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Vibrational surface thermodynamic functions of magnesium oxide
SURFACE
SCIENCE 26 (1971) 637-648 D North-Holland
VIBRATIONAL
SURFACE
FUNCTIONS
Publishing Co.
THERMODYNAMIC
OF MAGNESIUM
OXIDE*
K. H. RIEDER Institut
&terreichische Studiengesellschaft Metallurgic, Reaktorzentrum Seibersdorf,
fiir Atomenergie GmbH, Lenaugasse IO, Vienna VIII, Austria
Received 26 February
1971
The phonon frequency distributions of bulk and microcrystalline magnesium oxide samples have been determined from inelastic neutron scattering data. The resulting surface frequency distribution allowed the calculation of the following surface thermodynamic functions: vibrational surface energy, vibrational surface free energy, surface specific heat and surface entropy. In the calculations, anharmonic effects were taken into account approximately by use of the quasiharmonic approximation. The results were found to be in reasonable agreement with the results of calorimetric measurements as well as with the values derived from the theoretical model of Benson and Yun.
1. Introduction It has been discussed recently in several theoretical investigations1-4), that the existence of free surfaces of crystals causes certain kinds of atomic vibrational modes, which are localized near the surface and hence are called surface modes. In practice the existence of surface modes becomes important in two situations 5, : (A) In some scattering experiments, where only atoms near the surface of the single crystal specimen are sampled (low energy electron diffraction, thermal energy atomic and molecular scattering). (B) In microcrystals and thin films, where the ratio of the number of surface atoms to the total number of atoms is large. By using inelastic neutron scattering techniques, we have studied the phonon frequency distributions of magnesium oxide for bulk material and for microcrystals6). Because it is the distribution of the normal modes, which determines the vibrational contribution to the thermodynamic functions of a solid (at least in the quasiharmonic approximation), we could use our measured frequency spectra of the bulk and the microcrystalline MgOsamples to extract the vibrational contribution of the free surfaces to the * This work was presented in part at the meeting of the Austrian University of Vienna, September 1970. 637
Physical Society,
638
K.H.RIEDER
thermodynamic functions of magnesium oxide. In this paper we present the results of our work and compare them with the results derived from other experimental methods as well as from the theoretical model of Benson and Yun ‘). 2. Determination
of the phonon frequency distributions
In this section we discuss the results of the neutron spectroscopic measurements with two of our MgO samples. A bulk sample consisting of single crystal fragments with a mean grain size of 50 pm corresponding to a negligible specific surface of 0.03 m2/g was used as reference sample. The microcrystalline sample was produced by thermal decomposition of basic magnesium carbonate. The mean size of the MgO microparticles was 160 A. It was determined from electron micrographs and X-ray line broadening. From the diameter value we have estimated the specific surface area to be approximately 110 m’/g. The neutron scattering experiments were performed with a rotating crystal spectrometer installed at the light-water tank type reactor ASTRA of the reactor center Seibersdorf. The primary neutron energy was 22.2 meV, the scattering angle was chosen to be 100 degrees of angle. During the scattering experiments the samples were heated up to 800°K in order to avoid scattering contributions by adsorbed gases and water vapour. In evaluating the phonon frequency distributions from the scattering spectra we have used the incoherent approximations). In these computations the second order down scattering overlap, the multiphonon contribution, the detector efficiency and the instrumental resolution have been taken into account. The transmissions of the samples were chosen to be greater than 90 %, so that multiple scattering could be neglected. Although the incoherent approximation is not justified by theoretical arguments, we had several reasons, that - at least for our experimental conditions - its application should yield reasonable results 6,s). Today we are able to compare our result for the phonon frequency spectrum of the bulk MgO with the distribution derived recently by Sangster et a1.9) by fitting the parameters of a breathing shell model to measured dispersion relations of MgO. Fig. 1 shows both curves, The agreement is quite satisfactory, if we take into consideration, that the limited instrumental resolution of our experimental apparatus did not allow to resolve the fine structure in the high energy region of the spectrum. Fig, 2 exhibits both the frequency distributions of the bulk and the microcrystalline MgO samples together with the difference spectrum g,(u)= 9 (W),icro - 9 (~)bulk, which can be regarded as surface frequency distribution function. The difference spectrum has four peaks. Peak I arises from the
VIBRATIONAL
SURFACE
THERMODYNAMIC
FUNCTIONS
OF
MgO
639
2.10-14
12 w Crad/sec3
7540 ‘3
Fig. 1. Comparison of phonon frequency distributions of bulk MgO. The full line corresponds to the result of the present work; it was derived from inelastic neutron scattering data by use of the incoherent approximation. The broken line represents the distribution computed by Sangster et al. on the basis of a breathing shell model.
acoustical branch of surface modes, which yields the Rayleigh waves of the theory of eleasticity in the long wavelength limit. Whether peak II has to be attributed to acoustical or to optical surface modes cannot be decided at the present instant. The peaks III and IV originate from optical surface vibrations. It should be mentioned, that the frequency of the optically active surface mode w, derived from the expressions of Fuchs and Kliewerl”*il) lies within peak IV - for slab-shaped as well as for spherical particles (see fig. 2). A similar surface frequency distribution function as ours was derived for slab-shaped NaCl-crystals in the theoretical work of Tong and Maradudin 3).
640
K. H. RIEDER
9
(w
2,0.10-l
-
MgO - bulk
----
MgO -microcrystals
1 \ \ \ l,(
g&w
3
6
9
12.1013
radlsoc
II 0,5.10-’
O#E
Fig. 2. Phonon frequency distributions of bulk and microcrystalline Surface frequency distribution function gS(w)= g(cO)miero-g(~)b~lk
MgO (upper part). (lower part).
3. Debye temperatures Since the temperature dependence of the Debye temperature BD is very sensitive to the shape of the phonon frequency distribution function, we want to discuss in this section the Q,(T)- curves, which were computed from our g (o)-functions by equating the exact formula for the specific heat with the one derived from the Debye model. In all our calculations we approximately took into account anharmonic
VIBRATIONAL
SURFACE
THERMODYNAMIC
FUNCTIONS
OF
641
MgO
effects by using the quasiharmonic approximation. Within this approximation, the temperature dependence of a single phonon frequency is given by 0 (T) = C&-J (1 - 3cryT),
(1)
where o,, denotes the phonon frequency at absolute zero, c( stands for the linear expansion coefficient and y for the macroscopic Grtineisen parameter. The values LX = 14.5 x 10e6 deg-’ and y = 1.5 were taken from White and Andersonle). Fig. 3 shows the results of our calculations of the temperature dependence of the Debye Bo together with the results from the calorimetric measurements of Barron et a1.13) for MgO bulk material and those from Giauque and Archibaldlb) for a microcrystalline MgO-sample with a specific surface area
0, (“K) o Barron,
Berg, Morrison
* Giauquo. -present
~microcrystallino
Archibald work
material
200
300 T (“K)
Fig. 3. Temperature dependence of the Debye characteristic temperature for bulk and microcrystalline MgO. The curves are computed from the frequency distributions of fig. 2. The points are taken from specific heat data.
642
K.H.RIEDER
of 50 cm2/g. The agreement of the curves is good for the bulk MgO. For the microcrystalline material the main features of the 19,(T)-dependence derived from our measured frequency distribution are the same as those derived by Giauque and Archibald. The increase in the Debye temperature at low temperatures found for the microcrystals can be explained on the basis of our neutron spectroscopical results as an effect of the 0’ .9-dependence of the low energy part of the g (~)-function. This u’.9-dependence was found with our microcrystalline sample in contrary to the &-dependence, which is required by theory and which was indeed obtained with our bulk sample. However, the position and height of the low temperature peak of our B,(T)-curve for the microc~stalline sample do not agree very well with the results of Giauque and Archibald. This discrepancy possibly has to be attributed to the fact, that the quasiharmonic approximation is too crude to take into account the anharmonic behaviour of the MgO microcrystals. 4. Surface thermodynamic functions The lattice vibrational contribution at temperature T to any thermodynamic quantity P of one mole of a solid with the frequency distribution g (0) is given in the harmonic approximation by the following expression: corn
P(T)=3R
s
+)g(o)do,
ho
x=2kT.
(2)
0
R denotes the molar gas constant, k the Boltzmann constant. The particular expressions of p (x) are for the internal energy E, the Helmholtz free energy F, the specific heat at constant volume Co and the entropy S, respectively15) : e(x) = 2-x cothx,
(3)
f(x)=
(4)
Tln(2sinhx),
C,(x) = x2 cosech’ x , s(x) = x coth x - In (2 sinh x) .
(5) (6)
The surface contribution Ps (T) per unit area to the thermodynamic function P can be obtained directly from the surface frequency distribution function g,(o) by use of the following formula:
(7)
VIBRATIONAL
SURFACE
THERMODYNAMIC
FUNCTIONS
OF
643
MgO
-5 p
60
:.-w,.
‘.\.
.
-
‘.
----
Benson,
-.-.-
Benson,
.....
In w
.. Benson
-present
Yun ( BD T 776 ‘K 1 Yun (0, , Yun
.946
‘K)
( QD CT 1 from
Barron
1
work
I 200
400
600
600
1oW T
Fig. 4.
surface energy Es of MgO.
Vibrational
____
Ben=.
Yun
-
prosml
Work
(0,
:776’K)
200
400
600
800
1001 T
Fig. 5.
[.Kl
Vibrational
surface free energy FS of MgO.
[OKI
644
K.H.RIEDER
0
0
&mm.
Berg.
Morrison
----
0
Benson,
Yun
( Q.
-
present
work
a
_--__
----____
= 776 ‘K )
-----________ -7
200
400
600
eoo
100 T
Fig. 6.
[;K]
Surface specific heat at constant volume C,S of MgO.
2iIo
LOO
600
800
1000 T
Fig. 7.
[*Kl
Surface entropy Ss of MgO.
N denotes the molecular weight of the solid and A, the specific surface area of the microcrystalline material. In figs. 4-7 we present the results of the surface thermodynamic functions computed from the surface frequency distribution function g,(w) of fig. 2. These curves are shown together with those derived from the theory of Benson and Yun 7). This theory is based on the Debye model and essentially
VIBRATIONAL
results in a decrease
SURFACE
THERMODYNAMIC
of the Debye temperature
with surface area A compared to the Debye amount of bulk material. If the solid contains
FUNCTIONS
OF
MgO
645
OS,for a finely divided material temperature 8n of the same IZatoms in the volume V, 0;
is given by
es,= 8,
1-
(
“2 . n+u>
(8)
c is a constant depending on the elastic characteristics of the substance; its value can be taken in practice to be 0.138. Thus, effectively, a surface frequency distribution function with an extremely simple structure is used in this theory. It can be seen from figs. 4-7, that the curves for the thermodynamic surface quantities derived from the g,(w)-distribution of fig. 2 are in reasonable agreement with those calculated from the theory of Benson and Yun. However, contrary to other authors, who have used the value of the Debye temperature at absolute zero 8, (0) = 946 “K 7p16), our calculations were based on the high temperature limit of the Debye temperature e,(co) =776”K (ref. 13, see also fig. 3). It can be seen from fig. 4, that this assumption is reasonable. In this figure we have drawn the curves of the surface vibrational energy Es computed within the theory of Benson and Yun for both 0n (0) and 8, (co). A third curve (dotted) was computed on the basis of the formulas of Benson and Yun taking into account the exact en(T)-dependence given in ref. 13. As can be seen from fig. 4 this curve differs from that for 8, (co)only in a small region at low temperatures. In figs. 6 and 7 we have also included results from calorimetric measurements. The experimental points in fig. 6 originate from the specific heat measurements of Giauque and Archibald14) on microcrystalline MgO and of Barron et al.13) on bulk MgO. Taking into account, that the specific surface of the sample of Giauque and Archibald was estimated by a third author7) to be 50 m*/g, the agreement between our results and those from the specific heat measurements has to be regarded to be reasonable. In fig. 7 we have also included values of the surface entropy derived from the work of Jura and Garlandls), who have performed measurements of the heat of solution as well as of the specific heat on bulk and microcrystalline (A, =86m*/g) magnesium oxide samples. 5. Applicability
of neutron scattering techniques for studies of dynamical surface properties
It has been shown in the preceding sections, that inelastic neutron scattering experiments on bulk and microcrystalline samples of a given solid allow
646
K.H.RIEDER
the determination of the surface frequency distribution function, which governs the influence of free surfaces on the dynamical behaviour of the solid. Knowing the surface frequency distribution one can calculate those integral quantities, which depend on the vibrational properties of the surface-near atoms, for example the vibrational surface thermodynamic functions. Because of the weak interaction of neutrons with matter, neutron scattering experiments concerning surface problems have to be of type B in the classification of section 1. In practice they are restricted to microcrystals. The most serious troubles in the present work arose from the low intensity and the bad resolution of the neutron spectrometer used; to get scattering spectra with reasonable statistical accuracy, we were forced to perform spectrometer runs of six weeks per sample. Today, neutron spectrometers are built at high-flux-reactors, which allow to take scattering spectra of polycrystalline samples with good energy-resolution and high statistical reliability within a few days. Furthermore, these spectrometers are planned to allow simultaneous measurements of the inelastic scattering spectrum at many different scattering angles. This experimental possibility has an important consequence. In our work we had to use the incoherent approximation for the derivation of the phonon frequency distribution. The application of this approximation is convenient but doubtful, since the error introduced cannot be appreciated easilyis). Taking advantage of scattering data at many scattering angles, one can use the extrapolation method of Egelstaffzo) for an exact derivation of g(w). At last, within a reasonable time, scattering experiments can be performed at different sample temperatures, thus yielding detailed information on the anharmonicity of the sample material; consequently, the error introduced by using the quasiharmonic approximation in calculations of the temperature dependence of dynamic quantities can be avoided. It is obvious, that from scattering data on bulk and microcrystalline material reliable information can be drawn about the anharmonic behaviour of atoms near free surfaces. Doing neutron scattering experiments with microcrystals,we have to expect analogous difficulties as in calorimetric measurements : (i} The preparation of the samples has to be done very carefully in order to avoid uncontrollable falsifications of the results by adsorbed gases. In neutron scattering experiments one has to avoid especially contributions from adsorbed water vapour because of the large neutron scattering cross section of hydrogen. (ii) An essential point for the exact evaluation of the surface thermodynamic functions is the determination of the specific surface area of the microcrystalline samples. In our work we have used the geometrical surface area calculated from the measured mean size of the microcrystals. Jura and Gar-
VIBRATIONAL
landrs)
SURFACE
THERMODYNAMIC
FUNCTIONS
OF
MgO
647
refer in their work to the BET surface area. For our microcrystalline
sample we have determined the BET surface by adsorption of Ar at 86°K and have found it to be smaller than the geometrical surface area by a factor of about 2.6. Studies of the differences between both methods resulting in a standardization of the determination of specific surfaces would therefore be highly desirable. (iii) As has been proved in the theoretical studies of Allen et a1.1321) on noble-gas crystals, the dynamical behaviour of surface atoms is dependent on the crystallographic indices of the surface plane. Apart from a few cases, where it is possible to produce microcrystals with a simple and regular shape (for example MgO-smoke), there is no possibility in experiments with microcrystals to refer to surfaces with definite indices. Furthermore, one has to expect contributions of edge and corner atoms; these contributions can be investigated in principle by performing experiments with samples of different particle size. Acknowledgements The author is indebted to Doz. Dr. P. Koss for his interest in this work. He also wishes to express his thanks to Doz. Dr. E. M. Horl for many discussions and critical reading of the manuscript.
References 1) R. E. Allen, G. P. Alldredge and F. W. de Wette, Phys. Rev. Letters 23 (1969) 1285; 24 (1970) 301. 2) L. Dobrzynski, Surface Sci. 20 (1970) 99. 3) S. Y. Tong and A. A. Maradudin, Phys. Rev. 181 (1969) 1318. 4) S. W. Musser and K. H. Rieder, Phys. Rev. B2 (1970) 3034. 5) K. H. Rieder, Austrian-Danish-Polish Seminar on Neutron Scattering, Warsaw, 1969 (unpublished). 6) K. H. Rieder and E. M. Horl, Phys. Rev. Letters 20 (1968) 209. 7) G. C. Benson and K. S. Yun, in: The Solid-Gas Interface, Ed. E. A. Flood (Marcel Dekker, New York, 1967). 8) K. H. Rieder, Thesis, University of Vienna, 1968 (unpublished). 9) M. J. L. Sangster, G. Peckham and D. H. Saunderson, J. Phys. C (Solid State Phys.) 3 (1970) 1026. 10) R. Fuchs and K. L. Kliewer, Phys. Rev. 140 (1965) A 2076. 11) H. Boersch, J. Geiger and W. Stickel, Z. Physik 212 (1968) 130. 12) G. K. White and 0. L. Anderson, J. Appl. Phys. 37 (1966) 430. 13) T. H. K. Barron, W. T. Berg and J. A. Morrison, Proc. Roy. Sot. (London) A 250 (1955) 70. 14) W. F. Giauque and R. C. Archibald, J. Am. Chem. Sot. 59 (1937) 561. 15) A. A. Maradudin, E. W. Montroll and G. H. Weiss, in: Theory of Lattice Dynamics in the Harmonic Approximation (Academic Press, New York, 1963). 16) L. L. Levenson, CEA-BIB-12 (1969). 17) G. Jura, J. Chem. Phys. 17 (1949) 1335.
648
K. H. RIEDER
18) G. Jura and C. W. Garland, J. Am. Chem. Sot. 74 (1952) 6033. 19) F. W. de Wette and A. Rahman, Phys. Rev. 176 (1968) 784. 20) P. A. Egelstaff, in: Inelastic Scattering of Neutrons in Solids and Liquids (IAEA, Vienna, 1961) p. 25. 21) R. E. Allen and F. W. de Wette, J. Chem. Phys. 51 (1969) 4820.
SCIENCE 26 (1971) 637-648 D North-Holland
VIBRATIONAL
SURFACE
FUNCTIONS
Publishing Co.
THERMODYNAMIC
OF MAGNESIUM
OXIDE*
K. H. RIEDER Institut
&terreichische Studiengesellschaft Metallurgic, Reaktorzentrum Seibersdorf,
fiir Atomenergie GmbH, Lenaugasse IO, Vienna VIII, Austria
Received 26 February
1971
The phonon frequency distributions of bulk and microcrystalline magnesium oxide samples have been determined from inelastic neutron scattering data. The resulting surface frequency distribution allowed the calculation of the following surface thermodynamic functions: vibrational surface energy, vibrational surface free energy, surface specific heat and surface entropy. In the calculations, anharmonic effects were taken into account approximately by use of the quasiharmonic approximation. The results were found to be in reasonable agreement with the results of calorimetric measurements as well as with the values derived from the theoretical model of Benson and Yun.
1. Introduction It has been discussed recently in several theoretical investigations1-4), that the existence of free surfaces of crystals causes certain kinds of atomic vibrational modes, which are localized near the surface and hence are called surface modes. In practice the existence of surface modes becomes important in two situations 5, : (A) In some scattering experiments, where only atoms near the surface of the single crystal specimen are sampled (low energy electron diffraction, thermal energy atomic and molecular scattering). (B) In microcrystals and thin films, where the ratio of the number of surface atoms to the total number of atoms is large. By using inelastic neutron scattering techniques, we have studied the phonon frequency distributions of magnesium oxide for bulk material and for microcrystals6). Because it is the distribution of the normal modes, which determines the vibrational contribution to the thermodynamic functions of a solid (at least in the quasiharmonic approximation), we could use our measured frequency spectra of the bulk and the microcrystalline MgOsamples to extract the vibrational contribution of the free surfaces to the * This work was presented in part at the meeting of the Austrian University of Vienna, September 1970. 637
Physical Society,
638
K.H.RIEDER
thermodynamic functions of magnesium oxide. In this paper we present the results of our work and compare them with the results derived from other experimental methods as well as from the theoretical model of Benson and Yun ‘). 2. Determination
of the phonon frequency distributions
In this section we discuss the results of the neutron spectroscopic measurements with two of our MgO samples. A bulk sample consisting of single crystal fragments with a mean grain size of 50 pm corresponding to a negligible specific surface of 0.03 m2/g was used as reference sample. The microcrystalline sample was produced by thermal decomposition of basic magnesium carbonate. The mean size of the MgO microparticles was 160 A. It was determined from electron micrographs and X-ray line broadening. From the diameter value we have estimated the specific surface area to be approximately 110 m’/g. The neutron scattering experiments were performed with a rotating crystal spectrometer installed at the light-water tank type reactor ASTRA of the reactor center Seibersdorf. The primary neutron energy was 22.2 meV, the scattering angle was chosen to be 100 degrees of angle. During the scattering experiments the samples were heated up to 800°K in order to avoid scattering contributions by adsorbed gases and water vapour. In evaluating the phonon frequency distributions from the scattering spectra we have used the incoherent approximations). In these computations the second order down scattering overlap, the multiphonon contribution, the detector efficiency and the instrumental resolution have been taken into account. The transmissions of the samples were chosen to be greater than 90 %, so that multiple scattering could be neglected. Although the incoherent approximation is not justified by theoretical arguments, we had several reasons, that - at least for our experimental conditions - its application should yield reasonable results 6,s). Today we are able to compare our result for the phonon frequency spectrum of the bulk MgO with the distribution derived recently by Sangster et a1.9) by fitting the parameters of a breathing shell model to measured dispersion relations of MgO. Fig. 1 shows both curves, The agreement is quite satisfactory, if we take into consideration, that the limited instrumental resolution of our experimental apparatus did not allow to resolve the fine structure in the high energy region of the spectrum. Fig, 2 exhibits both the frequency distributions of the bulk and the microcrystalline MgO samples together with the difference spectrum g,(u)= 9 (W),icro - 9 (~)bulk, which can be regarded as surface frequency distribution function. The difference spectrum has four peaks. Peak I arises from the
VIBRATIONAL
SURFACE
THERMODYNAMIC
FUNCTIONS
OF
MgO
639
2.10-14
12 w Crad/sec3
7540 ‘3
Fig. 1. Comparison of phonon frequency distributions of bulk MgO. The full line corresponds to the result of the present work; it was derived from inelastic neutron scattering data by use of the incoherent approximation. The broken line represents the distribution computed by Sangster et al. on the basis of a breathing shell model.
acoustical branch of surface modes, which yields the Rayleigh waves of the theory of eleasticity in the long wavelength limit. Whether peak II has to be attributed to acoustical or to optical surface modes cannot be decided at the present instant. The peaks III and IV originate from optical surface vibrations. It should be mentioned, that the frequency of the optically active surface mode w, derived from the expressions of Fuchs and Kliewerl”*il) lies within peak IV - for slab-shaped as well as for spherical particles (see fig. 2). A similar surface frequency distribution function as ours was derived for slab-shaped NaCl-crystals in the theoretical work of Tong and Maradudin 3).
640
K. H. RIEDER
9
(w
2,0.10-l
-
MgO - bulk
----
MgO -microcrystals
1 \ \ \ l,(
g&w
3
6
9
12.1013
radlsoc
II 0,5.10-’
O#E
Fig. 2. Phonon frequency distributions of bulk and microcrystalline Surface frequency distribution function gS(w)= g(cO)miero-g(~)b~lk
MgO (upper part). (lower part).
3. Debye temperatures Since the temperature dependence of the Debye temperature BD is very sensitive to the shape of the phonon frequency distribution function, we want to discuss in this section the Q,(T)- curves, which were computed from our g (o)-functions by equating the exact formula for the specific heat with the one derived from the Debye model. In all our calculations we approximately took into account anharmonic
VIBRATIONAL
SURFACE
THERMODYNAMIC
FUNCTIONS
OF
641
MgO
effects by using the quasiharmonic approximation. Within this approximation, the temperature dependence of a single phonon frequency is given by 0 (T) = C&-J (1 - 3cryT),
(1)
where o,, denotes the phonon frequency at absolute zero, c( stands for the linear expansion coefficient and y for the macroscopic Grtineisen parameter. The values LX = 14.5 x 10e6 deg-’ and y = 1.5 were taken from White and Andersonle). Fig. 3 shows the results of our calculations of the temperature dependence of the Debye Bo together with the results from the calorimetric measurements of Barron et a1.13) for MgO bulk material and those from Giauque and Archibaldlb) for a microcrystalline MgO-sample with a specific surface area
0, (“K) o Barron,
Berg, Morrison
* Giauquo. -present
~microcrystallino
Archibald work
material
200
300 T (“K)
Fig. 3. Temperature dependence of the Debye characteristic temperature for bulk and microcrystalline MgO. The curves are computed from the frequency distributions of fig. 2. The points are taken from specific heat data.
642
K.H.RIEDER
of 50 cm2/g. The agreement of the curves is good for the bulk MgO. For the microcrystalline material the main features of the 19,(T)-dependence derived from our measured frequency distribution are the same as those derived by Giauque and Archibald. The increase in the Debye temperature at low temperatures found for the microcrystals can be explained on the basis of our neutron spectroscopical results as an effect of the 0’ .9-dependence of the low energy part of the g (~)-function. This u’.9-dependence was found with our microcrystalline sample in contrary to the &-dependence, which is required by theory and which was indeed obtained with our bulk sample. However, the position and height of the low temperature peak of our B,(T)-curve for the microc~stalline sample do not agree very well with the results of Giauque and Archibald. This discrepancy possibly has to be attributed to the fact, that the quasiharmonic approximation is too crude to take into account the anharmonic behaviour of the MgO microcrystals. 4. Surface thermodynamic functions The lattice vibrational contribution at temperature T to any thermodynamic quantity P of one mole of a solid with the frequency distribution g (0) is given in the harmonic approximation by the following expression: corn
P(T)=3R
s
+)g(o)do,
ho
x=2kT.
(2)
0
R denotes the molar gas constant, k the Boltzmann constant. The particular expressions of p (x) are for the internal energy E, the Helmholtz free energy F, the specific heat at constant volume Co and the entropy S, respectively15) : e(x) = 2-x cothx,
(3)
f(x)=
(4)
Tln(2sinhx),
C,(x) = x2 cosech’ x , s(x) = x coth x - In (2 sinh x) .
(5) (6)
The surface contribution Ps (T) per unit area to the thermodynamic function P can be obtained directly from the surface frequency distribution function g,(o) by use of the following formula:
(7)
VIBRATIONAL
SURFACE
THERMODYNAMIC
FUNCTIONS
OF
643
MgO
-5 p
60
:.-w,.
‘.\.
.
-
‘.
----
Benson,
-.-.-
Benson,
.....
In w
.. Benson
-present
Yun ( BD T 776 ‘K 1 Yun (0, , Yun
.946
‘K)
( QD CT 1 from
Barron
1
work
I 200
400
600
600
1oW T
Fig. 4.
surface energy Es of MgO.
Vibrational
____
Ben=.
Yun
-
prosml
Work
(0,
:776’K)
200
400
600
800
1001 T
Fig. 5.
[.Kl
Vibrational
surface free energy FS of MgO.
[OKI
644
K.H.RIEDER
0
0
&mm.
Berg.
Morrison
----
0
Benson,
Yun
( Q.
-
present
work
a
_--__
----____
= 776 ‘K )
-----________ -7
200
400
600
eoo
100 T
Fig. 6.
[;K]
Surface specific heat at constant volume C,S of MgO.
2iIo
LOO
600
800
1000 T
Fig. 7.
[*Kl
Surface entropy Ss of MgO.
N denotes the molecular weight of the solid and A, the specific surface area of the microcrystalline material. In figs. 4-7 we present the results of the surface thermodynamic functions computed from the surface frequency distribution function g,(w) of fig. 2. These curves are shown together with those derived from the theory of Benson and Yun 7). This theory is based on the Debye model and essentially
VIBRATIONAL
results in a decrease
SURFACE
THERMODYNAMIC
of the Debye temperature
with surface area A compared to the Debye amount of bulk material. If the solid contains
FUNCTIONS
OF
MgO
645
OS,for a finely divided material temperature 8n of the same IZatoms in the volume V, 0;
is given by
es,= 8,
1-
(
“2 . n+u>
(8)
c is a constant depending on the elastic characteristics of the substance; its value can be taken in practice to be 0.138. Thus, effectively, a surface frequency distribution function with an extremely simple structure is used in this theory. It can be seen from figs. 4-7, that the curves for the thermodynamic surface quantities derived from the g,(w)-distribution of fig. 2 are in reasonable agreement with those calculated from the theory of Benson and Yun. However, contrary to other authors, who have used the value of the Debye temperature at absolute zero 8, (0) = 946 “K 7p16), our calculations were based on the high temperature limit of the Debye temperature e,(co) =776”K (ref. 13, see also fig. 3). It can be seen from fig. 4, that this assumption is reasonable. In this figure we have drawn the curves of the surface vibrational energy Es computed within the theory of Benson and Yun for both 0n (0) and 8, (co). A third curve (dotted) was computed on the basis of the formulas of Benson and Yun taking into account the exact en(T)-dependence given in ref. 13. As can be seen from fig. 4 this curve differs from that for 8, (co)only in a small region at low temperatures. In figs. 6 and 7 we have also included results from calorimetric measurements. The experimental points in fig. 6 originate from the specific heat measurements of Giauque and Archibald14) on microcrystalline MgO and of Barron et al.13) on bulk MgO. Taking into account, that the specific surface of the sample of Giauque and Archibald was estimated by a third author7) to be 50 m*/g, the agreement between our results and those from the specific heat measurements has to be regarded to be reasonable. In fig. 7 we have also included values of the surface entropy derived from the work of Jura and Garlandls), who have performed measurements of the heat of solution as well as of the specific heat on bulk and microcrystalline (A, =86m*/g) magnesium oxide samples. 5. Applicability
of neutron scattering techniques for studies of dynamical surface properties
It has been shown in the preceding sections, that inelastic neutron scattering experiments on bulk and microcrystalline samples of a given solid allow
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the determination of the surface frequency distribution function, which governs the influence of free surfaces on the dynamical behaviour of the solid. Knowing the surface frequency distribution one can calculate those integral quantities, which depend on the vibrational properties of the surface-near atoms, for example the vibrational surface thermodynamic functions. Because of the weak interaction of neutrons with matter, neutron scattering experiments concerning surface problems have to be of type B in the classification of section 1. In practice they are restricted to microcrystals. The most serious troubles in the present work arose from the low intensity and the bad resolution of the neutron spectrometer used; to get scattering spectra with reasonable statistical accuracy, we were forced to perform spectrometer runs of six weeks per sample. Today, neutron spectrometers are built at high-flux-reactors, which allow to take scattering spectra of polycrystalline samples with good energy-resolution and high statistical reliability within a few days. Furthermore, these spectrometers are planned to allow simultaneous measurements of the inelastic scattering spectrum at many different scattering angles. This experimental possibility has an important consequence. In our work we had to use the incoherent approximation for the derivation of the phonon frequency distribution. The application of this approximation is convenient but doubtful, since the error introduced cannot be appreciated easilyis). Taking advantage of scattering data at many scattering angles, one can use the extrapolation method of Egelstaffzo) for an exact derivation of g(w). At last, within a reasonable time, scattering experiments can be performed at different sample temperatures, thus yielding detailed information on the anharmonicity of the sample material; consequently, the error introduced by using the quasiharmonic approximation in calculations of the temperature dependence of dynamic quantities can be avoided. It is obvious, that from scattering data on bulk and microcrystalline material reliable information can be drawn about the anharmonic behaviour of atoms near free surfaces. Doing neutron scattering experiments with microcrystals,we have to expect analogous difficulties as in calorimetric measurements : (i} The preparation of the samples has to be done very carefully in order to avoid uncontrollable falsifications of the results by adsorbed gases. In neutron scattering experiments one has to avoid especially contributions from adsorbed water vapour because of the large neutron scattering cross section of hydrogen. (ii) An essential point for the exact evaluation of the surface thermodynamic functions is the determination of the specific surface area of the microcrystalline samples. In our work we have used the geometrical surface area calculated from the measured mean size of the microcrystals. Jura and Gar-
VIBRATIONAL
landrs)
SURFACE
THERMODYNAMIC
FUNCTIONS
OF
MgO
647
refer in their work to the BET surface area. For our microcrystalline
sample we have determined the BET surface by adsorption of Ar at 86°K and have found it to be smaller than the geometrical surface area by a factor of about 2.6. Studies of the differences between both methods resulting in a standardization of the determination of specific surfaces would therefore be highly desirable. (iii) As has been proved in the theoretical studies of Allen et a1.1321) on noble-gas crystals, the dynamical behaviour of surface atoms is dependent on the crystallographic indices of the surface plane. Apart from a few cases, where it is possible to produce microcrystals with a simple and regular shape (for example MgO-smoke), there is no possibility in experiments with microcrystals to refer to surfaces with definite indices. Furthermore, one has to expect contributions of edge and corner atoms; these contributions can be investigated in principle by performing experiments with samples of different particle size. Acknowledgements The author is indebted to Doz. Dr. P. Koss for his interest in this work. He also wishes to express his thanks to Doz. Dr. E. M. Horl for many discussions and critical reading of the manuscript.
References 1) R. E. Allen, G. P. Alldredge and F. W. de Wette, Phys. Rev. Letters 23 (1969) 1285; 24 (1970) 301. 2) L. Dobrzynski, Surface Sci. 20 (1970) 99. 3) S. Y. Tong and A. A. Maradudin, Phys. Rev. 181 (1969) 1318. 4) S. W. Musser and K. H. Rieder, Phys. Rev. B2 (1970) 3034. 5) K. H. Rieder, Austrian-Danish-Polish Seminar on Neutron Scattering, Warsaw, 1969 (unpublished). 6) K. H. Rieder and E. M. Horl, Phys. Rev. Letters 20 (1968) 209. 7) G. C. Benson and K. S. Yun, in: The Solid-Gas Interface, Ed. E. A. Flood (Marcel Dekker, New York, 1967). 8) K. H. Rieder, Thesis, University of Vienna, 1968 (unpublished). 9) M. J. L. Sangster, G. Peckham and D. H. Saunderson, J. Phys. C (Solid State Phys.) 3 (1970) 1026. 10) R. Fuchs and K. L. Kliewer, Phys. Rev. 140 (1965) A 2076. 11) H. Boersch, J. Geiger and W. Stickel, Z. Physik 212 (1968) 130. 12) G. K. White and 0. L. Anderson, J. Appl. Phys. 37 (1966) 430. 13) T. H. K. Barron, W. T. Berg and J. A. Morrison, Proc. Roy. Sot. (London) A 250 (1955) 70. 14) W. F. Giauque and R. C. Archibald, J. Am. Chem. Sot. 59 (1937) 561. 15) A. A. Maradudin, E. W. Montroll and G. H. Weiss, in: Theory of Lattice Dynamics in the Harmonic Approximation (Academic Press, New York, 1963). 16) L. L. Levenson, CEA-BIB-12 (1969). 17) G. Jura, J. Chem. Phys. 17 (1949) 1335.
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18) G. Jura and C. W. Garland, J. Am. Chem. Sot. 74 (1952) 6033. 19) F. W. de Wette and A. Rahman, Phys. Rev. 176 (1968) 784. 20) P. A. Egelstaff, in: Inelastic Scattering of Neutrons in Solids and Liquids (IAEA, Vienna, 1961) p. 25. 21) R. E. Allen and F. W. de Wette, J. Chem. Phys. 51 (1969) 4820.