Mӧssbauer Spectroscopy UCY Lectures 2015

Mӧssbauer Spectrometry
A course for Undergraduates and Graduate students
Dr. Vlassis Petousis
1

Born: January 31, 1929, Munich, Germany
Died: September 14, 2011, Grünwald, Germany
http://www.nobelprize.org/mediaplayer/index.php?
id=880 3
Rudolf Mössbauer
PhD Work 1958
Nobel Prize in Physics 1961

Mössbauer Spectrometry
4
•  Mössbauer spectrometry provides unique measurements of electronic, magnetic, and
structural properties within materials.
•  Mössbauer spectrometry is based on the quantum mechanical “Mössbauer effect,” which
provides a nonintuitive link between nuclear and solid-state physics.
•  A Mössbauer spectrum is an intensity of γ-ray absorption versus energy for a specific
resonant nucleus such as 57Fe or 119Sn.
•  Mössbauer spectrometry looks at materials from the “inside out,” where “inside” refers to
the resonant nucleus.
•  For one nucleus to emit a γ-ray and a second nucleus to absorb it with efficiency, both
nuclei must be embedded in solids, a phenomenon known as the “Mössbauer effect.”

Mössbauer Spectrometry
5
•  Give quantitative information on “hyperfine interactions,” which are small energies
from the interaction between the nucleus and its neighboring electrons.
•  The three important hyperfine interactions originate from the electron density at
the nucleus
1.  the isomer shift (IS)
2.  the gradient of the electric field (the nuclear quadrupole splitting – EQS)
3.  the unpaired electron density at the nucleus (the hyperfine magnetic field – HMF).
•  Over the years, methods have been refined for using these three hyperfine interactions
to determine valence and spin at the resonant atom.
•  Even when the hyperfine interactions are not easily interpreted, they can often be used
reliably as “fingerprints” to identify the different local chemical environments of the
resonant atom, usually with a good estimate of their fractional abundances.

Mössbauer Spectrometry and its Sensitivity
6
•  For the most common Mössbauer isotope, 57Fe the linewidth is: 5x10-9 eV
•  Compared to the Mössbauer γ-ray energy of 14.4keV this gives a resolution of 1 in 1012
This is equivalent of a small speck of dust on the back of an elephant
or
One sheet of paper in the distance between the Sun and the Earth.

Principles of the Method – Nuclear Excitations
7
The nucleus can undergo transitions between quantum states much like the electrons in the
atom. Doing that we have large changes in energy.

8
The state of a nucleus is described in part by the quantum numbers :
E – Energy
I – Nuclear Spin
IZ – Nuclear Spin along the z - axis
In addition to these, 3 internal nuclear coordinates, to understand the Mössbauer effect also
needs spatial coordinates X, for the nuclear center of mass as the nucleus moves through
space or vibrates in a crystal lattice.
These center-of-mass coordinates are decoupled from the internal excitations of the nucleus.

9
The internal coordinates of the nucleus are mutually coupled.
For Example
The excited state of the 57Fe has spin:
I=3/2
IZ = -3/2 , -1/2, +1/2, +3/2
The round state of the 57Fe has spin:
I=1/2 and 2 allowed values for IZ
In the absence of hyperfine interactions to lift the energy degeneracies of spin levels,
all allowed transitions between these spin levels will occur at the same energy, giving a
total cross-section σ0 for nuclear absorption, of 2.57 × 10–18 cm2.
Iz=2I+1 states

10
•  Although σ0 is smaller by a factor of 100 than a typical projected area of an atomic electron
cloud, σ0 is much larger than the characteristic size of the nucleus.
•  It is also 100s of times larger than the cross-section for scattering a 14.41 keV photon by
the atomic electrons at 57Fe.

11
The characteristic lifetime of the excited state of the 57Fe nucleus (which is relatively long):
τ = 141 ns
The time uncertainty of the nuclear excited state, τ , is related to the energy uncertainty (due
to Heinseberg) of the excited state ΔE, calculated through the uncertainty relationship:
ħ ~ ΔE τ
For τ =141 ns, the uncertainty relationship provides: ΔE = 4.7 × 10−9 eV
This is remarkably small — the energy of the nuclear excited state is extremely precise.
A nuclear resonant γ-ray emission or absorption has an oscillator quality factor, Q = 3x1012.
The purity of phase of the γ ray is equally impressive.

12
20
1
1 ( )
/ 2
L
p p
w
=
−
+
Lorentzian (Cauchy) function
The HWHM (w/2) is 1.

13
•  For a single type of nuclear transition, the energy dependence of the cross-section
for Mössbauer scattering is of Lorentzian form, with a width determined by the
small lifetime broadening of the excited state energy:
0
0 2
( )
1 ( )
/ 2
j
j
j
p
E
E E
σ
σ =
−
+
Γ
For 57Fe
Γ = ΔE = 4.7 × 10−9 eV
Ej is the mean energy of the nuclear level transition (14.41 keV).
pj is the fraction of nuclear absorptions that will occur with energy Ej .

14
In the usual case where the energy levels of the different Mössbauer nuclei are
inequivalent and the nuclei scatter independently, the total cross section is:
A Mössbauer spectrometry measurement is usually designed to measure the energy
dependence of the total cross-section, σ(E), which is often a sum of Lorentzian functions
of natural line width Γ.
( ) ( )j
j
Eσ σΕ = ∑

15
•  It is sometimes possible to measure coherent Mössbauer scattering.
•  Here the total intensity I(E), from a sample is not the sum of independent intensity
contributions from individual nuclei.
•  One considers instead the total wave Ψ(r,E), at a detector located at r.
•  The total wave Ψ(r,E), is the sum of the scattered waves from individual nuclei j
Ψ(
!
r,E) = ψ j
(
!
r,E)
j
∑

17
The Mössbauer Effect
To gain an insight into physical basis of the Mӧssbauer effect importance of recoils
emission of γ-rays we must consider the interplay a variety of factors:
•  Energetics of free - atom recoil and thermal broadening.
•  Heisenberg natural linewidth.
•  Energy and momentum transfer to the lattice.
•  Recoil – free and Debye-Waller factor.
•  Cross – section for resonant absorption.
•  The Mӧssbauer Spectrum

18
Energetics of free - atom recoil and thermal broadening
Lets consider an isolated atom in the gas phase and define the energy difference between the
ground state of the nucleus Eg and its excite state Ee

19
The total energy of the system is:
Doppler Effect Energy
Recoil Energy

20
The mean kinetic energy per translation degree of freedom with random thermal motion is:
The mean broadening is:
Broadening by twice the geometric mean of the recoil
Energy and the average thermal energy
The distribution is Gaussian :

21
The values ER and ED can be more conventionally expressed in terms of γ-ray Energies Eγ :

22
Discussing ER and ED and Resonant Emission and Absorption.
Fundamental radiation theory tells us that the proportion of absorption determined by the
overlap between the exciting and exited distributions. In this case the γ – ray has lost energy
ER due to recoil.
In the reverse case, where a γ – ray
reabsorbed by a nucleus, a further
increment of energy ER is required
since the γ – ray must provide both
the nuclear excitation energy and the
recoil energy of the absorbing atom

23
Heisenberg Natural Linewidth
One of the most important influences on the γ – ray energy distribution is the lifetime of the
excited state.
The ground state nuclear level has an infinite lifetime and hence a zero uncertainty in energy.
The excited state has mean life time τ of a microsecond or less.
This gives a spread in energy of the γ – ray of width ΓS at half height :

24
Cross – section for Resonant Reabsorption
The probability of recoilless emission from a source is fS this recoilless radiation has a
Heisenberg width at half height of ΓS and the distribution of energies about the energy Eγ leads
to a Lorentzian distribution. The number of transitions N(E) with energy between (Eγ - E) and
(Eγ - E +dE) is given by:

25
The number of transitions N(E) with energy between (Eγ - E) and (Eγ - E +dE) is given by

26
Γa Heisenberg width at half height
σ0 effective cross section
Ie Nuclear spin at the excited state
Ig Nuclear spin at the ground state
a Internal conversion coefficient of
the γ – ray of the wavelength λ

27
Γa Heisenberg width at half height
σ0 Effective cross section
Ie Nuclear spin at the excited state
Ig Nuclear spin at the ground state
a Internal conversion coefficient of the γ – ray of the wavelength λ
(the ratio of conversion electrons to the γ – ray photon emission)

28
The Mössbauer Spectrum
A completely general evaluation of the problem is impossible, but useful result are
obtained if we assume that both source and absorber have the same linewidth Γ = ΓS =
Γa) Margulies and Ehrman showed that γ–ray transmission thought a uniform resonant
absorber :
γ - Transmission
Decrease of Transmission
ε – Energy
displacement between
source - absorber

29
If γ – ray from a source which has a substantial recoil-free fraction passed through an
absorber of the same material, the transmission of the γ – rays in the direction of the
beam will be less than expected because of their resonant reabsorption and subsequent
reemission over a 4π solid angle.
It was shown in the I(E) relation that the decrease in transmission is affected by the
the difference in the relative values of Eγ for the source and the absorber relative to
each other with velocity υ.
Using an external applied Doppler Effect.
If the effective Eγ are exactly matched at the certain Doppler velocity, resonance will
be at the maximum and the count rate a minimum.

30
The Doppler effect can be used to change the energy of the γ-ray by moving the source
relative to the absorber E(υ)=E0(1+υ/c) where υ is relative velocity and c the speed of
light.

32
The Mössbauer Techniques
•  Velocity Modulation of γ-rays
•  Constant Velocity Drives
•  Repetitive Velocity – scan Systems
•  Pulse Height Analysis Mode
•  Time Mode Spectrometers
•  Our Mössbauer Spectrometer

33
Velocity Modulation of γ-rays
The recoil free γ-ray energy of a typical MS transition is so precisely defined that its
Heisenberg width corresponds to the energy change produced by an applied Doppler velocity
of the order of 1 mm sec-1
It is possible to imagine a particular relative velocity between source and absorber at which
the γ-ray energy from the source will precisely match the nuclear energy level gap in the
absorber and resonant absorption will be at the maximum.
For a source and absorber which they are chemically identical this relative velocity will
be zero.
Application of additional velocity increment will lower the resonant overlap and
decrease the absorption. Application of a sufficient large relative velocity will destroy
the resonance completely.

34
We have already seen that the resonant absorption curve for an ideally thin source and
absorber has a width at half height Γr which is twice the Heisenberg width of the emitted γ-
photon

35
The two general approaches to the measurements of γ-ray transitions at different Doppler
velocities are:
1.  Measurement of the total γ-photons in a fixed time at constant velocity followed by
subsequent counts at other velocities. In this way the spectrum is scanned stepwise
one velocity at the time.
2.  Rapid scanning through whole velocity range and subsequent numerous repetitions
of this scan. Next we do is the accumulation of all the data for the individual
velocities essentially simultaneously.

36
Constant –Velocity Drives
To move an object at constant velocity with high reproducibility and stability when its
restricted by both a relatively small amplitude of movement and the necessity for repetitive
motion is a difficult problem in applied mechanics.
Several constant-velocity spectrometers have been described and here are briefly classified:
•  Lathe and Gears
•  Lathe and inclined plane
•  Hydraulic Device
•  Cams and Gears
•  Pendulum
•  Spinning Disk
•  Electromechanical Drive
•  Piezoelectric Drive

37

38

39
The advantages offered by a constant velocity spectrometer include the ability to
examine a small velocity range not centered on zero velocity and to calibrate the
instrument directly in terms of absolute velocity.
The disadvantages are that such an instrument it extremely tedious to operate unless it
has been fully automated. It is difficult to machine cams and lead screws to the
precision required to give an accurately linear drive. Mechanical wear and vibrations
are also a problem.

40
Repetitive Velocity-scan Systems
Using repetitive scanning techniques: The Doppler motion is provided by an
electromechanical drive system which is controlled by a servo-amplifier.
The amplified is fed with a reference voltage waveform which repeats itself exactly with the
frequency of between 5-40 Hz
The actual drive of transducer embodies two coils, one of which produces a voltage
proportional to the “actual” velocity of the shaft.
The servo amplifier compares this signal to the reference waveform and applies corrections
to the drive coil in order to minimize the difference.
In this way, the centre shaft which is rigidly connected to the source executes an accurate
periodic motion.

41
Repetitive Velocity-scan Systems
Sine wave: Is less demanding on the mechanical
adjustment of the transducer. Non liner scale on the final
spectrum.
Asymmetric double ramp: Executes 80% of the motion
with constant acceleration. à Linear velocity. At high
frequencies cause some difficulties.
Symmetric double ramp: Scans with constant acceleration
in opposite direction (mirror image) à Linear velocity. Can
be folded on order to give an additional check on linearity.
How its γ-ray can be used to produce a pulse with amplitude
characteristics of the instantaneous velocity.

42
Pulse-height Analysis Mode

43
Pulse-height Analysis Mode
Disadvantages:
1.  The ADC process is slow and each time a pulse is counted it imposes a variable “dead
time” up to the maximum of about 100µsec during which no other pulse can be
detected.
•  The “dead time” must be fixed by the operator at a value at least as great as the
maximum time for the storage. Otherwise one would register faster counting rates
at the lower channel address numbers.
2.  Poor linearity and stability of the ADC.

44
Time-mode (Multiscalar-mode) Spectometer

45
Our Mössbauer Spectrometer

46
Our Mössbauer Spectrometer

48
Mössbauer and Hyperfine Interactions
An Overview of Hyperfine Interactions
Given the existence of the Mössbauer effect, a question arises, what it can do ?
•  The answer is given in two parts:
1.  What are the phenomena that can be measured ?
2.  What do these measurables tell us about materials ?
•  There are 4 standard measurable quantities which can be categorized as:
1.  The Isomer Shift (IS).
2.  The Electric Quadrupole Splitting (EQS).
3.  The Magnetic Hyperfine Splitting (MHS).

49
The Isomer Shift (IS)
•  Is the easiest hyperfine interaction to understand.
•  It is a direct measure of electron density.
The IS changes with the valence of the Mössbauer atom such as 57Fe or 119Sn.
It is possible to use the IS to estimate the fraction of Mössbauer isotope in different
valence states, which may originate from different crystallographic site occupancies or
from the presence of multiple phases in a sample.
Unfortunately, although the isomer shift is in principle sensitive to local atomic
coordinations, it has usually not proven useful for structural characterization of materials,
except when changes in valence are involved. The isomer shifts caused by most local
structural distortions are generally too small to be useful.

50
The peaks in a Mössbauer spectrum undergo observable shifts in energy when the
Mössbauer atom is in different materials.
These shifts originate from a hyperfine interaction involving the nucleus and the inner
electrons of the atom.
These “isomer shifts” are in proportion to the electron density at the nucleus.
Two possibly unfamiliar concepts underlie the origin of the isomer shift:
1.  Some atomic electron wave-functions are actually present inside the nucleus.
2.  The nuclear radius is different in the nuclear ground and excited states.

51
The IS depends directly on the s-electrons and
can be influenced by the shielding electrons.
From the measured Delta Shift (δ) there is
information about the valance state of the
absorbing atom.

52
•  The IS is defined as a displacement of the frequency of the nuclear γ-transition in the
absorber nucleus ∆Eγa with respect to the source nucleus ∆Eγs.
•  The variation of the nuclear volume, ie, the nuclear charge radius, during the
γ-transition is responsible for the occurrence of IS, because the atomic nucleus is not a
point-like object but an object of a finite spatial extent.
•  The different nuclear charge radius in the excited and ground state induce different
electron-nuclear interactions therein, hence the frequency of the γ-transition in the
nucleus immersed in a specific electronic environment is different than in the bare
nucleus.

53
1.1. Introduction
of the energy of the resonance γ quantum between the
the absorber (a) nuclei, thus there appears a dependen
ergy of resonance γ quantum on the electronic environm
the given nucleus is immersed. The MIS, δ measured in
Doppler velocity necessary to achieve resonance is give
δ =
c
Eγ
(∆Ea
γ − ∆Es
γ )
where c is the velocity of light and Eγ is the energy of th
•  If one measures the change of the energy of the
resonance γ quantum between the source Es and the
absorber Ea nuclei, thus there appears a dependence of
the energy of resonance γ quantum on the electronic
environment in which the given nucleus is immersed.
•  The δ measured in terms of the Doppler velocity
necessary to achieve resonance is given:
•  The most valuable information derived from isomer
shift data refers to the oxidation state and spin state
of the active atom, its bond properties etc.

54
Electric Quadrupole Splitting (EQS)
Often correlated to IS.
•  The existence of an EQS requires an asymmetric (i.e., noncubic) electronic
environment around the nucleus, however, and this usually correlates with the local
atomic structure.
•  Again, like the isomer shift, the EQS has proven most useful for studies of oxides and
minerals.
•  The EQS is more capable of providing information about the local atomic coordination
of the Mössbauer isotope.
•  For 57Fe, the shifts in peak positions caused by the EQS tend to be comparable to, or
larger than, those caused by the isomer shift.

55
•  EQS in the Mössbauer spectrum occurs when a nucleus with an electric quadrupole
moment experiences a non-uniform electric field.
•  The nuclear charge distribution deviates from the spherical symmetry for a nucleus that
has spin quantum number I > 1/2 and thus has a non zero electric quadrupole moment.
•  The magnitude of the quadrupole moment may change in going from one state of
excitation to another.
•  The sign of the electric quadrupole moment, Q indicates the shape of the deformation.

56
•  The Q is constant for a given Mössbauer nucleus.
•  Changes in the quadrupole interaction energy
observed can only arise from the changes in the
Electric Field Gradient (EFG) generated by the
surrounding electrons and other nuclei.
•  The interaction between the electric quadrupole moment of the nucleus and EFG at the
nuclear position give rise to a splitting in the nuclear energy levels into sub-states, which
are characterized by the absolute magnitude of the nuclear magnetic spin quantum number
|mI|.

57
Electric Quadrupole Splitting (EQS) The Quadrupole Moment Q is :
1.  Negative for a flattened (pancake-shaped)
nucleus.
2.  Positive for an elongated nucleus
(cigar-shaped).

58
The separation between the lines, ∆EQ, is known as the
EQS and is written as:
of Figure 1.7 shows the quadrupole splitting of the
vels of 57
Fe, where the absoption line is split due to
the nuclear quadrupole moment with non-zero EFG
he separation between the lines , ∆EQ, is known as
plitting and is written as,
∆EQ =
1
2
qQVzz
1 + η2
3
1/2
(1.6)
ctrical charge, Q is the nuclear quadrupole moment,
tric ﬁeld gradient due to the total electron density
harges. V can be decomposed into three principal
Vyy, and Vxx, in descending order of magnitude, and
Where Vzz is the electric field gradient due to the
total electron density plus all nuclear charges.
The EQS provides information on the symmetry of the
coordination sphere of the resonating atom.

59
Magnetic Hyperfine Splitting (MHS) due to Hyperfine Magnetic Fields (HMF)
IS shifts are universal, Hyperfine Magnetic fields (HMF) are confined to ferro-, ferri-, or
antiferromagnetic materials.
While IS shifts tend to be small, HMFs usually provide large and distinct shifts of Mössbauer
peaks.
Because their effects are so large and varied, HMFs often permit detailed materials
characterizations by Mössbauer spectrometry.

60
The spins can be oriented with different projections along a magnetic field.
The energies of nuclear transitions are therefore modified when the nucleus is in a magnetic
field.
The energy perturbations caused by this HMF are sometimes called the “Nuclear Zeeman
Effect,” in analogy with the more familiar splitting of energy levels of atomic electrons when
there is a magnetic field at the atom.

61
A hyperfine magnetic field lifts all degeneracies of the spin states of the nucleus, resulting in
separate transitions identifiable in a Mössbauer spectrum.
Zeeman Effect

62
(+1/2→+1/2) (+1/2→+3/2)}. The allowed transitions are
shown in Figure 1. Notice the inversion in energy levels of
thenucleargroundstate.
Figure 2. Mössbauer spectrum from bcc Fe. Data were
electron w
electron an
electron w
the elect
gyromagne
nuclear sp
case for an
do not hav
57
Fe is larg
moment is
energy by
effectivem
where the
density at

63
•  In ferromagnetic iron metal, the magnetic field at the 57Fe nucleus, the HMF, is
33.0 T at 300 K.
•  When an external magnetic field is applied to a sample of Fe metal, there is a
decrease in magnetic splitting of the measured Mössbauer peaks.
•  This observation shows that the HMF at the 57Fe nucleus has a sign opposite to that
of the lattice magnetization of Fe metal, so the HMF is given as −33.0 T.

64
A nuclear state with spin I > 1/2 possesses a
magnetic dipole moment µ.
The magnetic field splits the nuclear level of spin I
into (2I + 1) equispaced nondegenerate substates
characterized by the magnetic spin quantum
numbers mI.
Therefore for 57Fe :
Excited state I = 3/2 is split into 4
Ground state with I = 1/2 into 2

65
The energies of the sublevels are given from first-order perturbation theory:
βN is the nuclear Bohr magneton.
µ is the nuclear magnetic moment.
mI is the magnetic spin quantum number.
gN is the nuclear gyromagnetic-factor.
(Is a proportionality between nuclear spin and nuclear magnetic moment. Unlike the
electron case the ground state and the excited state have not the same value on gN).

66
The magnetic hyperfine splitting enables one to determine the effective magnetic field
acting at the nucleus.
The total effective magnetic field is the vector sum of externally applied magnetic filed
and the internal magnetic field:
Heff = Hext + Hint and Hint = HL + HD + HC
•  HL is the contribution from the orbital motion of the electrons.
•  HD is the contribution of the magnetic moment of the spin of the electrons outside the
nucleus (spin-dipolar term).
•  HC is the contribution of the spin-density at the nucleus (Fermi contact term).

67
•  The magnetic hyperfine interaction gives a clear understanding of the magnetic
properties of materials.
•  In compounds with unpaired electrons the Mössbauer spectroscopy enables one to
distinguish between the high-spin and low-spin states, spin density at various nuclei in
a molecule, study the magnetic ordering, etc.

68

Mӧssbauer Spectroscopy UCY Lectures 2015

Recommandé

Recommandé

Contenu connexe

Tendances

Tendances (20)

En vedette

En vedette (8)

Similaire à Mӧssbauer Spectroscopy UCY Lectures 2015

Similaire à Mӧssbauer Spectroscopy UCY Lectures 2015 (20)

Dernier

Dernier (20)

Mӧssbauer Spectroscopy UCY Lectures 2015