Download Magnetic Neutron Scattering

Magnetic Neutron Scattering
Martin Rotter, University of Oxford
Martin Rotter
NESY Winter School 2009
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Contents
• Introduction: Neutrons and Magnetism
• Elastic Magnetic Scattering
• Inelastic Magnetic Scattering
Martin Rotter
NESY Winter School 2009
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2
Neutrons and Magnetism
Macro-Magnetism:
Solution of Maxwell´s
Equations – Engineering
of (electro)magnetic
MFM image
devices
Micromagnetism:
Domain Dynamics,
Hysteresis
Micromagnetic
10-1m
10-3m
Hall
Probe
VSM
SQUID
10-5m
MOKE
10-7m
MFM
simulation.
NMR
FMR
µSR
10-11m NS
10-9m
Atomic Magnetism:
Instrinsic Magnetic
Properties
Martin Rotter
NESY Winter School 2009
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I believe my talk is the only about magnetic properties of matter and therefore this
slide gives a quick overview about the different orders of magnitude and
corresponding methods of investigation of magnetic properties. With neutrons, the
topic of this talk, we usually probe magnetism on an atomic scale. However, recent
experiments show, that magnetic domains can successfully be studied by small
angle scattering techniques. This method might be in future of importance for the
study of antiferromagnetic domains, which are not easy to study by any other
method.
3
Bragg’s Law in Reciprocal Space (Ewald Sphere)
O
2π/λ
k
c*
2θ
τ=Q
Q = 2 sin (Θ ) k
k‘
S
N e cat
ut ter
ro ed
n
In
N e co
ut min
ro g
n
θ
a*
4
Single Crystal Diffraction
E2 – HMI, Berlin
k
Q
O
Martin Rotter
NESY Winter School 2009
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Many other talks cover the principle of single crystal diffraction. In this example,
the flat cone diffractometer E2, it is demonstrated, how a plane in reciprocal space
can be mapped by neutron diffraction by rotating the sample and recording the
scattered intensity on a position sensitive detector. Neutrons penetrate (usually) the
sample and allow a rotation of 360 degrees.
5
The Scattering Cross Section
Scattering Cross Sections

Number of scattered neutrons per sec  time −1
=
= area 
−1
−1
Incident neutron flux
 time area

Total
σ tot =
Differential
dσ Number of scattered neutrons per sec into angle element dΩ
=
dΩ
Incident neutron flux . dΩ
Double Differential
dσ
Number of ... and with energies between E' and E'+dE'
=
dΩdE '
Incident neutron flux . dE' dΩ
Scattering Law
dσ
k'
= S (Q, ω )
dΩdE ' k
Units:
S .... Scattering function
1 barn=10-28 m2 (ca. Nuclear radius2)
Martin Rotter
NESY Winter School 2009
6
The next transparencies give an overview on the calculation of the neutron scattering cross section. The formalism which has
been developed is beautiful and covers a lot of different cases and therefore I present in detail. On this slide the definitions of
the different types of cross sections are given. These can be measured by well calibrated instruments and compared to the
results of theoretical model calculations. Let me emphasize, that in order to facilitate such a comparison, neutron
instrumentation and theoretical modelling should be done on an absolute scale, i.e. in units of barn. It is a common practice,
that experimental data is not well calibrated (in intensity) and theoretical models predictions are done in arbitrary units. To see
how common this practice is, please just take the conference booklet and look at the plots in different talks, you will frequently
find “arbitrary units” … I believe, that with relatively small effort scattering data can be put on an absolute scale and in this
way it will be of much more use. I met several cases, where people were looking for an excitation in there data and discussed
tiny peaks as probably crystal field levels – however, an inspection on an absolute intensity scale shows immediately, that the
crystal field peaks should have a factor of 100 more intensity. We should try to avoid such an obvious misinterpretation of
experimental data.
Another caveat which comes to my mind in this context is the frequent practice to recalculate the x-axis (energy, scattering
angle) in a plot without proper transformation of the intensity: I use now the symbol “I” for the differential cross section - if
I(Ө) is measured as a function of scattering angle Ө, it has to be plotted as I(Ө) vs Ө. If Ө is to be transformed to the scattering
vector Q, then the proper transformation has to conserve the intensity, i.e. I(Ө)dӨ=I(Q)dQ. Therefore I(Q)=I(Ө) dQ/dӨ. Very
frequently the factor dQ/dӨ is ignored, we should avoid such bad practice !
Let me start now with the theoretical evaluation of the double differential cross section, which is the most detailed quantity
which can be studied without polarisation analysis of the neutrons (for polarized neutrons we would have to consider in
addition change of neutron polarisation in the scattering process):
the nominator is the number of scattered neutrons per second into angle element dΩ with energies between E’ and E’+dE’ and
can be written as W(k,k’)dN’, where
W(k,k’) … transition probability per time from plane incoming wave with vector k to scattered wave with vector k’, it is given
by Fermis golden rule as 2π/ћ Σif,sn’ δ(E+Ei-E’-Ef)PiPsn |<i|<k|<sn||Hint|sn’>|k’>|f>|2
dN’ ………number of scattered states in energie and Ω element dE’dΩ around k’, this can be calculated by imaging the
neutron states as plane wave states in a (large) periodicity volume V=L3
the possible states are plane wave states with wave vectors (kx,ky,kz)=(nx,ny,nz) 2π/L, the nx,ny,nz being integer
numbers.Any summation over these plane wave states Σnx,ny,nz(…) can for large periodicity Volume be written as an integral
using the relation 1=L∆kx/ 2π L∆ky/ 2π L∆kz/ 2π, which in the integral becomes ∫(…)Vd3k/(2π)3, from this we conclude that
dN’= Vd3k/(2π)3=Vk’2dk’dΩ/(2π)3. Now we
need to transform the momentum interval to an Energy interval,dE’= ћ2k’dk’/2M
The denominator consists of dE’dΩ times j,
j …. The neutron flux of incoming neutrons, which can be written as j=v/V=ћk/MV (v is neutron velocity, M neutron mass)
these ingredients we use to obtain the Master equation on the next slide …
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neutron mass
wavevector
Spin state of
the neutron
Psn
Polarisation
|i>,|f> Initial-,finalstate of the
targets
Ei,Ef Energies of –‘‘thermal
Pi
population
of state |i>
Hint
Interaction
-operator
M
k
|sn>
S. W. Lovesey „Theory of Neutron Scattering from
Condensed Matter“,Oxford University Press, 1984
d 2σ
k ' M 
=MartinRotter 2 
d ΩdE ' k  2π = 
2
∑
if , sn sn '
G
Q ) | sSchool
>|2 δ ( =ω + Ei 7− E f )
Psn Pi |< sn ; i | H
NESY
int (Winter
n '; f 2009
(follows from Fermis golden rule)
In the Master equation the Fourier transform of the interaction Hamiltonian has
been defined, i.e. the integration over the spatial neutron coordinates in the wave
function has been carried out (see formula on the next slide).
We now proceed with unpolarised neutron beams, so the probabilities for all spin
states Psn are equal, we will use
*) Psn=1/2
*) 1=Σsn |sn><sn|
*) Σsn <sn|snαsnβ|sn>=δαβ/4, here αβ refer to spatial coordinates x,y,z of the spin
vector sn
Thus we will get rid of the spin coordinates in the Master equation in the next two
slides …
7
Interaction of Neutrons with Matter
G G
G
G
H (Q) ≡ ∫ e − iQ⋅rn H (rn )d 3 rn
H int = H nuc + H mag
G G
G
G ~
2π= 2 j
H nuc (rn ) = ∑
(b + bNj I j ⋅ sn )δ (rn − R j )
M
j
2
G ~
G
G G
−iQ⋅R j 2π=
H nuc (Q) = ∑ e
(b j + bNj I j ⋅ sn )
M
j
2
G 2 1  G
G
G G
1 G e G
eG 
H mag (rn ) = ∑
A
A
A
P
P
−
+
+
+
 e
 e
n
e 
e  + 2 µ B se ⋅ B n
c
c 
 2m 
e 2m 
G
G
G
G ˆ
ˆ
H mag (Q) = 8πµ B ∑ { 12 gF (Q )} j e −iQ⋅Rj µN g n sn ⋅ Q
×J j ×Q
(
j
Martin Rotter
)
(
)
G
G
G G
H int ( Q ) = βˆ ( Q ) + 2 αˆ ( Q ) ⋅ s n
NESY Winter School 2009
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Here we write down explicit expressions for the interaction of the neutron with matter. Neutrons interact with the nuclei and
with the electrons.
The nuclear interaction is given by the nuclear potential, which in comparison to the neutron wavelength is well localised in
space and can be replaced by a deltafunction times a constant. These constant, however, may depend on the orientation of the
nuclear spin relative to the neutron spin I due to an exchange interaction. Probably the concept of exchange interaction is not
familiar to some of you and I mention here without proof that exchange interactions result from the requirement that the wave
function of identical particles should be antisymmetric in the particle coordinates (Pauli principle). If one is lazy and does not
want to do this antisymmetrisation of the wave function (in our case we are lazy and do not want to bother that the wave
function of the nuclear neutrons and that of our neutron being scattered have to be antisymmetric), Heisenberg has invented a
nice trick: we can calculate everything with product wave function, just as if the neutrons were distinguishable, but we have to
introduce in addition to the other energies involved an exchange interaction, which is given by the product of the spins times a
constant. This constant is determined by the overlap of the wave functions. More details can be found in any textbook or
lecture on magnetism, e.g. on my homepage …
The nuclear interaction is easily Fourier transformed due to the deltafunction involved.
Let me turn now to the magnetic interaction, the expression shown here is given in terms of (generalized) electronic
momentums in the sample (Pe) and the magnetic vector potential A. It can be derived
In the following way: we consider the change of electronic energy due to the magnetic field created by the neutron (one can
also look at the change of the neutron energy due to the magnetic field created by the electrons and get the same result). A
magnetic field will give rise to a Zeeman term (last term in the above equation) involving the electronic spin. In addition the
magnetic field created by the neutron will change the spatial movement of the electrons and has to be added to the magnetic
Vector potential in the kinetic energy term. Please note, that in Hamilton mechanics the kinetic energy term in the presence of
magnetic field has to be written in terms of a generalized momentum P (which is the Lagrange conjugate variable of the space
coordinate) and is nothing but the momentum of the electrons minus e/c times the magnetic Vector potential A. Thus inserting
a magnetic field will not change the kinetic energy (as known from electromagnetism). The difference between kinetic energy
term with the field created by the neutron and without gives the corresponding term in the interaction Hamiltonian. Note the
sum is over all electrons “e” in the sample.
In doing the Fourier transform a some approximations are involved
-Strong magnetism usually involves an unfilled electronic shell localized at an magnetic ion, thus the sum over electrons is
split into a sum over different ions j and over the different electrons in the ion
-Summation over the electrons can be done exact or in the so called dipole approximation (valid for small scattering vector Q),
we show here the ion results in dipole approximation which enables the magnetic interaction to be written in terms of the total
electronic angular momentum J of the ion j times the formfact F(Q) times the Landefactor g. note the gJ=M=the magnetic
moment.
In the next step in a formal way the total interaction is writte in terms of a neutron spin dependent part (blue) and a spin
independent part (yellow) and inserted into the master equation of the previous slide. Then the spin coordinates are eliminated
using the formulas given on the previous slide remarks resulting in the first expression on the next slide …
8
Unpolarised Neutrons - Van Hove Scattering
function S(Q,ω)
d 2σ
k'  M 
2
+
= 
δ (=ω + Ei − E f )∑ Pi (|< i | βˆ | f >| + < i | αˆ | f > ⋅ < f | αˆ | i >)
dΩdE ' k  2π= 2 
if
• for the following we assume that there is no nuclear order - <I>=0:
2
G
G
d 2σ
k '  =γe 2 
ˆ Q
ˆ )S αβ mag (Q, ω ) + N k ' S (Q, ω )

= N 
(δ αβ − Q
α
β
nuc
2  ∑
dΩdE '
k  mc  αβ
k
∞
G ~
G ~
G
1
iQ ⋅ R j ' ( 0 )
− i Q⋅ R j ( t )
iωt 1
1
1
{
}
{
}
S αβ mag (Q, ω ) =
dte
gF
(
Q
)
gF
(
Q
)
<
J
(
t
)
e
J
(
0
)
e
>T
∑
jα
j 'β
j 2
j'
2π= −∫∞
N jj ' 2
∞
G ~
G ~
G
*
*
1
1
− iQ ⋅ R j ( t ) iQ ⋅ R j ' ( 0 )
S nuc ( Q, ω ) =
dte iωt ∑ (b j b j ' + bNj bNj ' 14 δ jj ' I j ( I j + 1)) < e
e
>T
∫
2π= −∞
N jj '
G
G
~
R j (t ) = R j + u j (t )
Splitting of S into elastic and inelastic part
el
S nuc = Snuc + Snuc
el
inel
S mag = S mag + Smag
inel
S nuc = δ (=ω )
el
S mag = δ (=ω )
el
1
N
1
N
∑ (b
*
b j ' + bNj bNj ' 14 δ jj ' I j ( I j + 1))e
G G
G G
− iQ ⋅ R j + iQ ⋅ R j '
e
−W j −W j '
jj '
∑{
1
2
jj '
j*
gF (Q)}j < J jα >T
G G
G G
{12 gF (Q)}j ' < J j 'β >T e −iQ⋅R +iQ⋅R
j
j'
e
−W j −W j
We now turn to analyse the double differential cross section in terms of vanHove
scattering functions S for the magnetic and the nuclear part, which both can be
written as a sum of elastic and inelastic contributions. In doing this we write the
matrix elements in the Heisenberg representation of quantum mechanics, which
takes the time dependence from the state vector and puts it to the operators, in our
case R and J (see textbook on quantum mechanics for further details). The next step
is to write the positional operators Rj(t) in terms of the equilibrium position (Rj, a
number vector) and the deviation uj(t) (a Vector operator). When considering
elastic scattering function, we can consider the effect of the uj by assuming an
uncorrelated movement of the different atoms (i.e. neglecting phononic
correlations) in an harmonic oscillator potential. Even without the more rigorous
proof (which is available but more complicated) we can see that
<exp(-i uj Q) > ~ 1 - < i uj Q > - 0.5 < (uj Q) (uj Q) > + … ~ 1 – 0.5 <(uj
Q)2>~exp(-<(uj Q)2>/2)=exp(-Wj)
In case of isotropy of the Einstein oscillators the Wj can be simplified to Wj=<(uj
Q)2>/2=Q2<uj2>/6
9
L/2
∞
A short
inx 2π / L
f ( x) = ∑ f n e
...with... f n = ∫ f ( x' )e −i 2πnx '/ L dx'
Excursion
n =0
−L / 2
to Fourier
...
L/2
1 ∞
and Delta
f ( x) = ∫ ∑ einx 2π / L f ( x' )e −i 2πnx '/ L dx'
Functions ....
L n =0
−L / 2
1 ∞ in ( x − x ') 2π / L
∑e
L n =0
δ ( x)
δ (cx) =
c ∞
2π
qa = 2πx / L...∑ eiqna =
δ (q)
a
n =0
δ ( x − x' ) =
it follows by extending the range of x to more than –L/2 ...L/2 and
going to 3 dimensions (v0 the unit cell volume)
∑e
kk '
Martin Rotter
G G
G G
− iQ⋅G k + iQ⋅G k '
(2π )3
= NG
v0
∑
G
rez .latt . τ
G
G
δ (Q − τ )
NESY Winter School 2009
10
10
Neutron – Diffraction
1
el
S nuc = δ (=ω ) 
N
∑ b j b j 'e
*
G G
G G
− iQ ⋅ R j + iQ ⋅ R j '
e
−W j −W j '
jj '
+
1
N
∑| b
j 2 1
N
4
|
j

I j ( I j + 1)

G
G
G
R j = Gk + Bd
Lattice G with basis B: j = ( kd )........
Latticefactor
Structurefactor |F|2
 1

el
S nuc = δ (=ω ) ∑ δ QG , τG 
 N B
 τ
2
1
2
+ δ (=ω )
b
−
b
∑d d +
NB d
+ δ (=ω )
1
NB
∑b
one element(NB=1):
d 2 1
N
4
G G
G
*
− iQ⋅( B d − B d ' ) −Wd −Wd ' 

b
b
e
e
∑
d d'
d , d '=1

NB
Independent of Q:
„Isotope-incoherent-Scattering“
I d ( I d + 1)
d
σ nuc el inc = 4π
„Spin-incoherent-Scattering“
(
σi
σ c = 4π | b |2
2
dσ nuc inc
= N ⋅ 4π b 2 − (b ) + (bNd ) 2 14 I d ( I d + 1)
dΩ
el
)
Using the formulas of the previous slide we discuss now the elastic scattering, first
the nuclear contribution.
A speciality of neutron scattering already mentioned by Helmut Rauch is the
incoherent scattering contribution which arises from averaging the scattering cross
section over different possible isotope distribution in the sample. This averaging
process we indicate by a horizontal bar above the symbols. We note that the average
of a product of scattering lengths has to be done with great care, because for the
same atom the average of the square of the scattering length is not equal to the
scattering length average squared (i.e. <b>2≠<b2>), therefore, <bj* bj’>=<bj>*
<bj’> - δjj’ (<bj2>-<bj>2). This leads to the isotope incoherent scattering. Even in the
case of one isotope there is in addition the spin incoherent scattering arising from
the nuclear spins, which are not oriented (unless there is nuclear order at very low
temperatures, which may occur below a few mK ).
11
Magnetic Diffraction
S nuc
el
coh
S mag = δ (=ω )
el
1
= δ ( =ω )
N
1
N
∑b
j*
G
j ' − iQ ⋅ R j
b e
e
G
iQ ⋅ R j '
jj '
G
G
− iQ ⋅ R iQ ⋅ R
∑ {12 gF (Q)}j < J jα >T {12 gF (Q)}j ' < J j 'β >T e e
j
j'
jj '
d 2σ
k '  =γe 2 

= N 
dΩdE '
k  mc 2 
2
ˆ Q
ˆ )S αβ
(δ αβ − Q
∑
α
β
αβ
mag
G
G
k'
(Q, ω ) + N S nuc (Q, ω )
k
Difference to nuclear scattering:
Formfactor
{ 12 gF (Q)} j
Polarisationfactor
... no magnetic signal at high angles
ˆ Q
ˆ
(δαβ − Q
α β ) ... only moment components
normal to Q contribute
Martin Rotter
NESY Winter School 2009
12
12
Atomic Lattice
Magnetic Lattice
ferro
antiferro
Martin Rotter
NESY Winter School 2009
13
13
Atomic Lattice
Magnetic Lattice
ferro
antiferro
Martin Rotter
NESY Winter School 2009
14
14
Atomic Lattice
Magnetic Lattice
ferro
antiferro
Martin Rotter
NESY Winter School 2009
15
15
The Nobel Prize in
Physics 1994
In 1949 Shull showed the magnetic structure of the MnO crystal, which led
to the discovery of antiferromagnetism (where the magnetic moments of
some atoms point up and some point down).
16
Formfactor
Q=
Dipole Approximation (small Q):
Martin Rotter
F (Q ) =< j0 (Q) > +
NESY Winter School 2009
2− g
< j2 (Q) >
g
17
… in Shulls data we see that the magnetic intensity is strongest at small scattering
angle and decays at higher angles due to the magnetic form factor.
17
Arrangement of Magnetic Moments in Matter
Paramagnet
Ferromagnet
Antiferromagnet
And many more ....
Ferrimagnet, Helimagnet, Spinglass ...collinear, commensurate etc.
Martin Rotter
NESY Winter School 2009
18
18
GdCu2 T = 42 K
M ⊥[010]
TR= 10 K q = (2/3 1 0)
Magnetic Structure from
Neutron Scattering
N
Experimental data D4, ILL
Calculation done by McPhase
Rotter et.al. J. Magn. Mag. Mat. 214 (2000) 281
Martin Rotter
NESY Winter School 2009
19
In the next few slides I show, what nowadays is usually done automatically by a
computer program. First, the size of the magnetic supercell of an antiferromagnet
has been determined from the position of magnetic satellite reflections, which
appear below the Neel temperature. Then the moment orientation can be determined
by fitting the magnetic diffraction pattern. The computer calculates for every
magnetic reflection the magnetic structure factor and from this the magnetic
inensity. Then by convolution with the instrumental resolution a diffraction pattern
is calculated which can be compared to the experimental pattern for different
moment orientations … have a look at the subsequent slides on how this works.
19
GdCu2 T = 42 K
M ⊥[010]
TR= 10 K q = (2/3 1 0)
Magnetic Structure from
Neutron Scattering
N
Experimental data D4, ILL
Calculation done by McPhase
Rotter et.al. J. Magn. Mag. Mat. 214 (2000) 281
Martin Rotter
NESY Winter School 2009
20
20
GdCu2 T = 42 K
M ⊥[010]
TR= 10 K q = (2/3 1 0)
Magnetic Structure from
Neutron Scattering
N
Experimental data D4, ILL
Calculation done by McPhase
Rotter et.al. J. Magn. Mag. Mat. 214 (2000) 281
Martin Rotter
NESY Winter School 2009
21
21
GdCu2 T = 42 K
M ⊥[010]
TR= 10 K q = (2/3 1 0)
Magnetic Structure from
Neutron Scattering
N
Experimental data D4, ILL
Calculation done by McPhase
Rotter et.al. J. Magn. Mag. Mat. 214 (2000) 281
Martin Rotter
NESY Winter School 2009
22
22
GdCu2 T = 42 K
M ⊥[010]
TR= 10 K q = (2/3 1 0)
Magnetic Structure from
Neutron Scattering
N
Experimental data D4, ILL
Calculation done by McPhase
Rotter et.al. J. Magn. Mag. Mat. 214 (2000) 281
Martin Rotter
NESY Winter School 2009
23
23
GdCu2 T = 42 K
M ⊥[010]
TR= 10 K q = (2/3 1 0)
Magnetic Structure from
Neutron Scattering
N
Rpnuc = 4.95%
Rpmag= 6.21%
Experimental data D4, ILL
Calculation done by McPhase
Goodness of fit
Rotter et.al. J. Magn. Mag. Mat. 214 (2000) 281
Martin Rotter
R p ≡ 100
∑
hkl
I calc (hkl ) − I exp (hkl )
∑
NESY Winter School 2009
hkl
I exp (hkl )
24
The quality of a fit is usually described by parameters such as the Rp value …
24
NdCu2 Magnetic Phasediagram
(Field along b-direction)
4
FM
µ0H (T)
F2
2
F1
0
0
AF3
AF1 AF2
2
4
6
8
T (K)
Martin Rotter
NESY Winter School 2009
25
Another example of magnetic diffraction: when complex magnetic phases have to
be studied, nowadays cryoamagnets can be mounted on diffractometers and
spectrometers. Here I show a vertical field magnet for extremely high fields. In such
a split coil system fields up to 15/17 T may be produced. Using such a setup, the
diffraction can be done on single crystals and the different magnetic structures
which can be stabilised by variation of temperature and magnetic field can be
studied.
25
Complex Structures
µ0Hb=2.6T
AF1
µ0Hb=1T
µ0Hb=0
Q=
Experimental data TAS6, Riso Loewenhaupt, Z. Phys. (1996) 499
Martin Rotter
NESY Winter School 2009
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26
Complex Structures
µ0Hb=2.6T
F1
µ0Hb=1T
µ0Hb=0
Q=
Experimental data TAS6, Riso Loewenhaupt, Z. Phys. (1996) 499
Martin Rotter
NESY Winter School 2009
27
27
Complex Structures
µ0Hb=2.6T
F2
µ0Hb=1T
µ0Hb=0
Q=
Experimental data TAS6, Riso Loewenhaupt, Z. Phys. (1996) 499
Martin Rotter
NESY Winter School 2009
28
28
NdCu2 Magnetic Phasediagram
H||b
F1 ↑ ↑ ↓
F3 ↑↑
c
F1 ↑↑↓
a
b
AF1 ↑↑↓↓↑↓↓↑↑↓
Lines=Experiment
Colors=Theory
Calculation done by McPhase
Martin Rotter
NESY Winter School 2009
29
29
A caveat on the Dipole
Approximation
∑< ˆ
jj '
jα
>T < ˆ
Q
1
N
Q
el
Q
S mag = δ (=ω )
ˆ
jα
j 'β
>T e
≡−
G
− iQ ⋅R j
1
2µ B
e
G
iQ ⋅R j '
M jα (Q)
Q
>T ~ {12 gF (Q )}j < J jα >T
2− g
F (Q ) =< j0 (Q) > +
< j2 (Q) >
g
Dipole Approximation (small Q): < ˆ
jα
Q
E. Balcar derived accurate formulas for the
< ˆ
jα
>T
S. W. Lovesey „Theory of Neutron Scattering from
Condensed Matter“,Oxford University Press, 1984
Page 241-242
Martin Rotter
NESY Winter School 2009
30
A recent investigation during my first months in Oxford concerned the dipole
approximation of the magnetic form factor. In the dipole approximation only the
total dipolar moment on each magnetic ion enters the cross section. However, in
many atoms the dipolar moment is not homogenously distributed in the unfilled
shell. Due to the crystal electric field generated by neighboring atoms the magnetic
moment density is usually highly anisotropic. The neutrons in general are sensitive
to the Fourier transform of this magnetic moment density in the atom, here I denote
it by M(Q). In the literature frequently also the scattering operator Q is used, which
must not be confused with the momentum transfer.
30
E. Balcar
M. Rotter & A. Boothroyd
2008
did some calculations
Martin Rotter
NESY Winter School 2009
31
In the following I will show what effect going beyond the dipole approximation has
in the interpretation of magnetic diffraction pattern.
31
bct ThCr2Si2 structure
Space group I4/mmm
Ce3+ (4f1) J=5/2
TN=8.5 K
q=(½ ½ 0), M=0.66 µB/Ce
Goodness of fit:
Rpdip=15.6%
Rpbey=8.4 %
(Rpnuc=7.3%)
Martin Rotter
(|FM|2-|FMdip|2)/ |FMdip|2 (%)
CePd2Si2
Comparison to
experiment
Calculation done by McPhase
M. Rotter, A. Boothroyd, PRB, submitted
NESY Winter School 2009
32
Going beyond the dipolar approximation a clear improvement of the description of
the experimental data can be obatained, as is shown by the lower Rp value.
32
NdBa2Cu3O6.97
superconductor TC=96K
orth YBa2Cu3O7-x structure
Space group Pmmm
Nd3+ (4f3) J=9/2
TN=0.6 K
q=(½ ½ ½), M=1.4 µB/Nd
... using the dipole approximation may
lead to a wrong magnetic structure !
M. Rotter, A. Boothroyd, PRB, submitted
Martin Rotter
Calculation done by McPhase
NESY Winter School 2009
33
In NdBaCuO a wrong magnetic structure was published, because A. Boothroyd
used the dipole approximation for the form factor in the fit of the experimental data.
Therefore in future we should definitely use always the exact calculation to fit
magnetic diffraction data and avoid misinterpretation of the experimental data !
33
Inelastic Magnetic Scattering
• Dreiachsenspektometer – PANDA
• Dynamik magnetischer Systeme:
1. Magnonen
2. Kristallfelder
3. Multipolare Anregungen
Martin Rotter
NESY Winter School 2009
34
Here I show again the different methods for investagion of excitations in matter.
The red ellipses indicate that for magnetic excitations many methods are not
applicable and we are left with neutrons, NMR (nuclear magnetic resonance) and
muSR (muon spin resonance). Thus for investigation of magnetism in matter
neutrons are absolutely necessary. This is frequently ignored by many people
discussing on whether a new synchotron or a new neutron source should be built.
Please note that magnetism is important for almost all elements in the periodic
table, even the form of the periodic table itself cannot be explained without
considering the spin of the electrons. It is tempting to neglect the spin degrees of
freedom in diamagnetic materials, which we frequently find in nature, i.e. in
biology and chemistry and beautiful science can be done without spin. However, at
some stage also in these studies one may find, that spin is important, and then we
need neutrons to investigate it.
34
Three Axes
Spectrometer (TAS)
k
Q
Ghkl
k‘
q
(=k )
=ω =
2
2
(
=k ')
−
2M
2M
G G G G
G
Q = k − k ' = G hkl + q
Martin Rotter
NESY Winter School 2009
35
Here the principle of a three axis spectrometer is shown …
35
PANDA – TAS for Polarized Neutrons
at the FRM-II, Munich
Martin Rotter
NESY Winter School 2009
36
36
PANDA – TAS for Polarized Neutrons at the
FRM-II, Munich
beam-channel
monochromatorshielding with platform
Cabin with
computer work-places
and electronics
secondary spectrometer
with surrounding
radioprotection,
15 Tesla / 30mK Cryomagnet
Martin Rotter
NESY Winter School 2009
37
37
Martin Rotter
NESY Winter School 2009
38
… some parts of PANDA
38
Movement of Atoms [Sound, Phonons]
Brockhouse 1950 ...
The Nobel Prize in
Physics 1994
E
π/a
Phonon Spectroscopy: 1) neutrons
2) high resolution X-rays
Martin Rotter
NESY Winter School 2009
Q
39
39
Movement of Spins - Magnons
153
H =−
1
∑ J (ij )Si ⋅ S j
2 ij
MF - Zeeman Ansatz
(for S=1/2)
Martin Rotter
T=1.3 K
NESY Winter School 2009
40
The simplest model for magnetic excitations in solids – however it is not correct, as
experiments on the next slide show … the reason is, that changing one spin will
effect the neighbors, which tend to be oriented parallel to this spin because of the
exchange interaction, next slide shows what I mean.
40
Movement of Spins - Magnons
153
H =−
1
∑ J (ij )Si ⋅ S j
2 ij
T=1.3 K
Bohn et. al.
PRB 22 (1980) 5447
Martin Rotter
NESY Winter School 2009
41
…. The q dependence of magnetic excitations shows dispersion, this confirms the
spinwave model for magnetic excitations, see next slide
41
Movement of Spins - Magnons
H =−
1
∑ J (ij )Si ⋅ S j
2 ij
153
a
T=1.3 K
Bohn et. al.
PRB 22 (1980) 5447
Martin Rotter
NESY Winter School 2009
42
42
Movement of Charges - the Crystal Field Concept
+
+
+
+
+
4f –charge density
+
+
+
E
+
+
Hamiltonian H cf =
∑B
m
l
Olm (J i )
lm ,i
Martin Rotter
Q
NESY Winter School 2009
43
We put charges in the vicinity of a magnetic ion, this creates an electric field, the so
called crystal field. It deforms the charge density associated with the magnetic
electrons. In quantum mechanics this leads to a splitting of the degenerate magnetic
ground state. Neutrons can trigger excitations between these split states and thus
change the charge density ( although the interaction between the neutron and the
electron is only magnetic), note that with each state a different charge density is
associated – see next slides for an example…
Note that this is a simple picture. It may happen that the change in the charge
density on one ion will change the crystal field on the neighboring ion and
consequently also the charge density on the neighbor. This can be described by an
two ion interaction involving higher order (e.g. quadrupolar) moments. As in the
case of magnetic excitations, such an interaction will lead to dispersive modes
characterising a correlated change in the charge density. I will show later how this
idea leads to the concept of orbital excitations, orbitons.
43
NdCu2 – Crystal Field Excitations
orthorhombic, TN=6.5 K, Nd3+: J=9/2, Kramers-ion
Gratz et. al., J. Phys.: Cond. Mat. 3 (1991) 9297
Martin Rotter
NESY Winter School 2009
44
A crystal field spectrum of a magnetic ion … using this spectrum, the crystal
electric field has been determined …
44
NdCu2 - 4f Charge Density
< ρˆ (r ) >=| R4 f (r ) |2
∑ ec
n =0, 2, 4, 6
m = 0 ,..., n
θ < Onm (J ) >T Z nm (Ω)
nm n
T=100
T=40
T=10 K
K
K
Martin Rotter
NESY Winter School 2009
45
For each state the charge density is different…. The different crystal field states are
populated when increasing the temperature. Thus at high temperature the charge
density gets spherical. This leads to an anisotropic contribution to the thermal
expansion of the solid. Thus neutrons may help to understand bulk properties.
45
Calculate Magnetic Excitations and the Neutron
Scattering Cross Section
H = ∑ BlmOlm ( J i ) − ∑ g Ji µ B J i H −
lm ,i
i
1
∑ J i J (ij )J j
2 ij
2
G
d σ
k '  =γe 
ˆ Q
ˆ )S αβ mag (Q, ω )

(δ αβ − Q
= N 
α
β
2  ∑
k  mc  αβ
dΩdE '
G
inel
S αβ mag ( Q , ω ) = 2 π =1N ∑ { 12 gF ( Q )} d { 12 gF ( Q )} d ' e − iκ ⋅( B
2
2
b
{
dd '
}
1 αβ
αβ
χ dd
χ dd ' ( z ) − χ dβα'd ( − z*)
' ' ' ( z) ≡
2i
(
G
G
S = 2=
d
−B d ' )
G
αβ
e − W d −W d ' S dd
' (Q , ω )
1
χ
1 − e − = ω / kT
''
)
−1
χ (Q , ω ) = χ 0 (ω ) 1 − χ 0 (ω ) J (Q ) Linear Response Theory, MF-RPA
χ 0αβ (ω ) = ∑
< i | J α − < J α > H ,T | j >< j | J β − < J β > H ,T | i >
ij
ε j − ε i − =ω
( ni − n j )
.... High Speed (DMD) algorithm: M. Rotter Comp. Mat. Sci. 38 (2006) 400
Martin Rotter
NESY Winter School 2009
46
Inelastic scattering is complicated. This slide outlines the basic concept for the
calculation of magnetic excitations. A central equation is the so called “fluctuation
dissipation theorem” which relates the dynamical susceptibility chi to the scattering
function S.
46
F3 ↑↑
F3: measured
dispersion was
fitted to get
exchange
constants J(ij)
NdCu2
F1 ↑↑↓
Calculations done by McPhase
AF1 ↑↑↓↓↑↓↓↑↑↓
An example of a study of magnetic excitations in different magnetic phases by
applying magnetic a magnetic field on a three axis spectrometer. The
dispersionplots show many modes, the number of modes is equal to the number of
magnetic ions in the magnetic supercell times the transitions between the levels on
these ions. For Nd at low temperature only the crystal field groundstate doublet has
to be considered, thus in this case we have only one transition per ion and the
number of modes is equal to the number of Nd ions in the magnetic supercell. All
data can be described by a unique set of exchange interaction parameters for this
system. The calculation of the dispersion is far from trivial and has been done using
the formalism outlined on the previous slide. The green lines show the results of
these model calculations, the point size corresponds to the intensity of the mode.
Experimental data was obtained by energy scans at constant wave vector transfer Q.
From the maxima in the spectra the position of the modes in energy was determined
and is shown by the symbols.
47
1950
Movements of Atoms [Sound, Phonons]
1970
Movement of Spins [Magnons]
?
Movement of Orbitals [Orbitons]
aa
ττorbiton
orbiton
Description: quadrupolar
(+higher order) interactions
Martin Rotter
H Q = − ∑ C (ij ) ⋅ Olm (J i ) ⋅ Olm (J j )
ij ,lm
NESY Winter School 2009
48
Coming back to what I have mentioned along with the introduction of the crystal
field concept I show, that besides phonons and spinwaves I believe in the existence
of orbital excitations. Currently it is still a matter of believe, because we do not have
enough experimental evidence for this type of excitations. The reason is, that it is
difficult to distinguish between orbitons and magnons. This is in contrast to phonons
and magnons, where the two types of excitations can clearly be distinguished due to
the different interaction (nuclear, electronic) leading to a different q dependence of
intensity. Also polarised neutron experiments can be used to distinguish between
phonons and magnons. For orbitons, the only difference to magnons is, that the
interaction which leads to the dispersion is not bilinear but of higher order for the
orbitons. However, in some cases it may be possible to establish an orbiton, this is
when we observe a large dispersion of magnetic excitations and know, that the
magnetic interaction must be small (this we may conclude from a very low
magnetic ordering temperature). Then only higher order interactions can be the
reason for the large dispersion. Some cases in literature and some own experiments
on PrCu2 indicate the existence of orbitons. Further research is strongly needed to
learn more about this type of excitations.
48
Summary
• Magnetic Diffraction
• Magnetic Structures
• Caveat on using the Dipole Approx.
•
•
•
•
Martin Rotter
Magnetic Spectroscopy
Magnons (Spin Waves)
Crystal Field Excitations
Orbitons
NESY Winter School 2009
49
49
Martin Rotter, University of Oxford
Martin Rotter
NESY Winter School 2009
50
50
McPhase - the World of Magnetism
McPhase is a program package for the calculation of magnetic properties
! NOW AVAILABLE with INTERMEDIATE COUPLING module !
Magnetization
Magnetic Phasediagrams
Magnetic Structures
Martin Rotter
Elastic/Inelastic/Diffuse
Neutron Scattering
Cross Section
NESY Winter School 2009
51
This software suite is the result of more than 10 years programming. It allows the
evaluation of magnetic properties including neutron scattering cross sections.
51
Crystal Field/Magnetic/Orbital Excitations
McPhase runs on
Linux & Windows
it is freeware
www.mcphase.de
Magnetostriction
and much more....
Martin Rotter
NESY Winter School 2009
52
52
Important Publications referencing McPhase:
• M. Rotter, S. Kramp, M. Loewenhaupt, E. Gratz, W. Schmidt, N. M. Pyka, B. Hennion, R.
v.d.Kamp Magnetic Excitations in the antiferromagnetic phase of NdCu2 Appl. Phys. A74
(2002) S751
• M. Rotter, M. Doerr, M. Loewenhaupt, P. Svoboda, Modeling Magnetostriction in RCu2
Compounds using McPhase J. of Applied Physics 91 (2002) 8885
• M. Rotter Using McPhase to calculate Magnetic Phase Diagrams of Rare Earth
Compounds J. Magn. Magn. Mat. 272-276 (2004) 481
Thanks to ……
M. Doerr, M. Loewenhaupt, TU-Dresden
R. Schedler, HMI-Berlin
P. Fabi né Hoffmann, FZ Jülich
S. Rotter, Wien, Austria
M. Banks, MPI Stuttgart
Duc Manh Le, University of London
J. Brown, B. Fak, ILL, Grenoble
A. Boothroyd, Oxford
P. Rogl, University of Vienna
E. Gratz, E. Balcar TU Vienna
Martin Rotter
University of Oxford
……. and thanks to you !
NESY Winter School 2009
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