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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
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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.
Contents of shortened instructional materials
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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.
Recommended references
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Nuclear resonance fluorescence
The process of nuclear resonance absorption followed by the nuclear resonance emission of - radiation.
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.
Nuclear resonance fluorescence – the method of investigating condensed matter.Preferences of the method:
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The nature width of absorption/emission line
tWHeisenberg uncertainty principle:
0 grgr Wt
ex
exex Wt
,0
Γ – natural width of absorption/emission line
Basic energetic nucleus state:
Excited nucleus state:
t
W
- time interval disposed to the measurement of one energy value,
- inaccuracy in measuring energy,
- modified Planck´s constant.
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W
L(W)
W0 - /2 W0 + /2
W0
1
1/2
12
0
2
1
WW
WL
Lorentz shape:
0
1dWWLC
2
1
20
WL
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.
C L(W) – density of probability of γ-quanta absorption or emission by the W energy of the given isotope.
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Absorption/emission lines of resonant nucleus
Rem W 0
a) Free nucleus emission line,
b) fixed nucleus absorption/emission lines,
c) fixed nucleus absorption line.
Rab W 0
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Comparison of the atomic and nucleus fluorescence parameters
Observation: good difficult
Selected action parameter
Atomic fluorescence (in general)
Nucleus fluorescence (57Fe isotope)
W0 [eV] ~ 10 14,4·103
τex [ns] ~ 4,5 97,0
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
The nucleus reverse reflection energy:
2
20
2
22 cM
W
M
pW R
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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).
BR WW 1. - absorbing/emitting nucleus atom is ejected from the lattice,
keVWeVWB 15030,15
BR WW 2. - momentum accepted from the absorbed/emitted photone is transferred to the crystal by the nucleus.
Rudolf Mössbauer discovery
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.
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Probability of the process of recoil-free absorption/emission of -quantum
2
2
exp
xf
- modified wave length of -quantum,
x - nucleus oscillation amplitude in the direction of -quantum spreading.
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:
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.
Mössbauer-Lamb factor for Debye’s model of a solid:
2
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Nj 3,,1 Mean energy of j-th oscillator:
Mean photone number with ħωj energy:N – number of crystal atoms
Probability of recoil-free -quantum absorption/emission process:
DBk
maxTk
yB
Debye’s distribution function of oscillator frequencies:
substitution:
Mean atom shift from all oscillators:
max
max2
3max
,0
0,9
D
ND
jjj nW
2
1
1
1exp
Tkn
B
jj
dD
TkMN
n
MNr
B
N
j j
j
max
0
13
1
2 1exp2
121
T
DDB
D
dyy
yT
kMc
Wf
0
2
2
2
1exp1
4
3exp
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61exp1exp105
2
00
dyy
ydy
y
yT
T
D
D
Mössbauer-Lamb factor – low-teperature approximation
22
2
2
61
4
3exp
DDB
T
kMc
Wf
or
12
DD
TT
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).
DBkMc
Wf
2
2
4
3exp
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Mössbauer isotopes
So far 110 isotopes have been examined; their application in Mössbauer spectro-scopy 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.
isotope host W [keV] f
57Fe Fe 14,4 0,91
191Ir Ir 129 0,06
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.
Comparison of the properties of the most applied isotope and the isotope on which Mössbauer’s discovery was performed:
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Transmission arrangement of Mössbauer spectrometer
velocity control
unitvibrator
- source
absorbator
detector amplifier multichannel
analyser
0
t
t
a
+vmax
0
-vmax
v
Doppler effect:Activity mode with constant acceleration W
c
vW
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Numeric processig and evaluation of Mössbauer spectra
Theoretical model of a complex Mössbauer spectrum:
KkdxgxvFxpgvFCvC LkL
L
l
x
x
lklk ,,1,,,, 11
1
max
min
kv - average Doppler velocity assigned to the k-th spectrometer channel,
C - background; the number of impulses scanned at the velocity far from resonance absorption (v → ∞),
LlgvF lkl ,,1,,
- theoretical model of the l-th non-distributed subspectrum,
1,,1, Llg l
- vectors of unknown non-distributed parameters of all subspectra,
KkvC k ,,1, - teoretical number of impulses scanned in the k-th spectrometer channel,
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xp - distribution function of distributed parameter x satisfying a normalisation condition:
max
min
1)(
x
x
dxxp
maxmin
maxmin
,0)(
,0)(
xxxforxp
xxxforxp
11 ,, LkL gxvF
- teoretical model of the only distributed subspectrum
.,,,1,, 111 JJjjjj xppJjxxxforxpp
KkgxvFphgvFCvCL
l
J
j
LjkLjlklk ,,1,,,,1 1
11
Modified teoretical model of Mössbauer spectrum:
Distribution function is searched by fitting process as a table of values:
equidistant nodes: Jjxxh jj ,,1,1
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1
2
211
1
2 2J
j
jjj
K
k
kkek pppvCvCwS
1. optimalisation procedure step: minimalization of the functional given by the weighted sum of residue squares and a smoothing member:
Frank-Wolfe quadratic programming method has been applied.
K
k
L
l
lklkek gvFCvCwS1
21
1
,
1,,1, Llg l
Kk
vCw
kek
,,1
,1
2. optimalisation procedure step: functional minimalization:
1,,1,, JjpC j
( Tabulated values of the distribution function are given in the preceding step.)
In order to estimate the unknown parameters:
In order to estimate the unknown parameters:
Levenberg-Marquardt optimalisation method has been applied.
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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.
• 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).
Examples of applying the method:
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Elektrical and magnetic hyperfine interactions
Attention is given to the three types of hyperfine interactions:electrical monopole,electrical quadrupole,magnetic dipole.
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.
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Energy of electrical interaction of a nucleus with its charged environment
V - charged nucleus volume,
- nuclear charge density in position
- electrical field potential of a charged surrounding of nucleus.
Decomposition of the electrical field potential into the Taylor series:
zx
yx
xx
3
2
1
3,2,10
0
ix
r
rii
3,2,1,0
0
2
jixx
r
rjiij
3
1,
3
1
02
100
ji
ijji
i
ii xxxr
dVrrW
V
nuE
r
rnu
r
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For a nuclear charge, it holds:
dVrqZ
V
nue
1. Total nuclear charge:
2. Dipole nuclear moment vector – law of parity conservation:
3,2,1,0 idVxrM
V
inui
3. Quadrupole nuclear moment tensor:
V
nu
V
nu
dVr
dVrr
R
2
24. Effective nuclear charge radius R:
V
jinuij jidVxxrQ 3,2,1,,
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Electrical monopole interaction – shift of energy levels of nuclei
Energy increase of nuclear states:
Energy change of a nuclear shift:
222
0
2
0 06
1grex
e RRqZ
W
22
0
2
06
1R
qZW e
EM
- superposition of wave functions of surrounding charges with rel
r
r
density, forming the field having
3
1
3
1,
000i ji
ijijiieE QMqZW The energy of electrical nuclear interaction in approaching the first three members of a series:
2
00r
qrr eel
For an electron charge, the
Poisson equation holds:
potential.
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Electrical monopole interaction – isomer shift of spectrum
- v
+ v
v [mm / s]v
0
0- v
+ v
1/2 1/2
v
Difference in energies of the same W0 transition in a source (S) and in an absorbator (A):
nuclear factor atomic-molecular factor
vδ –isomer shift of spectrum
2222
000 00
6
1SAegrexeSA qRRqZWW
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Mössbauer spectroscopy of hydrogenated Fe91Zr9 amorphous alloys.
Isomer shift (IS) provides valuable physical-chemical information about absorber properties.
Journal of Magnetism and Magnetic Materials 128 (1993), 365-368.
It is influenced by:
- electron structure of an atom,- atom valency, chemical bond,- charge states Fe2+ and Fe3+
(they are differ significantly in ).
→
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Electrical quadrupole interaction
31
2 eqQA
Parameter of non-homogeneous electrical field at nucleus:
e
zz
zz
yyxx
0 zzyyxx
← quadrupole nuclear moment
A tensor of an electrical field gradient at the proper choice of coordinate system:
For diagonal non-zero elements the Poisson equation hold:
)3,2,1,(0 jijiij
,11 xx ,22 yy zz 33
asymmetry parameter →
The nucleus with non-spheric distribution of a charge in a non-homogeneous electrical field.
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124
13,
2
II
IImQAmIW I
IEQ
Electrical quadrupole interaction energy
IIIIm I ,1,,1,0,1,,1, Magnetic quantum number of a nucleus:
021,0 QII
120 IQ - multiple degeneration of energetic levels
Spin quantum number of a nucleus: 021 QI
Proper energy values:
Hamiltonian of an electrical quadrupole interaction:
EQEQ
ˆˆˆˆ
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Quadrupole splitting of Mössbauer spectrum
1/2 AQ
v
mI=±1/2
I=3/2
I=1/2
mI=±3/2
mI=±1/2
42
3,
2
3 AQW
42
1,
2
3 AQW
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Quadrupole interaction – angular dependence of lines intensities
3
2sin,
3
1cos 22 0 2
2
1,
2
1
2
1,
2
3
angle between the direction of -quantum emission and the main axis of a crystal symmetry
2
1,
2
1
2
3,
2
3 2cos12
3
2sin2
31
Transition from the excited to ground state
Angular dependence
of linesintensities
Relative lines intensities
polycrystal monocrystal
1
3 3
1
1 5
II mImI ,,
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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 non-homogeneous electric field.
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, …
There are two principal sources of non-homogenity of the internal electric field at nucleus:
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Proceedings of 7-th European Symposium on Thermal Analysis and Calorimetry – ESTAC 7, Balatonfüred 1998.
[Fe(CN)5NO]2- [Fe(CN)6]3- [Fe(CN)6]4-
Na+, K+, K+
[dipyPhBCl]+
[(Et2N)2PhBCl]+
[Me2PhS]+
Room temperature transmission Mössbauer spectra of new boronium cyano complexes.
Cations ↓
Anions →
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Magnetic dipole interaction
Nucleus with non-zero magnetic moment in an effective magnetic field.
Dipole magnetic nuclear moment: Ignu
nu g- nuclear magnetone, - gyromagnetic factor
Effective magnetic field at the nucleus: hfloc HHH
- hyperfine field.locH hfH
Main sources of a hyperfine field:
- local field,
1. contact Fermi nuclear interaction with s-electrons,2. dipole-dipole nuclear interaction with electrons having non-zero charge
density at nucleus.
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I
mHmIW I
IMD ,Corresponding proper energy values:
Hamiltonian of magnetic dipole interaction:
- gives the energy change of a nuclear state, if a nucleus is found in the magnetic field.
Energy of magnetic dipole interaction
1. magnetic structure, 2. magnetic phase changes,3. phase analysis, …
HMDˆˆˆ
Studying condensed substances, the magnetic hyperfine spectrum strukture provides valuable physical information about:
MDW
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Zeeman splitting of Mössbauer spectrum
v 0
14,4keV
I=3/2
I=1/2
mI
+1/2
+3/2
-1/2
-3/2
-1/2
+1/2
2/3exH
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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:
.. groexc 3
2sin,
3
1cos 22 2
2
1,
2
1
2
3,
2
3
2
1,
2
1
2
3,
2
3
2cos14
9
2
1,
2
1
2
1,
2
3
2
1,
2
1
2
1,
2
3
2sin3
2
1,
2
1
2
1,
2
3
2
1,
2
1
2
1,
2
3
2cos14
3
Transition angular
dependence of a line intensity
3 3 3
2 0 4
1 1 1
0relative intensity relative intensity
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Structure and properties of the ball-milled spinel ferrites.
Materials Science and Engineering A226-8, (1997), 22-25.
ZnMgNiMeOFeMe ,,42
24
32
2321 OFeMeFeMe
(A)-tetra
[B]-okta
24
32
21max
33
24
31
21
31
,0,
,10,0
,1,0:
OFeNitt
BFeAFet
OFeNiFetNi
Redistribution of cations Fe3+
induced by high energetic mechanical milling
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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.
→
↓
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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.
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Properties of the nanocrystalline Finemet alloys prepared by mechanochemical way.
Acta physica slovaca 48 (1998), 703-706.
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.
Initial material for milling:
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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 spectruminto subspectra.
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MIMOS on the Mars Exploration Rovers – Spirit and Opportunity
Rover Traces on the Martian Surface
Images Credit: NASA/JPL-Caltech
Extraterestrial Applications of Mössbauer Spectroscopy
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