Transcript Mössbauer spectroscopy

Pavol Jozef Šafárik University in Košice, Faculty of Science
Supportive Textbooks in Course:
Methods of Condensed Matter Spectroscopy –
Mössbauer Spectroscopy
Teacher: Pavol Petrovič
Study programme: Physics of Condensed Matter
The ESF project no. SOP HR 2005/NP1-051, 11230100466
The project is cofinanced with the support of the European Union
Contents of shortened instructional materials
I.
Physical principles of the Mössbauer effect.
II.
Methodology of experiments.
III. Hyperfine interactions - electrical monopole, electrical quadrupole and
magnetic dipole interactions.
IV. Physical information involved in hyperfine spectrum parameters.
V.
Processig and evaluation of Mössbauer spectra.
VI. Results obtained at the study of properties of new materials by means of
Mössbauer spectroscopy.
Note: all spectra presented in the abridged materials have been
published in scientific publications in which the author of these
materials has participated as an coauthor.
Recommended references
1. Dickson D.P.E, Berry F.J.: Mössbauer Spectroscopy. Cambridge
University Press, Cambridge, 1986.
2. Goldanskij V.I., Herber R.H.: Chemical Applications of Mössbauer
Spectroscopy. Academic Press, New York, 1968.
3. Gonser U.: Mössbauer Spectroscopy. Springer Verlag, Berlin, 1975.
4. Long G.J., Grandjean F.: Mössbauer Spectroscopy Applied to Magnetism
and Materials Science. Vol. 2. Plenum Press, New York, 1996.
5. Maddock A.G.: Mössbauer Spectroscopy. Principles and Applications of
the Techniques. Horwood Publishing, Chichester, 1997.
6. Ovchinnikov V.V.: Mössbauer Analysis of the Atomic and Magnetic
Structure of Alloys. Cambridge Inter. Sci. Publ., Cambridge, 2006.
7. Vértes A., Korecz L. Burger K.: Mössbauer Spectroscopy. Akadémiai
Kiadó, Budapest, 1979.
8. Wertheim G.K.: Mössbauer Effect – Principles and Applications.
Academic Press, New York, 1964.
Nuclear resonance fluorescence
The process of nuclear resonance absorption followed by the nuclear
resonance emission of  - radiation.
Nuclear resonance fluorescence – the method of investigating condensed
matter.
Preferences of the method:
1. deep penetration of -radiation into condensed substances,
2. small relative width of absorption and emission lines,
3. high spectrum parameters sensitivity to the internal and external factors
of the examined substance.
The nature width of absorption/emission line
W t  
Heisenberg uncertainty principle:
W - inaccuracy in measuring energy,
t
- time interval disposed to the measurement of one energy value,

- modified Planck´s constant.
Basic energetic nucleus state:
Excited nucleus state:
t   gr    W gr  0
t   ex  0,
W ex 

 ex

Γ – natural width of absorption/emission line
Analytical form of elementary absorption/emission line
Dependence of the number of emitted/absorbed γ-quanta by certain isotope
per time unit on energy or frequency of γ-radiation.
L(W)
1
Lorentz shape:

1/2

1

L W 0 

2
2


 W W


0
LW   1  



2






2
1




W
W0
W0 - /2

C
0
LW  dW  1
W0 + /2
C L(W) – density of probability of γ-quanta absorption or
emission by the W energy of the given isotope.
Absorption/emission lines of resonant nucleus
a) Free nucleus
emission line,
b) fixed nucleus
absorption/emission
lines,
c) fixed nucleus
absorption line.
 em   0  W R
  ab    0  W R
Comparison of the atomic and nucleus fluorescence parameters
Atomic
fluorescence
(in general)
Nucleus
fluorescence
(57Fe isotope)
W0 [eV]
~ 10
14,4·103
τex [ns]
~ 4,5
97,0
Selected action
parameter
Γ [eV]
~
10-7
4,5·10-9
Γ / W0
~ 10-8
3,1·10-13
WR [eV]
~ 5·10-10
1,9·10-3
WR / Γ
~ 5·10-3
4·105
Observation:
good
difficult
The nucleus
reverse reflection
energy:
WR 
p2
2M

W 02
2M c 2
Rudolf Mössbauer discovery
1955 – Max Planck Institute, Heidelberg
post-graduate study devoted to nucleus fluorescence 191Ir
1958 – publishing PhD results in Zeitschrift für Physik 151 (1958), 124-143,
(Kernresonzflureszenz von Gammastrahlung in Ir191) and Naturwissenschaften 45 (1958), 538.
1961 – Nobel prize award for Physics
Mössbauer explained his experimental results by the manifestation of recoil-free
nuclear resonance fluorescence whose existence was justified by the analogy with
the existence of elastic scattering of X-ray and slow neutrons in crystal (Lamb 1939).
The efficient cross-section of X-ray scattering by the atomic nucleus lattice is
substantially influenced by the energy WB of an elastic atomic bond in a
crystalline lattice of solid.
1. WR  WB - absorbing/emitting nucleus atom is ejected from the lattice,
WB  15, 30 eV  W  150 keV
2. W R  W B - momentum accepted from the absorbed/emitted photone
is transferred to the crystal by the nucleus.
Probability of the process of recoil-free
absorption/emission of -quantum
Recoil-free process – photone absorption by an absorber as a whole, without
any change of its internal energy.
Probability of this process is given by Mössbauer-Lamb factor:

x
f  exp  2
 
2





2
- modified wave length of -quantum,
x - nucleus oscillation amplitude in the
direction of -quantum spreading.
Mössbauer-Lamb factor for Debye’s model of a solid:
A solid – isotropic flexible medium capable of performing internal oscillations;
system of 3N bound quantum oscillators 
internal crystal energy is quantized 
probability of recoil-free process is no-zero.
1

W j   n j    j
2

Mean energy of j-th oscillator:
j  1,    , 3N
   j  
  1
n j  exp 

  k B T  
Mean photone number with ħωj energy:
N – number
of crystal
atoms
1
Mean atom shift from all oscillators:
r2 

NM
3N

j 1
n j 1 2
j


NM
 max 


 1  exp      1
 2   k B T  
0 

1 
 D  d
 

Debye’s distribution function of oscillator frequencies:
D  
9N
3
 max
D   0 ,
 2 , 0     max
   max
k
 max  B  D

substitution: y 
Probability of recoil-free -quantum absorption/emission process:

2
W
3


f  exp 
2
 4 Mc k B  D


  T
1  
  D




2 D T

0


y
dy  
exp  y   1  


kB T
Mössbauer-Lamb factor – low-teperature approximation
T  5  10   D
D T

0
y
dy 
exp  y   1
2

W
3

f  exp 
 4 Mc 2 k 
B D

T   D
0
y
2
dy 
exp  y   1
6
  2   T
1  

  6   
 D
 
2
or

 T 
  1 
 
 D 



2 



 3

W2

f  exp 
 4 Mc 2 k B  D 


Influence of recoil-free f fraction by a choice of:
1. isotope as a source and an acceptor of radiation (M, W),
2. host substance involving the isotope (D).
Mössbauer isotopes
There are approximatelly 200 nuclear transitions with parameters suitable for the
application in Mössbauer spectroscopy:
1. transition energy less than the energy of an elastic atomic bond in a lattice,
2. life span of an excited nuclear level within range of 10-5 s up to 10-13 s.
So far 110 isotopes have been examined; their application in Mössbauer spectroscopy is as follows (according to MEDC UNC, April 2007, 46 028 publications):
1. 57Fe – 64% papers, 3. 151Eu – 3% papers,
2. 119Sn – 18% papers, 4. 197Au – 2% papers,
5. other 106 transitions – 13% papers.
Comparison of the properties of the most applied isotope and the
isotope on which Mössbauer’s discovery was performed:
isotope
host
W [keV]
f
57Fe
Fe
14,4
0,91
191Ir
Ir
129
0,06
Transmission arrangement of Mössbauer spectrometer
absorbator
 - source
velocity
control
unit
detector
vibrator
amplifier
multichannel
analyser
0
Doppler effect:
a
Activity mode
with constant
acceleration
t
v
+vmax
0
-vmax
t
W  
v
W
c
Numeric processig and evaluation of Mössbauer spectra
Theoretical model of a complex Mössbauer spectrum:
C v k   C  
L

l 1

Fl v k , g l  
xmax


p x  F L1 v k , x, g L1  dx ,
k  1,    , K
xmin
C v k  , k  1,    , K
vk
C
- teoretical number of impulses scanned in the k-th
spectrometer channel,
- average Doppler velocity assigned to the k-th spectrometer channel,
- background; the number of impulses scanned at the
velocity far from resonance absorption (v → ∞),

Fl v k , g l  , l  1,    , L - theoretical model of the l-th non-distributed
subspectrum,

g l , l  1,    , L  1
- vectors of unknown non-distributed parameters
of all subspectra,

FL1 v k , x, g L1 
- teoretical model of the only distributed subspectrum
p x  - distribution function of distributed parameter x satisfying
a normalisation condition:
p( x )  0
for
p( x )  0
for
xmax
 p( x) dx  1
x  x min , x max
x  x min , x max
xmin
Distribution function is searched by fitting process as a table of values:
 
pj  p xj
for

x  x j , x j 1 ,
p J 1  p x J 1 .
j  1,  , J ,
equidistant nodes: h  x j 1  x j , j  1,, J
Modified teoretical model of Mössbauer spectrum:
C v k   C  
L

l 1
J


Fl v k , g l   h
j 1



p j F L1 v k , x j , g L1 ,
k  1,    , K
1. optimalisation procedure step: minimalization of the functional given
by the weighted sum of residue squares and a smoothing member:
K
S

w k C e v k   C v k 2  
k 1
J 1
2


p

2
p

p
 j 1 j j 1
wk 
k  1,  , K
j 2
In order to estimate the unknown parameters:
C ,
pj,
1
,
C e v k 
j  1, , J  1
Frank-Wolfe quadratic programming method has been applied.
2. optimalisation procedure step: functional minimalization:
L1

 
S
w k C e v k   C  
Fl v k , g l 


k 1
l

1


K


2
In order to estimate the unknown parameters:
( Tabulated values of the
distribution function are given
in the preceding step.)

g l , l  1,    , L  1
Levenberg-Marquardt optimalisation method has been applied.
Combined method for the analysis of
complex Mössbauer spectra including a
distribution in hyperfine interactions.
Nuclear Instruments and Methods in
Physics Research B72 (1992), 462-466.
Examples of applying the method:
• Sharp absorption lines of crystalline
iron with admixture (Fig.2).
• Broaded absorption line of the
amorphous alloy with one distributed
parameter (Fig.3).
• Spectrum decomposition of the
nanocrystalline alloy into subspectra six sharp sextets and one distributed
sextet (Fig.4).
Elektrical and magnetic hyperfine interactions
 Additional interactions between the nucleus and its charged surrounding
result from the fact that the nucleus is not any structureless body, but a
set of very close, moving charged and neutral particles having a certain
spatial arrangement in a final volume.
 Mössbauer effect facilitated visualisation and quantification of hyperfine
interaction parameters.
 Attention is given to the three types of hyperfine interactions:
 electrical monopole,
 electrical quadrupole,
 magnetic dipole.
Energy of electrical interaction of a nucleus with its
charged environment
WE 

 
 nu r  r  dV
V - charged nucleus volume,

 nu r 

- nuclear charge density in position r

 r  - electrical field potential of a charged
surrounding of nucleus.
V
Decomposition of the electrical field potential into the Taylor series:
x1  x
x2  y
x3  z

  r 
 i 0  


x
i  r  0


 r    0 
3

i 1
i  1, 2, 3
1
x i  i 0 
2
  2 r  
 ij 0  

 xi x j  r 0
3
x
i , j 1
i
x j  ij 0    
i, j  1, 2, 3
For a nuclear charge, it holds:
1. Total nuclear charge:
Z qe 


 nu r  dV
V
2. Dipole nuclear moment
vector – law of parity
conservation:
3. Quadrupole nuclear
moment tensor:
Mi 


 nu r  x i dV  0 , i  1, 2, 3


 nu r  x i x j dV , i , j  1, 2, 3
V
Qij 
V
4. Effective nuclear charge radius R:
R2 
V
 2
 nu r  r dV
V

 nu r  dV
Electrical monopole interaction – shift of energy levels of nuclei
The energy of electrical
nuclear interaction in
approaching the first three
members of a series:
For an electron charge, the
Poisson equation holds:
W E   0 Z qe 
3
3
 0 M  
i
i
i 1

 r   

 el r 
0
ij
0Qij
i , j 1

qe
0

r 
2


 r  - superposition of wave functions of surrounding charges with  el r 

density, forming the field having  r  potential.
Energy increase of nuclear states:
Energy change of a nuclear shift:
W EM
2
1 Z qe

0
6 0
2
R2
1 Z qe2
2
2
2 
W0 
 0   Rex
 Rgr



6 0
Electrical monopole interaction – isomer shift of spectrum
Difference in energies of the same W0 transition in a source (S) and in an
absorbator (A):
1 
2
2 
2
2   

  W0 A  W0 S 
Z qe  Rex  Rgr   qe  A 0  S 0  


  

60 
nuclear factor
v
-v
atomic-molecular factor
v [mm / s]
0
+v
1/2
1/2
v
-v
0
+v
vδ –
isomer
shift of
spectrum
Isomer shift (IS)  provides valuable physical-chemical information about
absorber properties.
It is influenced by:
- electron structure of an atom,
- atom valency, chemical bond,
- charge states Fe2+ and Fe3+
(they are differ significantly
in ).
Mössbauer spectroscopy of
hydrogenated Fe91Zr9
amorphous alloys.
Journal of Magnetism and
Magnetic Materials 128
(1993), 365-368.
→
Electrical quadrupole interaction
The nucleus with non-spheric distribution of a charge in a non-homogeneous
electrical field.
A tensor of an electrical field gradient at
the proper choice of coordinate system:
 ij  0  i  j (i, j  1,2,3)
For diagonal non-zero elements the Poisson equation hold:
11   xx ,
Q
 zz
qe
 22   yy ,
 xx   yy   zz  0
33   zz
← quadrupole nuclear moment
asymmetry parameter →
Parameter of non-homogeneous
electrical field at nucleus:

 xx   yy
 zz
A  Q qe 1 
2
3
Electrical quadrupole interaction energy
ˆ

EQ
Hamiltonian of an electrical
quadrupole interaction:
3 m I2  I I  1
W EQ I , m I   A Q
4 I 2 I  1
Proper energy values:
Magnetic quantum number
of a nucleus:

m I   I , I  1, ,1,0,1, , I  1, I
I  0, I 1 2  Q  0
Spin quantum number
of a nucleus:
Q0


ˆ
ˆ

ˆ
 Q   E 


I 1 2  Q  0
2 I  1
- multiple degeneration of energetic levels
Quadrupole splitting of Mössbauer spectrum
3  AQ
3
W ,   
2
4
2
mI=±3/2
AQ
1
3
W ,    
2
4
2
mI=±1/2
I=3/2
mI=±1/2
I=1/2
1/2 AQ
v
Quadrupole interaction – angular dependence of lines intensities
  angle between the direction of -quantum emission and the main axis
of a crystal symmetry
Transition from
the excited to
ground state
Angular
dependence
of lines
intensities
I , m I   I , m I 


3 3 1 1
 2 ,  2    2 ,  2 
3
1  cos 2 
2
3 1 1 1
 2 ,  2    2 ,  2 
3
1  sin 2 
2
Relative lines intensities
polycrystal
1
cos2   ,
3
sin2  
monocrystal
2
3
0
  2
1
3
3
1
1
5
Quadrupole splitting of spectrum – physical information
The existence of quadrupole splitting of spectrum is evidence that at the place
of the Mössbauer atom with non-zero quadrupole moment there is a nonhomogeneous electric field.
There are two principal sources of non-homogenity of the internal electric
field at nucleus:
1. electron charges of incompletely occupied electron levels
in a particular atom,
2. ion charges surrounding the nucleus, if their symmetry is
lower then cubic.
Quadrupole splitting provides highly valuable information about:
•
•
•
the structure of an electron shell,
chemical bond,
overall crystal or molecule architecture, …
Room
temperature
transmission
Mössbauer
spectra of new
boronium cyano
complexes.
Anions →
Cations
↓
Na+, K+, K+
[dipyPhBCl]+
Proceedings of
7-th European
Symposium on
Thermal
[(Et2N)2PhBCl]+
Analysis and
Calorimetry –
ESTAC 7,
Balatonfüred
[Me2PhS]+
1998.
[Fe(CN)5NO]2- [Fe(CN)6]3-
[Fe(CN)6]4-
Magnetic dipole interaction
Nucleus with non-zero magnetic moment in an effective magnetic field.

  nu g I

Dipole magnetic nuclear moment:
 nu
g - gyromagnetic factor
- nuclear magnetone,
Effective magnetic field at the nucleus:
H loc - local field,
H hf
H  H loc  H hf
- hyperfine field.
Main sources of a hyperfine field:
1. contact Fermi nuclear interaction with s-electrons,
2. dipole-dipole nuclear interaction with electrons having non-zero charge
density at nucleus.
Energy of magnetic dipole interaction
Hamiltonian of magnetic dipole interaction:
Corresponding proper energy values:
W MD
ˆ

MD
ˆ ˆ
   H
mI
W MD I , m I     H
I
- gives the energy change of a nuclear state, if a nucleus is
found in the magnetic field.
Studying condensed substances, the magnetic hyperfine spectrum strukture
provides valuable physical information about:
1. magnetic structure,
2. magnetic phase changes,
3. phase analysis, …
Zeeman splitting of Mössbauer spectrum
mI
+3/2
2/3exH
+1/2
-1/2
-3/2
I=3/2
14,4keV
-1/2
I=1/2
+1/2
0
v
Dipole interaction – angular dependence of lines intensities
  angle between the direction of emitting -quantum and a vector
of an effective magnetic field at the nucleus:
exc.  gro.
angular
dependence of
a line intensity
3 3
1 1
 2 ,  2    2 ,  2 
3 3
1 1
 2 ,  2    2 ,  2 
9
1  cos 2 
4
1
1
3
1
 2 ,  2    2 ,  2 
1
1
3
1
 2 ,  2    2 ,  2 
3 sin 2 
1
1
3
1
 2 ,  2    2 ,  2 
1
3 1
1
 2 ,  2    2 ,  2 
3
1  cos 2 
4
Transition




relative intensity
1
cos   ,
3
2
sin  
3
2
2
relative intensity
0
 
  2
3
3
3
2
0
4
1
1
1
Structure and properties of the
ball-milled spinel ferrites.
Materials Science and
Engineering A226-8, (1997),
22-25.
Me Fe2 O4
Me
2
1
Me  Ni, Mg , Zn


Fe3 Me2 Fe23 O42
(A)-tetra
[B]-okta
Redistribution of cations Fe3+
induced by high energetic
mechanical milling
Ni : t  0,   1,
 Fe  Ni
3
1
2
1

Fe13 O42
t  0, 0  1, Fe 3  A  Fe 3 B
  
t  t max ,   0, Ni12 Fe23 O42
Hydrogen induced changes on the hyperfine
magnetic field of amorphous Fe-Ni-Zr alloys.
Key Engineering Materials 81-83 (1993),
357-362.
→
Influence of hydrogenation on the magnetic
properties of amorphous Fe-Co-Zr alloys.
Journal of Magnetism and Magnetic
Materials 112 (1992), 334-336.
↓
The Structure and
Magnetic Properties of
Fe-Si-Cu-Nb-B Powder
Prepared by
Mechanochemical Way.
Physica status solidi 189
(2002), 859-863.
Coexistence of three
phases containing Fe:
1. -Fe crystalline grains,
2. granule bounds and
intergranule area,
3. superparamagnetic
particles.
Properties of the
nanocrystalline Finemet
alloys prepared by
mechanochemical way.
Acta physica slovaca 48
(1998), 703-706.
Initial material for
milling:
1. nanocrystalline tape,
2. alloy in the atomic
relation of elements:
Fe : Cu : Nb : Si : B
73,5 : 1 : 3 : 13,5 : 9,
3. pure elements in the
given atomic relation.
Influence of annealing on the
crystallographic structure and
some magnetic properties of
the Fe-Cu-Nb-U-Si-B
nanocrystalline alloys.
Journal of Materials Science
33 (1998), 3197-3200.
Phase analysis
of nanocrystalline system
Fe73.5Cu1Nb3-xUxSi13.5B9
(x=1, 2, 3 at.%)
by decomposition
of complex spectrum
into subspectra.
Extraterestrial Applications of Mössbauer Spectroscopy
MIMOS on the Mars Exploration Rovers – Spirit and Opportunity
Images Credit: NASA/JPL-Caltech
Rover Traces on the Martian Surface