Mӧssbauer Spectroscopy UCY Lectures 2015

Mӧssbauer Spectrometry

A course for Undergraduates and Graduate students

Dr. Vlassis Petousis

1

1st Course

2

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

Principles of the Method – Nuclear Excitations •  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

Principles of the Method – Nuclear Excitations 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 pw

= −+

Lorentzian (Cauchy) function

TheHWHM(w/2)is1.

13

Principles of the Method – Nuclear Excitations •  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

jj

j

pE E E

σσ = −

+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.

( ) ( )jj

Eσ σΕ =∑

15

Principles of the Method – Nuclear Excitations •  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∑

2nd Course

16

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

The Mössbauer Effect 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

The Mössbauer Effect 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

The Mössbauer Effect 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

The Mössbauer Spectrum 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 Mössbauer Spectrum 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.

3rd Course

31

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

The Mössbauer Techniques 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

The Mössbauer Techniques 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

The Mössbauer Techniques 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

The Mössbauer Techniques 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

The Mössbauer Techniques Pulse-height Analysis Mode

43

The Mössbauer Techniques 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

The Mössbauer Techniques Time-mode (Multiscalar-mode) Spectometer

45

The Mössbauer Techniques Our Mössbauer Spectrometer

46

The Mössbauer Techniques Our Mössbauer Spectrometer

4th Course

47

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 Isomer Shift (IS) 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 Isomer Shift (IS) 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 Isomer Shift (IS) •  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


The Isomer Shift (IS)

1.1. Introduction 11

of the energy of the resonance γ quantum between the source (s) and

the absorber (a) nuclei, thus there appears a dependence of the en-

ergy of resonance γ quantum on the electronic environment in which

the given nucleus is immersed. The MIS, δ measured in terms of the

Doppler velocity necessary to achieve resonance is given in Eqn. 1.4,

δ =c

Eγ(∆Ea

γ −∆Esγ) (1.4)

where c is the velocity of light and Eγ is the energy of the γ quantum.

Figure 1.7: The isomer Shift and Quadrupole Splitting of the nuclear energylevels and corresponding Mossbauer spectra.

This shift appears in the spectrum as the difference between the

position of the baricenter of the resonance signal and zero Doppler

velocity as shown in Figure 1.7 [5, 6, 12]. Traditionally, the energy dif-

ferences ∆Ea/sγ are calculated within the framework of perturbation

•  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


Electric Quadrupole Splitting (EQS)

•  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


Electric Quadrupole Splitting (EQS) •  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


Electric Quadrupole Splitting (EQS) The separation between the lines, ∆EQ, is known as the EQS and is written as:

1.1. Introduction 13

and other nuclei. Therefore, the interpretation of quadrupole split-

tings requires the knowledge of the EFG. 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 nu-

clear magnetic spin quantum number |mI |.

As the Mossbauer spectroscopy involves the absorption of the γ-

rays to promote a nucleus from the ground state to an excited state,

the quadrupole Hamiltonian has to be solved for each energy level if

both levels have nuclear spin greater than 1/2. For 57Fe, the ground

state has nuclear spin I = 1/2 and the lowest excited state has I = 3/2.

The second part of Figure 1.7 shows the quadrupole splitting of the

nuclear energy levels of 57Fe, where the absoption line is split due to

the interaction of the nuclear quadrupole moment with non-zero EFG

at the nucleus. The separation between the lines , ∆EQ, is known as

the quadrupole splitting and is written as,

∆EQ =1

2qQVzz

!

1 + η2

3

"1/2

(1.6)

where e is the electrical charge, Q is the nuclear quadrupole moment,

and V is the electric field gradient due to the total electron density

plus all nuclear charges. V can be decomposed into three principal

components, Vzz, Vyy, and Vxx, in descending order of magnitude, and

η is the asymmetry parameter defined as (Vxx - Vyy)/Vzz. For the

substates with axially symmetric EFG (η = 0), the energy separation

∆EQ is,

∆EQ =1

2qQVzz (1.7)

The quadrupole splitting provides information on the symmetry of

the coordination sphere of the resonating atom.

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


Magnetic Hyperfine Splitting (MHS) due to Hyperfine Magnetic Fields (HMF) 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


Magnetic Hyperfine Splitting (MHS) due to Hyperfine Magnetic Fields (HMF) 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


Magnetic Hyperfine Splitting (MHS) due to Hyperfine Magnetic Fields (HMF)

and excited states are set by selection rules. For the M1 magnetic dipole radiation for 57Fe, six transitions are allowed: {(−1/2→−3/2) (−1/2→−1/2) (−1/2→+1/2) (+1/2→−1/2) (+1/2→+1/2) (+1/2→+3/2)}. The allowed transitions are shown in Figure 1. Notice the inversion in energy levels of the nuclear ground state.

Figure 2. Mössbauer spectrum from bcc Fe. Data were acquired at 300 K in transmission geometry with a constant acceleration spectrometer (Ranger MS900). The points are the experimental data. The solid line is a fit to the data for six independent Lorentzian functions with unconstrained centers, widths, and depths. Also in the fit was a parabolic background function, which accounts for the fact that the radiation source was somewhat closer to the specimen at zero velocity than at the large positive or negative velocities. A 57Co source in Rh was used, but the zero of the velocity scale is the centroid of the Fe spectrum itself. Separation between peaks 1 and 6 is 10.62 mm/s.

In ferromagnetic iron metal, the magnetic field at the 57Fe nucleus, the HMF, is 33.0 T at 300 K. The enormity of this HMF suggests immediately that it does not originate from the traditional mechanisms of solid-state magnetism. Furthermore, 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 latter 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.

It is easiest to understand the classical contributions to the HMF, denoted Hmag, Hdip and Horb.! The contribution Hmag!is the magnetic field from the lattice magnetization, M, which is 4πM/3. TO this contribution we add any magnetic fields applied by the experimenter, and we subtract the demagnetization caused by the return flux. Typically, Hmag<+0.7 T. The contribution Hdip is the classical dipole magnetic field caused by magnetic moments at atoms near the Mössbauer nucleus. In Fe metal, Hdip!vanishes owing to cubic symmetry, but contributions of +0.1 T are possible when neighboring Fe atoms are replaced with nonmagnetic solutes. Finally, Horb originates with any residual orbital magnetic moment from the Mössbauer atom that is not quenched when the atom is a crystal lattice. This contribution is about +2 T (Akai, 1986), and it may not change significantly when Fe metal is alloyed with solute atoms, for example. These classical mechanisms make only minor contributions to the HMF.

The big contribution to the HMF at a Mössbauer nucleus originates with the “Fermi contact interaction.” Using the Dirac equation, Fermi and Segre discovered a new term in the Hamiltonian for the interaction of a nucleus and an atomic electron

HFC = – 8π/3 gegnµeµnI.S δ(r) (23) Here I and S are spin operators that act on the nuclear and electron wavefunctions, respectively,! µe and µN are the electron and nuclear magnetons, and δ(r) ensures that the electron wavefunction is sampled at the nucleus. Much like the electron gyromagnetic ratio, ge, the nuclear gyromagnetic ratio, gN, is a proportionality between the nuclear spin and the nuclear magnetic moment. Unlike the case for an electron, the nuclear ground and excited states do not have the same value of gN; that of the ground state of 57Fe is larger by a factor of −1.7145. The nuclear magnetic moment is gN µNI, so we can express the Fermi contact energy by considering this nuclear magnetic moment in an effective magnetic field, Heff!,!defined as

(24)

where the electron spin is ±1/2, and |ψ(0)|2 is the electron density at the nucleus. If two electrons of opposite spin have the same density at the nucleus, their contributions will cancel and Heff will be zero. A large HMF requires an unpaired electron density at the nucleus, expressed as |S| > 0.

The Fermi contact interaction explains why the HMF is negative in 57Fe. As described above (see Isomer Shift), only s electrons of Fe have a substantial presence at the nucleus. The largest contribution to the 57Fe HMF is from 2s electrons, however, which are spin-paired core electrons. The reason that spin-paired core electrons can make a large contribution to the HMF is that the 2s↑ and 2s↓ wavefunctions have slightly different shapes when the Fe atom is magnetic. The magnetic moment of Fe atoms originates primarily with unpaired 3d electrons, so the imbalance in numbers of 3d↑ and 3d↓ electrons must affect the shapes of the paired 2s↑ and 2s↓, electrons.

These shapes of the 2s↑ and 2s↓ electron wavefunctions are altered by exchange interactions with the 3d↑ and 3d↓ electrons. The exchange interaction originates with the Pauli exclusion principle, which requires that a multielectron wavefunction be antisymmetric under the exchange of electron coordinates. The process of antisymmetrization of a multielectron wavefunction produces an energy! contribution from the Coulomb interaction between electrons called the “exchange energy,” which is the! expectation value of the Coulomb energy for all pairs of electrons of like spin exchanged between their wavefunctions.

The net effect of the exchange interaction is to decrease the repulsive energy between electrons of like spin. In particular, the exchange interaction reduces the Coulomb repulsion between the 2s↑ and 3d↑ electrons, allowing the more centralized 2s↑ electrons to expand outward away from the nucleus. The same effect occurs for the 2s↓ and 3d↓ electrons, but to a lesser extent because there are fewer 3d↓ electrons than 3d↑ electrons in ferromagnetic Fe. The result is a higher density of 2s↓ than 2s↑ electrons at the 57Fe nucleus. The same effect occurs for the 1s shell, and the net result is that the HMF at the 57Fe nucleus is opposite in sign to the lattice magnetization (which is dominated by the 3d↑ electrons). The 3s electrons contribute to the HMF, but are at about the same mean radius as the 3d electrons, so their spin unbalance at the

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

Education

Transcript of Mӧssbauer Spectroscopy UCY Lectures 2015