in Reciprocal Space (Ewald Sphere)
Scatte
red
Neutr
on
Bragg’s Law in Reciprocal Space (Ewald Sphere)
Neu
tron
C
τ
O
Reflecting Plane
sin OB τ= sin = O τ /OB = O τ /(2/
sin = (O τ/2)But since τ is a reciprocal lattice point, the length O τ is by definition equal to 2/dhkl
sin = (1/2dhkl)
2dhkl sin =
sin = (1/2dhkl) = (1/2)(1/dhkl) = (1/2)hkl
Pulver- diffraktometrie
D1B
guide hall n°1, thermal guide H22
monochromator
take-off angle 44.22°
crystal pyrolytic graphite (002)
. wavelength 2.52 Å
. flux at sample/n cm-2s-1 6.5 x 106
crystal Germanium (311)
. wavelength 1.28 Å
. flux at sample/n cm-2s-1 0.4 x 106
max beam size 5 x 2 cm2
angular range 2 -20° ... 144°
detector
3He multidetector containing 400 cells
angular range 2 80°
radius of curvature 1.525 m
detector efficiency 60 % at = 2.52 Å max diameter / mm available around the sample
600
sample environment
cryostat 1.7 ... 300 K
furnace < 800 °C
furnace < 2500 °C by special arr
electromagnet 1 T; 22 mm vertical or horizontal gap
GdCu2In
|κ|[Å-1] 0 1 2 3 4 5 6
I(κ)
[co
unts
]
2||)( 21
SFTLI SF.. StrukturfaktorL ... Lorentzfaktor (betont kleine Winkel) Einkristall: 1/sin2θ Pulver Zyl.:1/(sin2θ.sinθ)
T Transmissionskoeffizientγ Korrektur für Extinktion
Pulver-diffraktometrie
)sin(4
2θ.... Streuwinkel Detektor
dIdI )()(
7C2, LLB Saclay
GdCu2In#lambda= 0.58 A#thetamax=18#nat=4 nonmagnetic atoms in primitive crystallographic unit cell:#[atom number] x[a] y[b] z[c] dr1[r1] dr2[r2] dr3[r3] [Gd] 0 0 0 0 0 0[Cu] 0.25 0.25 0.25 0.25 0.25 0.25[Cu] 0.25 0.25 0.75 0.75 0.75 -0.25[In] 0.5 0.5 0.5 0.5 0.5 0.5# a=6.62 b=6.62 c=6.62 alpha= 90 beta= 90 gamma= 90# r1x= 0 r2x= 0.5 r3x= 0.5# r1y= 0.5 r2y= 0 r3y= 0.5 primitive lattice vectors [a][b][c]# r1z= 0.5 r2z= 0.5 r3z= 0# nofatoms=1 number of atoms in primitive unit cell
)(1
)()2(
'
)('
*
..0
3''
dd
WWidd
Blattrez
cohel
nucdddd eebb
NvS BBκ
τ
τκ
Gitterfaktor Strukturfaktor
h k l d[A] |kappa|[A^-1]2theta Ikern imag itot |sf| lpg } 1.000 1.000 1.000 3.82206 1.64389 8.703 11.661 0.000 11.661 24.966 87.101 -1.000 -1.000 -1.000 3.82206 1.64389 8.703 11.661 0.000 11.661 24.966 87.101 -1.000 1.000 1.000 3.82206 1.64389 8.703 11.661 0.000 11.661 24.966 87.101 1.000 -1.000 -1.000 3.82206 1.64389 8.703 11.661 0.000 11.661 24.966 87.101 1.000 -1.000 1.000 3.82206 1.64389 8.703 11.661 0.000 11.661 24.966 87.101 -1.000 1.000 -1.000 3.82206 1.64389 8.703 11.661 0.000 11.661 24.966 87.101 1.000 1.000 -1.000 3.82206 1.64389 8.703 11.661 0.000 11.661 24.966 87.101 -1.000 -1.000 1.000 3.82206 1.64389 8.703 11.661 0.000 11.661 24.966 87.101 0.000 2.000 0.000 3.31000 1.89820 10.053 0.025 0.000 0.025 1.336 65.389 0.000 -2.000 0.000 3.31000 1.89820 10.053 0.025 0.000 0.025 1.336 65.389 0.000 0.000 2.000 3.31000 1.89820 10.053 0.025 0.000 0.025 1.336 65.389 0.000 0.000 -2.000 3.31000 1.89820 10.053 0.025 0.000 0.025 1.336 65.389 2.000 0.000 0.000 3.31000 1.89820 10.053 0.025 0.000 0.025 1.336 65.389 -2.000 0.000 0.000 3.31000 1.89820 10.053 0.025 0.000 0.025 1.336 65.389 0.000 2.000 2.000 2.34052 2.68446 14.235 83.558 0.000 83.558 109.804 32.822 0.000 -2.000 -2.000 2.34052 2.68446 14.235 83.558 0.000 83.558 109.804 32.822 2.000 2.000 0.000 2.34052 2.68446 14.235 83.558 0.000 83.558 109.804 32.822 -2.000 -2.000 0.000 2.34052 2.68446 14.235 83.558 0.000 83.558 109.804 32.822 -2.000 0.000 2.000 2.34052 2.68446 14.235 83.558 0.000 83.558 109.804 32.822 2.000 0.000 -2.000 2.34052 2.68446 14.235 83.558 0.000 83.558 109.804 32.822 2.000 0.000 2.000 2.34052 2.68446 14.235 83.558 0.000 83.558 109.804 32.822 -2.000 0.000 -2.000 2.34052 2.68446 14.235 83.558 0.000 83.558 109.804 32.822 -2.000 2.000 0.000 2.34052 2.68446 14.235 83.558 0.000 83.558 109.804 32.822 2.000 -2.000 0.000 2.34052 2.68446 14.235 83.558 0.000 83.558 109.804 32.822 0.000 -2.000 2.000 2.34052 2.68446 14.235 83.558 0.000 83.558 109.804 32.822 0.000 2.000 -2.000 2.34052 2.68446 14.235 83.558 0.000 83.558 109.804 32.822
Beispiel 2
• In einem elastischen Streuexperiment beträgt die Einfallsenergie 63 meV. Die Gitterkonstante der kubischen Probe beträgt 0.314 nm. Kann der (430) Reflex in diesem Streuexperiment vermessen werden ?
Neutron – PhotonStreuquerschnitte
Vorteile von Neutronen:1. Kontrast bei benachbarten Elementen – man sieht z.B. Überstrukturen2. Leichte Elemente besser nachweisbar3. Isotope können unterschieden werden
Laue Methode• Einkristalle •„weißer“ Strahl• Film oder Flächendetektor hinter der Probeschnelles Erkennen der Symmetrie -wird zum Orientieren von Einkristallen benutzt
VIVALDI
very-intense vertical-axis Laue diffractometer
4-Kreismethode•Einkristalle •monochromatischer Strahl• ein Detektor• EK in beliebige Richtungen orientierbar (Eulerwiege)
D10ILL
φχ
ω
Flugzeitmethode
Spallationsquelle (gepulst)
Probe
Det
2θ
•Streuwinkel fest (Vorteil z.B. bei Druckzellen)• |k| wird variiert (kein Monochromator) über die Zeit (zuerst kommen die raschen, dann die langsameren Neutronen) • bessere Nutzung der Quelle (keine Monochromator-verluste)•Auflösung umso besser, je größer Abstand zur Quelle (HRPD: 90m)
Time-of-flight
Bragg equation - 2dhklsin =
Two basic equations:
mv
h
t
Lv
where m,v = mass, velocity of neutron
L = length of flight path t = time of flight of neutron
Time-of-flight equation
Combine: sind2
mL
ht
sindh
mL2t
L is a constant for the detector, h, m are constants so:
t d
d-spacings are discriminated by the time of arrival of the neutrons at the detector
The biggest error in the experiment is where the neutrons originate
This gives an error in the flight path, L
typical value ~5cm
d
d
t
t
L
L
Hence as L increases, error in d is reduced - resolution of the instrument is improved
e.g. instrument at 10m compared to instrument at 100m
100m = HRPD, currently highest resolution in the world
HRPD, GEM
Sample area collimators and detectors on HRPD. Neutrons enter via the yellow flight tube on the left.
GEMGeneralpurposeMaterialsDiffraktometer
p-dichlorobenzene (DCB)
refined structure
.
a/h
c/l
b/k
A
C
dhkl
C = ah
bk
-
h ka b -C . hkl= ( ( . (h a * + k b * + l c *)=
bh ka -
. (h a * + k b * + l c *) (h a * + k b * + l c *) =.
hh
+ 0 + 0 – (0 + kk
+ 0) = 1 – 1 = 0
Therefore hkl is perpendicular to C . In the same way one can showthat it is perpendicular to A, therefore perpendicular to the plane
hkl = h a * + k b * + l c *
|hkl| n = ha* + kb* + lc* ha* + kb* + lc*
|hkl|n =
ahdhkl = cos = a
h. n =
ah
ha* + kb* + lc*
|hkl|. =
|hkl|
