- Email: [email protected]
Longitudinal lattice modes of mixed valence-mixed metal compounds of platinum and palladium
.Chem@l Physics 45 (1980) 393-399 0 North-Holland Publishing Company
_:
._.
‘. :: .. ,:
..”
... :
.:LONGITUDINAL OF PLATINUM
..
:..
. LATTICE MODES OF MIXED VALENCE-MIXED AND PALLADIWM
METAL COMPOUNDi
..
C.E. PARASI&VAIDIS* Physics Laboratory
A, National
Technical Universiry,
At&s,
Greece
and’ C. PAPATRIANTAFILLOU N.R.C. Democriros,
Received
Aghia Parask&
Arcikis, Greece
20 July 1979
Mixe&vaIence-mixed metal compounds of platintim and palladium are complicated ionic crystals containing independent parallel chains, and a linear unit cell of four masses containing both Pt and Pd -each-in a given state of oxidation II or IV. The natural vibrations of such systems contain longitudinal lattice modes and almost independent local modes. Their Raman spectra contain indistinguishably the frequencies of both kinds of modes. A study of these vibrations requires some method of distinguishing unambiguously between these two kinds of modes, not presently available. We present here a detailed study of the longitudinal lattice modes. Our study develops criteria for distinguishing with some certainty between the twa kinds of modes and we assip accordin_ely their observed Raman frequencies. Furthermore, we _ propose distinction by performing Raman spectroscopy _ a method for their unambiguous on a disordered version of these compounds.
1. Introduction Mixed-valence compounds have been the subject ofextensive but not all systematic study [I]_ These compounds contain the same element (or elements on the same column of the periodic table) in two different states of oxidation. They are classified by Robin and Day [I] in three classes. In class I systems the ions of differing valence are placed on sit& of very different,symmetry and ligand field strength, resulting in extreme * Most of this work was conducted at N.R.C. Democritos under a grant of the National Hellenic Research Foundation @I.E.).
localization
and in of those of their constituent parts, while in class III systems no distinction can be made between the ions of different valence, since they are placed on sites of equivalent symmetry. Class II systems are an intermediate case, where although their valence electron wavefunctions are delocalized, the sites of the ions of different valence are distinguishable, and the properties of these systems usually differ from the sum of the properties. of [email protected] parts. For example ‘. they show intense absorption in the visible iegiqn of the spectrum.which‘is not present in compounds consisting of only one of the two valence states. electronic
of the valence shell electrons
properties
which
are the sum
.
C.E. Paraskecaidis,
394
Mixed-valence palladium A striking
compounds
C. Papnrrianra~lbujLongitudinaI
of platinum
and
are class II mixed-valence compounds. feature of these compounds is that their
crystal structure contains chains of alternating octahedrally coordinated M(IV) and square-planar coordinated M’(U) ions, where M, M’ are either Pt or Pd. The II and IV coordinated ions are at the center of structures that are formed off the chain. A halogen ion is placed on the chain between the IV and II sites, lying closer to the quadrivalent ion. These compounds cf Pt and Pd are diamagnetic and exhibit highly anisotropic solid state behavior. They are semiconductors with room temperature conductivity along the chain at least 300 times larger than that perpendicular to the chain [2]_ The lattice vibrations of these compounds exhibit a similar anisotropic behavior and the collective excitations of the lattice are basically longitudinal (along the chain) modes, while the independent modes, being localized_ are insignificantly affected by this anisotropy
131. The lattice vibrations have been studied from the respective infrared (IR) and resonance Raman (RR) spectra. Such experiments are the only experimental studies of the lattice vibrations of these compounds and they give limited information about them. Important quantities such as the density of vibrational states (or their dispersion relations equivalently) have been deduced for those of the compounds that have the same metal ion (Pt or Pd) on the chain [4] from their respective IR and RR spectra, but a direct experimental confirmation has yet to be reported. In the present work we present a study of the longitudinal lattice modes of those compounds that have both Pt and Pd along the chain, each with different valence (i.e. Pt(IV), Pd(II) or vice versa) and in section 2 we present the formalism for such a study. The RR and IR spectra of these compounds .%.I
22 .
Q
9r :
A2
~__c;-___~_-_~,--L-_-‘~ r--
----
02 .
P d.? d
UNlr
CELL
Fig I. Unit cell stracture. Inter-ion distances d, c dl. Masses: mgl = mg2 mass of halogen ion (Cl or Br), rrt,, = mass of the complex ion with Pt (or Pd) IV at its center, nlAA2 = mass of the complex ion with Pd (or Pt) II at its center, ,I**, f )>td2.
locrice modes ofPc and Pd
are harder to interpret because their unit cell consists of four masses having three different values (see fig. 1). The interest of these mixed valencemixed metal ion compounds, beyond the present study of their ordered version, lies in the possibility they present to be prepared in a form where the Pt or Pd have both valences II and IV at random. In that case the lattice is disordered and all its vibrations are localized [5J. Then, the density of vibrational states can be deduced, and the independent modes distinguished from the collective ones, by analyzing their Raman spectra. The above are discussed to a larger extent, along with the results of the present study, in the concluding section. 2. The one-dimensional model We consider an infinitely long linear chain which consists of three types of masses. The mass of the complex ion with the octahedrally coordinated (IV) Pt (or Pd) ion at its center, mxI, the mass of the complex ion with the square-planar coordinated (II) Pd (or Pt) ion at its center, mAz, and the mass of the halogen ion, ma, p laced between the octahedrally and square-planar coordinated ions [I]. (See fig. 1.) The halogen ions are placed at a distance d, from the Pt (or Pd) IV ions, and at a distance dZ (>d,) from the Pd (or Pt) II ions. Two effective spring constantsf, andf, are introduced, corresponding to d, and d,, respectively (i.e_f, connects the octahedrally coordinated ion with the two halogen ions and _f2 connects the square-planar coordinated ion with the two halogen ions). The unit cell consists of four masses and two different spring constants. Its length is 2(d, + d,). Since the tinit cell consists of four masses arranged one-dimensionally, the equation of motion will give four longitudinal lattice modes - one acoustical and three optical. The equation for the normal modes is
Dk = wz(fI +ft
- w’m~
C.E. Parurkecnidis,
For convenience tion:
we introduce
the following
395
lattice modes OJPJ and Pd
C. Papatriantafill~u/LongitudiM~
where
nota-
A0 = v’(Y - 1 - 28), y = 0.? = (27#,
(2)
g =fi
+_L,
(3)
_fi =
af,. -
(4)
mB= @hlr
(5)
m,, = vh,,
(6)
and eq. (1) becomes: D, = y(g - may) yz - + 1 +
g2 --gj
*u
+
B
‘I +
(
1 ++x 1 + 28 ‘J LX+1 >
VW I
(29)’ -
a2
3
sin2k(d,
(l+b2q
AZ = y’(y’ -
+ d,) = 0.
+ 1)1+
.&I + 1 + 2h)},
1 - 2Bq),
0 2
<
’ < pil
’ uw
rl +
28~)
2 + 8h + 1)
VI
(7) and
1+2!k< rlfl
, I
(9)
y1 = s/f+, ,,+[1+2/3(+$3]
33~1 < q+l
(8)
’
0!z2<*+?p
forq
> 1;
(164
vz z c 1 + 2,f311, for 11C 1, 0<
(16b)
- >
VI
where yi = (Zxv,)‘, and we have taken into account the fact that y&i < p2Jy1 in assigning vJvr and vJvl to relations (15) and (16) respectively. These relations, (15) and (16) along with eq. (12) have to be obeyed in order to have consistent results for vi, v2, vj and a Since eq. (13) is quadratic in a, the two real and positive solutions are a_ and a+ (a_ G a+). The physically correct choice for CLis LX-. (See discussion in ref. [4].) Having these considerations in mind, inequalities ( 15) and (16) become:
111 >I'
x 1+*-(I y3 +-[I +w(g3] l/2 w(l+~+w9) x[I)I ( [
(
8pa( 1 +-9 + 2pq) +a)‘[1 +2/9[(1 fricz)/(l +cC)]}2
(10)
l-(l+Cr)2{l+2j3[(~+~)M1ia)1~Z
_ (11)
The k z 0 frequencies vi, v2, vj obey, according
of the three optical modes to eqs. (9)-(11) the relations
tiz + .i&r
+ IraM
= 1 + 28Ul
+ CI)].
(12)
If the three k 2 0 optical mode frequencies vlr v2, v3 are considered known, one can obtain a quadratic equation for a by substituting eq. (9) into eqs. (IO) or (11). A2u2 + Ala
(14b)
and */ is either (yJyJ or (y&r). Since the two effective spring constantsf, andf, are real and positive, a ( =f2/fi) has to be real and positive. Hence we seek the conditions under which eq. (13) gives a real and positive. The conditions for both solutions of eq. (13) for a to be real and positive reduce to the following two inequalities for the three k z 0 longitudinal lattice mode frequencies: o
The spectra for these compounds are obtained by optical methods. We are therefore interested in the k = 0 modes. These are given by the k = 0 solutions of eq. (7), which are: YA =
A, = 2WCy’ - 1 -KS
(14a)
+ A0 = 0,
(13)
4@1(1 + Pq)(l - V) zz 2s(l + Bs) + (1 + 2Pl)(l - PI) < 0 VI < 8(V + 1 +2/G)
2 +I%+
1) ’
for ‘7 < 1.
7
z
(174
(17b)
C.E. Paraskecaidis, C. PuparriantajilioujLongitudinal lattice modes of Pt and Pd
396
2p_
i
for rl > 1.
(l8b)
The conditions for only one of the two real solutions to be positive are met if the following inequalities for y’ are satisfied: 1 tZj?
1+2fltr,
1 + 28Q < 7’ i
1 + 28,
forq>
1;
(19a)
for ~7< 1.
(19b)
If the values of the masses mA, and Q, close, i.e. (g - 1) < 1, then the inequalities (18) become approximately
o<($)2a(1 +F),
are very (17) and
for rl > 1;
octahedrallv coordinated constituent molecule vibrates independently. This is consistent with eq. (23). It gives the ratio of the squares of the two optical frequencies of a system with three masses connected linearly with spring constants of the same;:: value, where the two masses at the end have the same value, which is j3 times the value of the mass in the middle. Therefore, the inequalities that the three k 1 0 optical longitudinal lattice modes vr , .v2,vg have to obey are the inequalities (17) and (18). These along with eq. (12) show the consistent way in which vrr v2, vs have to be chosen from the experimental resonance Raman and infrared spectra.
(2Oa) 3. Results and discussion (2Ob) and 1+0(1-~)<(r;i)&?P, for iI > I: (21a) 1
+/fl _!+)<(!2)2<
1+2PC1
-(l-d.
fort1 c 1 (21b) and inequalities 1+2~<~‘<1
(19) become t.2#?[1
+ (rf - I)],
for Jj > 1;
- (1 - t7)] c y’ < I + 28,
for 4 c 1.
G=) 1 + 2fl[l
(226) Note that in the limit ofq = I relations (20) and (21) reduce to relations (39) of ref. [4]. Special mention has to be made in this limit for the case where only one of the two solutions is positive. In this case the positive solution of eq. (15) goes to zero (IL--f 0) and eqs. (19) [or (22)] reduce to the limiting condition f = 1 t2j.X
(23)
The fact that CC--f 0 implies that the effective spring constant that is responsible for the connection between the octahedrally coordinated constituent molecule and the square-planar one goes to zero: (f2 4 0). Therefore, this is the case where the
We applied our model, discussed in section 2, to the experimental optical (resonance Raman and infrared) spectra of four mixed metal-mixed valence compounds of Pt and Pd [6,7]. Their mass distribution within the unit cell as well as the inequalities the three longitudinal k z 0 lattice mode frequencies have to obey [eqs. (17) and (IS)] are given in table 1. The k z 0 optical frequency that is readily observed from these experimental spectra is vr _ The system vibrates with frequency vr when the masses of the two halogen ions (tna) on the chain in the unit cell vibrate against each other with displacements that are equal in magnitude and opposite in direction, while the heavier ions on the chain (rnA,, mA2)remain motionless [4]. Once this frequency, vIr is known we determine the sum of the two effective spring constants g = f, -i-Jr from eq. (9) as well as the inequalities (17) and (18) that the other two k s 0 longitudinal lattice mode frequencies (vZ, vs) have to obey. These results are presented in table 2. The dispersion relations for these compounds can be calculated from eq. (.l) when the two effective spring constants Jr andfi are determined [4]. The frequencies of the independent modes of these compounds remain practica!ly the same as those of the corresponding modes of the free (not in the chain) constituent molecules, while collective longitudinal modes that can be associated with modes of the free molecules (not in the chain) may
t fi
323 196 325 315
“I
< < < <
v3 v3 “3 vj
c < < c
359.14 240.6 359.69 348.62
14.60095 12.11917 14.78232 13.88664
< < < c
344.66 223.04 345.61 335.44
v1 v2 vz v2
for Y,, v2, v3
and the
x lo7 x IO7 x IO7 ~‘10~
OC(P~/V,)~ <0.1386293 0<(1~,/~,)~<0.294926G O<(~,/v,)~
Inequalilies
f, + (2
< < < <
1.4196473 1.3914248 1.3914248 0.7186878
‘1
v1
0.\181399 0.2534347 0.1124298 0.1564375
II
is v’ = c/J, where c is tlic speed of ligllt and I is the wavelength,
Cl Br Cl Cl
B,
for v,, v2, YJ
120.26 106.44 117.55 115.31
Pd+2E+2CI Pd+IF Pd+4F Pt+4F
A,
0 0 0 84.76
“3
Cl Br Cl Cl
BI
vi is given in cm-‘,f, in amu~cm2. v is the wavenumber I/k The actual frequency actual spring constant isf = c2f(amu/s2).
I 2 3 4
Compound label
Table 2 VI, ~2, VJ andf,
E = NH,,
F = CH,NII,.
Pt+2E+2CI Ptl-4F Ptt4F Pd+4F
1
2 3 4
A,
in n unit cell and the inequalities
Compound lube1
Table 1 Mass distribution
1.138625)3+,/v,)‘< 1.2949266+,/v,)‘< I.1308321 <(v~/v,)~c l.1340069c(v,/v,)2c
~ 1.2362698 1.5068694 1.2248596 1.2248596 -
9
2
%
$
g 2:
5 6 ri: x f
j
2 9 F;
a
r, b b 9 e P B E -. *;
398
C.E. Paraskevnidis. C. PopatriuntajifloujLongit~dinaI
have quite different frequencies [3,4]_ When the spectra of both constituent free molecules are available along with those of the mixed valence compound, the above observation enables one to distinguish the independent modes as those that practically have the same frequency in both spectra. In order to illustrate this we take an example from the RR spectra ofcompound 1 and its constituent free molecules [6]_ For the constituent tranr[Pt’v(NH,),CI,] we take the two observed \(Pt-Cl) frequencies, 327 cm- ’ and 343 cm-‘. For the compound [Pd’t(NH~)2C12Pt’V(NHJ2Cl~] these frequencies become 323 cm-’ and 344 cm-‘, respectively_ This indicates that the first frequency corresponds to a Iongitudinal lattice mode, while the second one to an independent transverse mode. This is more clearly illustrated in the case of Wolffram’s red [IS] (with Pt in both oxidation states) where the frequency of such a mode changes from 344 cm-’ in* the constituent free molecule to 316 cm-’ in the compound. (See discussion in ref. [4].) The preceding discussion of our results showed that the 1R and RR spectra of these compounds do not always give adequate information for their lattice vibrations, and thererare the assignment of the various observed frequencies to independent modes or collective longitudinal ones, and the resulting values for the spring constants, is not unambiguous. However, as briefly discussed in section 1, Raman spectroscopy on these systems should be able to give further information provided that sampIes can be prepared in which both Pt and Pd are found in valence II as well as valence IV states at random. For example, in ref. [7] they study the periodic compound [Pt”(en,)Pd’v(en)ZXZ] (ClO&. This compound could be possibly prepared in a “random’* version which would read: {[Pt”‘(en)~Pt’V(en)lX,],[Pt”(en),Pd’v(en)lX2],. [Pd”(en)2Pd’v(en)lX2]~ [Pd”(en),Pt”(en),X,T1,E
(CIO,),,
where _Yf y + r + nr = 1. We are not aware of any experiments with such “random” samples. In this version, the random sequence of Pt and Pd on the chain wouId introduce IittIe eIectronic randomness, since it can maintain its valence
lattice modes oJPt and Pd
periodicity, its unit cell being of the form M”(en),M”V(en), X,, where MM’ = Pt or Pd. On the other hand, the compound would have.a considerable lattice disorder since the mass ratio of the vibrating Pt or Pd structures is around 2 : 1 (a case of moderate disorder). In that case the lattice vibrations would consist of independent modes which would be the same as those of the ordered chain, and collective longitudinal modes which would be of the “localized phonon” type. The Raman spectra of such compounds should contain the same distinguishable frequencies as those of their ordered versions, corresponding to the independent modes superimposed over a much smoother background, which is a measure of the collective vibrational states [9, lo]. The information we can obtain this way is twofold. First we can distinguish between the independent and the collective modes of those compounds, something not possible in their ordered version, and second we can obtain the density of their longitudinal states, which, as established for other moderately disordered systems [ 1 I], is remarkably similar to the corresponding density of their ordered version. These extra pieces of information are necessary for a complete study of the Iattice vibrations of all such compounds, and an experimental confirmation of the above predictions is, we believe, of considerable interest.
Ackninowledgement We wish to thank Dr. G.C. Papavassiliou for supplying us with his results prior to publication. We also wish-to thank Dr. A. Theophilou and Professor E. Anastassakis for helpful discussions. One of us (C.E.P.) wishes to acknowledge the hospitality of the theoretical physics group of N.R.C. Democritos, where a large part of this work was done.
References rll -- M.B. Robin and P. Day, Advan. Inorg. Chem. Radiochem. 10 (1967) 247. [2] L.V. Interrante, K.W. Browall and F.P. Bundy, Inorg. Chem. 13 (1974) 1158.
i
C-E. PdraskeGaidis,C.
PapatrianrofilloirlLongiflrdinnl
[3] C.E. Paraskevaidis and C. Papatriantafillou, Chem. Phys. Letters 58 (1978) 301. [4] C.E. Paraskevaidis, C. Papatriantatillou-and G.C. Papavassiliou, Chem. Phys. 37 (1979) 389. [S] R.E. Borland, Proc. Phys. Sot. 78 (1961) 926; Proc: Roy. Sot. A 274 (1963) 529. (61 R.J.H. Clark and W.R. Trumble, Inorg. Chem.‘lS (1976) 1030. [7] G.C. Papavassiliou, D. Layek and Theophanides. submitted to J. Raman Spectry.
j
lattice
modes ofPr atid Pd
399
.. -181 R.J.H. Clark, ML. Frank and W-R1 Trumble. Chem. Phys. Letters 41 (1976) 287. [P] J.E. Smith Jr., M.H. Brodsky, B.L. Crowder, M-1. Nathan and A. Pinczuk, Phys. Rev. LeTte?s-26 (1971) 642. [IO] F-L. Galeener and G. Lucovsky, Phys. Rev. Letters 37 (1976) 1474. -. [ 1 l] R. Shuker and R.W. Gainmon, Phys. Rev. Letters 25 (1970) 222.
_:
._.
‘. :: .. ,:
..”
... :
.:LONGITUDINAL OF PLATINUM
..
:..
. LATTICE MODES OF MIXED VALENCE-MIXED AND PALLADIWM
METAL COMPOUNDi
..
C.E. PARASI&VAIDIS* Physics Laboratory
A, National
Technical Universiry,
At&s,
Greece
and’ C. PAPATRIANTAFILLOU N.R.C. Democriros,
Received
Aghia Parask&
Arcikis, Greece
20 July 1979
Mixe&vaIence-mixed metal compounds of platintim and palladium are complicated ionic crystals containing independent parallel chains, and a linear unit cell of four masses containing both Pt and Pd -each-in a given state of oxidation II or IV. The natural vibrations of such systems contain longitudinal lattice modes and almost independent local modes. Their Raman spectra contain indistinguishably the frequencies of both kinds of modes. A study of these vibrations requires some method of distinguishing unambiguously between these two kinds of modes, not presently available. We present here a detailed study of the longitudinal lattice modes. Our study develops criteria for distinguishing with some certainty between the twa kinds of modes and we assip accordin_ely their observed Raman frequencies. Furthermore, we _ propose distinction by performing Raman spectroscopy _ a method for their unambiguous on a disordered version of these compounds.
1. Introduction Mixed-valence compounds have been the subject ofextensive but not all systematic study [I]_ These compounds contain the same element (or elements on the same column of the periodic table) in two different states of oxidation. They are classified by Robin and Day [I] in three classes. In class I systems the ions of differing valence are placed on sit& of very different,symmetry and ligand field strength, resulting in extreme * Most of this work was conducted at N.R.C. Democritos under a grant of the National Hellenic Research Foundation @I.E.).
localization
and in of those of their constituent parts, while in class III systems no distinction can be made between the ions of different valence, since they are placed on sites of equivalent symmetry. Class II systems are an intermediate case, where although their valence electron wavefunctions are delocalized, the sites of the ions of different valence are distinguishable, and the properties of these systems usually differ from the sum of the properties. of [email protected] parts. For example ‘. they show intense absorption in the visible iegiqn of the spectrum.which‘is not present in compounds consisting of only one of the two valence states. electronic
of the valence shell electrons
properties
which
are the sum
.
C.E. Paraskecaidis,
394
Mixed-valence palladium A striking
compounds
C. Papnrrianra~lbujLongitudinaI
of platinum
and
are class II mixed-valence compounds. feature of these compounds is that their
crystal structure contains chains of alternating octahedrally coordinated M(IV) and square-planar coordinated M’(U) ions, where M, M’ are either Pt or Pd. The II and IV coordinated ions are at the center of structures that are formed off the chain. A halogen ion is placed on the chain between the IV and II sites, lying closer to the quadrivalent ion. These compounds cf Pt and Pd are diamagnetic and exhibit highly anisotropic solid state behavior. They are semiconductors with room temperature conductivity along the chain at least 300 times larger than that perpendicular to the chain [2]_ The lattice vibrations of these compounds exhibit a similar anisotropic behavior and the collective excitations of the lattice are basically longitudinal (along the chain) modes, while the independent modes, being localized_ are insignificantly affected by this anisotropy
131. The lattice vibrations have been studied from the respective infrared (IR) and resonance Raman (RR) spectra. Such experiments are the only experimental studies of the lattice vibrations of these compounds and they give limited information about them. Important quantities such as the density of vibrational states (or their dispersion relations equivalently) have been deduced for those of the compounds that have the same metal ion (Pt or Pd) on the chain [4] from their respective IR and RR spectra, but a direct experimental confirmation has yet to be reported. In the present work we present a study of the longitudinal lattice modes of those compounds that have both Pt and Pd along the chain, each with different valence (i.e. Pt(IV), Pd(II) or vice versa) and in section 2 we present the formalism for such a study. The RR and IR spectra of these compounds .%.I
22 .
Q
9r :
A2
~__c;-___~_-_~,--L-_-‘~ r--
----
02 .
P d.? d
UNlr
CELL
Fig I. Unit cell stracture. Inter-ion distances d, c dl. Masses: mgl = mg2 mass of halogen ion (Cl or Br), rrt,, = mass of the complex ion with Pt (or Pd) IV at its center, nlAA2 = mass of the complex ion with Pd (or Pt) II at its center, ,I**, f )>td2.
locrice modes ofPc and Pd
are harder to interpret because their unit cell consists of four masses having three different values (see fig. 1). The interest of these mixed valencemixed metal ion compounds, beyond the present study of their ordered version, lies in the possibility they present to be prepared in a form where the Pt or Pd have both valences II and IV at random. In that case the lattice is disordered and all its vibrations are localized [5J. Then, the density of vibrational states can be deduced, and the independent modes distinguished from the collective ones, by analyzing their Raman spectra. The above are discussed to a larger extent, along with the results of the present study, in the concluding section. 2. The one-dimensional model We consider an infinitely long linear chain which consists of three types of masses. The mass of the complex ion with the octahedrally coordinated (IV) Pt (or Pd) ion at its center, mxI, the mass of the complex ion with the square-planar coordinated (II) Pd (or Pt) ion at its center, mAz, and the mass of the halogen ion, ma, p laced between the octahedrally and square-planar coordinated ions [I]. (See fig. 1.) The halogen ions are placed at a distance d, from the Pt (or Pd) IV ions, and at a distance dZ (>d,) from the Pd (or Pt) II ions. Two effective spring constantsf, andf, are introduced, corresponding to d, and d,, respectively (i.e_f, connects the octahedrally coordinated ion with the two halogen ions and _f2 connects the square-planar coordinated ion with the two halogen ions). The unit cell consists of four masses and two different spring constants. Its length is 2(d, + d,). Since the tinit cell consists of four masses arranged one-dimensionally, the equation of motion will give four longitudinal lattice modes - one acoustical and three optical. The equation for the normal modes is
Dk = wz(fI +ft
- w’m~
C.E. Parurkecnidis,
For convenience tion:
we introduce
the following
395
lattice modes OJPJ and Pd
C. Papatriantafill~u/LongitudiM~
where
nota-
A0 = v’(Y - 1 - 28), y = 0.? = (27#,
(2)
g =fi
+_L,
(3)
_fi =
af,. -
(4)
mB= @hlr
(5)
m,, = vh,,
(6)
and eq. (1) becomes: D, = y(g - may) yz - + 1 +
g2 --gj
*u
+
B
‘I +
(
1 ++x 1 + 28 ‘J LX+1 >
VW I
(29)’ -
a2
3
sin2k(d,
(l+b2q
AZ = y’(y’ -
+ d,) = 0.
+ 1)1+
.&I + 1 + 2h)},
1 - 2Bq),
0 2
<
’ < pil
’ uw
rl +
28~)
2 + 8h + 1)
VI
(7) and
1+2!k< rlfl
, I
(9)
y1 = s/f+, ,,+[1+2/3(+$3]
33~1 < q+l
(8)
’
0!z2<*+?p
forq
> 1;
(164
vz z c 1 + 2,f311, for 11C 1, 0<
(16b)
- >
VI
where yi = (Zxv,)‘, and we have taken into account the fact that y&i < p2Jy1 in assigning vJvr and vJvl to relations (15) and (16) respectively. These relations, (15) and (16) along with eq. (12) have to be obeyed in order to have consistent results for vi, v2, vj and a Since eq. (13) is quadratic in a, the two real and positive solutions are a_ and a+ (a_ G a+). The physically correct choice for CLis LX-. (See discussion in ref. [4].) Having these considerations in mind, inequalities ( 15) and (16) become:
111 >I'
x 1+*-(I y3 +-[I +w(g3] l/2 w(l+~+w9) x[I)I ( [
(
8pa( 1 +-9 + 2pq) +a)‘[1 +2/9[(1 fricz)/(l +cC)]}2
(10)
l-(l+Cr)2{l+2j3[(~+~)M1ia)1~Z
_ (11)
The k z 0 frequencies vi, v2, vj obey, according
of the three optical modes to eqs. (9)-(11) the relations
tiz + .i&r
+ IraM
= 1 + 28Ul
+ CI)].
(12)
If the three k 2 0 optical mode frequencies vlr v2, v3 are considered known, one can obtain a quadratic equation for a by substituting eq. (9) into eqs. (IO) or (11). A2u2 + Ala
(14b)
and */ is either (yJyJ or (y&r). Since the two effective spring constantsf, andf, are real and positive, a ( =f2/fi) has to be real and positive. Hence we seek the conditions under which eq. (13) gives a real and positive. The conditions for both solutions of eq. (13) for a to be real and positive reduce to the following two inequalities for the three k z 0 longitudinal lattice mode frequencies: o
The spectra for these compounds are obtained by optical methods. We are therefore interested in the k = 0 modes. These are given by the k = 0 solutions of eq. (7), which are: YA =
A, = 2WCy’ - 1 -KS
(14a)
+ A0 = 0,
(13)
4@1(1 + Pq)(l - V) zz 2s(l + Bs) + (1 + 2Pl)(l - PI) < 0 VI < 8(V + 1 +2/G)
2 +I%+
1) ’
for ‘7 < 1.
7
z
(174
(17b)
C.E. Paraskecaidis, C. PuparriantajilioujLongitudinal lattice modes of Pt and Pd
396
2p_
i
for rl > 1.
(l8b)
The conditions for only one of the two real solutions to be positive are met if the following inequalities for y’ are satisfied: 1 tZj?
1+2fltr,
1 + 28Q < 7’ i
1 + 28,
forq>
1;
(19a)
for ~7< 1.
(19b)
If the values of the masses mA, and Q, close, i.e. (g - 1) < 1, then the inequalities (18) become approximately
o<($)2a(1 +F),
are very (17) and
for rl > 1;
octahedrallv coordinated constituent molecule vibrates independently. This is consistent with eq. (23). It gives the ratio of the squares of the two optical frequencies of a system with three masses connected linearly with spring constants of the same;:: value, where the two masses at the end have the same value, which is j3 times the value of the mass in the middle. Therefore, the inequalities that the three k 1 0 optical longitudinal lattice modes vr , .v2,vg have to obey are the inequalities (17) and (18). These along with eq. (12) show the consistent way in which vrr v2, vs have to be chosen from the experimental resonance Raman and infrared spectra.
(2Oa) 3. Results and discussion (2Ob) and 1+0(1-~)<(r;i)&?P, for iI > I: (21a) 1
+/fl _!+)<(!2)2<
1+2PC1
-(l-d.
fort1 c 1 (21b) and inequalities 1+2~<~‘<1
(19) become t.2#?[1
+ (rf - I)],
for Jj > 1;
- (1 - t7)] c y’ < I + 28,
for 4 c 1.
G=) 1 + 2fl[l
(226) Note that in the limit ofq = I relations (20) and (21) reduce to relations (39) of ref. [4]. Special mention has to be made in this limit for the case where only one of the two solutions is positive. In this case the positive solution of eq. (15) goes to zero (IL--f 0) and eqs. (19) [or (22)] reduce to the limiting condition f = 1 t2j.X
(23)
The fact that CC--f 0 implies that the effective spring constant that is responsible for the connection between the octahedrally coordinated constituent molecule and the square-planar one goes to zero: (f2 4 0). Therefore, this is the case where the
We applied our model, discussed in section 2, to the experimental optical (resonance Raman and infrared) spectra of four mixed metal-mixed valence compounds of Pt and Pd [6,7]. Their mass distribution within the unit cell as well as the inequalities the three longitudinal k z 0 lattice mode frequencies have to obey [eqs. (17) and (IS)] are given in table 1. The k z 0 optical frequency that is readily observed from these experimental spectra is vr _ The system vibrates with frequency vr when the masses of the two halogen ions (tna) on the chain in the unit cell vibrate against each other with displacements that are equal in magnitude and opposite in direction, while the heavier ions on the chain (rnA,, mA2)remain motionless [4]. Once this frequency, vIr is known we determine the sum of the two effective spring constants g = f, -i-Jr from eq. (9) as well as the inequalities (17) and (18) that the other two k s 0 longitudinal lattice mode frequencies (vZ, vs) have to obey. These results are presented in table 2. The dispersion relations for these compounds can be calculated from eq. (.l) when the two effective spring constants Jr andfi are determined [4]. The frequencies of the independent modes of these compounds remain practica!ly the same as those of the corresponding modes of the free (not in the chain) constituent molecules, while collective longitudinal modes that can be associated with modes of the free molecules (not in the chain) may
t fi
323 196 325 315
“I
< < < <
v3 v3 “3 vj
c < < c
359.14 240.6 359.69 348.62
14.60095 12.11917 14.78232 13.88664
< < < c
344.66 223.04 345.61 335.44
v1 v2 vz v2
for Y,, v2, v3
and the
x lo7 x IO7 x IO7 ~‘10~
OC(P~/V,)~ <0.1386293 0<(1~,/~,)~<0.294926G O<(~,/v,)~
Inequalilies
f, + (2
< < < <
1.4196473 1.3914248 1.3914248 0.7186878
‘1
v1
0.\181399 0.2534347 0.1124298 0.1564375
II
is v’ = c/J, where c is tlic speed of ligllt and I is the wavelength,
Cl Br Cl Cl
B,
for v,, v2, YJ
120.26 106.44 117.55 115.31
Pd+2E+2CI Pd+IF Pd+4F Pt+4F
A,
0 0 0 84.76
“3
Cl Br Cl Cl
BI
vi is given in cm-‘,f, in amu~cm2. v is the wavenumber I/k The actual frequency actual spring constant isf = c2f(amu/s2).
I 2 3 4
Compound label
Table 2 VI, ~2, VJ andf,
E = NH,,
F = CH,NII,.
Pt+2E+2CI Ptl-4F Ptt4F Pd+4F
1
2 3 4
A,
in n unit cell and the inequalities
Compound lube1
Table 1 Mass distribution
1.138625)3+,/v,)‘< 1.2949266+,/v,)‘< I.1308321 <(v~/v,)~c l.1340069c(v,/v,)2c
~ 1.2362698 1.5068694 1.2248596 1.2248596 -
9
2
%
$
g 2:
5 6 ri: x f
j
2 9 F;
a
r, b b 9 e P B E -. *;
398
C.E. Paraskevnidis. C. PopatriuntajifloujLongit~dinaI
have quite different frequencies [3,4]_ When the spectra of both constituent free molecules are available along with those of the mixed valence compound, the above observation enables one to distinguish the independent modes as those that practically have the same frequency in both spectra. In order to illustrate this we take an example from the RR spectra ofcompound 1 and its constituent free molecules [6]_ For the constituent tranr[Pt’v(NH,),CI,] we take the two observed \(Pt-Cl) frequencies, 327 cm- ’ and 343 cm-‘. For the compound [Pd’t(NH~)2C12Pt’V(NHJ2Cl~] these frequencies become 323 cm-’ and 344 cm-‘, respectively_ This indicates that the first frequency corresponds to a Iongitudinal lattice mode, while the second one to an independent transverse mode. This is more clearly illustrated in the case of Wolffram’s red [IS] (with Pt in both oxidation states) where the frequency of such a mode changes from 344 cm-’ in* the constituent free molecule to 316 cm-’ in the compound. (See discussion in ref. [4].) The preceding discussion of our results showed that the 1R and RR spectra of these compounds do not always give adequate information for their lattice vibrations, and thererare the assignment of the various observed frequencies to independent modes or collective longitudinal ones, and the resulting values for the spring constants, is not unambiguous. However, as briefly discussed in section 1, Raman spectroscopy on these systems should be able to give further information provided that sampIes can be prepared in which both Pt and Pd are found in valence II as well as valence IV states at random. For example, in ref. [7] they study the periodic compound [Pt”(en,)Pd’v(en)ZXZ] (ClO&. This compound could be possibly prepared in a “random’* version which would read: {[Pt”‘(en)~Pt’V(en)lX,],[Pt”(en),Pd’v(en)lX2],. [Pd”(en)2Pd’v(en)lX2]~ [Pd”(en),Pt”(en),X,T1,E
(CIO,),,
where _Yf y + r + nr = 1. We are not aware of any experiments with such “random” samples. In this version, the random sequence of Pt and Pd on the chain wouId introduce IittIe eIectronic randomness, since it can maintain its valence
lattice modes oJPt and Pd
periodicity, its unit cell being of the form M”(en),M”V(en), X,, where MM’ = Pt or Pd. On the other hand, the compound would have.a considerable lattice disorder since the mass ratio of the vibrating Pt or Pd structures is around 2 : 1 (a case of moderate disorder). In that case the lattice vibrations would consist of independent modes which would be the same as those of the ordered chain, and collective longitudinal modes which would be of the “localized phonon” type. The Raman spectra of such compounds should contain the same distinguishable frequencies as those of their ordered versions, corresponding to the independent modes superimposed over a much smoother background, which is a measure of the collective vibrational states [9, lo]. The information we can obtain this way is twofold. First we can distinguish between the independent and the collective modes of those compounds, something not possible in their ordered version, and second we can obtain the density of their longitudinal states, which, as established for other moderately disordered systems [ 1 I], is remarkably similar to the corresponding density of their ordered version. These extra pieces of information are necessary for a complete study of the Iattice vibrations of all such compounds, and an experimental confirmation of the above predictions is, we believe, of considerable interest.
Ackninowledgement We wish to thank Dr. G.C. Papavassiliou for supplying us with his results prior to publication. We also wish-to thank Dr. A. Theophilou and Professor E. Anastassakis for helpful discussions. One of us (C.E.P.) wishes to acknowledge the hospitality of the theoretical physics group of N.R.C. Democritos, where a large part of this work was done.
References rll -- M.B. Robin and P. Day, Advan. Inorg. Chem. Radiochem. 10 (1967) 247. [2] L.V. Interrante, K.W. Browall and F.P. Bundy, Inorg. Chem. 13 (1974) 1158.
i
C-E. PdraskeGaidis,C.
PapatrianrofilloirlLongiflrdinnl
[3] C.E. Paraskevaidis and C. Papatriantafillou, Chem. Phys. Letters 58 (1978) 301. [4] C.E. Paraskevaidis, C. Papatriantatillou-and G.C. Papavassiliou, Chem. Phys. 37 (1979) 389. [S] R.E. Borland, Proc. Phys. Sot. 78 (1961) 926; Proc: Roy. Sot. A 274 (1963) 529. (61 R.J.H. Clark and W.R. Trumble, Inorg. Chem.‘lS (1976) 1030. [7] G.C. Papavassiliou, D. Layek and Theophanides. submitted to J. Raman Spectry.
j
lattice
modes ofPr atid Pd
399
.. -181 R.J.H. Clark, ML. Frank and W-R1 Trumble. Chem. Phys. Letters 41 (1976) 287. [P] J.E. Smith Jr., M.H. Brodsky, B.L. Crowder, M-1. Nathan and A. Pinczuk, Phys. Rev. LeTte?s-26 (1971) 642. [IO] F-L. Galeener and G. Lucovsky, Phys. Rev. Letters 37 (1976) 1474. -. [ 1 l] R. Shuker and R.W. Gainmon, Phys. Rev. Letters 25 (1970) 222.