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

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

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

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:

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.

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.

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

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

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.

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

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

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

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:

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

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,

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

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.

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:

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.

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

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,,

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.

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

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 ).

→

Electrical quadrupole interaction

31

2 eqQA

Parameter of non-homogeneous electrical field at nucleus:

e

zz

qQ

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.

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

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

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 ,,

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:

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 →

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.

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

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

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

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

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.

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:

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.

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