Proton mobility in solids. I. Hydrogenic vibration modes and proton

PROTON MOBILITY IN SOLIDS

cule reactions ocmr a t all in the liquid, the sequence of reactions 22, 23, and 24 would be more likely than reactions 25 and 26. I n this study, a solution of 0.43 mole % CaHs in the liquid methane-& was irradiated, and the product ethane was analyzed. The results were: G(C2De) = 1.47; G(C2DsH) = 0.154; G(CzHzD4) = 0.015; G(CH3CD3) = 0.018; G(C2HsD) = 0.037; G(C2Hs) = 0.017. A statistical production of CD2H. from the small CD3H impurity in the CD4 would lead to G(CzD5H) = 0.084, if all the ethane comes from methyl radicals. It can be concluded that the ion-

1097

molecule reactions 24 and 26 make a t most a minor contribution in the liquid phase. Of course, nothing can be concluded from the data about the occurrence of reactions 22,23, and 25. In this experiment, there were appreciable yields of isobutane and 2,3-dimethylbutane, indicating that some of the hydrogen atoms had abstracted from propane. A calculation shows that this is in accord with log A = 13.0 - 6300/2.3RT,43 with A in cc mole-' sec-l. (43) I(. Yang, J . Am. Chem. Soc., 84, 719 (1962).

Proton Mobility in Solids. I. Hydrogenic Vibration Modes and Proton Delocalization in Boehmite

by J. J. Fripiat,l* H.Boamans, and P. G. Rouxhetlb Laboratoire de Physieo-Chimk Minkrale, I n s t h t Agronomique, Univerdv of Louvain, Heverlee-Louvain,Belgium (Received November 7, 1966)

When boehmite is heated progressively to elevated temperatures, a continuous decrease in the intensity of the fundamental OH vibration bands is observed. The effect is more pronounced for the deformation and torsion modes than for the stretching modes, but it is perfectly reversible on cooling when the temperature has not exceeded 400". To explain these observations, a proton delocalization process was postulated. In terms of this process, the observations may be explained by the disappearance of discrete vibrational energy levels above a specified level where the probability of the proton tunneling through the potential energy barrier is close to the absorption frequency. This interpretation is in agreement with the large band widths and with their increase upon heating. The frequency shifts accompanying the loss of intensity are due to the thermal expansion of the crystal unit cell. A complete assignment of the hydrogenic modes has been attempted in relation to the hydrogenic structure proposed by Wickersheim and Korpi.

I. Introduction Upon heating minerals which contain constitutional hydroxyls, such as micas, a progressive loss in intensity of the OH stretchkg infrared bands has been observed and has been assigned to a proton delocalization process.2 The decrease in intensity was fully reversible below the dehydroxylation temperature. The

delocalization process was considered as a translation of protons in the mica framework but their compli(1) .(a) University of Louvain and M.R.A.C. (Tervuren); (b) h ~ l r a n au t F.N.R.S. (2) (a) J. J. Fripiat, P. Rouxhet, and H. Jacobs, Am. Mineralogist, 50, 1937 (1965); (b) J. J. Fripiat, H. Jacobs, and P. Rouxhet,

SiZieates I&., 31,469 (196s).

Volume 71, Number 4

March 1067

J. J. FRIPIAT, H. BOSMANS, AND P. G. ROUXHET

1098

cated structure did not favor a thorough analysis of the phenomenon. Aluminum monohydrate boehmite offers an interesting opportunity to extend these investigations to a wellknown and rather simple s t r u c t ~ r e ~(Figure -~ 1 ) . The natural as well as the synthetic forms of this mineral are polycrystalline. The random orientation of the particles averages the beam polarization by the lattice and avoids any pleochroism in the absorption. An interesting solution to the problem of hydrogenic structure in boehmite has recently been developed through application of infrared spectroscopy. According to Wickersheim and Korpi,Gan infinite OH-OH zigzag chain seems to be responsible for the hydrogenic modes of vibration. The presence of an OH chain and of relatively strong hydrogen bonds between two adjacent oxygen atoms would favor proton tunneling and would enhance greatly an eventual proton delocalization process. Indeed, a rapid decrease in the intensity of' the two stretching modes and of the three deformation modes has been observed for boehmite AlOOH, as well as for AlOOD: the change was completely reversible below 400". A rather complete treatment to explain these observations has been attempted on the basis of the computation of the probability of the proton tunneling through the potential barriers between lattice points. The infrared absorption has been experimentally characterized by the measurement of the integrated band intensity. For sufficiently thin layers, the absorbance of a fundamental infrared band of frequency vnm is

A(v,,) =

hvnrnN0

___ c

CNwBnrn(1 - Nwj) w

(M",")

=

f+,M+,* d7

(2)

If the dipole moment M is split kt0 its three components along 2, Y, and X , for sufficiently small displacement~One Obtains (Mzn*m)= Mozf+n+rn*dT

+ c (5)

f+ntt+m*

2

6St

0

dr

(3)

St is the normal coordinate corresponding to the fundamental mode of frequency vnm and +,grn* is set for the complete series of +,(ti) and The subscript zero indicates the equilibrium position. Because of the orthogonality of the eigenfunctions, relationship 3 reduces to eq 4 and 5.

(1)

where B,, is the Einstein coefficient of absorption and N , and NU,,are the fraction of the No OH oscillators in the initial and in the final energy levels, respectively. N , represents the relative population of the initial level w (quantum number n) while (1 - N,!) accounts for the lack of occupancy of the final level w' (quantum number m). The vibrational quantum numbers m and n of the considered normal mode v,,are related by the selectivity rule m = n + 1. The summation included in relationship 1 is theoretically extended to all the possible energy levels of the oscillator. No anharmonicity effect-will be considered since, according to Pimentel and McClellan,' there is no evidence that these effects are stronger in hydrogen-bonded systems than in other systems. The transition probability between the w and w' levels, represented by Brim, is proportiona1 to the square of the transition moment, defined as The Journal of Phgsical Chemistry

Figure 1. Structure of boehmite:S small circle, aluminum; large circle, oxygen 01 hydroxyl; dashed line, hydrogen bonds.

w.

J.

(3) v. Milligan J. L. McAtee, Phys, Chem., 60, 2,3 (1956). (4) H. Bosmans and P. Michel, Compt. Rend., 249, 1532 (1959). (5) H.Bosmans, to be published. (6) K. A. Wickersheim and G. K. Korpi, J. Chem. Phye., 42, 579 (1965). (7) G . C. Phentel and A. McClellan, #'The Hydrogen Bond," W. H.Freeman and Co., San Francisco, Calif., 1960.


1099

(4) and 819

B,, = -((MnB"))2 3h2

(5)

It will be shown that the decrease in intensity of the fundamental modes of vibration observed on heating is accounted for by a tunnel effect and by an increase in the populations of the higher vibrational energy levels. 11. Infrared Spectra at Room Temperature 1. Band Assignment. The infrared spectra obtained for boehmite and partially deuterated boehmite diluted in KBr pellets are shown in Figure 2, as recorded with a Beckman IR-12 spectrometer. It is observed that the 3297- and 3090-cm-l OH stretching bands are replaced in a 95% deuterated sample by one single 3297) cm-I = 3193 em-'. Acband a t 1/2 (3090 cordingly, the two OD stretching bands are replaced by one band a t 2400 cm-' in a 10% deuterated sample. This suggests that these bands are coupled stretching modes and from these observations, Wickersheim and Korpis have explained the hydrogenic modes of vibration on the basis of an infinite OH. .OH. .OH zigzag chain similar to structures proposed by Hornig and Osberg and Hornig and Hieberte for crystalline hydrogen halides or HX-DX mixed crystals. Extending these investigations into the lower frequency region, we found three other hydrogenic modes for AlOOH at 1160, 1080, and 736 em-'. Partial deuteration causes the 1160- and 1080-cm-l bands to collapse into one component as shown in Figure 1. Therefore, these bands may probably be assigned to coupled bending bands. Neglecting interactions between the chain and the rest of the structure, Wickersheim and Korpi6 anticipated the six optically active hydrogenic modes, shown in Figure 3. The "unit cell" contains two coplanar OH groups, and the bisectors of the angle formed by two adjacent 011 bonds are parallel or perpendicular to the chain direction. The factor group of the chain is then isomorphous with the point group CzV. Wickersheim and Korpi6assigned the high-frequency stretching mode S3to the 3297-cm-I band and the low-frequency stretching mode SI to the 3090-cm-l band. Accordingly, we assign the S 4 bending mode to the 1160cm-1 band and the S2 bending mode to the lOSO-cm-' band although the former has only one-third of the integrated intensity of the latter. This is supported by the following considerations.

+

According to eq 4 and 5 , the absorption coefficient B,, is proportional to the square of the partial derivative of the dipole moment components x, y, and z, with respect to the normal coordinates. Mixing the symmetry coordinates to produce the actual normal modes is slight because of the appreciable frequency separation between the various motions belonging t o different symmetry coordinates within each irreducible representation. The partial derivatives with respect t o the symmetry coordinates may thus be used to approximate the intensity factor. Let SI and S3 be parallel to the 0-0 links and the angle 2 a between two adjacent links be slightly dzerent from 90'. S2and S4 are normal to SI and S3. Assume that 6 is the angle between the direction of the OH bond and the 0-0 link. For the two stretching modes, it is easily seen that

(g)2

+

r$)2

+ ptv2 = 4p2 cos2 a

= pz2

=

ptz2

pv2

= 4p2 sin2 a

where p is the absolute value of the change of the dipole. Accordingly, but with other values for pz, pv, and p E , one obtains for the bending modes

(g)2

= pz2

+

pv2

= 4p2 cos2 (a

+ 6) (7)

From (6) and (7), the intensity ratios for the two stretching modes and the two bending modes are Sa/& = cotg2 a and 8 4 / 8 2 = tg2 (a 6), respectively. From the X-ray structure data,5 2 a = 85'22'; &/SI should then be equal to 1.17. This is in perfect agreement with the experimental ratio of 1.17 f 0.05 found for the integrated intensities of the two OH stretching bands. Using a = 42'41', the ratio 8 4 / 8 2 may be computed for various values of 6. For 6 equal to - 5 , -10, and -15', the following values are obtained for S4/S2: 0.594, 0.41, and 0.274. This shows that for an approximate deviation of 12" outward from the 0-0 link, the difference observed in the intensity ratio of the S4 and S2 bands is of the right order of magnitude. Such a strong deviation has also been found in diaspore by neutron difTraction.lo The intensity

+

(8) D.F. Hornig and W. E. Oaberg, J. Chem. Phys., 2 3 , 662 (1955). (9) D.F.Hornig and G. L.Hiebert, ibid., 27,752 (1957). (10) W.R. Busing and H. A. Levy, Acta Cryst., 11, 798 (1958).

Volume 71, Number 4 March 1967

J. J. FRIPIAT, H. BOSMANS, AND P.G.ROUXHET

1100

I

I

I

4000

3600

3200

2800

2400

2000

1600

IS00

1400

1200

1000

800

600

-

400 300

Cm-1.

Figure 2. Infrared spectra of boehmite samples in KBr (dilution: 0.75%). One division of the ordinate axis = 10% absorbance. Deuterated samples were obtained by synthesizing boehmite from bayerite a t 150’ under pressure in presence of a mixture of HzO-DtO. Below: FeOOH, lepidocrocite.

ratio S4/S2as well as the tendency for these two bands to collapse into one band on partial deuteration supports the assignment of S4 and S2 to the two bending modes in the x-y plane. The remaining hydrogenic mode at 755 cm-l will be assigned to 8 6 and it will be assumed that it occurs The Journal of Phyeieal Chemistry

at the same frequency as the infrared-inactive mode SS. The bands at 618 and at 522 cm-l present in both AlOOH and AlOOD are probably the v 3 and v4 modes, respectively, of an Aloe octahedron. They are also present in aluminum trihydrates: gibbsite, bayerite, and nordstrandite.6 Weaker spectral components at

PROTONMOBILITY IN SOLIDS

1101

410 and 372 cm-l are assigned to skeletal modes as similar bands were found in trihydrates. The bands observed a t 326 cm-l in AlOOH and a t 305 cm-l in AlOOD, which have an isotopic ratio lower than 1.35, might be AI-(OH) or A1-(OD) mode. Table I summarizes these assignments; they are supported also by the infrared spectrum of lepidocrocite (FeOOH). I n particular, the five active hydrogenic modes are found at approximately the same positions with comparable intensities while the v3 and v4 of FeO6 octahedron are shifted toward lower frequencies, at 480 and 360 cm-l, respectively. 2. The Potential Energy Function. Consider the symmetry coordinates of species AI, BI, Az, and B2 shown in Figure 3, the only ones for which genuine normal vibrations occur for the symmetry point groups Cav of the zigzag chain. The orientation of the Cartesian axes has been chosen such that the y axis

Table I: Aasignments of the Infrared Bands (cm-1)

a

n

+ 2C12SlSZ + czzs22 + csasa2 + 2C84S84 + f + diiSiz + 2dizSiSz + &Sz2 + &Sa2 + 2d34SaS4 + d4494' d& + des&2

2T =

where Sf = dSJdt.

Assignments

3297 3090 1160 1080 755 618 522 410 372 326

2470 2350

Hydrogenic mode SS Hydrogenic mode SI Hydrogenic mode Sd Hydrogenic mode SZ Hydrogenic mode Ss va of A106 octahedron v4 of AI06 octahedron Skeletal mode of A I 4 layers Skeletal mode of Ai-0 layers AI-(OH) or AI-(OD) mode

?

790 540" 625 530" 394 366 305

Broadened bands.

sz + s4 + sa = 8 6 - 8s =

xz =

y1 = SI

yz =

s1 - Sa sz - s4

21

22

86

21

2T = 2mp(Sz2

(8)

0

cas - d3aX ~ 4 3 d4ah 0 0

+ S4' + + Sa2 + Ss2+ SB~)(11) $1'

A

tances 0-H and of the angles HOO. Let Q1and Qa be

0 0 0 0

0

0 0 0

86

From (11) and (8), d11 = dB = d33 = d 4 = dss = da = 2 m and ~ d12 = d34 = 0. The potential energy will be expressed in terms of valence force coordinates, Le., as a function of the change of the internuclear dis-

cS&Li2

The secular determinant is then

0 0 0 0

+

since the most general displacement is simply a superposition of the symmetry coordinates. For the kinetic energy, it follows that

C11Sl2

C44s4'

AlOOD

In order to express the cf5 and df, coefficients in terms of more familiar quantities, the potential and kinetic energies have to be translated into other systems of coordinates. As shown in Figure 3, the hydrogen atom displacements are in Cartesian coordinates

corresponds to the 0-0 axis. The 0-0-0 angle in the zigzag chain will be considered as exactly equal to 90" and the 0-H bond will be assumed to coincide with the 0-0 axis. These assumptions bring a considerable simplification since the normal and the Cartesian coordinates are mutually orthogonal. The potential and the kinetic energies expressed in terms of symmetry coordinates are then, according to the above

2v =

AlOOH

0

- d34X cu - d44X 0 0 ca4

It can be immediately factored into six equations where Xf

=

css

0 0 0 0 0

- dsd 0

C66

=o

(9)

- dssh

47r2vt2


1102


and of the bending vibrations (k14 = k32). If kl is the restoring force in the line of OH bond, and if k2 and k5 are the restoring forces opposing the deformation of the OH bonds in the plane (“bending”) or out of plane (“torsion”), respectively, the potential energy function becomes I

2v =

X

Ai

Si

Xl

+

Q32)

k2(Q22

s3

Bl

+ 2ki3QiQ3+ 2k14(&1&4+ &3&2)+ + Q42) + 2k24Q2Q4+ ks(Q2 +

kl(Qi2

X3

Q62)

(12)

or, in terms of symmetry coordinates

2T‘

=

+

+

+ +

+

2(ki ki3)Si2 2(k2 k24)S22 4k14SlS2 - 4k14s384 2(ki - k13)S3~ 2(k2 - k ~ > S 4 ~2kjSj2 2k5S~~(13)

+

s2 XZ

Ai

B1

Xz

s1,

Y f

I

X

I

inactive in I.R. A2

ss

As

X

02

A6

s6

Figure 3. Hydrogenic modes in boehmite according to Wickersheim and KorpiaE The symmetry coordinates Sl , SE,the roots of the secular equation XI AS, and the Cartesian displacement coordinates are indicated with respect to the symmetry species AI, B1, A2, and Ba.

..

...

+

+

+

From (13) and (S), it follows that c11 = 2(k1 k13); CIZ = 4h4; ~ 2 2= 2(k2 k24); ~ 3 3= 2(k1 - k13); c34 = -4ki4; C44 = 2(kz - k 2 4 ) ; Cj5 = C66 = kg. Using the ctj and dt, coefficients, (10) is solved in terms of the restoring forces. The system of simultaneous equations has been treated as follows. First k24 has been put equal to zero; k13 is then found equal to -0.41. Secondly, k14 has been neglected; k13 was then equal to -0.36; k13 was thus essentially unaffected by this change in assumptions. An average value of -0.385 was then taken and (10) was solved for the other coefficients. Table I1 contains the results. The calculated frequencies are within a few per cent of the experimental data. If, instead of mH, the reduced mass is used, all the restoring forces should be multiplied by 0.942.

+

Table 11: Restoring Forces Obtained for AlOOH

the displacement of the H atoms, 1 and 2, along the 0-0 axes Qi

=

si + s3;

Q3

=

Si

- S3

The deformation coordinates are Q2 Qj

= rod1 =

SZ

+

rom

8 5

- Sa;

=

=

S4;

Q4

= ro&

= 82

&e = ~ O H Y=~ 85

- S4

+ 8s

where TOH is the internuclear distance 0-H and where 61 and 62 are the deformation angles of 0-H1and 0-H2 in the x-y plane of the zigzag chain and y1 and y2 are the deformation angles of 0-HI and 0-H2 along the x axis. Three additional coefficients will be considered in order to take into account (1) the coupling of the valence vibrations (kls), (2) the coupling of the bending vibrations (k24, and (3) the coupling of the stretching The Journal of Physical Chemiatry

Restoring forces ( X 106 dynes

Stretching kl 5.714 1.036 Bending h 0.318 Torsion k5 Coupling stretch- -0.385 ing-stretching, kia Coupling stretch- &0.630 ing-bending, k14

Coupling bending-bending,

Frequencies--

r

Obsd

Calcd

om-1)

v1 = 3100 cm-l

u4

1075 cm-’ = 3290 cm-l = 1150 cm-1

v1 = YZ = vs = v4 =

3090 cm-l 1080 cm-l 3297 cm-l 1160 cm-1

~5

=

740 cm-’

v5 =

755 cm-1

~2 YS

=

-0.025

k24

The negative sign of k13 is rather surprising a t first sight. It may be understood as follows. Suppose that Q1 and Q3 are both positive. Coupling term 2k13Q1Q3


9700

31

I

3! 3

1103

3400

Figure 4. Stretching modes Sa and SIa t increasing temperatures.

31

One division of the ordinate scale = 10% absorbance.

IO

Volume 71, Number 4

March 1967

1104


25'C

I

3Q'C

92"c

!7!5OC

178c

1400

1200

1300

n

0

Figure 6. Bending modes Sqand 82 and torsion mode Ssa t increasing temperatures. One division of the ordinate scale = 10% absorbance.

forces hydrogen atom 1 to approach the oxygen atom to which hydrogen atom 2 is attached, while the latter approaches the opposite oxygen atom. This reinforces hydrogen bonds and decreases the potential energy. Therefore, ( k l ( Q ~ ~ Q?) 2k&&3) < kl(QI2 Q37. It follows that 2k13&1&3 < 0 and that k13 < 0 if Q1Q3 > 0. A negative interaction force constant has been obtained for other H-bonded systems, such as 13(0H)311 and ice.12 The coupling of the stretching and of the bending vibrations is also characterized by a high restoring force k14. It has been

+

+

+

n

shown6 that A100 angles are close to the tetrahedral

A

value 109'28' while the 000 angle in the chain is noticeably smaller (2a = 85'30'). When hydrogen atom 1 approaches the opposite oxygen atom, the positive electrical field arising from Ala+ and acting on hydrogen atom 2 is somewhat attenuated. Therefore, the force opposing the bending of OH toward the A1-0 bond must be decreased to some extent.

111. Infrared Spectra at Increasing Temperatures Samples of boehmite diluted in thin KBr pellets The Journal of Physical Chemistry

(0.75%) were heated and the spectra were recorded at increasing temperatures, using the double-beam IR-12 Beckman spectrometer. Figures 4, 5 , and 6 show some examples of the changes observed for the stretching, bending, and torsion modes. Three main features are observed a t increasing temperatures: a shift of the two stretching modes toward higher frequencies and of the bending modes toward lower frequencies, a progressive broadening, and a marked decrease in intensity for all the hydrogenic modes. Whatever the changes, all the observed phenomena are reversible on cooling the samples to room temperature, provided the limit of 400' has not been exceeded. Below this temperature, dehydroxylation of boehmite does not occur. 1. Frequency Shifts. From Table 111, it may be observed that the S 3 and 81 frequencies for AlOOH and AlOOD become progressively closer to each other as the temperature is increased. According to equations in (lo), the coupling force constant k13 decreases (11) D.F.Hornig and R. C. Plumb, J. Chem. Phus., 26,637 (1956). (12) C. Haas and D. F. Hornig, ibid., 32, 1763 (1960).


1105

Table I11 : Frequeiicy Shifts with Respect t o the Temperature

OK

300 391 439 541 576 652

-MOOD-

AlOOH

-AlOOH-

Temp,

s2,

81, -1

Temp.

84,

om

Sa, cm -1

OK

cm-1

em-'

3089 311.2 3123 3155 3170 321 5

3297 3320 3330 3345 3355 3375

300 372 412 465 502 548 600 643

1162 1158 1160 1150 1150 (1145) (1140) (1145)

1080 1080 1080 1073 1070 1070 1060 1058

accordingly. The weakening of the hydrogen bond is probably a t the origin of this, since both the Sa and S1 bands shift simultaneously toward higher frequencies. As expected, the effect of temperature on the frequencies of the bending modes Sz and 8 4 is less pronounced and the shift is in the opposite direction.' The frequency of the So "torsion" mode is apparently not temperature sensitive. This is to be expected since parallel OH chains are not hydrogen bonded. C~ggeshall'~ has attempted to calculate the frequency shift and the intensity of absorption of a hydroxyl group which is hydrogen bonded to a neighboring oxygen atom on the basis of an electrostatic interaction. If the change of the potential energy of the OH group due to the polarization is introduced into the Schrodinger equation with a hIorse potential energy function, the energy levels :%reshifted by a term -

- qE,/a)(v

+

'/2)

~(21.1) "'(De - qE,/2~)'" /-

where a = 1/2V2kl/De is the constant of the Morse function given by (j' = De(l - e--a(U-v'JH) 12 (14) and De is the dissociation energy of the O-H bond. q is the charge unbalance for the OH group and E ,

is the electric field component parallel to the direction of the valence bond. Therefore, for a 0 -+ 1 transition, the frequency (v,) of the stretching band of the perturbed OH oscil1:ttor is shifted with respect to that for the unperturbed oscillator (vo) as

Temp,

81,

cm -1

OK

cm-1

cm-1

755 755 755 755 7.55 (750) 750 (750)

300 373 436 487 569 629

2347 2350 2375 2375 2375 2390

2475 2469 2475 2475 2490 2493

SO,

SS,

the polarization force qE, from 0.324 X to 0.265 X esu. According to Bosmans,6 the coefficient of thermal expansion along the direction normal to the OH-0 chain and parallel to the zigzag plane is 2.81 X As the width of the chain is about 2 A, an increase in temperature of the order of magnitude of 300°K lengthens the 0-0 distance by about 0.02 A. Such a change corresponds to a reasonable decrease of the potential energy, amounting to 0.85 kcal mole-'. The lengthening of the 0-0 distance is also in agreement with the frequency shift predicted by the onedimensional model for hydrogen bonding proposed by Lippincott and S~hroeder.'~An average stretching frequency of 3193 cm-I (at 300°K) corresponds to an 0-0 bond length of 2.72 A, while the value observed at 650"K, i.e., 3295 cm-l, is in agreement with a bond length of 2.74 A. The X-ray structural data6 a t 300°K indicate an 0-0 distance of 2.73 A. I t may be therefore concluded that the frequency shift observed a t increasing temperature accounts for the thermal dilatation of the crystal unit cell. 2. Broadening of the Absorption Bands. Table IV shows the half band widths observed for the SI, SS,SZ,and Se modes a t two temperatures. According to the Heisenberg uncertainty principle, the half band width A v is approximately equal to the sum of the halfwidths of the upper and lower states, or, for a transition from the energy level w to w'

where r W is the lifetime of the proton a t the energy level w. Pw is then the probability of the proton Between 300 and 660"K, the shift of the mean stretching vibration vS := 1/2(v1 vg) is of the order of magnitude of 100 cm-'. Assuming the "free" OH stretching frequency vo to be 3750 cm-' and D e to be 110.2 kcal,I4 a shift of 100 cm-' corresponds to a decrease of

+

~~

~

(13) N . Coggeshall, J. Chem. Phys., 18, 978 (1950) (14) L. Pading, "The Nature of the Chemical Bond," 3rd ed, Cornel1 University Press, Ithaca, N . Y.,1960. (15) E. R. Lippincott and R. Schroeder, J . Phys. Chem., 61, 921

(1957).

Volume 71,Number 4 March 1967

1106


leaving the state w per unit time. Table IV shows that Pw is always slightly lower than the vibration frequency v but that it is much greater than the Einstein transition coefficients. It may then be assumed that P, represents a probability of the proton leaking out of the potential well at the energy level w.

Table IV : Observed Half Band Widths and “Leaking Out” Probability Values a t 300°K and a t Approximately 600°K for AlOOH Temp, OK

-Vibrational

s1

Half band width, cm-’

300 180

P W I Y

300

Half band width, cm-l

600 210

PUIY

600

0.37

0.41

-AlOOH---.--, Temp, OK

300 39 1 439 54 1 576 652

AlOOD-Stretch. bands, SI

+ ss

1.00 0.91 0.865 0.79 0.755 0.65

OK

Stretch. bands, SI ss

300 373 436 487 569 629

1.00 0.85 0.76 0.72 0.59 0.56

Temp,

+

modesSI

120 0.23 210 0.41

s 2

42 0.24 61 0.36

Sa

0.82 0.68 133 ~1

For the OD stretching modes, SI and S3, pw/v deduced from the observed band widths is only 0.175 a t 300°K as compared to 0.37 and 0.23 for the corresponding bands in A100H. The results shown in Table IV are only approximate because of the overlapping of the SI and Ss and of the S2 and S q modes. It must be emphasized that the actual broadening is probably higher than that reported in Table IV because of the deformation of the high-frequency side of the bands. This is more apparent for the low-frequency modes SZ, SA, and S a than for the stretching modes as shown in Figures 4 , 5 , and 6. If the leak through the potential barrier is attributed to a tunnel effect of protons, the increase of Pw with temperature may be due to an increase in the populations of vibrational levels closer to the top of the energy barrier. 3. Loss in Intensity of Hydrogenic Modes. At low dilution in KBr, it is assumed that the integrated band intensity represents adequately the characteristic absorbance. The relative absorbance A, is then defined as the ratio of the integrated intensity a t the temperature T to the integrated intensity a t the temperature To = 300°K; k, being the absorption coeficient, it follows that

I n order to express the loss of intensity shown in Table V, the two SBand SI stretching bands were integrated together because the delineation between them is more hazardous at high temperature. The ratios The Journal of Physical Chemistry

Table V : Change of Relative Absorbances with Respect to the Temperature

of the integrated intensities Sa/$ were calculated between 300 and 500°K for AlOOH and A100D. They do not change by more than *4%. I n agreement with eq 6 one may conclude that the angle 2a between two adjacent 0-0 links is not noticeably modified. The radiant energy emitted by the hot sample has to be considered. In the instrument used in this work, the infrared beam is chopped first between the Kernst glover and the sample, and secondly, between the sample and the detector. The energy emitted by the hot sample is thus added to the radiant energy transmitted through the sample and therefore it contributes to decrease the over-all absorbance. On the contrary, when the 2d chopper is stopped, the additional energy is not detected as it is not modulated. By recording the hydrogenic fundamental bands a t increasing temperatures under this condition and by comparing with those obtained with the instrument in “normal” operation, n o significant diference was observed below 650°K f o r the stretching modes. However, for the lower frequency bending and torsion modes, at the same temperature, the contribution of the radiant energy emission decreases the relative absorbances by about 0.15. Actually, the relative absorbances of the AlOOH bending and torsion bands are equal, within the experimental error (*0.05), to those shown in Table V for the AlOOH stretching bands. When log (1 - A,) is plotted against the inverse of the temperature, the activation energies of the process responsible for the intensity losses are close to 2 kcal, whatever the vibrational modes. Before going further, it seems necessary to examine a possible explanation for the intensity loss observed upon heating. It is well known that hydrogen bonding provokes an enhancement of the intensity of the OH stretching bands.’ On the contrary, as the thermal dilatation reduces the strength of the OH. . . O bond, it might be assumed that this effect contributes to de-

1107


Table VI Energy,

kcsl

Level

Multiplicity

300

400

Temp, O K 600

1

600

700

A. Population of the Energy Levels ( X 104) in AlOOH 0 2.11 3.08 3.30 4.22 5.19 5.41 6.16 6.33 6.38 6.60 7.30 7.52 8.27 8.44 8.49 8.71 8.89 9.24 9.41 9.43

1 2 1 1 3 2 2 1 4 1 1 3 3 2 5 2 2 1 1 4 1

9330.6 546.7 53.5 37.7 24.0 3.1 2.2 0.3 0.9 0.2 0.1 0.1 0.1

8320.7 1175.7 173.5 133.8 124.6 24.5 18.9 3.62 11.7 2.79 2.15 2.59 1.98 0.52 1.05 0.4 0.3 0.12 0.08 0.24

7117.0 1708.5 323.8 262.4 307.6 77.7 63.0 14.7 49.2 11.9 9.68 13.99 11.4 3.50 7.5 2.86 2.3 0.95 0.67 2.24 0.56

5949.5 2026.7 450.8 380.3 517.8 153.6 129.6 34.1 117.6 28.8 24.3 39.2 32.7 13.2 25.5 9.81 8.2 3.5 2.6 8.91 2.2

4879.5 2155.9 535.3 460.7 714.4 236.5 203.5 58.7 210.4 50.5 43.5 78.4 66.9 26.0 58.0 22.33 19.0 8.4 6.6 23.08 5.70

3845.6 2117.1 580.9 505.0 874.2 316.0 278.0 87.0 312.0 76.0 66.3 129.0 112.5 48.3 106.0 41.4 36.0 15.3 13.0 11.8

2925.2 1928.0 578.9 513.4 953.1 380.0 338.4 113.0 412.0 102.0 90.1 186.0 165.0 75.5 170.0 66.0 59.0 26.0 22.5 20.5

B. Population of Energy Levels ( X 104) in AlOOD 0 1.55 2.26 2.42 3.10 3.81 3.97 4.52 4.65 4.68 4.84 5.36 5.52 6.07 6.20 6.23 6.39 6.59 6.78 6.90

1 2 1 1 3 2 2 1

4 1 1 3 3 2 5 2 2 1 1 1

8219.8 1233.3 185.7 140.3 138.8 27.6 21.0 4.1 13.2 3.1 2.39 3.0 2.3 0.62 0.75 0.50 0.40 0.10 0.10

crease the absorption coefficients. In this case, the matrix element (Mol1)of the dipole moment for the transition between the ground and first excited states calculates to bel3 (MOJ) = (bo

- 2)””/(bo - 1)

For an unperturbed Morse function, bo is given by bo = 4?r(2pDe)’”/(ah - l), while bo for the Morse function perturbed by the electrical polarization qEp is

6574.2 1889.4 384.0 311.3 407.3 108.0 89.5 21.7 74.0 17.8 14.7 23.4 19.2 6.4 14.0 5.4 3.8 1.4 1.3 1.1

5067.8 2151.1 523.5 441.7 684.8 216.0 187.5 52.5 186.0 45.0 38.5 69.0 58.5 22.96 50.0 19.6 16.8 6.8 5.6 5.0

According to eq 5, the ratio of the absorption coefficients is equal to

Using the same Y , , Y O , D e , and a values as in eq 15, this ratio decreases from 1.166 a t 300’K to 1.137 at Volume 71, Number 4

March 1067


1108

650’K. This effect is much smaller than that shown in Table V and the proposed explanation does not hold. I n order to interpret the loss in intensity of the OH fundamentals a t increasing temperatures, we will assume that energy levels above a limiting value do not contribute to the discrete absorption of radiant energy. It may be considered that a transition will not contribute to the absorption band when it takes place toward a level 20’ for which P,,i = P,t/vf 1! 1, Purlf,being the tunneling probability per period v f - l of the ith vibration mode. The physical meaning of this hypothesis will be given later. The phenomenon should become more noticeable with increasing populations of vibrational states characterized by higher P, values. This hypothesis is in agreement with the reported broadening of the absorption bands. Therefore, the summation of eq 1 cannot be extended beyond the energy level w’ for which the tunneling probability per vibration period is close to 1. For a transition from w toward w’ = w hvr

+

able population shift toward slightly higher vibrational states affects the intensity of the absorption bands. In order to emphasize this point, let us proceed empirically as follows. The average temperature coefficient obtained by plotting log (1 - A,) against T-1 being about 2 kcal for the stretching, bending, and torsion modes, it may be assumed that for AlOOH the vibrational levels above the fundamental (Table VIA) do not contribute to the discrete absorption of radiant energy. This means that the energy levels for which PWjf+ 1 are approximately equal to 11, 5 , and 4 kcal for the stretching, bending, and torsion OH modes, respectively. Under these conditions, eq 21 gives the relative absorbances (computed) shown in Table VII. They are compared with the values deduced from the best-fitting curves plotted with the experimental values (obsd) of Table V. The differences are approximately within the experimental errors. Table VI1 : Computed and Observed Relative Absorbances AlOOH

The relative absorbance defined previously by eq 17 becomes

where v is the vibrational quantum number a t the 1) is introenergy level w. The square root of (v duced to take into account the change of the B absorption coefficient above the fundamental level. The relative population of a vibrational state 20, as defined in eq 1, is given by the relationship

+

N, = q Q - 1e

-w/kT

(22)

where w = Zivfhvi, v1 being the vibrational quantum numbers and v f the corresponding frequencies. The statistical weight g is introduced to take into account the assumed mechanical degeneracy of the S5 and Se 1 with 216 = v5. From the vibramodes: g = ve tional characteristic temperatures et, N, may be easily computed as shown in Table VI, the partition function being

+

&-1

=

n(1 - e - @ f / T1 i

(23)

The activation energies determined above for the process involved in the loss of intensity of the OH fundamentals are of the order of magnitude of the first energy levels. This is an indication that a noticeThe JOUTW~ of Physical Chemistry

Temp, OK

Si

+ Sa,

S2

+ 6’4,

Sa,

Obsd

comp

comp

comp

300

1

400 500 600

0.91 0.82 0.71

1 0.89 0.76 0.64

1 0.88 0.74 0.61

0.83 0.67 0.54

1

-A100D-SI Obsd

1 0.83 0.70 0.57

+ SP-

Comp

1 0.85 0.72 0.60

For the OD stretching modes, the computed data of Table VI1 were obtained using the relative populations of the two first vibrational levels (Table VIB). The limiting energy level is then located a t about 9 kcal.

IV. Discussion on the Tunnel Effect It is now necessary to verify whether the assumed leak of proton out of the potential well may be accounted for by an acceptable potential barrier. An evaluation of the dissociation energy is necessary to obtain the Morse function of the OH bond. The A1-0 bond energies in analogous crystalline substances (corundum and boehmite) were selected in order to allow for the electrostatic long range forces. The energy of the hydrogen bonds in boehmite has been estimated to 4.5 kcal mole-’ according to Lippincott and Schroeder.15 The calculation is made as shown hereafter using the enthalpy data proposed by Latimer.16 (16) W. Latimer, “Oxidation Potentials,” 2nd ed, Prentice-Hall Inc., New York, N. Y., 1952.


1109

+ 2O2(g) + Hz(g) = A1203.Hz0(boehmite) = (6A1-0 + 20-H + 20s.H) (AlZO3):= 6(A1-0) = 2A1 + 3/zOz(g)(corundum)

2A1

10(3p2)= ~ / ~ o ~ ( g ) 2H('S,/,) = H2(g) 2 0 . €IO = 2 0 nonhydrogen bonded

20(3P2)

+ 2H(%Sl/,) = 20-H

+ 2H0

= -471.80 kcal/mole

AH AH AH AH

= 399.09 kcal/mole = 59.16 kcal/mole = -104.18 kcal/mole = 9.00 kcal/mole

-

MI = -227.05 kcal/mole

hydrogen bonded

Thus AH for 0-H = -113.5 kcal/mole. A similar calculation gives 115.3 kcal/mole for the 0-H bond in gibbsite, A1(OH)3, and 120.3 kcal/mole for brucite, Mg(0H)z. It must be understood that AH represents the difference between the energies of the free H and 0 atoms and that of a OH group in the solid. The dissociation energy DO of OH is therefore referred to the lowest energy W and DO = 113.5 kcal. The potential function along the 0-0 axis has been constructed using the Morse function given by eq 14. The height of the potential barrier may be computed if the positions t3f the minima are determined with respect to the energy scale. If, as usual, the fundamental level is considered as the zero level, the minimum of U(y) is lowered below this level by 4.6 kcal, ie., the average energy of '/z hvl and l/2hv3 and De = D O 4.6 kcal. For the deuterated system, the corresponding energies were introduced into the calculation. The distance between two adjacent V(y) minima, 210 = 0.698 A, was obtained by subtracting from the 0-0 distance (2.73 A in boehmiteb), twice the length rOH of the OH bond (1.016 A). I n the decomposition of the hydrogenic motions, oxygen atoms were supposed to be immobile; this assumption holds so far as the 0-0 and OH frequency domains are very different. For the calculation of the tunneling probability P,, a distribution function of the 0-0 distances should be considered and the average quantum mechanical value (Pw) should be computed. An alternative less sophisticated treatment may be proposed in order to correct the width (2Z0) and consequently the height of the energy barrier E* for the 0-0 vibration. Let R be the distance between two oxygen atoms and Ro the equilibrium value (Ro = 2.73 A). The mean quadratic value of ( R - Ro)is easily obtained from

AH

where H , is the Hermite orthogonal function and a = 4n2mvo/h, yo being the fundamental 0-0 frequency. Averaging ( ( R - RO)u2) over all the energy levels produces

( a ~ > - l $(v u-0

+ l/z)e-"hvo/kT

If kT/hvo is small enough, allowing for a continuous sequence of energy levels

+

( ( R - Ro)2)= (a&)-'s,' ve-"h"O~kTdv (2a)-'

=

+

( R - RO)'+~*(R - Ro)dR and

=

(

v

3:

+- -

Under the same conditions, the partition function Q is equal to kT/hvo. Therefore (25)

Assuming vo = 200 cm-l, in agreement with the frequency proposed by Haas and HorniglZ for the 0-0 vibration in ice, the square root of ( ( R - RO)2) changes with respect to the temperature as shown in Table VIII. The correction due to the oxygen vibration is thus quite appreciable. The height E* of the potential barrier is obtained graphically for each lo value. At the equilibrium position R = Rot 210 = 0.698 A and

Table VI11 : Square Roots of the Mean Quadratic Values of the 0-0 Stretching Amplitude. Maximum and Minimum Widths (210)of the Potential Barrier along the 0-0 Axis and Corresponding Barrier Heights ( E * ) d ( R - Rd*, A

Plomax,

E*max,

Blomin,

E*min,

OK

A

kea1

A

koa1

300 400 500 600 700

0.181 0.200 0.217 0.233 0.248

0.879 0.898 0.915 0.931 0.946

35.3 36.4 37.4 38.1 39.1

0.517 0.498 0.481 0.465 0.450

16.6 15.6 14.7 14.0 13.0

T,

(24)

1 (-h o + i)

1 kT

( ( R - R o ) ~=) -

Volume 7 1 , Number 4

March 1067


1110

E*

= 26 kcal above the minimum of the potential energy curve. Eckart" has given an exact solution of the Schrodinger equation for a particle of mass m and of total energy w* approaching a single energy barrier of height E* and of decreasing thickness. w* = w hvS/2 and E* are measured with respect to the minimum of the potential curve. Hence the probability of tunneling per stretching period is

9 1

I

n

+

Figure 7. Tunneling probability Pws through the potential barrier along the 0-0 link. w = energy level above the fundamental. Barrier height: E* = 21.2 kcal.

where

From (26), it follows

From eq 27, it is found that the barrier height has to be equal to 21.2 kcal for Pws,the tunneling probability along the 0-0 axis, being very close to 1 a t the energy level w = 11 kcal. Under these conditions, eq 21 gives the computed relative absorbance shown in Table VI1 for the OH stretching modes. A barrier height of 21.2 kcal is still in the range predicted by Table VIII, taking into account the vibration of the oxygen atoms. The tunneling probabilities Pw* for AlOOH and AlOOD are reproduced in Figure 7. For the latter, Pus approximates 0.94 at the limiting energy level of 9 ltcal used for computing the corresponding data of Table VII. It follows that the estimated variation of the tunneling probability along the 0-0 axis with respect to the energy levels allows one to account for the loss in intensity of the stretching modes a t increasing temperatures. ItJmight be suggested that the barrier permeability defined as the tunneling probability per unit time, Pwsvs= P,, is representative of the leaking process out of the specified potential well. If this assumption is accepted, Pwb and P w cthe tunneling probabilities per period of the bending or of the torsion vibration, may be derived from the relationship

Pwbvb = P,'V,

=

Pw'Vv,

(28)

where vs, v b , and v C are the fundamental stretching, bending, and torsion frequencies, respectively (Table I). The summation involved in eq 21 breaks down, consequently, at the energy levels corresponding to PWtb 1 or Puts N 0.352 for the bending absorption 'v 1 or Pwps N 0.237 for bands (S2 SI) and to PWft

+

The Journal of Physical Chemistry

the torsion absorption band (8s). One may easily check on Figure 7 that P,J' reaches 0.237 a t 4.3 kcal and 0.352 a t 5 kcal. This is in excellent agreement with the empirical limiting energy levels used for computing the relative absorbances in Table VII. I n summary, the loss in intensity observed upon heating may be explained as follows: (1) there is a domain of energy levels, below the top of the potential energy barrier, where the tunneling probability is close to unity; (2) any optical transition leading to this domain does not contribute to the absorption band; (3) the shift of the population toward vibrational levels, from which such an optical transition is possible, accounts quantitatively for the observed variations of the relative absorbances. The physical meaning of point 2 will be discussed now.

V.

Delocalization Processes In a static model of a perfect crystal, every proton has a well-defined position and all the protons are oriented in a regular fashion, as shown for instance in Figure 3. The wave function of the whole protonic system is then defined by the product of individual proton wave functions in their different vibrational states. For such an independent particle model, the total transition probability between energy levels results from transitions of individual protons within their own potential well. For an imperfect crystal, at increasing temperatures, it may be assumed that the movement of each proton is independent of the movement of the other protons and that each proton has a given probability to move to neighboring but less stable positions. Such a movement may be performed either along the directions of a stretching, bending, or torsion vibration. The individual wave function will no longer correspond to (17) C. Eckart, Phys. Rev., 35, 1303 (1930).


that of a simple harmonic oscillator. Assuming that the general form of the potential well is that represented in Figure 8A, it can be easily deduced that the spacing between energy levels will vary rather abruptly as soon as the proton displacement becomes very probable. One may assume that optical absorption a t the fundamental frequency is obtained only for transitions between levels, below some critical energy value. When the temperature is raised, one would thus expect a decrease of the fundamental absorption while secondary absorptions should appear at different frequencies. These absorptions may be weak enough to be masked in the background. Another possibility exists that a proton stays in the neighborhood of one given oxygen but that it produces two configurations by “flipping” between two equilibrium positions as shown in Figure 8B. The passage from one configuration to another may result from displacements along the directions of the stretching or the bending vibrations, but all the displacements have to be coherent. This means a breakdown of the independent particle model. As far as the radiant energy absorption is concerned, this model would produce consequences similar to those discussed above. The mechanisms represented by these two first models cannot be described as true delocalization processes since the proton movement is restricted to a narrow domain. One might also consider a third model that is fundamentally different from the previous ones. Instead of considering a proton that moves around in the vicinity of some specified oxygen ion, it might be assumed that all the protons belong collectively to all oxygen ions as schematically represented in Figure 8C. The proton would thus be considered in a way similar to electrons in a metal. In order to construct such a wave function, it would be necessary to consider the protonic movements as independent of the movements of electrons and heavy ions in the lattice. The energy bands that arise from the vibrational levels in independent potential wells should be narrow a t the bottom since protons have only a small tunneling probability. They become quite large when the proton may pass easily through or over the potential barriers between neighboring crystallographic positions. At increasing temperatures, the absorption bands would be flattened out so much on the high-

1111

Figure 8. Schematic models proposed to explain the delocalization processes (see text).

frequency side that they are partially lost in the background. This last model should correspond to what may be considered as a true delocalization process. The deductions exposed above support the concept that tunneling of protons through potential barriers causes the decrease in intensity of the OH fundamental obtained upon heating. The simultaneous deformation of the high-frequency side of the bands might be in favor of the third model. However, it would be too hazardous to conclude whether the proton movement is restricted to a narrow domain or is extended through the whole crystal.

Acknowledgments. We wish to acknowledge the part taken by Professor A. Meessen of the University of Louvain in the preparation of this manuscript and especially in the discussion of the various models proposed for the delocalization processes.


Proton mobility in solids. I. Hydrogenic vibration modes and proton

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