vibration occurring in the molecules or complex ions of interest. Thus ... presents the principle of infrared and Raman scattering on an elementary level. ..... Raman methods including polymers, inorganic and organic compounds, and biological ...
C.M. Julien et al. Materials Science for Energy Storage Published by Anna University, Chennai, India (2010)
Chapter 12
BASIC CONSIDERATIONS ON VIBRATIONAL SPECTROSCOPY C.M. Julien Université Pierre et Marie Curie, Paris, France
1. Introduction The lithium-ion technology has opened a wide research field both in material physics and chemistry to achieve a class of materials for high voltage cells. At present, the lithium-ion battery technology seems to have the potential to satisfy all requirements of high energy, high power applications, and to meet the low cost demands. However, there remain several key materials issues such as the structural stability over several hundreds of cycles, which need to be solved. In the area of positive electrode materials, extensive investigations on the requirements of optimum-ideal electrode system have shown that, layered oxides, spinels and polyphosphates are the most promising systems with great potential for structural improvements for long cycle life [1-2]. The characterization of the new materials and the ultimate goal of correlating structural characteristics with physical and chemical properties, demands the use of a broad range of techniques that could provide the information to establish the sought correlation’s. Vibrational spectroscopy is one of the most powerful techniques available for materials characterization [3]. As local probes, Raman scattering and Fourier transform infrared spectroscopy are useful tools when XRD is ineffective for amorphous substances. The vibrational spectrum of a molecule, as observed in both infrared absorption and Raman studies, is direct manifestation of forces arising from mutual interactions of electrons and nuclei. By vibrational spectroscopy is meant the determination of the energy levels of various fundamental modes of vibration occurring in the molecules or complex ions of interest. Thus, from the observed vibrational frequencies, we can glean information about the force constants which hopefully can then be correlated with electronic structure and bonding theories. This lecture is devoted to an introductory presentation of vibrational spectroscopy. Section 2 presents the principle of infrared and Raman scattering on an elementary level. Since numerous textbooks are involved in theoretical aspects and instrumentation as well, no attempt has been made to cover any sophisticated aspects. Analysis methods for frequency assignments are presented in Section 3. The concept of group frequencies and the factor group analysis are considered.
2. Principle of infrared and Raman spectroscopy Vibrational spectroscopy of molecules can be relatively complicated. Quantum mechanisms requires that only certain well-defined frequencies and atomic displacements are allowed. These are known as the normal modes of vibration of the molecule. There are several types of motion that contribute to the normal modes (Fig. 1) such as (i) symmetric and asymmetric stretching motion between two bonds (symbol s), (ii) bending motion between three atoms connected by two bonds (symbol ), (iii) internal rotation about single bonds, and (iv) out-of-plan deformation modes that change an otherwise planar structure into a non-planar one.
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2.1. Infrared activity Chemical bonds are found to behave like Hooke's law springs, i.e. F=-Kx, for small displacements. The classical physical description of energy involved described as two masses M and m connected by a spring (Fig. 1). In order to absorb radiation in going from one energy state to another, there must be a change in dipole moment. During a normal vibration of a molecule, the dipole moment can be expressed to the first order:
( k ) zd X k
X k ,
(1)
where o=zd is the permanent dipole moment and X is the normal coordinate describing the molecule motion. z is the charge. If / X k is different from zero, the infrared-active vibration must fall under the irreducible representation containing the translation Tx, Ty and Tz. Derivation of these selection rules based on symmetry are given by the character tables for the point group (see below).
FIG. 1. The bending and asymmetric stretching vibrations cause a change in the dipole moment of the molecule and thus are infrared active. The symmetric stretching vibration causes no change in dipole moment and so is infrared inactive.
The larger the change in dipole moment, the stronger is the absorption. For a harmonic oscillator the frequency of vibration is independent of both the amplitude and the energy of the oscillation and is given by an equation derived from Hooke's law:
1 2c
K M m , Mm
(2)
2.2. The Born-von Karman model Virtually every discussion of lattice dynamics begins with the analysis of the motion of a one-dimensional array of identical particles. This problem is one of the oldest having first treated by Newton in 1686. The treatment of the dynamics of crystal lattices is usually considered to originate with the model of Born and von Karman [4] applied to sodium chloride. Consider a chain consisting of two types of atoms arranged alternatively, as shown in Fig. 2, representing an ionic crystal, for example. The force constants K and k have been used to represent bonds. The forces on the atoms of mass M and m are given, respectively, by:
F2n K ( x2n x2n 1 ) k ( x2n 1 x2n ) M
2 x2 n
,
t 2 2 x2n 1 . F2n 1 k ( x2n 1 x2n ) K ( x2n 2 x2n 1 ) M t 2 Periodic solutions in time and space are proposed of the form:
186
(3) (4)
x 2 n X 1 exp 2 i t n a ,
(5)
x2n 1 X 2 exp 2 t n 1 / 2a .
(6)
a K
xn-1
k
m
M
xn
xn+1 xn+2
FIG.2. The linear diatomic chain. The distance a is the dimension of the unit cell which contains two types of atoms.
1 2
Frequency,
2
2k d a
2k s
longitudinal optic
1 2ks 2 M
transverse optic
1 2kd aM
longitudinal acoustic
1 2ks 2 m 1 2kd am
transverse acoustic
0
/a Wave vector, q
2/a
FIG. 3. Dispersion curves for the longitudinal and transverse modes of the diatomic chain where is the reduced mass =mM(m+M)-1, ks is associated with a covalent bond, kd represents the much weaker intermolecular forces. With ks>>kd the "secular" equation can be approximated.
Substitution of these expressions in Eqs (3) and (4) leads to the pair of linear, homogeneous equations for the amplitudes X1 and X2: [422M - (K+k)]X1 + [Keia + ke-ia]X2=0,
(7)
[keia + Ke-ia]X1 + [422m - (K+k)]X1 =0.
(8)
The roots of the secular equation are given by: 1/ 2
4 2 2
2 2 K k 1 1 K k 1 1 4kK sin 2 a 2 M m 2 M m mM
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.
(9)
For the special case in which all bonds are identical, K=k, which implies equal spacing of the atoms, and Eq. (9) reduces to: 1/ 2 2 . 4 1 1 1 1 2 4 K sin a mM M m M m 2
(10)
2
In this discussion of the diatomic chain, only longitudinal vibrations have been treated. If the potential energy of displacement of atoms perpendicular to the chain is determined by a bending force constant, kd, the results described the dispersion curves for longitudinal and transverse modes as shown in Fig. 3. The generalization to three dimensions of equations of motion such as Eqs. (3) and (4) is straightforward, but complicated. This problem is discussed in detail by Born and Huang [5]. 2.3. The Raman scattering process The Raman effect arises when a photon is incident on a molecule and interacts with the electric dipole of the molecule. It is a form of electronic (more accurately, vibronic) spectroscopy, although the spectrum contains vibrational frequencies. The dipole moment field,
P , induced in a molecule by an external electric
E , is proportional to the field as: P E ,
(11)
where is the polarizability tensor of the molecule. The electric field due to the incident radiation is a time-varying quantity of the form Ei=Eo cos(2it).
virtual state photon out
vibrational energy level photon in
photon out
photon in
h ground state
h
INFRARED ABSORPTION
RAYLEIGH SCATTERING
STOKES LINES
ANTI-STOKES LINES
RAMAN SCATTERING
FIG. 4. The energy level diagram involved in the infrared absorption and the Raman scattering spectroscopy.
In classical terms, the interaction can be viewed as a perturbation of the polarizability: o
X k X k higher .terms ,
(12)
k
analogous to Eq. (1) for the permanent dipole moment. For a vibrating molecule, the polarizability is also a time-varying term that depends on the vibrational frequency of the molecule, as: =o+m cos(2t).
(13)
Thus, the oscillating dipole can be expressed as:
188
P o Eo cos( 2 i t ) m Eo
cos( 2 i t ) cos( 2t ) , X k
E cos 2 ( i )t cos 2 ( i )t ) . P o Eo cos( 2 i t ) m o 2 X k
(14) (15)
It can be seen from Eq. (15) that the induced dipole moment varies with three components frequencies, i, (i-), and (i+) and can therefore give rise to Rayleigh scattering, Stokes and antiStokes frequencies, respectively (see energy diagram in Fig. 4). The incremental difference from the frequency of the incident radiation, I, are by the vibrational frequencies of the molecule . The ratio of the intensity of the Raman anti-Stokes and Stokes lines is predicted to be: 4
h I A i exp I S i k BT
.
(16)
Raman spectroscopy is sensitive to even symmetry (gerade) modes of vibration while IR is sensitive to odd (un-gerade) modes. The frequencies and symmetry are uniquely determined by the symmetry, type and spatial arrangement of the atoms that make up the structure. Thus, the Raman spectra are a unique fingerprint of the atomic arrangement of the compound. Just about any compound can be studied with Raman methods including polymers, inorganic and organic compounds, and biological specimens. One of the major advantages of Raman over infrared spectroscopy is the low sensitivity toward interference by water. Thus, aqueous samples can be tested by Raman that cannot be examined with IR methods. Because of the small sample requirement, the low sensitivity to the interference by water, the spectral detail, and the conformational and environmental sensitivity, Raman methods are widely used in the field of biology and chemistry.
117 (A1)
-InSe
400
177 (E)
200 100 0
-200
-100
0
100
199 (A1) 211 (E) 227 (A1)
300 41 (E)
Raman intensity (cps)
500
200
Raman shift (cm-1) FIG. 5. Raman spectrum of the layered -InSe phase showing the Stokes and anti-Stokes lines recorded at room temperature using a 514.5 nm Ar+ laser line at 30 mW power excitation. The Boltzmann exponential factor is the dominant term in Eq. (16), which makes the anti-Stokes features of the spectra much weaker than the corresponding Stokes lines. In the example spectrum, i.e. layered -InSe phase, notice that the Stokes and anti-Stokes lines are equally displaced from the Rayleigh line (Fig. 5). Resonance Raman scattering occurs when one looks at virtual states very close to an electronic state producing very intense spectrum. Being sensitive to the nature of substituents as well as geometry, Raman spectroscopy yields extremely useful information about functional groups, which lie at the heart of chemistry and biology. By using the tunable IR wavelengths (700-1000 nm) of an Argon-ion pumped Ti:sapphire laser, one is able to avoid the fluorescence that is often seen when biological samples are excited by visible light. Also, for ease of observation of scattering it is advantageous to have the incident frequency close to a transition frequency of the scattering medium. In this case, known as resonance Raman scattering, the ability to tune the laser to a select frequency is important, hence the need for UV, Visible and tunable IR sources.
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3. Analysis and assignment of frequencies Infrared and Raman are complementary techniques because the selection rules are different. We can classify molecules into three classes according the spectroscopic activity. For example: (i) Homonuclear diatomic molecules do not have an IR absorption spectrum, because they have no dipole moment, but do have a Raman spectrum, because stretching and contraction of the bond changes the interactions between electrons and nuclei, thereby changing the molecular polarizability. (ii) For highly polyatomic molecules possessing a center of inversion (such as benzene) it is observed that bands that are active in the IR spectrum are not active in the Raman spectrum (and vice-versa). (iii) In molecules with little or no symmetry, modes are likely to be active in both IR and Raman spectra. Analysis of vibrational spectra of solid-state phases is a difficult task because a fundamental frequency will not appear by itself in the spectrum but will appear as a combination band, arising from two or more simultaneous vibrational transitions. There are different ways to assign these frequencies. The interpretation of the vibrational spectrum of a solid may be considered from two points of view, which of course do not exclude each other: either determines the symmetry properties of the vibrational bands or try to assign the observed frequencies to vibrations of define atoms or groups of atoms. In the latter analysis one can consider local vibrations and mixed vibrations. A local vibration corresponds to a vibrational mode of localized atom or group of atoms (such as a complex anion), which is enough decoupled from the rest of atoms in the lattice for easy determination. Mixed vibration is not localized on a given atom or small group of atoms. Most popular techniques are the group frequency analysis and the factor group method. The first approach is nearly a trivial one if only general assignments are proposed. This occurs when the groups, which may interact, have distinctly different individual frequencies, or are very distant in the crystal [6]. These assignments can be made by comparison, but it may become much more difficult if specific features or details are to be understood and correlated with the structure. Even in apparently simple cases such as spinels, since the bonding forces (the so-called force constants) of local vibrations are not known with sufficient accuracy. A more fundamental difficulty comes from the existence of mixed vibrations, namely vibrations which are not localized, but on the contrary imply the simultaneous participation of several, different atoms (cations and/or anions). According to group theory, such interactions are allowed only between vibrational modes which belong to the same class of symmetry; but when this condition is fulfilled, no detailed prediction can be made about the importance of these interactions. They may be strong when the atoms or groups in interaction have similar frequencies and are near-neighbors. The second approach will give, through a group theoretical analysis, the number and the infrared- and Ramanactivity of the vibrations, and their distribution among the different symmetry classes. This analysis is possible only if the detailed structure is known, and a full use of its results needs the study of single crystals in polarized light. It must be emphasized that, apart from simple cases, this analysis does not give the assignments of the bands to vibrations of given atoms: the group theory is dealing with symmetry properties, and not with bonding forces. 3.1. The group frequency analysis The concept of group frequencies is quite useful in deciding upon the correct frequency assignment. This concept is simply that a given chemical entity, e.g. pyramidal XY3 unit, XY4 tetrahedron, trigonal bipyramidal XY5 group, XY6 octahedral, etc., will have roughly the same frequency in different compounds. It is quite useful when the frequency of the group is well separated from frequencies arising from other groups. For example, a CH stretching vibration occurs in the neighbourhood of 3000 cm-1, a C≡N stretch occurs around 2050 to 2200 cm-1, a C≡O stretch around 1950 to 2150 cm-1, etc. Some very useful group frequency charts are given by Nakamoto [6].Let consider the case of XY4 tetrahedron and XY6 octahedron which are the two main entities building the lattice of oxides used as electrode materials for lithium batteries. Tetrahedral XY4 molecules have the Td symmetry. Figure 6 illustrates the four normal modes of vibration of a tetrahedral XY4 molecule. All four vibrations in Td symmetry are Raman active, whereas only 3 and 4 are infrared active (Table 1). Fundamental frequencies of a tetrahedral unit have trends as follows: 3 > 1 and 2 > 4,
(17)
that holds for the majority of the compounds. In the same family of compounds, the stretching frequency decreases as the mass of the X atom decreases. Normal coordinate analyses of tetrahedral XY4 molecules have been carried out by a number of investigators [6]. The vibrational frequencies of tetrahedral transition metal oxides, XO4-type, are listed in Table 2. The general frequency rules 3>1 and 2>4 hold for the majority of compounds. It should be noted that frequencies 2 and 4 are often too close to be observed as separate bands in Raman spectra. A gradual decrease in frequency is also seen in isoelectronic series such as [CrO4]2->[VO4]3->[TiO4]4-. Although these trends are obvious in terms of the frequency, the same results are expected in terms of the force constant since the mass effect is nonexistent or very small in these series.
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TABLE 1. Activity of the normal modes of XY4 molecule in the Td symmetry.
Mode 1 2 3 4
Symmetry A1 E F2 F2
1(A1) s(M-O)
Activity R R R, IR R, IR
Type of vibration symmetric stretching asymmetric stretching symmetric bending asymmetric bending
2(E) d(O-M-O)
3(F2) d(M-O)
4(F2) d(O-M-O)
FIG. 6. Normal modes of vibration of tetrahedral XY4 molecules.
A second application of analysis of XY4 molecules is the investigation of normal spinels with a highvalency tetrahedral cation such as normal II-IV germanates, I-II-V vanadates and normal I-VI molybdates and tungstates. These spinels show that the high-frequency band of their IR spectrum must be assigned to a vibration of the tetrahedral group [7]. This is strongly supported by the very good correspondence between the actual frequencies observed for each type of compound, and the characteristic stretching frequency of the corresponding tetrahedral anion in non-spinel compounds. Across the periodic table, the stretching frequencies increase as the oxidation state of the central atom becomes higher. The effect of lowering the oxidation state is clearly seen in a series such as [MnO4]n- with n=1,2,3. As in many cases, the higher the oxidation state, the higher is the frequency. Octahedral XY6 molecules have the Oh symmetry. Figure 7 illustrates the six normal modes of vibration of an octahedral XY6 molecule. Vibrations 1, 2, and 5 are Raman active, whereas only 3 and 4 are infrared active. Since 6 is inactive in both, its frequency is estimated from an analysis of combination and overtone bands. The order of the stretching frequencies is: 1 > 3 >> 2 or
1 < 3 >> 2,
(18)
depending on the compound. The order of the bending frequencies is 4>5>6 in most cases. In the same family of the periodic table, the stretching frequencies decrease as the mass of the central atom increases. TABLE 2. Vibrational frequencies (cm-1) of tetrahedral XO4 type compounds. Group [SiO4]4[VO4]3[TiO4]4[CrO4]2[MnO4]2[PO4]3-
1
2
3
4
819 826 761 846 812 938
340 336 306 349 325 420
956 804 770 890 820 1017
527 336 371 378 332 567
191
1(A1g) (M-O)
2(Eg) (O-M-O)
3(F1u) (M-O)
4(F1u) (O-M-O)
5(F2g) (O-M-O)
6(F2u) (O-M-O)
FIG. 7. Normal modes of vibration of octahedral XY6 molecules.
The trend in 1 directly reflects the trend in the stretching force constant (and bond strength) since the central atom is not moving in this mode. In 3, however, both X and Y atoms are moving, and the mass effect of the X atom cannot be ignored completely. Across the periodic table, the stretching frequencies increase as the oxidation state of the central atom becomes higher. As in many cases, the higher the oxidation state, the higher is the frequency. The Raman intensity of an XY6 molecule normally follows the order I(1)>I(2)>I(5). Table 3 lists the activity of the normal modes of octahedral XY6 molecule in the Oh symmetry. On the other hand, some anomaly may occur such as in the spinel Li1+Mn2O4 because the static Jahn-Teller effect of the Mn3+ ion causes a tetragonal distortion, i.e. a strong elongation of the octahedron in the spinel structure. The configuration of the trivalent manganese-ions in the spinel (t32ge1g, high spin) tends to be stabilized in the D3h symmetry rather than the octahedral symmetry [8]. Also if one considers all the Mn3+ ions accommodated in octahedral MnO6 environments in the spinel structure, the regular Oh symmetry transforms in the lower D4d symmetry through the scheme represented in Fig. 8. Thus, the F1u modes are normally split into (A2u + Eu) infrared active components. Therefore, IR modes having the (A2u + Eu) symmetry are intense and the combination bands appear with similar frequencies, which induce broad bands in the IR spectrum.
D3d Compression Oh Elongation D3d FIG. 8. Structure of the distorted octahedral XY6 groups and their IR-active stretching modes. The tetragonal distortion induces the D4h symmetry whereas the trigonal distortion produces the D3d symmetry.
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TABLE 3. Activity of the normal modes of XY6 molecule in the Oh symmetry.
Mode
Symmetry
Activity
A1g Eg F1u F1u F2g F2u
R (stretching) R (stretching) IR (stretching) IR (bending) R (bending) inactive
3.2. The factor group method The factor group method of classifying fundamental vibrational modes of crystals, as developed by Bhagavantam and Venkatarayudu [9], is certainly the most powerful method of treating simple crystal structure types. Yet it is little used and only a small number of the simplest structure types have been completely analyzed. Secondly there has been discussion of the method in the spectroscopy literature. There are 3N-6 internal degrees of freedom for a crystal containing N atoms. However, since the wavelength of infrared radiation is very large compared with the dimensions of the unit cell, the vibrations of many unit cells will move in phase. Thus of the very large number of possible vibrations, only the vibrations of one unit cell will appear as distinct modes of the 3N (where N is the number of atoms in the unit cell) degrees of freedom, 3 represent pure translations and appear as the acoustic modes involved in the propagation of sound waves through the crystal. The remaining 3N-3 modes are distributed between internal (molecular) modes, translatory (lattice vibration) modes, and rotatory modes (which would become free rotations in the limit of zero interaction of groups within the structure). In the factor group analysis, the unit cell modes are classified according to the way in which they transform under the operations of the factor group [10-12]. The factor group is obtained from the space group by defining all operations involving the translation part of the space group as identity. The factor group is isomorphous with the corresponding point group. For instance, normal spinel belongs to Oh7 and thus its normal vibrations are classified according to the irreducible representations of point group Oh. The equilibrium position of each atom of a solid lies on a site, i.e. Wyckoff's position that has its own symmetry. This site symmetry, a subgroup of the full symmetry of the Bravais unit cell, must be ascertained correctly for each atom. It is easy to do in some cases, difficult in others [12]. 3.2.1. TiO2 crystal Let us consider the case of TiO2 as the first example. The anatase form of TiO2 crystallizes in the tetragonal system with I41/amd space group or D4h19 spectroscopic symmetry. There are two molecules per Bravais unit cell (Z=2). Therefore, there are two equivalent titanium atoms and four equivalent oxygen atoms in this Bravais cell. Titanium cations are located in a Wyckoff positions (0,0,0; 0,½,½) and oxygen anions occupy the e Wyckoff positions (0,0,u; 0,0,u; 0,½,u+¼;0,½,¼-u). The determination of the irreducible representation of the TiO2 crystal gives the number of lattice vibration. Using the correlation method, the different steps are as follows [12]. (1) The corresponding site symmetries for the D4h19 space group which contains both the infrared and Raman active modes are given by: 2D2d(2), 2C2h(4), C2v(4), 2C2(8), Cs(8), C1(16),
(19)
where each site-group symbol is preceded by an integer indicating the number of distinct sites of each symmetry and followed by the multiplicity in parentheses. The bold-face type indicates the site of the atom. These site symmetries appear in the alphabetical order of site position with the Wyckoff notation.
193
Obviously the two equivalent Ti atoms can be accommodated only on the D2d(2) site, while the four equivalent oxygen atoms can be located on the C2v(4) site. TABLE 4. Character table for C2v (oxygen) and D2d (titanium) site group. The bold-face indicates species that contain the translations Tx, Ty, and Tz. The species of the rotations Rx,Ry, and Rz, and the polarization tensor elements are also given.
C2v A1 A2 B1 B2
E 1 1 1 1
C2 1 1 -1 -1
D2d A1 A2 B1 B2 E
E 1 1 1 1 2
2S4 1 1 -1 -1 0
v(zx) 1 -1 1 -1 C2 1 1 1 1 -2
v(yz) 1 -1 -1 1 2C'2 1 -1 1 -1 0
Tz Rz Ty,Ry Tx,Rx
2d 1 -1 -1 1 0
xx,yy,zz xy xz yz
Rz
xx+yy,zz
Tz (Tx,Ty);(Rx,Ry)
xx-yy xy xzyz
(2) When the species of each site group is identified for each excursion of an equivalent set of atoms, one identifies the species of the translations Tx, Ty, and Tz from the character table. Table 4 indicates the presence of the titanium lattice vibrations designated as degrees of freedom in species B2 and E, while the oxygen lattice vibrations designated as degrees of freedom in species A1, B1 and B2. (3) The next step is to correlate the B2 and E species of the site group D2d to the D4h factor group species. Similarly, the correlation of A1, B1 and B2 species of the site group C2v to the D4h factor group species is made for oxygen atoms. Table 5 shows this correlation and identifies the species of the lattice vibration in the crystal. (4) Summing up the individual species related to each equivalent set of atoms, the total irreducible representation of the crystal is constructed. This representation contains both optical and acoustic vibrations. By group factor analysis, one obtains the irreducible representation of degree of freedom for atoms, denoted by (i): for 2Ti:
(Ti) = B1g + A2u + Eg + Eu,
for 4O:
(O) = A1g + B1g + 2Eg + A2u + 2Eu.
(20)
The total representation of the crystal, vibr , can be calculated by utilizing : cryst
cryst Γvibr = eq set 1 + eq set 2 + …-acoust = atom - acoust.
(21)
Excluding the acoustic modes, i.e. they have zero frequency at the centre of the Brillouin zone ( q 0 ), acoust = A2u + Eu, the factor group analysis of D4h19 yields the total active vibrational modes as:
TiO2 A1g 2 B1g 3E g A2u 2 Eu .
(22)
There are three fundamental vibrational modes allowed in IR, one A2u and two degenerate Eu, while the Raman spectrum contains six fundamental lattice vibrations, one A1g, two B1g, and three degenerate Eg modes. One vibration, B2u, will be inactive in both the infrared and Raman spectrum. This procedure needs only minor modification to include the intramolecular vibrations and librations
194
for molecular crystal. Here the irreducible representation of a molecular crystal can be defined as: mol .cryst cryst vib vib mol .vib lib acoust ,
(23)
which assumes the separation of the vibrations into internal and external modes.
TABLE 5. Correlation for the lattice vibrations of TiO2. (R) and (IR) indicates the Raman- and infraredactive species, while (in) indicates inactive species.
3.2.2. LiFePO4 molecular crystal A natural division in applying the correlation method to a molecular crystal, such as the phosphor-olivine LiFePO4, can be made as follows. (i) Derive the lattice vibration of the PO 34 ions and Fe2+, Li+ ions. (ii) Calculate the libration, i.e. rotation, of the PO4 group in the crystal. (iii) Use the correlation technique to predict the number of intramolecular vibrations of the groups. By combining the irreducible representation obtained from parts (i), (ii) and (iii) and using Eq. (23), the total representation for LiFePO4 is constructed (Table 6) [13]. The olivine structure is characterized by the Pnma space group (D2h16 spectroscopic symmetry). This structure is usually described in terms of a hexagonal close-packing of oxygen with Li and Fe ions located in one-half of the octahedral sites (4a and 4c Wyckoff positions, respectively) and P atoms in one-eighth of the tetrahedral 4c Wyckoff position. There three equivalent set of sites for the oxygen atoms: O(1) and O(2) in 4c, and O(3) in 8d Wyckoff positions. The FeO6 octahedra share four corners in the cb plane being cross-linked along the a axis by PO4 groups, whereas the Li ions are located in rows, running along a, of edge-shared LiO6 octahedra that appear between two consecutive [FeO6]∞ layers. After summing up, the irreducible representation corresponding to the olivine crystal is as follows after taking into account that the Au modes are Raman and IR inactive, and subtracting the three acoustic modes (B1u+B2u+B3u):
195
( LiFePO4 ) 11Ag+7B1g+11B2g+7B3g+13B1u+9B2u+13B3u.
(25)
There are 36 Raman-active vibrations and 35 infrared-active modes. One interesting point is evidenced by this treatment: because lithium atoms are located in sites with inversion symmetry, there are no Raman-active species related to Li atoms located in 4a sites, so that we expect only to see modes involving lithium vibrations in infrared.
TABLE 6. Factor group analysis for olivine LiFePO4.
References [1] C.M. Julien, NATO-ASI Series 3-85 (2000) 1. [2] C.M. Julien, Mater. Sci. Eng. R 40 (2003) 47. [3] C.M. Julien, Solid State Ionics 136-137 (2000) 887. [4] M. Born and T. von Karman, Physik Zeit. 13 (1912) 297. [5] M. Born and K. Huang, Dynamical Theory of Crystal Lattices, Oxford, 1954. [6] K. Nakamoto, IR and Raman of Inorganic and Coordination Compounds, John Wiley & Sons, New York, 1977. [7] J. Preudhomme and P. Tarte, Spectrochim. Acta 28A (1972) 69. [8]. C.M. Julien and M. Massot, J. Phys. Condens. Matter 2003) 15: 3151. [9] S. Bhagavantam and T. Venkatarayudu, Proc. Indian Acad. Sci. 9 (1939) 224. [10] R.O. Kagel, in Handbook of Spectroscopy, vol. II, ed. J.W. Robinson, Heyden & Sons, London, 1974, p. 107 [11] G. Turrell, Infrared and Raman Spectra of Crystals, Academic Press, London, 1972. [12] W.G. Fateley, F.R. Dollish, N.T. McDevitt and F.F. Bentley, Infrared and Raman Selection Rules for Molecular and Lattice Vibrations: the Correlation Method, Wiley Interscience, New York, 1972. [13] M.T. Paques-Ledent and P. Tarte, Spectrochim. Acta 29A (1973) 1007.
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