The low frequency Raman spectrum in relation to the phase transition of KH3(SeO3)2

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The low frequency Raman spectrum in relation to the phase transition of KH3(SeO3)2

M. Krauzman, R.M. Pick

To cite this version:

M. Krauzman, R.M. Pick. The low frequency Raman spectrum in relation to the phase transition of KH3(SeO3)2. Journal de Physique, 1980, 41 (12), pp.1441-1446.

�10.1051/jphys:0198000410120144100�. �jpa-00208970�

The low frequency ^Raman spectrum ^{in relation to the} phase ^transition

of KH3(SeO3)2

M. Krauzman and R. M. Pick

Département ^de Recherches Physiques (*), ^Tour 22, Université P.-et-M.-Curie, 75230 Paris Cedex 05, France

(Reçu ^{le 28 mai} 1980, accepté le 22 août 1980)

Résumé.

Le spectre ^{Raman de basse} fréquence ^{de KTS a} été mesuré et analysé au-dessus de Tc ^{à 1 bar et 8 kbars,}

dans la géométrie B3g correspondant ^{à la} déformation statique ^{de basse} température. ^Les spectres ^{montrent claire-}

ment l’existence d’un mode de proton au-dessous de 50 cm-1 1 et l’analyse ^établit que les effets quantiques ^{et le} couplage ^des protons ^{avec un mode} optique ^{vers 40 cm-1 ne} jouent guère ^{de rôle dans le mécanisme de la transi-}

tion. En revanche, il s’avère que l’interaction entre les protons ^{est de} type classique, que ^ces derniers ont une

dynamique relaxationnelle (avec ^{un temps de} relaxation 03C4

(27 cm-1)-1) ^et que leur couplage ^{avec la} déformation

B3g ^{est très} important : ^{toutes ces} propriétés soulignent ^la grande différence entre KH2PO4 ^{et KTS} malgré ^la grande sensibilité, ^{à la} deutération, ^{de leurs} températures critiques Tc.

Abstract. 2014 The low frequency ^Raman spectrum ^of KTS has been recorded and analysed ^above Tc ^{at 1 bar and}

8 kbars, ^{in the} B3g ^{geometry which} corresponds ^{to the low} temperature static deformation. The spectra clearly

show the existence of a proton mode below 50 cm-1; ^the analysis shows that quantum effects and the proton coupling ^{with an} optic ^{mode around 40 cm-1} play ^minor roles in the transition mechanism. On the contrary, the interaction between the protons ^{turns out to be} classical in nature, their dynamics ^{is of the} relaxation type

(with ^a relaxation time 03C4

(27 cm-1)-1) ^{and their} coupling ^{with the} B3g deformation is _very important : ^{all those}

effects stress the large difference between KH2PO4 ^{and KTS in} spite ^{of the} large sensitivity, ^to deuteration, ^{of their} critical temperature Tc.

Classification Physics ^Abstracts

63.50

78.30

61.50K - 63.20H

1. Introduction.

Potassium trihydrogen ^selenite

(KTS) belongs ^{to the} large ^{class of} crystals ^{where an} hydrogen bond is essential in a phase ^transition (a ^{survey is} given ⁱⁿ [1]).

This is evidenced by ^two observations :

- first, ^the phase transition occurs at a tempera-

ture Tr (213 ^{K for} KTS) ^{which is} very sensitive to deuteration (Tc

^{287 K} for the deuterated KTS) ; - secondly ^the importance ^{of the} hydrogen ^is

understood from structural considerations : at a tem-

perature ^{T above} T,,, ^KTS belongs ^{to the space}

group Dà: ^{(Pbcn) [2] with Z}

⁴ formula units.

The point group D2h is of order 8 so that the _{12 pro-} tons and the 24 oxygens ^are displayed ^{into two} types of hydrogen ^bonds [2, 3] (Fig. 1). ^{In one} type ^contain- ing ^{8 bonds} (01H1O3;

^2.60 À), ^the hydrogen ^{is in a} simple general position ^{close to} oxygen 03 ^while

in the other type (02H202

^2.567 Â), ^the large length ^{of the 4} remaining ^bonds requiring ^{also a double}

well potential [4] conflicts with the space group requir-

(*) ^{L.A. 71.}

Fig. ^1.

Projection ^{on the ac} plane ^{of the} T> Tc’ D2b ^structure

of KH3(Se03)2 (from ^Ref. [3]). ^{The K atoms are not} represented.

They ^{are located on} the twofold b axis passing through ^{the middle}

of the 0 2 H 2 0 2 bonds. The disordered H2 ^{atoms are} also omitted.

ing ^the hydrogen ^to be in the middle of the bond. As a

result, ^the _proton is disordered and statistically

distributed between two symmetrically ^related posi-

tions along ^{the bond. Below} Tc, ^{all the} H2 protons order [5]. ^The corresponding lowering ^{of the} sym-

metry (C2’h, P21/b’ ^Z

^{4) [5] allows a distortion} belonging ^{to the} B31 ^(q

^{0) symmetry} representation

of the D2h group. The growing (for decreasing ^tem-

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:0198000410120144100

1442

peratures) ^{of a} spontaneous ^{shear strain s,} [6] ^{and the} vanishing (at ^T

T) ^{of the related} C44 ^elastic

constant are observed [7, 8]. ^Other published ^investi- gations ^on dielectric [9, 101, birefringence [10], ^thermal expansion [11, 12], specific ^heat [11], ^{sound atte-}

nuation [13] properties ^are consistent with the above arguments.

In some respects, ^the situation ^is comparable ^{to the}

case of KH2po4 (KDP), ^{the first} crystal of this class which has been widely studied and understood.

However, the latter is actually characterized by ^two aspects :

-

First, ^{the T} > Tc paraelectric phase ^belongs

to the space group Dl’ (I42d) ^and transforms into

a ferroelectric phase ^of symmetry C29 ^{(Fdd2) when}

T Tc, allowing ^{for a} spontaneous electrical pola-

rization P. along ^the C2 ^axis (which ^was originally

the high temperature S4 ^{axis). The} large sensitivity

of Tc ^on deuteration (Tc

^{122 K to} Tc

²²⁰ K)

establishes here too that the ordering ^{of the} protons is responsible ^{for the} transition, ^{but as all the} hydro-

gen bonds are almost perpendicular ^{to the} preserved C2 ^{axis, the} strength ^of P. and other ferroelectric

properties ^cannot be accounted for by ^the proton motion alone. An intrinsic coupling ^{between the}

proton ordering ^{and the lattice atoms} (i.e. optic phonons) ^does necessary exist.

-

Secondly, ^{there are} four 0-H-0 bonds (disor-

dered at T > Te) ^{connected to each} (P04)- - - ^ion.

In order to balance the electrical neutrality ^{of each} (K-P04)- - group, ^a jump ^{of a} proton ^{from one site}

to the other along ^the bond, forces the jump ^{of other}

protons. As the shortest topological ^closed path along hydrogen ^{bonds connects 6} P04 ions, ^there

exists strong ^non local correlations in the proton motions.

The properties ^{of KTS are much} simpler ^{in these}

two respects :

-

First, ^{none of the two} phases ^is ferroelectric

(they possess ^an inversion centre) ^{and the} proton-

optic phonon coupling ^{may be weak.}

-

Secondly, ^each 02H202 disordered bond is included in a single KH3(Se03)2 group and no corre-

lation between the proton motions is needed to maintain the electrical neutrality.

On the other hand, Makita et al. [7] studying C44(T ) ^above Tc ^{concluded that the} primary ^transition parameter (that ^we assign ^{to be the} proton ordering)

is strongly coupled ^{to the} acoustic branch and would

induce, ^if uncoupled, ^a phase transition at a tempe-

rature To

Tc - ^{137 K.} This - 137 value has to be taken only ^{as a} rough estimate, ^as they neglect ^all

the thermal anharmonicities arising ^{in the} vicinity

of the melting point and also because it ^seems to exist some internal unconsistencies in the numbers

quoted by ^{these authors} [7, 8].

2. Experiments.

2. 1 EXPERIMENTAL SETTING.

We have measured the Raman bc tensor element

(the only ^one belonging ^{to the} B3g representation)

from T = 4 K to T

332 K at room pressure and from T

80 K to T

298 K at P

8 kbars. Deute- rated KTS Raman spectra have also been recorded from T

4 K to T

293 K at room pressure.

We used a Coderg ^T800 triple spectrometer ^with

1.5 cm-1 slit width. An _argon laser 5 145 A line was

focussed on the sample ^{with a} power of 0.2 W or

unfocussed with a power of 0.4 W. The _pressure cell is approximately ^{a cube of 11.5 cm} side made of

Maraging steel. The pressure fluid (helium) ^{was sent} through Maraging ^steel capillary pipes. ^{The cell}

was thermally ^{insulated inside a} polystiren ^{box with} plexiglass windows and cooled by liquid nitrogen flowing through ^two copper coils in close contact with two opposite faces of the cell. The thermocouple

remains at room pressure and measures the tempe-

rature inside an hole drilled in the cell. Four sapphire

windows were mounted, allowing ^a f /4.5 ^aperture

for a light ^{beam of 6 mm in diameter.} Assuming ^that

a set of a plexiglass ^window plus ^a sapphire ^window

acts as an elliptical polarizer, ^{we used the} following

method to get rid of this effect : birefringent ^mica

blades were placed ^{out of the} cryostat, producing ^an elliptical polarization opposite ^{to that} of the windows.

By ^a trial and error methpd ^{on selecting the thick-}

ness and orientation of the mica blades, ^the proper linear polarizations ^{on the} sample ^was achieved,

both for the entering ^laser light ^{and for the scattered} light ^with depolarization ratios of the order of 0.05.

This number is the depolarization ratio checked

on a few internal vibrations selected according ^to

their selection rules.

The area under a KTS mode located approximately

at 60 cm-1 ^was used as an internal monitor to cali- brate the intensities of all spectra. ^The temperature dependence of the latter (expressed ⁱⁿ squared ^arbi- trary amplitude ^units (a.u.)2) ^{can be determined}

assuming this mode follows a (n ⁺ 1) Bose-Einstein law.

2.2 RESULTS.

Figure ² shows the effect of tem-

perature ^above Tc ^{on the low} frequency spectrum in the a(bc) b geometry ^{at room} pressure and under the highest pressure (P

⁸ kbars) ^we could reach.

This geometry ^involves light scattering through ^the only ^{Raman tensor} component (bc) showing ^{a low} frequency wing growing continously ^from T Tc

to reach a maximum intensity ^{at T}

Tc. ^This wing

interacts clearly ^{with an} optic phonon ^{at m - 40} cm-1,

but the spectrum ^below 35 cm-1 is always ^mono- tonic, ^{i. e. no} underdamped soft mode is observed,

even at P

8 kbars. Figure 3 shows that no quali-

tative change ^{results from} deuteration.

The absence of a marked pressure effect (except

for an approximate - ⁵ K/kbar ^{shift of} Tc) ^{is in}

clear contrast with KDP where at P

6.5 kbars,

a low frequency ^soft wing (coupled ^{to an} optic ^mode

at co - 170 cm-1) ^becomes underdamped ^when

T > 7c ^{+ 50 K} [14]. ^{In order to} go deeper ^{into the}

comparison ^{of the} properties ^{of the two} crystals,

we recall in the next section the theory of the ferro- electric properties ^{of KDP.}

3. Phase transition of KDP. - The experiments ^on

KDP support ^{a series} of theoretical models which

can be summarized as follows :

i) ^A pseudospin S is attached to each proton belonging ^{to an} 0-H--0 bond.

ii) ^The pseudospin part ^of the energy is approxi-

mated by [15]

where 2 Q is the _energy splitting ^{of the two} tunnelling

states of a proton ^{in a} double well potential ^{at site} i, and Jij describes the interaction between protons ^at sites i and j.

iii) The interaction of the proton ^or pseudospin

system with the lattice (essentially ^{via an} optic phonon)

is introduced and the total hamiltonian is solved

thermodynamically and/or dynamically [16] ^{or alter-} natively, ^the pseudospin ^{system described} by equa- tion (1) is first solved [17] ^{and then} coupled ^{to an} optic phonon [18, 19]. ^Both ways give basically ^the

same results : for instance when equation (1) ^alone

is solved in a molecular field approximation ^[17], sz > decreases when the temperature ^increases and vanishes for T >, Tp. Spin fluctuations occur at a

well defined frequency (Dp which vanishes ^at T

Tp

and for q

^{0 :}

where

This collective proton mode is then coupled directly

to a Raman active optic ^{mode at} coo [14], ^and directly (or undirectly through ^this optic mode) ^{to an acoustic}

mode [20]. This leads to two temperature dependent optic ^modes cô (T) ^and roÕ(T) ^{and a} temperature dependent ^sound velocity v(T). ^One easily shows, by solving ^for v(Tc)

^{0 and} wÕ(T_)

^{0 that}

so that the phase transition takes place ^at Tc, ^{with the} vanishing ^{of an elastic constant.} Tc ^is given, ^{in the}

R.P.A. as the solution of

J* > J being the result of the coupling ^{of the} pseudo- spin ^{with the} optic and acoustical modes.

In fact, ^the above model has serious drawbacks :

a) The molecular field treatment is only ^an approxi-

mate solution of equation (1).

b) ^{In a Raman} experiment, light is scattered only by ^the protons ^{and the} optic mode. As far as the former

are concerned, ^an anharmonic contribution to the

proton ^mode dynamics ^{comes from an exact treat-}

1444

ment of (1), ^a second contribution arises from relaxa- tion processes, ^not included in (1) which would exist even in the absence of the tunnelling energy Q.

Finally, ^there also exists anharmonic contributions to the optic ^mode dynamics ^{and those are not} explicit- ly induced in the usual phonon Hamiltonian.

c) The model is physically meaningful provided

the tunnelling energy is restricted to a rather narrow

range i.e. on the one hand Q must be large enough

in order to rise out of the anharmonic part ^{of the} dynamics (underdamping) ; ^{on the other hand,} equa- tion (3) implies ^Q J*/4 in order for a phase ^transi-

tion with proton ordering ^{to take} place.

In KDP, ^{it turns out} that the latter conditions (c)

are satisfied ; ^the anharmonic contributions are always

added phenomenologically (following ^a procedure proposed ⁱⁿ [21]) by the various authors who have discussed either the Raman spectrum (wP ^and rot

coupling) [14], ^or the Brillouin spectrum (with ^the

additional coupling ^{to an acoustic} branch) [20].

We shall here follow the ^same procedure ^{for des-} cribing ^{our Raman} spectra.

4. Analysis.

^The spectra ^show clearly ^{the inter-}

action of a wing-shaped mode, ^{with a} well defined

optic ^{mode at - 40 cm - 1. In order} to extract the characteristics of the proton mode, ^a two-coupled

mode analysis ^{has been tried for T} > Tc ^{at P}

^{1 bar}

and P

8 kbars. Following [14], ^{the Raman inten-} sity ^is given by

where n is the Bose-Einstein factor

R; (i

^1, 2) ^{is the} polarizability derivative related to the mode i and Gij ^are elements of the matrix G defined by

where

pp/cp ^and po/mÉ ^are the static susceptibilities ^{of the} proton ^mode (i

1) ^{and the} optic ^mode (i

2).

A(w) ^{is the} coupling between these two modes. The results of our numerical analysis ^{lead us to take} A(co) ^{as a constant} real number. As a consequence

(cf. [21]) ^the resulting parameters ^are characteristic of the uncoupled modes, i.e. the proton ^{and the} optic ^mode parameters ^are directly ^labelled by ^the

P and 0 indices in equations (5), (6) ^and (7).

On the other hand, ^{for a} given ^{mode, the} pola- rizability derivative R and the strength p are redun- dant and cannot be both determined from the Raman

scattering experiment. A realistic assumption ^{is that}

the Ri ^{do not} depend ^{on T. In order to follow the}

temperature dependence of pP, ^{we assumed} Ri

¹

in all what follows. Similarly, expecting ^a very small T dependence ^{of the} optic ^mode characteristics, ^we arbitrarily ^used po

1. (Any other choice would lead to results equivalent ^{to ours,} provided ^the coupling ^{A includes} the proper factor.)

Under the above assumptions, all fits of equation (4)

to our experiments give ^the following ^{results :} 1) ^The proton ^{mode was found} overdamped. ^This produces well-known difficulties due to strong ^cor- relations in the fitting procedure ^{and leads to} large

uncertainties or divergences mainly in the deter- mination of the parameters pP, cop, TP and d . Examin-

ing carefully ^{the fit} convergences when one or two of these parameters ^were kept fixed, ^{we found that :} 2) ^{The fits are} significantly improved ^when rop is given ^{a value} larger ^{than the} highest ^recorded frequency (50 cm - 1). Best fits are obtained for cop

taking any value above - 100 cm - 1.

As TP > rop > m, the set of equations (4) ^to (7)

with the ro2 term neglected ⁱⁿ equation (6) ^and Ri

^po

^{1 can} be rewritten as

with

where j

In other words, fp > rop > ro implies ^{that the} proton ^mode dynamics is described by the relaxation term gr, ^a point which will be further discussed in the next section.

Equation (8) has been fitted to our Raman spectra

at several temperatures ^above Tc, ^{both at 1 bar and}

8 kbars (where Tc

^171.7 K). ^There still remain strong correlations between ^some parameters, ^i.e.

very small changes ⁱⁿ X2 result from a variation of Li up ^{to a} factor of 2 if all the other parameters ^are allowed to readjust. ^{In order to} follow their variation

versus T, ^{A was assumed to be constant.} The results for the five remaining ^fitted parameters of equa- tions (8)-(12) ^are listed in table I where a.u. stands for arbitrary amplitude unit and the uncertainties

are deduced from the confidence _range given by ^the

fit routine. The r.m.s. relative error on the spectra

was typically 1 % ^{and never above} 1.4 %. ^These

errors do not take into account the uncertainties

on the experimental intensity determination. Let us

note that the (a.u.) is related to the scattering ^inten-

sity ^{of the 60 cm-1} mode. The latter _can, in prin-

ciple, vary upon deuteration : the (a.u.) may thus

have a different magnitude in the lower part ^of

table 1.

Table I.

Parameters taken from equations (8), (9), (10), fitted ^{to the Raman} spectra.

Upper part : ^P

^{8 kbars,} T,,

^{171.7 K, L1}

¹⁰ cm-l/a.u.

Middle part : ^P

¹ bar, Tc

^{213 K, L1}

⁵ cm-l/a.u.

Lower part : deuterated KTS, ^P

1 bar, Tc

^{287 K, L1}

4.6cm-l/a.u.

Finally, inspection ^{of table 1} shows that :

3) ^The optic ^{mode is not or} only weakly tempe-

rature dependent ^at all presures.

4) ^The relaxation time ^r is also rather stable.

5) ^The strength ^or susceptibility pr of the proton mode is the only physical quantity showing ^a signifi-

cant temperature dependence ^{as is shown on} figure ^3.

6) ^The deuteration changes ^{the value of i} by ^a

factor of 2 but does not affect the frequency mo and the linewidth To of ^the optic ^mode.

5. Discussion. - The preceding analysis ^{can be}

looked at from two différent points of view : the

validity ^{of a mean field treatment} of (1) ^{for the} pseudo- spin dynamics, ^{and the} consistency ^{of our} light scattering results with other recent experiments.

5. 1 As far as the first point ^is concerned, ^{our results} show that the proton dynamics ^is always overdamped.

This means that the meanfield approximation ^of (1)

which is sufficient ^to explain ^{the Raman} scattering experiment ^{in KDP, is not} appropriate ^{in our case.}

Due to its success in the former _case, it implies ^that

the intrinsic relaxation terms not included in (1) play ^an important ^{role, so that the} tunnelling ^must

not be considered as the leading ^{term of} (1) ^but

rather that the tunnel splitting ^is always ^{small com-} pared ^{with the} spin ordering energy J. This is in clear contradistinction with the KDP _case, but from

a chemical point ^{of view, is not a} surprise because the 0-H-0 bond in KTS is 2.56 À compared ^{to a 2.49} A

value in KDP. This perfectly agrees with the point

of view of Ichikawa [22] ^who finds that 2.47 A _repre-

sents the limit between a single ^{and a} double well

potential ^{for the} proton : ^{the tunnel} splitting ^{2 Q}

is thus expected ^{to decrease} enormously ^{from KDP}

to KTS as is actually ^found.

Consequently, ^the dynamics ^of (1) ^{is close to that}

of an Ising pseudospin ^{model. No} really satisfactory

treatment of such a model exists, and all the known methods are lead to assume, at ^some point, ^a pseudo- spin relaxation dynamics. ^This is indeed what we

find. Furthermore, ^these pseudospin treatments (cf.

e.g. refs. [23, 24]) show that the static susceptibility

p, should diverge ^{in a mean field} approximation ^as (T - Ts) - 1 ^where Tg ^{is the} pseudospin ordering temperature. ^{A linear} extrapolation ^of P; l(T) ^leads

to T.

^{85 K at 1 bar} (cf. caption ^of Fig. 4).

Fig. ^4.

Intensity Pr of the proton ^{mode versus} temperature. ^The lines represent ^fits of p,

^{A/(T -} Ts) ^to the data of table 1 (full circles) : a) ^P

¹ bar, A

756, Ts

^{85 K ;} b) ^P

⁸ kbars,

A

443, 7,

^{65.5 K.}

1446

5. 2 The preceding result is in line with the analysis

of the soft elastic constant C44 ^made by ^Makita

et al. [7] (cf. ^section 1). Assuming only ^an interaction of this constant with a soft feature, they ^{found that}

this soft feature would freeze around 76 K, ^{and that} it was its strong coupling ^{with the elastic constant}

which pushed 7c towards 213 K. Both results _agree

on this respect, ^{and on} the order of magnitude T. (1).

A recent neutron scattering study ^{of the same} coupled soft feature-acoustic phonon has also been

performed ^on deuterated KTS by Noda et al. [25].

This allowed to study ^{the soft mode at} higher q ^values.

The authors concluded that the soft mode was either real but heavily damped ^or relaxational. From their

experiments, ^{one can estimate a} value of r of - 1 meV

(i.e. ⁸ cm - l) ^{in the} _vicinity ^of _Tc’ ^which agrees with

our own determination (13.5 cm - l) in the deute- rated sample (see end of section 4).

6. Conclusion.

Our light scattering experiments

on KTS conducted at 0 and 8 kbars at various tem-

e) ^{The same} pseudospin ^treatment proposes ^{that T} would also

diverge ^at T.. ^{But, as we are far from} T,, ^{and as} the individual

pseudospin relaxing mechanism is certainly température dependent,’

the two effects _may easily cancel in the limited température range available.

peratures ^above Tc ^{indicate a classical} (and ^{not a} quantum mechanical) behaviour of the proton dyna- mics, ^dominated by ^a relaxation _process.

The coupling of this relaxation mode with an optic phonon ^{of the same} symmetry ^{can be considered}

as real, ^{and is never} _{very strong} even if it affects

markedly ^{the recorded Raman} spectra.

The protons by themselves do not tend to order at Tc ^{but at a much lower} temperature ^{as could be}

already suggested by the elastic constant measure- ments [7] ; ^{the direct} proton-elastic ^constant coupling plays ^an important role in the determination of the actual transition temperature.

All these aspects ^{are in clear} contradistinction with the KDP _case, where the role of tunnelling ^is impor-

tant, ^{as well as the} coupling ^{with an} optical pho-

non [14], ^{while the} coupling ^{of these} quantities ^with

the elastic constants of the same symmetry plays ^a

minor role.

The origin ^{of the} important influence of deute- ration on the transition temperature 7c ^{is thus not}

at all obvious and further studies are necessary ^to elucidate this point.

Acknowledgments.

^We would like to thank Pro- fessor C. T. Walker for _many fruitful discussions and also David Walker for adapting ^{a numerical} fitting program ^{to our} minicomputer.

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