Transcript Kein Folientitel

Neutrons
versus
X-rays
A. Magerl
Crystallography and Structural Physics
University of Erlangen-Nürnberg
Novosibirsk 10. Oct. 2011
Content
What is it all about?
YBa2Cu3O7
High temperature superconductor
Y3+
Ba2+
Cu
O2-
Information on an atomic/molecular scale:
From structure
and dynamics
to functionality
Why diffraction?
Interference
Let us do diffraction. We shine waves onto our sample and these are
scattered from each atom, i. e. each atom becomes a little (radio-) emitter.
These waves are coherent and they interfere with each other. The resulting
interference pattern is representative for the arrangement of the atoms
(phase relations between the waves) and for the kind of atoms present
(amplitudes of the waves).
scattering processes
I
I
w
w0
elastic
w0
inelastic
w
I
-D
+D w
w0
I
quasielastic
G
w0
w
Why
1
Å radiation
diffraction?
For particles with mass: Ekin = ½ mv2
p = 2Ekin / v = mv
For particles without mass: (photons)
photons
neutrons
E = hν = ћω,
p = hν/c = h/λ = ћk
In the following we will concentrate on X-rays and neutrons
Why diffraction?
Wave-particle
dualismus
Waves and particles are two representations of the same ‘thing’. Sometimes,
‘it’ appears quantized (photoelectric effect or counting in a detector),
sometimes ‘it’ appears as a wave (interference). The two representations are
connected by the de Broglie equation (de Broglie 1924), which relates the
momentum p (particle property) with the wavelength λ (wave property)
pλ=h
or
p=ћk
h = Planck‘s constant = 6,63 * 10-34 Js = 4.14 *10-15eVs
Note, it is important and elegant to chose the appropriate physics picture
when describing specific phenomena in natural science.
The brilliance of x-ray sources since their discovery in 1895
1027
The development
of the brilliance
of x-ray sources
LINAC driven sources
1024
(forth generation)
1021
undulators
third generation
1018
wigglers
bendingmagnets
1015
1012
second generation
first generation
1012
109
sun
106
109
rotating anode
Cray T90
lamp
106
x-ray tube
103
Cray 1
candle
ILL
Since 40 years the increase
in brilliance is 0.3% per day
FRM II
103
1900
1940
1980
year
2020
Whyneutron
The
diffraction?
case
Why diffraction?
Interaction
with matter
Absorption: The incoming radiation vanishes, the energy
becomes removed from the wave field, the material is
damaged or warms up.
Scattering: A new wave field is created without a change
of the energy contained in all wave fields (at least for
elastic scattering). In general the energy in the direction
of the incoming wave becomes reduced while an energy
flow in new direction(s) builds up.
X-rays:
σtot = σphotoabs + σCompton + σpair + σscattering
The dominant interaction is through σphotoabs. It depends on
the photon energy and on the element (characteristic
absorption edges).
The penetration of 1-Å x-rays into matter is limited by
absorption, samples have to be small, i. g. < 1mm.
Neutrons:
σtot = σnuclearabs + σscattering
Absorption and scattering are similar (with exceptions);
The scattering amplitude does not depend on the energy.
The penetration into matter is i. g. higher than for X-rays
Whyscattering
The
diffraction?
process
 = 1/r expi(k1r-wt+)


0 = expi(k0z-wt)
=2/k
Born approximation: An incident plane wave with amplitude A = 1 and wavenumber k0 is
partially scattered by an electron (nucleus, etc.) into direction k1 with |k0| = |k1|.
e
 
i ( k 0 z - wt )
 

 i (k z - wt )
e i ( k 1r - wt +  ) 
0
A e
+ f ( )

r


f(Ω) is called scattering length. It describes the amplitude of the scattered wave and it
contains all the physics. E. g. if f(Ω) is big the interaction is strong.
The amplitude factor A is needed for energy normalisation. We will set A=1 at present.
Note, an interaction in Fourier space is described as scattering. The scattering length f(Ω) = f(δ,φ)
is the Fourier transform of the interaction potential.
 iQ r 
f () ~  V(r )e
dr
with
 
  
4
Q  k1 _ k 0 
sin 
2

Why diffraction?
Amplitude
of scattered wave
Scattering
length
X-rays: Thomson scattering:
Every accelerated charge emits elmag. radiation. Electrons are
(loosely) bound with a binding frequency f « x-ray frequency ω0.
The electric field of the x-ray wave induces oscillations of the
electrons (the motion of the nucleus is neglected) which results
in dipole radiation. This can be calculated exactly within



electrodynamics: eE(t )  mx(t ) + mx (t ) + mf 2x
d(t) = e.x(t)
x(t)
R
for non-polarized light:
E(z,t)

k0
r0  e
2
2
d   f   2 
d
4
2 
1 
1
+
cos
 

2 4 2 


m c
2  
2 
 r0  1  1 + cos   

2 
e
= 2.82•10-15 [m] is the classical electron radius.
mc
Nota: re « 1 Å, i. e. we have a weak scattering process (Born approximation).
Neutrons:
No simple dependence on the atomic number, on mass, etc.; measure it!
Scattering cross sections
Neutrons
X-rays
(sin θ)/λ = 0
(sin θ)/λ = 0,5 Å-1
Atomic weight
Scattering cross sections
Why diffraction?
Beyond
Thomson
The resonance behavior of a classical pendulum:
damping 10%
x
x0
Visible region,
w0
why is the sky blue? 0
w
X-ray region,
re = constant

-
Absorption region with
dispersion corrections
f1 and f2
Beyond the Thomson approximation
The scattering length: f(Q) = f0(Q) + f1(E) + if2(E)
Cu:
Z=29
Ionisation energy = 8920 eV
from http://www-cxro.lbl.gov/optical_constants/asf.html
Anomalous dispersion makes specific atoms particularly visible
The (atomic) form factor f(Q)
Interference of first order: The interference pattern of the electrons within one atom
is called the form factor f(Q). It is the Fourier transform of the electron density
distribution in an atom ρ(r):

 iQ r 
f (Q)   (r ) e dr
For simplicity we assume a spherical symmetric charge distribution


f (Q)   (r ) eiQ rcos 2r 2dr sin d 

4  (r ) r 2
0
sin(Qr )
dr
Qr


with
1
f (Q  0)  4  (r ) r 2 dr  4 0 R 03   0 V0  Z
3
0
f(Q) describes the amplitude of the scattered wave at the wave vector transfer Q.
The form factor can be given as a function of Q or of (sinθ/λ).
Full interference is always obtained for Q=0. f(Q=0) equals the number of electrons of
the atom, each single electron scatters with the Thomson cross section.
Form factor f(Q)
Formfaktor
f of single
Form factor
für
Einzelelektronen
electrons
and total
und
Gesamtformfaktor
form factor
Ne
Na1+
K1+
F1Ne
Na
Atomic form factor for F (9 electrons),
Ne (10 electrons) and Na (11 electrons),
and for Na+ and F- (both 10 elektrons).
In addition, the
indicated for F.
thermal
factor
is
1s
2s
3s
Ensemble of atoms - unit cell
b
Interference of second order: adding up
the waves originating from a finite particle
ensemble (unit cell of a crystal): structure
factor (of the unit cell) F(Q):
J
for x-rays: F(Q )   f j (Q ) e

iQ r j
j 1
xj
a
yj
r2
r1




with the fractional coordinates rj  x ja + y jb + z jc
In crystals, the arrangement of the J atoms is named the motive (decoration) of the
unit cell, and F(Q) represents the FT of the charge density in the unit cell.
F(Q)
for x-rays
Q
In principle, this is all we want to know
Single crystal
b
r2
r2
a
r1
na = 0
nb = 0
r2
r1
na = 0
nb = 1
r1
na = 0
nb = 2
r2
r2
r1
na = 1
nb = 0
r2
r1
na = 1
nb = 1
r1
na = 1
nb = 2
Interference of third order: adding up the waves originating from a crystalline structure,
i. e. of a periodic arrangement of unit cells, with translation vectors a, b, and c. This is
called the lattice sum.
Na ,Nb ,Nc
The phase is

e




iQ n a a + n b b + n c c 
n a  0, n b  0, n c  0
Na
 e
na  0
 
iQn a a
Nb
 e
nb  0
 
iQn b a
Nc
 e
nc  0
 
iQn c a
Bragg peaks
In 1 dimension:
Geometrical
sum with
solution
Na
 
iQnaa
e
na 1
Na  
iQ n a a
e
n a 1
N
 
iQna
 e
n0
N -1  
i Qn a
  e
n0


iQa
1+ e
+e

2iQa
+e

3iQa
+ .... + e

NiQa




i NQa / 2  - i NQa / 2
i NQa / 2 
-e
e

1 - ei NQa e

 
 



i Qa / 2  - iQa / 2
iQa / 2 
1 - e iQa
e

sin(NQa / 2)

sin(Qa / 2)
e

-e

N
 
  Qa  n 2 


Discussion lattice sum
(sinNx / sinx)2
The squared lattice sum has within one period:
1 main maximum & (N-2) secondary maxima & (N-1)
zero points
phase x=Qr [rad]
All maxima have the same height
The width of the main maximum is 2/N, for secondary
maxima 1/N.
For larger crystals the width becomes rapidly very
small (Bragg peaks). Nanomaterials may retain a
finite width.
The lattice sum concentrates the diffracted intensity
into a few spots in reciprocal space (Bragg peaks),
which can be measured with sufficient intensity.
The lattice sum has finite values only if (Laue equ.):
Qa=n•2π & Qb=n•2π & Qc=n•2π
Diffraction pattern
The scattering pattern is the product of:
the lattice sum (the lattice determines the position of the Bragg peaks) and
the structure factor (the motive determines the intensity in the Bragg peaks)
f(Q)
for x-rays
0
2
a

a

a
Q
Form factor for atoms
Neutrons
X-rays
Neutrons see the nucleus
X-rays see the electrons
The scattering lengths are independent of Q The scattering lengths vary with Q and ħω,
and ħω (not for magnetic scattering), but
but are independent of the isotope
depend on the isotope (incoherent
scattering)
f(Q)
f(Q)
Q
Q
Neutrons see easily the light elements and
are ‘naturally’ sensitive to magnetism
(Very powerful developments of new
sources)
σ x-rays ≈ σ neutrons
Ensemble of atoms - unit cell
b
Interference of second order: adding up
the waves originating from a finite particle
ensemble (unit cell of a crystal): structure
factor (of the unit cell) F(Q):
J
for x-rays: F(Q )   f j (Q ) e
r2
a
r1

iQ r j
d
J
and for neutrons: F(Q )   b j e
j 1
j 1




with the fractional coordinates rj  x ja + y jb + z jc

iQ r j
In single crystals, the arrangement of the J atoms is named the motive (decoration) of
the unit cell, and F(Q) represents for x-rays the FT of the charge density, for neutron
the FT of the nuclear potentials
f(Q)
f(Q)
for neutrons
2π/d
for x-rays
2π/d
Q
This is all we want to know
Q