Spectroscopic investigations of some crystalline borates
Citation for published version (APA):Bronswijk, J. P. (1979). Spectroscopic investigations of some crystalline borates. Technische Hogeschool Eindhoven. https://doi.org/10.6100/IR141789
DOI:
10.6100/IR141789
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SPECTROSCOPIC INVESTIGATIONS
OF SOME CRYST ALLINE BORA TES
SPECTROSCOPIC INVESTIGATIONS
OF SOME CRYST ALLINE BORA TES
PROEFSCHRIFT
TER VERKRIJGING VAN DE GRAAD VAN DOCTOR IN DE TECHNISCHE WETENSCHAPPEN AAN DE TECHNISCHE HOGESCHOOL EINDHOVEN, OP GEZAG VAN DE RECTOR MAGNIFICUS, PROF. DR. P. VAN DER LEEDEN, VOOR EEN COMMISSIE AANGEWEZEN DOOR HET COLLEGE VAN DEKANEN IN HET OPENBAAR TE VERDEDIGEN OP
DINSDAG 24 APRIL 1979 TE 16.00 UUR
DOOR
JOHANNES PETRUS BRONSWIJK
PROMOTOR: PROF. DR. J.M. STEVELS CO-PROMOTOR: PROF. DR. R. PRINS ADVISEUR: DR. D.L. VOGEL
Het onderzoek beschreven in dit proefschrift ~erd
finan-cieel gesteund door de Nederlandse Organisatie voor
Zuiver-~etenschappelijk Onderzoek (ZWO), en ~erd uitgevoerd onder auspiciën van de Stichting Scheikundig Onderzoek in
aan mijn ouders,
DANKBETUIGING
Dit proefschrift is tot stand gekomen dankzij de medewerking van vele anderen.
In het bijzonder ben ik dank verschuldigd aan dr. D.L. Vogel, zonder wiens theoretische kennis van de spectroscopie vele berekeningen niet uitgevoerd hadden kunnen worden.
Ook ik mijn dank uit aan ir. E. Strijks, wiens vaardigheid in de Ramanspectroscopie een erg grote steun voor mi} is geweest.
Voor het maken van de vele Röntgenopnamen ben ik dank verschuldigd aan mevr. M. Duran, dr. ir. D.A.G. van Oeffelen, ir. J.P. v.d. Berg, dhr. P.C. Krüger en
dhr. H. de Jonge Baas.
Verder wil ik iedereen in de vakgroep Anorganische Chemie danken, die een bijdrage heeft geleverd aan de totstandkoming van dit proefschrift; in het bijzonder dhr. W. van Herpen en mevr. M. Kuijer.
Dr. ir. W.J.Th. van Gemert en ir. H.P.A.M. van der Staak dank ik voor de vele interessante discussies.
Tenslotte dank ik Ine van 't Blik-Quax voor het typen van het manuscript.
CONTENTS I Introduetion 1.2 References page 9 1 2 II General Remarks 13
II.1 Assignment of the Raman and infrared spectra
II.2 Normal Coordinate Analysis II.2.1 Theory
II.2.2 Schachtschneider's Computer Programs
II.2.3 Extensions of the Schachtschneider 13 1 5 15
19
Programs 21
II.3 Factor Group and Site Group Analysis 24
II.4 Experimental Work 25
II.S References 26
III Translational Vibration Coordinates 29
III.l Introduetion 29
III.Z Translational Vibrations 32
III.3 Normal Coordinate Analysis of Yttrium Vanadate
III.4 Conclusions and Discussion III.5 References
IV Spectroscopie Investigations of Lithium Metaborate 36 42 44 46 IV.l Introduetion 46
IV.2 Factor Group Analysis of LiB0
2 49
IV. 3 Experimental Work 52
IV.3.1 Preparation of the Samples 52
IV.3.2 Raman and Infrared Spectra 52
IV.3.3 Assignment of the Raman Spectra 60 IV.3.4 Normal Coordinate Analysis of
LiB02 62
IV.3.4.1 Internal Vibrations of LiB0
IV.4 References
V Normal Coordinate Analysis of a Netwerk of Pentaborate Groups V.1 Introduetion page 66 67 67 V.Z Experimental Work 70
V.3 Assignment of the Raman and Infrared
Spectra 71
V.4 Normal Coordinate Analysis of the Network of Pentaborate Groups
V.S References
77 82 VI Spectroscopie Investigations of the Diborate 83
VI.1 Introduetion 83
VI.2 The effect of hydragen-deuterium sub-stitution on the frequency of the vibra-tions of hydragen containing compounds 84 VI.3 Normal Coordinate Analysis of the
Di-borate group 90
VI.3.1 Factor Group and Site Group
analysis of the Diborate Group 90
VI.3.2 Experimental Work 93
VI.3.3 Assignment of the Raman Spectrum
Borax 94
VI.4 References 99
VII Structure of Vitreous B2
o
3 101VII.1 Introduetion 101
VII.2 Experimental Work 104
VII.3 Raman Spectrum of Crystalline B
2
o
3 104VII.4 References 106
page Appendix I 109 Appendix I I 1 1 1 Appendix I I I 11 3 Summary 116 Samenvatting 11 7
I ln.1:roduction.
In this thesis, a study is made of the Raman and infrared spectra of various crystalline alkali borates.
The structure of vitreous alkali borates has nat been known for many years. Bray (1) and Reekenkamp (2) made attempts to elucidate the structure of these systems. With their results, it was possible to construct a
diagram in which the fraction of the various structural units occurring in these compounds is plotted versus the alkali content (Figure 1.1).
Figure I . l The fraction
of the various struc-tural units in vitreous
0.6
0.2
10 20 30 40 50
__,. mole'/, alkali oxyde
alkali borates as a function of the al-kali content. N 4 is the fraction of Bo 4 -tetraheders, a is the fraction of Bo;-groups. and b is the fraction of B0
However, the arrangement of these structural units into larger groups, as they are well known in crystalline alkali borates, were not discussed by Bray and Beekenkamp.
With the advent of the laser in Raman spectroscopy it became possible to make Raman spectra of these glasses.
-1
In these spectra, two rather sharp bands occur at 806 cm and 770 cm-1 . The intensity of the band at 806 cm-1 de-creases and the intensity of the band at 770 cm-1 in-creases with increasing alkali content. The Raman spectrum
-1
of 100% B2
o
3 shows a band at 806 cm only, whereas this band is absent in the Raman spectrum of vitreous borates with 35% alkali content. As shown by depolarisation measurements, these bands are due to totally symmetrie vibrations.To explain these phenomena, Bril (3) and Konijnendijk
(4) made a Raman and infrared study to elucidate the structure of vitreous alkali borates. Konijnendijk has discussed the occurrence of borate groups in vitreous alkali borates by camparing the Raman spectra of borate glasses with those of crystalline borates, while Bril has performed a normal coordinate analysis of sodium meta-borate in order to determine the origin of the band at 770 cm-1
In this thesis, the work of Bril has been continued and some other crystalline borates have been investigated in order to get some insight into the origin of the
band at 770 cm-1. Some of the best known groups which are present in crystalline alkali borates have been given in Figure I.Z.
The work described in this thesis contains the following items.
(i) A normal coordinate analysis of LiB02. The simplicity of the structure makes it possible to calculate a set of force constants which can be used for other boron-oxygen configurations.
(ii) A normal coordinate analysis of a network of penta-borate groups. This explains the origin of the band
at 770 cm- 1 which occurs in vitreous alkali borates with a low alkali content.
(iii) A normal coordinate analysis of the diborate group. This explains the origin of the band at 770 cm-1 in vitreous borates with a h alkali content. (iv) A discussion of the or in of the band at 806 cm-1
by campar the Rarnan spectra of vitreous and crystalline 03. e' SORON Q OXYGEN DlBORA TE GROU? WETA8CRA'!E Na2o. s2o3.Kp Bp3 Rb20 8~,1-:_,
Figure 1.2 Various borate groups occurring in crystalline alkali borates.
The results of these investigations give more and better information about the Raman spectra of vitreous alkali borates and, therefore, of the structure of the corresponding glasses. With the results of the normal coordinate analysis of the compounds mentioned above it is possible to investigate the triborate group. This is of some importance since this group is present in vitreous alkali borates with 0-33 mole % alkali oxyde, as was shown by Konijnendijk.
1.1 REPERENCES
1) P.J. Bray and J.G. O'Keefe, Phys. Chem. Glasses, 1963, 4, 37.
2) P. Beekenkamp, Phil Res. Repts. Suppl. 1966, no. 4. 3) Th.W. Bril, Raman Spectroscopy of Crystalline and
Vitreous Alkali Borates. Thesis 1976, Eindhoven University of Technology.
4) W.L. Konijnendijk, The Structure of Borosilicate Glasses, Thesis 1975, Eindhoven Unive-rsity of Technology.
11 General Remark s
Generally speaking, it is rather difficult to inter-pretRaman and infrared spectra of crystalline coumpounds. Several steps have to be undertaken for understanding these spectra and the methods used have been discussed in this chapter.
I I. 1 ASSIGNMENT OF THE RAMAN AND INFRARED SPECTRA
With factor group analysis, a group-theoretical method, it is possible to determine the number of vibra-tions distributed over the various irreducible representa-tions of the point group which corresponds with the space group of the crystal (1, 2).
With this methad it is possible to get the following in-formation.
(i) The number of translational vibrations i.e. vibra-tions of , mainly ionically, bonded groups or atoms, and their distribution over the various symmetry species.
(ii) The number of internal (intramolecular) vibrations,
i.e. vibrations of, mainly covalently, bonded atoms, and their distribution over the various species. (iii) The number of rotational and acoustical vibrations
and their distribution over the various representa-tions of the point group.
With the aid of site group analysis (3) and correla-tion tables (4) some predictions can be made about the frequency and intensity of several bands in the Raman and infrared spectra.
To distinguish lattice vibrations from internal vibrations in crystals, we have made use of various crystalline compounds with identical molecular groups, but different kinds of alkali ions. Por instance, in LiB0 2, (BOz)00-chains are present which are connected by Li+-ions. The same ebains are present in CaB
2
o
4, but in this compound they are connected by ca2+-ions. Therefore, in the Raman and infrared spectra of these compounds, the internal vibrations will have nearly the same frequency and relative intensity, but the frequency and intensity of the lattice vibrations will differ.According to the vibration equation for two-atomic mole-cules (5)
-V31,62)~
ZncV
t
where f V ].! c=
force constant in mdyn/K, frequency in cm -1 reduced mass of the atoms in grams, and light velocity in cm s -1 •the lattice vibrations will usually have a lower frequency than the internal vibrations because of the lower force constant and the relative high mass of ionically bonded groups.
Additional information for the assignment of the various vibrations in the Raman and infrared spectra can be obtained by using isotopes, in our case, for instance, 10 B and nB (Boron with the natural abundance of isotopes) and 6Li and 7Li. Vibrations involving boron atoms will show a change in frequency by substituting one boron atom for another, according to the vibration equation.
Vibrations invalving atoms with a relatively higher mass will have a lower frequency. In many spectra of crystalline borates with a natural abundance of boron atoms it may be observed that certain bands are split up due to the isotape effect (6).
Most information necessary for the assignment of the Raman spectra is obtained from spectra of a polished,
oriented single crystal of the crystalline compounds (3, 7). The single crystal is irradiated with plane-polarised,
monochromatic light and the scattered radiation is ob-served and analysed for different polarisation directions. By measuring the depolarisation ratios of the several bands in the spectra it may be possible to distribute the observed bands over the various Raman active symmetry species of the point group, which underlies the space group of the crystal.
I I. 2 NORMAL COORDINATE ANALYSIS
The GF matrix method, developed by E.B. Wilson, is a methad for calculating the vibrational frequencies starting from the kinetic and potential energy of a vibrating
molecule, represented by the G and F matrix, respectively. A detailed derivation and description of this metbod has been given by Fadini (5). We shall confine ourselves toa short review of this method.
II . 2 . 1 Theory
A method to determine the vibrational frequencies, basedon internal coordinates, starts from the equations of motion:
where T is the kinetic energy. V the potential energy of the vibration. and Ri an individual internal coordinate. The general salution of this equation is given by
(2)
vi being the frequency of the ith normal vibration. The internal coordinates may be changes in bond distance (stretchings), bond angle (bendings), dihedral angle (torsions) and in the angle between a plane through the molecule and a bond between an atom and another atom lying in that plane (out of plane movements) These internal coordinates are related to the atomie displacements
Pa
bythe relation
N ... +
E st . p a.=1 a. a.
where the veetors sta. are called the Wilson s-vectors. The s-vectors may be found by a method described in
in the literature (4a).
The choice of coordinates for lattice vibrations is a more complicated case, since the molecular ions are supposed to move as rigid entities. In chapter III of this thesis this problem will be treated in more detail.
The kinetic energy, expressed in internal coordinates or cartesian coordinates, can be written as
~ -1 .
2T = R.G .R (4)
or
ZT
X.M.X
(5)respectively, where R represents the column vector of the internal coordinates, X is the column vector of the
cartesian coordinates of the atomie displacements. and M is a diagonal matrix of the masses of the atoms. The relation between R and X is:
R B.X
where B is a transformation matrix. From (S) and (6) it is seen that
1
-G
=
B.M .Bi ons ( 4) ,
In order to express the potential energy, V, in internal coordinates, V is expanded in a Taylor series:
(6) ( 7) V + [ i 0 .R. l + 1 i: i ' j .R .• R.+ •• 0 0 l J (i,j = 1,2, ... )
where the Ri are the individual internal coordinates. In the harmonie approximation (5), the higher derivatives can be neglected. The first term V0 is arbitrarily chosen equal to zero, and in the equilibrium position the first derivative with respect to an internal coordinate is zero, since the potential energy is then a minimum.
The force and interaction constants can be defined as fellows:
F ..
-lJ 0 (i,j 1 • 2' ... )
and the expression for the potential energy becomes (in matrix notation)
-2V
=
R.F.Rwhere F is the matrix of the force constants.
Substituting Equation (2) in (1) gives a set of linear equations in .•. Ar. This set of equations has only a non-trivial salution for the amplitude A. if the
l
secular equation
( 8)
0 ( 1 0) is satisfied, where E is the identity matrix. This equation may be obtained upon substitution of Eqs. (4), (7) and
(9) in (1).
The eigenvalues Ài are related to the frequencies vi of the normal vibrations:
À. l
2 2
4rr vi
However, by using the symmetry properties of the molecule or crystal it is possible to simplify Equation
(10). This is done by introducing the so-called internal symmetry coordinates. These coordinates are collected into a column vector S. The relation between S and R is given by
S
=
U.R ( 11)where U is a transformation matrix. The elements of the U matrix can be determined by the methad of Nielson and Berryman (8, 9). The G,
F
andG.F
matrices then factorise into blocks (according to the symmetry species of the point group corresponding with the factor group of the crystal) and the secular equation becomesIUGFÜ-
E.ÀI
=
o.
After eliminatien of the redundancies (Chapter II.2.3 of this thesis) the order of the secular equation becomes 3N - 6 for molecules and 3N - 3 for crystals. The great advantage of the use of internal symmetry coordinates is that the size of the problem is reduced and the secular equation is split up into a number of equations of
smaller order, making it possible to relate the calculated frequencies to their symmetry species.
Using the foregoing treatment, and given the masses, positions of the atoms and the force and interaction
constants, it is possible to determine the vibrational frequencies of a molecule or al. However, a great problem in carry out a normal coordinate analysis is that the force and interaction constauts are generally not wellknown and, therefore, the vibrational frequencies cannot be calculated with much accuracy.
By camparing the frequencies calculated from a set of estimated force constauts with the observed frequencies, it is possible to refine the force constants. However, in nearly all cases there are many more constauts than vibrational ies. Thus, the number of equations is generally less than the number of unknowns. However, for refining the force constauts many iteration procedures have been developed (5) where, e.g., the function
IF
observed- F
calculatedI
or the function
\v
observed- v
calculatedI
is minimised.
11.2.2 Schachtschneider's computer programs
Por performing a normal coordinate analysis with Wilson's GF matrix method, various sets of computer programs are available. Among them are the programs of Schachtschneider (10) and of Schimanouchi (11).
For our investigations, we used the programs of Schachtschneider, which will be described briefly.
(a) CART
This program calculates the cartesian coordinates (the X matrix) of the constitut atoms from the
known geometry of a molecule and, if desired, it also computes the moments of inertia of the molecule.
(b) GMAT
From the X matrix, calculated with CART, and the definition of the internal coordinates of the molecule, the B matrix is determined. With the given masses of the atoms and the B matrix, the G matrix can be cal-culated according to Eq. (7), After introducing the U matrix, the block-diagonalised G matrix is determined.
(c) VSEC
Program VSEC calculates the vibrational frequencies from the G and Z matrices (block-diagonalised or not). For practical reasons, the Z matrix takes the place of the force constant matrixFin Eq. (9). The two
matrices are related by the expression
N
L: Z. "k'q,k
k=1 lJ
(12)
The quantities q,k (k
=
1, ••• ,N) are the N force and interaction constauts to be defined for a particular problem.z ..
k is the coefficient with which thecon-lJ
stant q,k occurs in element , defined by Eq. (8), of the force constant matrix F.
With this program, it is also possible to cal-culate the amplitudes of the vibrations, the potential energy distribution over the force constants and the cartesian displacements for each normal coordinate.
(d) FPERT
This program is an iteration program. With the G and Z matrices and the initial values of the force constauts $k' vibrational frequencies are calculated and compared with the observed frequencies. The
iteration procedure is based on a method first proposed by King and Crawford (12, 13) and the function
Schachtschoeider has also written a program for calculating the G matrix for a polymer or crystal: GMATP. However, when applying it to crystals this program has several disadvantages. Therefore, this program has been adapted to crystals, and extended in order to include other advantages (vide infra).
II.2.3 Extensions of the Schachtschoeider programs In order to make the programs of Schachtschoeider applicable to crystals, various extensions have been made. Same of these extensions have been brie described in the literature (6) and (6a). Another one will bedes-cribed in this thesis.
These extensions are the following.
(a) GMOP, GMOP/SECONDVERSION and GMOP/THIRDVERSION Schimanouchi et aZ. (14) have described how the GF matrix methad can be applied to the optically active vibrations of crystals, when these are in phase in all primitive unit cells (i.e. in the so-called k 0 approximation). In the case of a crystal, the optical analogues G(opt) and F(opt) have to be substituted for the matrices G and F in the vibration equation. G(opt) is computed from the masses of the atoms and the matrix B(opt) = l:B . . , where B . . refers to a B matrix whose
j 1,) l , J
row and columns correspond with the internal coordinates of the ith primitive unit cell and the cartesian
coordinates of the jth primitive unit cell, respectively. Since the elements of B. . are zero for all cartesian
l , J
coordinates belonging to atoms, which do not form a part of an internal coordinate defined in the ith cell, only a very restricted number of atoms needs to be considered.
The calculation of B(opt) and G(opt) is carried out in an extension of programm GMAT, which is called GMOP and has been developed by Bril and Vogel.
SUPACR (GMOP/SECONDVERSION), developed by Vogel (15). The purpose of this subroutine is to eliminate the redundancies. The procedure is based on a theorem found by Sunn, Barrand Crawford (16), and allows of trans-forming the old set of symmetry coordinates to a set of new coordinates, which contains the redundancies as zero coordinates. These zero coordinates are deleted by
program GZCONVERSION.
A third extension of program GMAT is the subroutine TRAVIB (GMOP/THIRDVERSION). With this subroutine it is possible to make use of specially devised translational vibration coordinates, i.e. coordinates invalving ions which vibrate as rigid entities, without changing the bond distances and bond angles within the ions. A detailed description of the theory of this program will be given in chapter III of this thesis.
(b) GZCONVERSION
When transforming the old set of internal symmetry coordinates S into a new set S' containing the redundant coordinates as zero coordinates, the G and F matrices are transformed correspondingly. Por each zero coordinate, the new matrix G' contains a row and corresponding
column of mere zero's for each zero coordinate. Though this is not true for the new F matrix F', the product G'F' will contain the same rows and columns of zero elementsas does G'. These rows and columns may be deleted from the product matrix without loss of informa-tion. The secular equation
IG"F" - E.ÀI = 0 (13)
where G'' and F'' are equal to G' and F', respectively, without the rows and columns of zero elements, contains the samenon-zero roots Ài as does Eq. (9). The trans-formation from G into G' ', and F into F'' (or rather Z into Z'') is performed by program GZCONVERSION, which
has been described by Bril and Vogel (6a). (c) FLEPO
This program, written by Dikhoff and Vogel, is an iteration program to determine or refine force and interaction constants. A detailed description of the theory of this program has been given by Dikhoff (17). The iteration procedure is based on a minimisation procedure developed by Fletcher and Powell (18). The advantages of this methad over the minimisation
procedure used in FPERT is that it does not deverge and that the calculation is faster. The input of FLEPO consists mainly of the G'' and Z'' matrices, the ob-served frequencies, and a set of initial values (generally estimated beforehand) for the force con-stants ~i"'"'~N" The output consists mainly of a set of adjusted force constants.
With the theory and computer programs described above, it is possible to perfarm a normal coordinate analysis of crystals. Usually, we used GMOP/THIRDVERSION, GZCONVERSION and FLEPO for determining the force and interaction constants. However, as seen in Section
II.2.1, a great problem in performing a normal coordinate analysis is the great number of unknowns. Thus, with program FLEPO, it is not possible to modified all force constants at once, because of the fact that the number of equations is too small. Usually, some force constants have been held fixed, while the other have been modified.
This procedure is repeated until theforce constants have been refined.
Using the force and interaction constants obtained with the methad just described, it is possible to get further information about the normal vibrations with
program VSEC. This program provides us with the amplitudes of the vibrations and the cartesian displacements of the atoms in the various normal vibrations, so that the normal
vibrations are completely determined,
II.3 FACTOR GROUP AND SITE GROUP ANALYSIS
Factor group and site group analysis have been described by Fately (4). A short review of this methad will be given in this section.
When applying factor group analysis to crystals, the crystal structure has to be known. Use has to be made of the primitive unit cell or Bravais space cell. The
crystallographic unit cell may be identical with the Bravais space cell or be larger by some multiple. This depends of the crystal structure: an F-centered crystallo-graphic unit cell, for instance, contains four Bravais space cells. The factor group of the space group is the set of cosets obtained when the space group is decomposed relative to the group of all its primitive translations. The factor group is isomorphic with one of the 32 point groups, which notatien can be obtained from the Schoenflies-notation by dropping the superscript (3).
Since isomorphic groups have the same irreducible representations, we may use in the following the symmetry species of the isomorphic point group. With the methad of Bhagavantam and Venkatarayudu (9) it is possible to cal-culate the number of vibrations belonging to the symmetry species of the point group. The Wyckoff sites of the atoms have to be known.
In many cases it is worth while to make a distinction between lattice vibrations and internal vibrations. This can be done by site group analysis, The equilibrium position of each atom lies on a site that has its own symmetry. This symmetry is determined by all the symmetry elements of the point group which pass through that point. The site group is a subgroup of the full symmetry of the Bravais cell. In the case of lattice vibrations, the site of the centre of gravity of the group of atoms is con-sidered. This site symmetry is correlated to the factor
group of the crystal and tabulated in a correlation table (4) so that predictions can be made concerning the infrared and Raman act of each vibration.
II.4 EXPERTMENTAL WORK
Crystalline alkali and alkaline earth borates have been prepared by fusing tagether the corresponding oxides or carbonates with B
2
o
3 or B03 in stoechiometric amounts, foliowed by slow cooling. Several other compounds, like 6LiB0 2, 7LiB02, and Na2B4
o
7.1DH2o
have been prepared from aqueous solution. The following chemical compounds were used as raw materials.K 2
co
3, , Na 2co
3, Li2co
3,cs
2co
3, Bzo3, H 3Bo3, CaO, Rb 2co
3, 10 B203,Merck, Art.4928 p.a. Merck, Art.6392 p.a. Merck, Art.5680 p.a. Merck, Art.2040 p.a. Merck, Art. 163 p.a. Merck, Art. 166 p.a.
Riedel de Hahn, 30405 p.a. Merck, Art.7612
containing 93% 10B and 7% 11B, 20th Century Electranies Ltd., nE 4180
6LiOH.H
2
o,
containing 95.5%6Li and 4,5% 7Li, Oak Ridge National Labaratory
7
LiOH.H20, containing 99,9% 7Li, Oak Ridge National Laboratory.
After preparatien of the crystalline compounds, a Debye-Scherrer X-ray pattern was made and the measured d-values were compared with d-values from the
ASTM-system (19), or with calculated d-values. The latter were determined with computer program DVAL of Koster (20) from the known structure of the crystalline compound in question.
spectrometer. The spectrometer was equipped with an Ar+-LASER (blue line 488.0 nm, green line 514.5 nm) model 165-00 of Spectra Physics, and with a He-Ne LASER (red line 647.1 nm) model 165-01 of Spectra Physics.
The infrared spectra were made with a GRUBB~PARSONS
MK-3 double-beam grat spectrophotometer in the region 400-4000 cm- 1 and a HITACHI EPI-1 spectrophotometer in
-1 -1
the region 200-700 cm In the region 20-400 cm , the spectra were recorded on a Fourier Transfarm Infrared Spectrograph of BRUKER. For the region 200-4000 cm- 1 pellets were made with Csl as a matrix material (weight ratio compound: matrix 1:50), and for the region
20--1
200 cm with polyethylene as a matrix material (weight ratio compound: matrix= 1:5). The starting materials were grinded by hand as finely as possible. The mixture of crystalline powder and matrix material was vibrated during a quarter of an hour so as to get, a random dis-tribution of the compound in the matrix material after moulding. The pressure used for moulding was 10 kg/cm2 for Csi pellets and 2 kgf/cm2 for polyethylene pellets.
II.S REPERENCES
1) L.A. Woodward, Introduetion to the Theory of Molecular Vibrations and Vibrational Spectroscopy, Oxford
University Press, London, 1972.
2) K. Nakamoto, Infrared Spectra of Inorganic and Coordination Compounds, Sec. Ed., John Wiley
&
Sons (Wiley Interscience) New York, 1970.3) G. Turrell, Infrared and Raman Spectra of Crystals, Academie Press, London, 1972.
4) W.G. Fately, F.R. Dollish, N.T. McDevitt and F.F. Bentley, Infrared and Raman Selection Rules for Molecular and Lattice Vibrations: The correlation Method, John Wiley & Sons (Wiley Interscience),
New York, 1972.
4a) E.B. Wilson, J.C. Decius and P.C. Cross, Molecular Vibrations, Mc.Graw-Hill Book Company, New York, 1955.
5) A. Fadini, MolekUlarkraftkonstanten, Dr. Dietrich Steinkopff Verlag, Darmstadt, 1976.
6) Th.W. Bril, Raman scopy of Crystalline and Vitreous Alkali Borates, Thesis Eindhoven University of Technology, 1976.
6a) Th.W. Bril and D.L. Vogel, Proceedings of the Fifth International Conference un Raman Spectroscopy, held 2-8 Sept. 1976 at the University of Preiburg (Germany). Ed. by E.D. Schmid et al., H.F. Schulz Verlag, Preiburg im Breisgau, 1976.
7) H. Poulet et J.P. Mathieu, Spectres de Vibration et Symmétrie des Cristaux, Gordon
&
Breach, Paris, 1970. 8) J.P. Nielsen and L.H. Berryman, J. Chem. Phys. 194917, 659.
9) S. Bhagavantan and T. Venkatarayudu, Theory of Groups and its Applications to Physical Problems, Academie Press, New York, 1969.
10) J.H. Schachtschneider, Techn. Nos 231-64 (Vol I and II) Shell Development Company, Emeryville,
California, 1966.
11) T. Schimanouchi, M. Tsuboi and T. Migazawa, J. Chem. Phys., 1961, 351597.
12) W.T. King, Characteristic Group Vibrations, Ph.D Thesis, University of Minnesota, 1956, Diss. Abstr. 18, 2172, 1958.
13) W.T. King, I.M. Mills and B.L. Crawford, J. Chem. Phys., 1957' 27' 455.
14) T. Schimanouchi and M. Tsuboi, J. Chem. Phys., 1961, 35, 1957.
15) D.L. Vogel, unpublished. For a short description see 16) 17) Ref. C.E. 1949, 6). Sun, 1 7'
R.G. Parr, and L. Crawford, J. Chem. Phys., 840.
Th.G.M.H. Dikhoff, Structuuronderzoek van Wolframaat en Molybdaat glazen met Behulp van Raman en Infrarood-spectroscopie, Report Eindhoven University of
18) R. Fletcher and M.J.D. Powell, Computer Jüurnal, 1963, 6, 163.
19)
ZO)
American Society for Testing and A.S. Koster, Eindhoven Univers unpublished.
Materials. of Technology,
l i l Trans lational Vibration Coordinates
III.1 INTRODUCTION
As stated before, the computer programs of Schachtschneider are not very well suited if lattice vibrations have to be taken into consideration. These programs only permit of using stretching, bend , out-of plane and torsion coordinates invalving two, three, four and four atoms, respectively.
If we want to describe, for instance, a translational vibration, we might use for a coordinate the change in distance (stretching) between an appropriate atom A in one ion and an appropriate atom B in the other ion. However, in doing so, the rigidity of both ions is not guaranteed: if the distance between A and B changes, the distance between A and the other atoms of its ion and that between B and the other atoms of the second ion,
and also the various bond angles are automatically altered. Therefore, we will define other coordinates which will guarantee the rigidity. These translational vibra-tion coordinates, as they will be called, have been introduced already by Kumar and coworkers in their normal coordinate analysis and Raman intensity anlysis of sodium nitrate (1) and lithium nitrate (2), and of yttrium
vanadate and calcium tunstate (3). However, indeveloping the Wilson veetors associated with these coordinates and the individual atoms contained in each ion, these
authors did nat take into account the Eekart (or Sayvets) conditions (4).
Let us consider as an example the translational vibration of sodium nitrate in which the sodium atoms on the one hand and the nitrate ions on the other hand are rnaving in an antiparallel fashion and both groups of
atoms perpendicular to the trigonal axis connecting the eentres of gravity of the ions (Fig. 111.1.1). If the change ~v in the angle v is taken as the coordinate t of this translational vibration, the s veetors are as follows. For the Na+ ion~:
~/D
and for the N03 ions:
St
NO = -2~/D• 3
where +
è
are unit veetors D is the nearest-neighbour a nitrogen atom.(III.1.1)
(111.1.2)
in the vibration direction, and distance between a sodium and
;p
--~~~1f~~~~--~----~~--~/
c
3
~: SODIUM Q: OXYGENe:
NI TROGENFigure I I I . l . l A translational vibration coordinate in
sodium nitrate. The plane of the
No;
groups isper-pendicular to the trigonal axis.
The diagonal element, gt t' in the G matrix is then given
'
by Eq. (III.1.3):
....
I\l st
a a , ,a (III.1.3)
where the summation is over one N03 ion and two Na+ ions, yielding
,t
'
2
Na[ (III.1.4)
~x ( 1/mx) stands for the ree mass of ion X and
111
for the magnitude of vectort.
Eq (III.1.3) still applies if the atoms in ion
Nüi
are considered separately. However, the summation must then be taken over all individual atoms, yielding(III.1 .4) and(III.1.5) yield the expression:
(III.1.6)
From this relation, Kumar et at. have obtained
,N (III.1.7a)
and
(III.1.7b)
However, as long as only Eq III.1.6 is considered, there is an infinity of solutions for ~t' N and
,o·
The second relation to be taken into account is the first Eekart condition (4):0 (III.1.8)
where the summation is either over the ions or over all individual atoms, the displacements of which contribute to the coordinate t. The s veetors given by Eqs. (III.l .7a) and (III.1.7b) do not fulfil this condition.
The correct salution from Eqs. (III.1 .6) and
(III.1.8), taking Eqs. (III.1.1) and (III.1.2) into
account, is given by .... mN st N m
'
N0 3 (III.1.9a) and -+ st,O mNO 3 (III.1.9b)s veetors for the general case of translational vibrations, and a subroutine TRAVIB of program m10P for their cal-culation have been developed by D.L. Vogel and the author of this thesis and will be described in section III.Z.
III.Z TRANSLATIONAL VIBRATIONS
We have stated already that the ions in a crystal move as rigid entities when performing translational vibrations. The s-vector for each of the atoms of an ion can be calculated from Eq. (III.1.3) and taking the Eekart conditions into account.
Consider two ions N and M containing n and m atoms, respectively (Fig. III.2.2). In the general case, the translational vibration can be described by a change in distance of the eentres of gravity of the ions.
,-... -+ I \ -e I I NM~-1-·-·-·-·-·-· ' I I I
'
... /N
~ ...,
I \\ ... t e ·!-·~ NM I I \/
'...
_./M
Figure III.2 The s veetors of two ions N and M containing n and m atoms, respectively.
The s veetors of the translational vibration coordinate for the ions N and M as a whole are given by the
expressions _,. ·>" st N
,
-eNM (III.2.1) and....
_,. st M,
eNM (III.2.2)....
where + eNM are the unit veetors in the vibration direction. The contribution to these s-vectors of the ith atom of
ion N and of the jth atom of ion M may be written as:
_,. _,. s xi.st,N t,Ni (i 1, ... ,n) (III.2.3) and _,. + s yj.st,M t ,Mj (J 1 , ••• ,m) (III.2.4)
where x i (i
=
1, ... ,n) and y j (j=
1, ... ,m) are constants to be determined. The diagonal element, t' is given'
by the expression gt t ,
=
a a E)l st ,where )ln and llM are the reciprocal masses of the ions N and M. If the atoms of the individual atoms are
con-sidered separately, the summatien for element gt t becomes
'
n _,. 2 m 2
E 11 N !st NI + _E
ll.~vLI-;t
M.l
i=1 i ' i ]=1 J ' J
+ 2 n 2 + 2 m 2
ist NI E JlN .x.+ istMI E llM .y.
' i=1 i 1 , j =1 l j J
From Eqs. (III.1.5) and (III.1.6) it is seen that n 2 l: ]l .x. llN i=1 Ni l (III.2.7a) and m 2 E llM .y. llM j=1 j J (III.2.7b)
Taking into account the first Eekart condition
...
Es = 0
a t,a (III.2.8)
where the summatien is over the individual atoms of the two ions, it is possible to find another relationship for the xi's and yi's. Eqs. (III.2.3), (III.2.4) and (III.2.8) yield:
n _.. m _..
E x.st N + E y.st u 0
i=1 l ' j=1 J •'''
Therefore, with Eqs. (111.2.1) and (III.2.2) we obtain:
n l: x. i=l l M 2:
Y·
j 1 J (III.2.9)We have seen in the previous section that in case of a N0
3 ion, ~he x~'s are given by (see Eqs. (11I.1.9a) and
(III.1. 9b) mi llNO 3 (i N,O) x. mNO ll· l 3 l
mN. llN l X· 1. mN (III.2.10a) and m?vl. \.i
....J
-M Y· J mfvl llM. J (III.2.10b)Eqs. (III.2.7a), (III.2.7b) and (III.2.9) will be satisfied.
We may arrive at the same result by the following reasoning. The s-vector !t . for a coordinate t and an
,]
atom j will have
(i) a direction equal to that in which a given dis-placement of atom j will produce the greatest in-crease of t. This direction will coincide with the line connecting the eentres of gravity of the two ions.
(ii) a magnitude equal to the increase of t produced by a unit displacembent of atom j in that direction A unit displacement in any direction of an atom Mj with mass mM. and belonging to the polyatomic ion M will
J
cause a displacement of the centre of gravity of the ion equal to
in that direction, where
1s the mass of the polyatomic ion M. Taking this and
(i) and (ii) into account, and defining êNM as the unit vector directed from the centre of gravity of ion N to
ion M, we obtain for the Wilson s-vectors of an atom in ion N
mN.
-+ 1 -+
st,Ni mN eNM (i 1 , ••• n) (III.2.11a) and of an at om in ion M
....
mM. .... s + _1_ eNMt,Mj mM (j 1 , ••• m) (III.2.11b)
Camparing these equations with Eqs. (III.2.1) up to and including (III.2.4), we obtain Eqs. (III.2.10a) and
( II I. 2. 1 Ob) •
The translational vibration coordinates defined in this chapter do not interact with internal coordinates, since for any off-diagonal element of the G matrix
descrihing the interaction between a translational vibra-tion coordinate t and an internal coordinate i,
-+
LIJ st .s.
Cl. Cl. , Cl. 1, Cl.
This is zero since Z::
s.
= 0 . let. Cl. l l l ion N eNM • L a in ion N ->-s. 10:The translational vibration coordinates may be combined into adapted coordinates by the projection operator metbod in the same way as internal symmetry coordinates are
constructed from internal coordinates.
III.3 NORMAL COORDINATE ANALYSIS OF YTTRIUM VANADATE In order to check. the adequacy of our choice of translational vibration coordinates we decided to perform
a normal coordinate analysis of YV04. With these coordina-tes, it is possible to determine a set of force constants
for the internal vibrations of the VO~- ion in YV04,
which does not interact with the force constants of the translational vibration coordinates.
The crystal structure of YV04 has been published by
Baglio and Gashurov (6). YV04 crystallises inspace group
I 4
1/amd with two molecules per primitive unit cell
(Fig. 111.3.1). lts factor group is isomorphic with point
group D
4h.
With the aid of factor group analysis it is possible to determine the distribution of the vibrations over the
various irreducible representations of point group D
4h.
The structure of the vibrational representation,
r .b , is
Vl •
of the translational representation, rtrans.'
and of the librational representation, rlibr.,
All gerade symmetry species are Raman active, except species AZg' whereas Azu and Eu are infrared active. In the structure of the total representation given by Chaves
and Porto (7), there occur four B2g and one B
1g vibrations.
The B1g and Bzg symmetry species have been interchanged
by a different choice in orientation of the symmetry
operations 8~ and 8~' and C~ and C~'·
Since the assignment of the Raman and infrared spectra by Miller et al. (7) is not reliable, we considered only the assignment of the Raman spectrum given by Chaves and Porto (8). This assignment has also been used by Kumar,
and is represented in Table 111.3.1 Alg: 889 cm -1 B2g: 259 cm -1 -1 840 cm-1 375 cm E -1 g cm -l B1g: 816 cm ? ( 48 7) -1 260 cm- 1 487 cm 265 cm -1 1 61 cm -1 154 cm -1
Table III.3.1 Assignment of the Raman spectrum
given by Chaves and Porto
The bands of species Alg (at 889 and 375 cm-1) and
-1
species B2g (at 259 cm ) are internal vibrations, as is seen from the structure of the representations. Of the four bands of species Blg' one pair will consist of internal vibrations, and one pair of translational vi-brations. The pair of internal modes of species B1 will
g -1
correspond most probably to the bands at 816 and 487 cm . The bands at 265 and 154 cm-1 will then belong to the pair of translational vibrations of species B1g.
As far as the bands of species E are concerned, two g must pertain to internal vibrations, two to translational vibrations, and one to a degenerate pair of librations. The band at 840 cm- 1 will pertain to an internal vibra-tion. In the B
1 g representation we found two lattice -1
vibrations at 265 and 154 cm • The bands at 260 and 161 cm- 1 , belonging toE , will also be translational vibrations, since we
exp~ct
the librations of thevo~-
ion to lie at a lower frequency. This is because there is a larger change in potential energy, when avo~-
ionvibrates.against a Y3+ ion than when it is merely rotating in the crystal lattice: therefore, the force constant of the libration is expected to be lower than that for a
3-translation vibration of a
vo4
ion.From a correlation of the molecular symmetry D2d via the site symmetry D2d to the point group D4h is seen that the intensity of an internal vibration of species E
is determined by a yz or z , and may therefore he expected to he small. Besides, according to a calculation of Kumar
et aî. , this second internal vibration of species E must have zero intensity and 487 cm-1 for its
wavenumber~
We have adopted this value in our calculations.' ' , I I ~
w
! i--··-·i·--·
-·~-:.-=-r----___ 1 -·~-:.-=-r----___-·~-:.-=-r----____ • .
::~__
j __ _
' I ' / ': /
:.
---~0::--- -~-1 ' I I I I I I ' '•
YTTR !UM\/1
x
vo
3-4 - IONQ
x
vo"
3-- !ONFig. 111.3.1 Crystal Structure of Yvo
4
In order to carry out a normal coordinate analysis, including the translational vibration coordinates, we made use of the computer program GMOP/THIRDVERSION (which has been described in Chapter II.2.3 of this thesis). The position parameters of the atoms in the unit cell
(Fig. III.3.1) were taken from the refinement of the crystal structure of YV04 by Baglio and Gashurov.
The internal and translational vibration coordinates used in our analysis have been defined in Table III.3.2. From the X matrix (cartesian coordinates of the atoms),
the definition of the internal and translational vibra-tion coordinates and the U matrix, determined with the projection operator metbod (Appendix I), the G matrix and SUPACR U matrix have been calculated using program GMOP/THIRDVERSION. Program GZCONVERSION has been used to convert the G matrix and Z matrix, which has been set up by inspection, into their equivalents G'' and Z'', which correspond to a new set of symmetry coordinates without redundancies (cf. page 12).
Internal Coordinates: d: VO bond distance
a: ovo bending, top or bottorn angle of the vo~--ion
8: OVO bending, side angle of the vot--ion Translational Vibration Coordinates:
Attraction a: y3+ vo~- y3+(0,!,1)
3-vo 4 co,o,n
b: y3• vot- Y3•co,o,1J - vo4 3-co,o,n
c: y3• - vo3- CLLD - vo~-co,o.n
Repulsion d: vo4 -3- vo4 3- (O,O,D vo3
-o,o.n
e: y3+ - y3+ Y3+0,!,D y3+(0,!,1)
Table III.3.2 Internal and translational coordinates of YV0
4. Top and bottorn angles
are the angles which are bisected by the s
4 axis in the crystal (103°54
1
) , The side
angles are the four other angles of magnitude 113°54'.
It has been rather difficult to estimate the initial force constants ~i for YV0
4, especially for the transla-tional vibration coordinates. The trial values for the bond stretching and bending force constants of the
vo~
ion have been estimated on the ground of the values found for ions likeMoO~-
and Mno; (9, 10, 11). The magnitude of the values for the translational vibration coordinates has been estimated from the calculation by Morioka (12) of force constants for the Cd-Cl and Cd-Br latticeof these bonds is of the same magnitude as the Coulomb force between the ions in YV04.
With the assignment of the Raman spectrum of YV04, proposed by Chaves and Porto, and the initia! force constants ~-. it was possible to obtain a refined set of
1
force constants (Table II I. 3. 3). The refined set was calculated using program FLEPO, in a two-step procedure:
(i) perturbation of the force constants while the inter-action constants are held fixed.
(ii) Perturbation of the interaction constants while the force constants are held fixed.
We have defined five different force constants, f , a , f , fd, and f , for the translational vibration c e coordinates. Since no reliable infrared spectra were available, we could not assign frequencies to the
translational vibrations of symmetry species A and E ,
u u
but only to those of species ZB1 g and ZE . Therefore, we g had only four relations with five unknown parameters. Since all five TRAVIB force constants have been entered as variabie parameters in program FLEPO, the values obtained for these constants are somewhat arbitrary, since they
depend on the trial values chosen.
The mean fference between observed and calculated frequencies is about 0·01%. fd fa fB fa fb f c fd f e
Force constants _in mdyn/~
initial value final value 5· 0 5. 39 1. 0 0. 96 1. 0 0. 94 0 • B3 1. 25 0. 7 9 1 '05 0. 30 8. 53 -0·44 -0.13 -0·55 -0 ·03
Table III.3.3 The initia! trial and final
values of the force and interaction constauts for the internal and translational vibration coordinates of YV0
The values of the refined set of force constants for the internal force constants look very reasonable. For
3-the analogous ion Moü 4 , 3-the bond stretching force con-stant, fd, has been found to be equal to 4·85 mdyn/~, while the bending force constants vary from
o·s
to 1·2mdyn/~. for this ion. The fact that the
stretching-stretching interaction constant is larger than all stretching-bending and bending-bending interaction constants also comes up to what we expected, as well as the circumstance that interaction constants with a common bond are higher than those with a common atom only.
III.4 CONCLUSIONS AND DISCUSSION
As has been proved in Sectien III.2, the model for the translational vibration coordinates involves a zero interaction between lattice and internal vibrations. Therefore, there is no influence of the translational vibrations on the force constants of the internal vibra-tions. We have seen that the choice of our internal force field has been satisfactory.
We must now check whether the final values found for · the force constants of the translational vibrations may be considered to be reasonable. In order to do so, we should derive expressions for the potential energies of the Y3+ and the VO~- ions in the crystal field and take
the second derivatives with respect to the ion-ion
distances. Repulsive forces, Coulomb forces, polarizability
3-and dipole-dipole interactions of the vo4 ion should be taken into account for this purpose.
Therefore, the translational vibration coordinates of the type described seem to be very useful coordinates for a normal coordinate analysis of crystals which in-cludes the lattice vibrations.
Since we were not prepared for such a large task (since the investigation of YV04 is only a side line in
this thesis) we have chosen for a much simpler, though very qualitative check.
~orioka and Nakagawa (12) have stuclied the substances CdC12, CdBr2 and Cdi2• These consist of mono-atomie ions only and their vibrations are, therefore, only translational vibrations. Treating the ions as rigid, unpolarizable
antities and consiclering only repulsive and Coulomb
forces, they calculated the stretching force constants of the Cd---Cl, Cd---Br and Cd---I bands andrepulsivo force constantsof the nearest neighbour Cl---Cl, Br---Br and
I---I ion pairs by using the normal coordinate analysis for crystals as described by Schimanouchi et al. ( 13).
For the Cd---X force constants they found 0·540 (X Cl), 0·472 (X= Br) and 0'277 (X I)mdyn/~, respectively, and for the X-X repulsion 0•080 (X= Cl), 0·073 (X Br), and 0·081 mdyn/~ (X= I), respectively, These are of the same order of magnitude as we have found for fa• fb and fc on the one hand, and of fd and fe on the other hand.
It may be assumed that the force constant of a translational vibration related to two ions A and B will decrease with decreas Coulomb force between A and B. This provides us with a check for the relative magnitude of the force constants fa' fb' and fc for the ions Y3+ and
vo
3-4
•=
YTTRIUMO=OXYGEN
Clll =VANADIUM
Figure 111.4,1 The Coulomb force between two
. . 3+ d
3-l.Ons, ~.a, Y an V0 4 ,
The Coulomb force between a Y3+ ion and a VO~- ion
can be found by vector addition of the farces between
3-each of the atoms constituting the
vo
4 ion on the one hand and the Y3+ ion on the other, according to the
formula
FCoulomb
where qi and qy are the charges of atoms i and Y,
respectively, r. is the distance between atoms i and Y,
l
and ~i is a unit vector along the direction through atoms i and Y (see Fig. 111.4.1). Of course, this force is dependent on the charge distribution in the vol- ion. In Table 111.4.1, the Coulomb force has been calculated for two different charge distributions, viz. (in units of the electronic charge ZV 5, z 0 = 2 and zV
=
3, z0=
1·5.coulomb force coulomb force distance of the
force constant value zv • 5' zo 2 zv 3' zo 1 '5 eentres of gravity
attraction fa 1• zs 1· 02 .e2 0·831.e2 3•8934 at tra ct ion fb 1· os 0· 76.e 2 0·792.e z 3' 1460 attraction fc 0· 53 0·30.e2 0·30o.e2 5' 0367
Table III.4.1 Calculated force constants in
mdyn/~ (Column 2) and Coulomb forces for
the translational vibration coordinates (Columns 3 and 4)
1t is seen from Table 111.4.1, that , fb, and
fc fellow a trend which is in keeping with our assumptions mentioned above (i.c. there values decrease with decreasing Coulomb force, and not with increasing distance between
3+
3-the ion y and vo4 )
111.5 REPERENCES
1) S.P. Kumar, V.A. Padma and N. Rajeswara Rao, Indian
J. Pure Appl. Phys. 1972, 10, 275-278.
2) S.P. Kumar, V.A. Padma and N. Rajesware Rao, J. Chem. Phys., 1974, 60, 4156.
3) V. Buddha Addepalli, S.P. Kumar, V.A. Padma and N. Rajesware Rao, Indian J. Pure Appl. Phys. 1976, 14, 726.
4) R.J. Malhiot and Salvador M. Ferigle, J. Chem. Phys., 1954, 22, 717.
5) E.B. Wilson, J.C. Decius and P.C. Cross, Molecular Vibrations, Me Graw-Hill Book Company, New York, 1955.
6) J.A. Baglio and G. Gashurov, Acta Cryst., 1968, B24, 292.
7) S.A. Miller, H.H. Caspers and H.E. Rast, Phys. Rev., 1968, 168, 964.
8) A. Chaves and S.P.S. Porto, Solid State Communication, 1972, 10, 1075.
9) R.H. Busey, and 0.1. Keiler jr., J. Chem. Phys., 1964, 41, 215.
10) A. Müller and F. Königer and N. Weinstock, Spectrochim. Acta A, 1974, 30A(3), 641.
11) Paul Philippe Cord, Pierre Courtineet Guy Pannetier, Spectrochim. Acta A, 1972, 28(8), 1601.
12) Y. Morioka and I. Nakagawa, Spectrochim. Acta, 1978, 34A, 5.
13) T. Schimanouchi, M. Tsuboi and T. Miyazawa, J. Chem. Phys. 1961, 35, 1597.
IV
Spectroscopie
o:f
LithiuiD
IV.1 INTRODUCTION
In ve st:iga ti ons
Metaborate
CaB 2
o
4 and LiB0 2 are borates of known crystal structure. Both contain infinite ebains (BOz)00, whichare isolated in the sense that they are not bound by homapolar honds (Fig. IV.1.1), nor condensed into a three dimensional net\vork as in crystalline B
2
o
3 (1).• = boron
0
"oxygen chain axisFigure IV. I. 1 (Bo;) co -chains as they are present
in CaB
2o4 and LiBo2.
Lithium metaborate (LiB0 2) belongs to space group P2 1/c
(C~h),
and its factor group is isomorphic with point groupc
2h. There are four formula units LiB02 per primitive unit cell (Fig. IV.1 .2). Each primitive unit cell thus contains four Li+-ions and two B02-links of two
(BOz)00-chains. A complete structure determination has been carried out by Zachariasen {2). The most important bond distauces and bond augles are listed in Table IV.1 .1.
B(1)-0(1) 1 . 3 3 0
5\
<0(1) -B(l) -0(2) 116·37° B(l)-0(2) 1 . 4005\
<0(1)-B(l)-0(3) 126·11° 8(1)-0(3) 1 . 3 8 9ft
<0(2)-B(l)-0(3) 117·52° < B ( 1) -0 ( 2)- B ( 2) 132·80°Table IV.I.I Some bond distauces and bond angles in LiB0
2
Calcium metaborate (CaB 2o4) belongs to space group
P nca (D21b4) with four formula units per primitive unit ~ cell. lts factor group is isomorpbic with point group Dzh (3, 4). Eacb primitive unit cell contains four Ca2+
ions and two B0 2-links of each of four (80;)00-chains.
In botb LiB0 2 and CaB2
o
4 the (Bo;)oo-chains havea pseudo line symmetry consisting of a screw axis 21 along the chain axis (Fig. IV.1.1), and a mirror plane and
glide plane perpendicular to eachother and intersecting eacbother along the screw axis. This is the line symmetry numbered No. 20 by Bbagavantam and Venkatarayudu (5).
However, the real symmetry of the ebains in both compounds is somewhat lower: the boron and oxygen atoms of the ebains are not entirely coplanar so tbat, strictly speaking, there is no mirror plane in either of the
compounds. The only real symmetry element of the ebains is in LiB02 the twofold screw axis 2
1 along the chain axis, and in CaB2
o
4 the glide plane througb the chain axis. Tbis is illustrated by Figure IV.1.3 sbowing the projections of the boron and oxygen atoms - of the two B0 2-links of the same chain in a primitive unit cell in a plane perpendicular to the chain axis in a) LiB02, b) CaB2o
4 and c) the idealised symmetry of line group No. 20.In spite of the different symmetries of the chains in LiB0
2 and CaB2
o
4, and owing to only a slight difference in their geometrical arrangements, the normal vibrations of the ebains lvill be very much the same. A comparison of the vibrational spectra of these substances will,Figure IV.I.2 Primitive unitcellof LiB0 2
therefore, lead rather easily to a separation of the frequencies into those belonging to the internal vibra-tions (vibravibra-tions within the chains) and those pertaining to lattice vibrations (librations of the chains and
vibrations of the cations and chains relative to eachother). Since LiB02 has a simple structure (it contains only 16 atoms per primitive unit cell and has simple boron-oxygen chains, which are not interconnected) and its spectrum may be unravelled easily, not only by camparing it with the spectrum of CaB 2
o
4, but also due to the possibility of using Li6 and Li7 isotopes, it is a suitable compound for a normal coordinate analysis.a) I I I .I b) I """ ---~--e-or-s--o::;____
_ _
clFigure IV.1.3 Projection of a {B0
2) roup of a
chain on the plane perpendicular to the chain axis.
a) LiB0
2 : c2, <0 BO
-
170•66°, <BOO I 6 5 • 1 5b) CaB
2o4 :
c
s' <0-
BO 171•00°, <BOO 158•29c) idealised symmetry of the chain (Line group
No, 20)
IV.2 FACTOR GROUP ANALYSIS OF LiB0 2 Since the space group of
point group c2h' the vibrations pertain to the symmetry species
is isomorphic with of this compound will of Czh' Using factor group analysis we have determined the numbers of internal vibrations, translational vibrations, librations and acoustic vibrations per symmetry species of CZh' The result has been given in Table IV.2.l
tot al
czh internal translational rotations acoustic number cf
A B A B
vibrations vi brat i ons vibrations vibrations vibrations
g 7 4 1 0 12 g 7 5 0 0 12 u 7 I 1 12 u 7 3 0 2 12 total 48 (N " 16-+3N 48)
Table IV.2.1 Internaland lattice vibrations of LiB0
