Mössbauer showed that nuclear radiation can be emitted and absorbed by another nucleus of the same kind as the emitter without recoil if the nucleus placed in a solid [1] . When a free nucleus emits a γ-ray (E_γ) during transition from an excited state level with energy (E_e) to the ground state level with energy (E_g) it will recoil. This recoil energy and velocity are due to conservation of energy and momentum, resulting in the emitted γ-ray being of lower energy than the nuclear transition energy (E₀ = E_e − E_g). The same is observed where the absorbing nucleus recoils, meaning the energy of the resonantly absorbed photon is less than the transition energy. To understand this effect [7],assume a free nucleus of mass M with an excited state level at an energy E₀ and moving with a velocity V along the direction of the emission of E_γ energy (Figure 1).

The energy above the ground state at rest is E 0 + ( 1 / 2 ) M V 2 .When a γ-ray of energy E_γ is emitted,the nucleus recoils and has a velocity (V+υ) and a total energy of E 0 + ( 1 / 2 ) M ( V + υ ) 2 . By considering conservation of energy before and after the emission of the a γ-ray:

E 0 + 1 2 M V 2 = E γ + 1 2 M ( V + υ ) 2 (1)

The actual energy of the photon emitted is given by:

E γ = E 0 − 1 2 M υ 2 − M υ V = E 0 − E R − E D (2)

The energy E_γ of the emitted photon in the above equation is the excitation energy E₀ reduced by the recoil kinetic energy E_R of the nucleus and the thermal or Doppler energy (E_D = MυV) which is proportional to the initial velocity V from the thermal motion of the nucleus. The recoil nucleus kinetic energy E_R expressed by:

E R = 1 2 M υ 2 (3)

Equating the components of momentum before and after the emission process of γ-ray gives:

M V = M ( V + υ ) + E γ c (4)

where (E_γ/c) is the momentum of the photon and c is the speed of light. The recoil momentum of the γ-ray is:

M υ = − E γ c (5)

Inserting Equation (5) into Equation (3), the expression for the recoil energy E_R of the free nucleus after the emission of the γ-ray is given by:

E R = E γ 2 2 M c 2 (6)

Since the atoms of the emitter as well as the absorber have thermal energy, they move in certain velocities, and for each velocity the emitted wave’s frequency is changed according to the Doppler shift. The Doppler energy E_D is dependent on the thermal motion of the nucleus, and will therefore have a distribution of values which is temperature dependent. A mean value, Ē_D, can be expressed [7] which is related to the mean kinetic energy Ē_k per translational degree of freedom of a free atom.

E ¯ D = E γ 2 E ¯ k M c 2 ;   E ¯ k ≅ 1 2 k B T (7)

where k_B is Boltzmann’s constant and T is the absolute temperature. As a result, the statistical distribution in energy of the emitted γ-rays is displaced from the true excited state energy by -E_R and broadened by E_D into a Gaussian distribution of width 2Ē_D. The distribution for absorption has the same shape but is displaced by +E_R. Figure 2 illustrated schematically the statistical distributions for the emitting and absorbing atoms. These energy distributions must strongly overlap in order for resonant absorption to occur.

The recoil energy is much larger than the natural linewidth (Г₀ ≈ 10⁻⁸ eV) of the nuclear energy involved which is determined by the Heisenberg’s uncertainty principle as Г₀ = ħ/t, where ħ is Planck’s constant and t is the mean life time of

the excited state, e.g. for E_γ = 21.64 keV of ¹⁵¹Eu nucleus, E_R ≈ 10⁻³ eV. This results in a reduction of the E_γ energy, (beyond the Г₀ limit) therefore, nuclear resonant absorption of γ-ray emitted by the free nuclei can not occur.

Thus, the resonance emission and absorption can take place under the condition that 2E_R ≤ Г₀. This resonance condition is fulfilled when the recoil momentum or energy upon emission or absorption of a γ-ray is taken up by the solid as a whole rather than by a free nucleus. As the equations for recoil energy and Doppler broadening contain the reciprocal mass M, E_R and Ē_D become very small and much lower than Г₀ and thereby fulfill the resonance condition.

If the nucleus is bound in a solid, in which the movements of atoms are superpositions of quantized vibrational modes, then the recoil energy E_R must be almost as large as the energy of the vibration quanta phonon, in order to excite a phonon. In the Einstein model of lattice vibrations, there is only one phonon frequency ω. The lattice vibrations are quantized, and the recoil energy can only be transferred to the lattice if it takes place in integral multiples ħω. During the emission of a gamma quantum, part of the energy E₀ may be lost to the lattice by exciting a vibrational state. In this case the energy of the gamma quantum is reduced by the amount ( 0 , ± ℏ ω , ± 2 ℏ ω , ⋯ ) and the energy of the gamma quantum will then be too small to be resonantly absorbed by another nucleus. If the recoil energy E_R < ħω, then either zero-phonon event or one-phonon event (ħω) of vibrational energy may be transferred to the lattice. It has been shown [8] through a theoretical consideration that when an average is taken over many emission processes, the energy transferred per event is exactly equal E_R. Using this fact we may write an expression which leads directly to the fraction, f, of emission events which takes place without lattice excitation (zero-phonon transition), and a fraction (1 − f) transfers one phonon energy ħω:

f = ( 1 − f ) ℏ ω   or   f = 1 − E R ℏ ω (8)

Only these events (i.e. recoilless fraction f) give rise to resonance emission or absorption, i.e. to the Mössbauer effect. f is called Mössbauer-Lamb factor.

The probability of recoil-free fraction of γ-rays emitted or absorbed without loss of energy to the lattice vibrations is related to the Mössbauer-Lamb factor f which can be written as [3] :

f = exp ( − k γ 2 〈 x 2 〉 ) ;   k γ = E γ ℏ c = 2 π λ (9)

where k_γ is the wave number of γ-ray with the wavelength of the gamma ray (λ = 0.57 Å) in case of ¹⁵¹Eu.

〈 x 2 〉 is the mean square amplitude for thermal vibrations of the emitting or absorbing nucleus in the direction of γ-ray. The Mössbauer Lamb factor f value when it increases, leads to an increase the Mössbauer effect at relatively low γ-ray energy and at low temperature, 〈 x 2 〉 decreases with lowering temperature. Therefore, most Mössbauer spectroscopy experiments are performed at low temperature such as liquid nitrogen or liquid helium. For the Debye model of lattice vibrations Equation (9) has the form [3] :

f = exp ( − 3 E R 2 k B Θ D ( 1 + 4 ( T Θ D ) 2 ∫ 0 Θ D T x e x − 1 d x ) ) (10)

where Θ_D is the Debye temperature of the crystal in which the nucleus is bound, T is the absolute temperature of the crystal at which the experiment is performed and k_B is the Boltzmann constant. Since the fraction f is strongly temperature dependent and relative to the Debye temperature, decreases as the temperature increases and conversely, an approximations can be made for very low and high temperature conditions:

f = exp ( − 3 E R 2 k B Θ D )     for   T ≪ Θ D ; (11)

f = exp ( − 6 E R T k B Θ D 2 )   for   T > Θ D ; (12)

In the transmission geometry of Mössbauer experiment, a sample is exposed to a γ-ray source and a detector measures the intensity of the beam transmitted through the sample. This technique is achieved by moving a source relative to an absorber at a variable velocity. In this way, the energy of the emitted γ-rays is shifted due to the Doppler effect in order to have a resonance between a source and an absorber, the energy increases if a source moves toward an absorber and decreases when it moves away. Shifting and splitting of the nuclear energy levels between a source and an absorber can be measured at velocities in the order of mm/s. The resulting Mössbauer spectra is a plot of γ-ray transmission intensity as a function of the source velocity which can be converted to energy shift. A basic set-up for a Mössbauer experiment in transmission geometry and a typical result are shown Figure 3(a) and Figure 3(b) respectively. The Doppler energy shift (E_D) as a function of velocity (υ) is equal to [3] :

E D = E γ υ c (13)

Gamma ray is the most energetic radiation in the electromagnetic spectrum; it has the lowest wavelength and the highest frequency in the spectrum. γ-rays are sent out spontaneously by some radioactive nuclei and having energies in a range 10 - 100 keV. One of the most used Mössbauer isotopes among rare earth elements is ¹⁵¹Eu. The most popular source for Mössbauer spectroscopy with ¹⁵¹Eu is ¹⁵¹SmF₃. Such source has a high recoil free fraction even at room temperature and a single emission line with no appreciable line broadening. The nuclear

decay scheme for ¹⁵¹Sm, 21.64 keV level and the value of relevant parameters are shown in Figure 4. The half-life time of ¹⁵¹Sm is 87 years. It decays first to a metastable state of ¹⁵¹Eu by electron capture to the excited state I_e = 7/2 (21.64 keV) or to the ground state I_g = 5/2 by the γ-ray emission. The 21.64 keV state decays in turn to the ground state by γ-ray emission or internal conversion. The half-life of the 21.64 keV excited state is 9.7(3) ns, giving rise to energy uncertainty. A Mössbauer γ-ray then has a natural linewidth (2Г₀) of a resonant emission and absorption of 0.652 mm/s. Other nuclear parameters of the ¹⁵¹Eu isotope are shown in the Table 1.

The isomer shift arises due to the non-zero volume of the nucleus and it is directly proportional to s-electron density at the same nucleus. This leads to an electric monopole or Coulombic interaction, which is the interaction between electrons of the atom and the nucleus. The difference in s-electron density of the emitting and absorbing nuclei and the difference between the radius sizes of the nucleus in the ground and excited states affect this interaction, and lead to only a shift of the nuclear energy levels Figure 5(a). In the resulting Mössbauer spectrum, such as Figure 5(b), the isomer shift is the displacement of the absorption line away from zero velocity and is observed where the Doppler velocity applied to the source correctly modifies the source’s gamma-rays energy to account for the difference between the source and the absorber. This relation between charge electron density and nuclear radius is so called isomer shift (S) and is given by the equation [7] :

S = 2π 3 Z e 2 ( | Ψ a ( 0 ) | 2 − | Ψ s ( 0 ) | 2 ) ( 〈 r e 2 〉 − 〈 r g 2 〉 ) (14)

where | Ψ s ( 0 ) | 2 and | Ψ a ( 0 ) | 2 are the electron charge densities at the emitting and absorbing nuclei respectively ( | Ψ s ( 0 ) | 2 ≠ | Ψ a ( 0 ) | 2 ) . Z is the atomic number and e is the elementary electronic charge. 〈 r e 2 〉 and 〈 r e 2 〉 are the mean square nuclear radii of the excited and ground states respectively (r_e ≠ r_g). The isomer shift measurements can provide useful information on the electronic structure surrounding the nucleus and is good for probing the valence state of Mössbauer atom. As the wave function of the s-electrons penetrate into outer shells changes in these shells will directly alter the s-electron charge density at the nucleus. For example, Eu²⁺ and Eu³⁺ have electron configurations of (4f⁷) and (4f⁶) respectively. The Eu²⁺ have less s-electron density at the nucleus due to the greater screening effect of p, d, and f-electrons. This produces a positive isomer shift greater in Eu³⁺ than Eu²⁺. The isomer shift of Eu nuclei in compounds containing stable divalent and trivalent Eu ions falls in two well known separated regions and varies in the range between ≈−8 and ≈−14 mm/s for Eu²⁺ and between ≈ 0 and ≈ +4 mm/s for Eu³⁺ [12] . The ¹⁵¹Eu Mössbauer isomer shift can be used to investigate the valence state in Eu-based compounds, due to this larger difference (∆ ≈ 14 mm/s) between Eu²⁺ and Eu³⁺ compared to the linewidth of the resonance line (3 mm/s).

If the nucleus has a non-spherical charge distribution (nuclear spin quantum number I > 1/2), then it has electric quadrupole moment [7] . Therefore, the quadrupole moment can interact with an electric field gradient (EFG) at the nucleus due to the surrounding electrons, producing a splitting in the nuclear energy levels. The electric field gradient could have two components: one due to the asymmetry of the electron charge and another to the asymmetry of the lattice charge distribution. Such interaction depends on the squared nuclear spin quantum number and lift only partially the degeneracy of the nuclear levels. The coordinate system of the electric field gradient tensor is chosen such that | V x x | ≤ | V y y | ≤ | V z z | . The asymmetry parameter η is generally defined as:

η = V z z − V y y V z z ;   0 ≤ | η | ≤ 1 (15)

The electric field gradient can be specified by two parameters η and V_zz. The parameter η describes deviations of the electrical field gradient from rotational symmetry around the principal component (z-axis). The z-axis is often chosen to be the symmetry axis in describing a nucleus which is subjected to this hyperfine interaction, and is the axis along which the EFG is a maximum. There are two contributions to V_zz [13] :

V z z = ( 1 − γ ∞ ) ⋅ V z z l a t + ( 1 − R ) ⋅ V z z v a l (16)

where γ_∞ and R are Sternheimer antishielding and shielding factors respectively. V z z l a t is the contribution of the crystal lattice, i.e. it results from the ionic charges of the atoms which surround the Mössbauer isotope. V z z v a l describes the contribution of the localized electron of Mössbauer atoms, due to incompletely filled electron shells (valence electrons). In 4f systems this contribution is mostly dominated by a non-spherical charge distribution of the 4f electrons, however in some cases, other valence electrons (e.g. 6p or 5d) have a significant influence. The expression for the electric quardrupole interaction gives rise to the following energy levels scheme:

E Q = e Q V z z 4 I ( 2 I − 1 ) ( 3 M 2 − I ( I + 1 ) ) ( 1 + η 2 3 ) 1 2 (17)

with eQ being the nuclear quadrupole moment and I the nuclear spin of the nuclear state. V_zz is the max value of the electrical field gradient along a symmetry axis at the nucleus. E_Q dependent on the absolute value of the magnetic quantum number M and it is degenerate in respect of the sign of M. M = − I , − I + 1 , ⋯ , I , I_g = 5/2, I_e = 7/2. It means these two states configurations have the same energy. The magnetic quantum numbers determine the number of energy levels. Therefore, they determine the number of Mössbauer lines. The excited state level spin I_e = 7/2 is split into four M sublevels (±7/2, ±5/2, ±3/2, ±1/2), while the ground state level with spin I_g = 5/2 is split into three M sublevels (±5/2, ±3/2, ±1/2), as shown in Figure 6. The resultant spectrum is from transitions between ground and excited state sublevels subject to the selection rule ∆M = 0, ±1.

The magnetic hyperfine splitting is due to the interaction of magnetic moments μ of the ground and excited states of the nucleus with a magnetic field with the condition that the spin quantum number I > 0 [7] . Such interactions lift completely the degeneracy of the nuclear levels which correspond to different orientations of the spin I. The Hamiltonian for magnetic hyperfine dipole interaction is given as:

H m = − μ ⋅ B e f f = − g μ N I ⋅ B e f f (18)

where B_eff is referred to as the size of the magnetic hyperfine field at the position of the nucleus. (μ_N = 5.050 × 10⁻²⁷ JT⁻¹) is the nuclear magneton, and g is referred

to as the g-factor (g = μ/Iμ_N) of the nuclear state. The spin I and g are usually different for the excited and ground states. Due to this interaction, the magnetic field splits the nuclear energy levels of spin quantum number I into 2I + 1 equally spaced non-degenerate levels with an energy given by:

E m = − μ B I M = − g μ N B e f f M (19)

M is the magnetic quantum numbers and can take the values M = − I , − I + 1 , ⋯ , I − 1 , I . Transition can take place between sublevels subject to the selection rule that ∆M should change by 0, ±1 in the absorption process. Figure 7 shows the splitting of the excited state (I = 7/2) and ground state (I = 5/2) nuclear levels which gives rise to very complex ¹⁵¹Eu Mössbauer spectra with 18 allowed possible transitions but experimentally an eight-line pattern is seen

in the case of good counting statistics or an intense spectrum. Figure 8 shows the corresponding Mössbauer lines.

The total effective magnetic hyperfine internal field B_eff acting on the atomic nucleus originates primarily from the atom’s own electrons can be described in terms of different contributions [14] . This model has been used to analyze the different contributions to B_eff in several Eu intermetallic compounds [15] [16] .

B e f f = B c + B c e + B t h f + B d i p (20)

B_c is the Fermi contact field which is caused by the core polarization of the s-electron density at the nucleus. It is originating from the interaction of the S = 7/2 spin of the Eu²⁺ ion with its own (nonmagnetic) electron shells [18] . It has the largest contribution to the internal field B_eff with accepted value of (−34 T) [19] . In the nonrelativistic approximation, this effective field is regarded as being due to the difference of spin up and spin down densities m(0) at the nucleus.

B F = 8π 3 μ B m ( 0 ) (21)

B_F is produced by the polarization of electrons whose wavefunctions overlap the nucleus, i.e. s-electrons. This polarization is due to unpaired electrons in the f orbital and gives an imbalance in spin density at the nucleus from the difference in interaction between the unpaired electron with s-electrons of parallel or antiparallel spin to its own. This can be expressed formally as [20] :

B F = − 8π 3 μ B ∑ { | Ψ ↑ s ( 0 ) | 2 − | Ψ ↓ s ( 0 ) | 2 } (22)

where μ_B is the Bohr magneton and | Ψ ↑ s ( 0 ) | 2 and | Ψ ↓ s ( 0 ) | 2 are the spin up and spin down densities for s-electrons at the nucleus respectively. The summation is taken over all s-electrons. The large negative part of this field is owing to core polarization. This results from the different exchange interactions of the 1 s, 2 s, 3 s and 4 s electrons of parallel and antiparallel spin with the f shall [21] . The exchange interaction between the spin up polarized f shell and the spin up s-electron is attractive, while that between the f shell and a spin down s-electron is repulsive. As a result the radial parts of the two s-electron wave functions will be different, one being pushed toward the nucleus, the other pulled outward. Therefore, the spin densities at the nucleus no longer cancel and a Fermi contact interaction is observed [5] .

B_ce is the contribution from the conduction electron spin polarization produced by the 4f electron of the magnetic ion itself.

B_thf is the transferred hyperfine field produced at an Eu nucleus by the spin polarization of the conduction electrons by the 4f magnetic moment of the neighboring atoms. It has been found the transferred hyperfine fields are relatively small in Eu-intermetallic compounds [15] [16] [17] . However, |B_thf | is very sensitive to external pressure and determine the pressure dependent of |B_eff | [14] .

B_dip is contribution from the dipole interaction with the moment of the electron spin. The total B_dip dipolar field is the sum of the Lorentz field, the demagnetization field, and the contribution from neighboring magnetic dipoles; B_dip is usually very small [22] compared to the other contributions.