Orientational order parameter in CO₂-based alloys with rare gases from THEED data: pure CO₂

Using the transmission high-energy electron diffraction technique, we studied the temperature dependence of the diffraction reflection intensities from the solid phase of CO₂. To deduce absolute values of the orientational order parameter, we modified the existing reconstruction method to account...

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Datum:	2007
Hauptverfasser:	Danchuk, V.V., Solodovnik, A.A., Strzhemechny, M.A.
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Veröffentlicht:	Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України 2007
Schriftenreihe:	Физика низких температур
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spelling	irk-123456789-1217952017-06-17T03:03:17Z Orientational order parameter in CO₂-based alloys with rare gases from THEED data: pure CO₂ Danchuk, V.V. Solodovnik, A.A. Strzhemechny, M.A. Classical Cryocrystals Using the transmission high-energy electron diffraction technique, we studied the temperature dependence of the diffraction reflection intensities from the solid phase of CO₂. To deduce absolute values of the orientational order parameter, we modified the existing reconstruction method to account for the triatomic shape of the molecule. It is shown that for triatomics solids a higher-rank order parameter, n4, is important and cannot be neglected. Application of the modified approach yielded n and n₄ values as a function of temperature over the range from 6 to 70 K. Both orientational order parameters proved to be a nonmonotone function of temperature. 2007 Article Orientational order parameter in CO₂-based alloys with rare gases from THEED data: pure CO₂ / V.V. Danchuk, A.A. Solodovnik, M.A. Strzhemechny // Физика низких температур. — 2007. — Т. 33, № 6-7. — С. 783-789. — Бібліогр.: 16 назв. — англ. 0132-6414 PACS: 61.14.–x; 61.14.Dc; 75.40.Gb http://dspace.nbuv.gov.ua/handle/123456789/121795 en Физика низких температур Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
topic	Classical Cryocrystals Classical Cryocrystals
spellingShingle	Classical Cryocrystals Classical Cryocrystals Danchuk, V.V. Solodovnik, A.A. Strzhemechny, M.A. Orientational order parameter in CO₂-based alloys with rare gases from THEED data: pure CO₂ Физика низких температур
description	Using the transmission high-energy electron diffraction technique, we studied the temperature dependence of the diffraction reflection intensities from the solid phase of CO₂. To deduce absolute values of the orientational order parameter, we modified the existing reconstruction method to account for the triatomic shape of the molecule. It is shown that for triatomics solids a higher-rank order parameter, n4, is important and cannot be neglected. Application of the modified approach yielded n and n₄ values as a function of temperature over the range from 6 to 70 K. Both orientational order parameters proved to be a nonmonotone function of temperature.
format	Article
author	Danchuk, V.V. Solodovnik, A.A. Strzhemechny, M.A.
author_facet	Danchuk, V.V. Solodovnik, A.A. Strzhemechny, M.A.
author_sort	Danchuk, V.V.
title	Orientational order parameter in CO₂-based alloys with rare gases from THEED data: pure CO₂
title_short	Orientational order parameter in CO₂-based alloys with rare gases from THEED data: pure CO₂
title_full	Orientational order parameter in CO₂-based alloys with rare gases from THEED data: pure CO₂
title_fullStr	Orientational order parameter in CO₂-based alloys with rare gases from THEED data: pure CO₂
title_full_unstemmed	Orientational order parameter in CO₂-based alloys with rare gases from THEED data: pure CO₂
title_sort	orientational order parameter in co₂-based alloys with rare gases from theed data: pure co₂
publisher	Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
publishDate	2007
topic_facet	Classical Cryocrystals
url	http://dspace.nbuv.gov.ua/handle/123456789/121795
citation_txt	Orientational order parameter in CO₂-based alloys with rare gases from THEED data: pure CO₂ / V.V. Danchuk, A.A. Solodovnik, M.A. Strzhemechny // Физика низких температур. — 2007. — Т. 33, № 6-7. — С. 783-789. — Бібліогр.: 16 назв. — англ.
series	Физика низких температур
work_keys_str_mv	AT danchukvv orientationalorderparameterinco2basedalloyswithraregasesfromtheeddatapureco2 AT solodovnikaa orientationalorderparameterinco2basedalloyswithraregasesfromtheeddatapureco2 AT strzhemechnyma orientationalorderparameterinco2basedalloyswithraregasesfromtheeddatapureco2
first_indexed	2025-07-08T20:32:06Z
last_indexed	2025-07-08T20:32:06Z
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fulltext	Fizika Nizkikh Temperatur, 2007, v. 33, Nos. 6/7, p. 783–789 Orientational order parameter in CO2-based alloys with rare gases from THEED data: pure CO2 V.V. Danchuk, A.A. Solodovnik, and M.A. Strzhemechny B. Verkin Institute for Low Temperature Physics and Engineering of the National Academy of Sciences of Ukraine 47 Lenin Ave., Kharkov 61103, Ukraine E-mail: strzhemechny@ilt.kharkov.ua Received February 7, 2007 Using the transmission high-energy electron diffraction technique, we studied the temperature depend- ence of the diffraction reflection intensities from the solid phase of CO2. To deduce absolute values of the orientational order parameter, we modified the existing reconstruction method to account for the triatomic shape of the molecule. It is shown that for triatomics solids a higher-rank order parameter, �4, is important and cannot be neglected. Application of the modified approach yielded � and �4 values as a function of tem- perature over the range from 6 to 70 K. Both orientational order parameters proved to be a nonmonotone function of temperature. PACS: 61.14.–x Electron diffraction and scattering; 61.14.Dc Theories of diffraction and scattering; 75.40.Gb Dynamic properties (dynamic susceptibility, spin waves, spin diffusion, dynamic scaling, etc.). Keywords: electron diffraction, rare gases alloys, orientational order. 1. Introduction The problem of the orientational order in mixed cryo- crystals consisting of two components, one of which is a lin- ear molecule and the other is a rare gas atom, still remains an important issue for random systems. Usually, orientational order is quantitatively described by the orientational order parameter (OOP), which in a solid made up of linear mole- cules is defined as � �� P2(cos ) , (1) where P x2( ) is the Legendre polynomial and � is the angle the molecule in a specific sublattice of an orientationally ordered structure makes with the axis of the preferable ori- entation of this molecule. The resonance methods, which are traditionally used to determine �, fail for mixed sys- tems and, actually, the only constructive option is to utilize diffraction data for that purpose. In our earlier paper de- voted to this problem [1] we outlined the general approach for a specific case of Ar–CO2 mixed crystals. This ap- proach has been developed to a quite high precision for diatomics and then successfully applied for the reconstruc- tion of � in �-N 2 from our own powder x-ray diffraction measurements [2] and in the orientationally ordered almost pure para-deuterium [3] from the old neutron diffraction data of Yarnell et al. [4]. As we will show below, applica- tion of this approach even to symmetric triatomics requires further development; electron diffraction data were used to tentatively reconstruct � in the noncubic diatomic crystal of �-oxygen [5]. Since some new problems, which are in- herent even in pure triatomic CO2, need detailed consider- ation, in this paper we start with the orientational order in pure CO2 crystal. We expected that our approach might yield qualitatively new results compared to those that can be obtained on diatomic crystals. At the same time, consid- ering the fact that transmissive high-energy electron dif- fraction data were not so far obtained with the aim of ob- taining reliable integrated intensities, we made new accurate transmission high-energy electron diffraction measurements on solid CO2 over the temperature range from 5 to 70 K, close to the sublimation threshold. 2. Theory In the theory developed below we will make use of the fact that the structure of CO2 over the entire domain of its sold state is Pa3. The integrated intensity of scattered electrons can be represented in the form [6,7]. © V.V. Danchuk, A.A. Solodovnik, and M.A. Strzhemechny, 2007 I P f FT s s i s� � � � � � � ( ) ( ) ( ) ( )� � q q q qR e 2 2 , (2) where summation runs over positions R s occupied by all C and O atoms in the elementary cell; q is the momentum transfer vector; � is the diffraction angle; ( )� accounts for the geometric factor of data taking; P( )q is the repe- tition factor for a particular reflection; fT ( )q is the Debye–Waller temperature factor; and Fs is the structure amplitude. Let us first consider the scattering amplitude Fs . Tak- ing into account the fact that the carbons occupy the sites of a fcc array and summing over pairs of oxygen atoms in each molecule (sublattice) we obtain for the structure amplitude F f f i c c c( ) [ ( ) ( ) cos ( )]q q q qw qR� � e C O 2 2 � . (3) Here f C ( )q and f O( )q are the atomic scattering factors for the carbon and oxygen atoms, respectively; summa- tion runs over the four sublattices c of the Pa3 structure with Rc being the centers of the four molecules in the four sublattices; wc is the momentary direction of the unit vector along the respective molecular axis in sublattice c; �� 2 d a T/ ( ) , (4) where d �11599. � is the C � 0 bond length in the CO2 molecule [8] and a T( ) is the lattice parameter, which is temperature dependent. At temperatures close to zero, the powder x-ray lattice parameter [9] is a � 5 5542. � and �1 31214. . Since the summing in Eq. (3) goes over all el- ementarily cells, the integrated intensity in Eq. (2) will contain a self-averaged value of the cosine. This averaged cosine can be expanded [1] in spherical harmonics [10], which will yield for the scattering amplitude F e f i c c( ) [ ( )q q qR� � 2� C � � �� 2 2 1 0 f i l j ql l l l l cO even ( ) ( ) ( ) ( ( ) ( ) )]q C n C w . (5) Here the brackets denote thermodynamic averaging; j yl ( ) are the spherical Bessel functions; summing in the inner sum is over even l; C lm( )n and C nl ( ) are Racah’s spherical harmonics and the respective spherical tensors [8]; n and q are unit vector and the length of the momen- tum transfer: q n� q ; and ( ( ) ( )) ) ( )C n C w (n wl l c lm m l lm cC C� � � � , (6) is the scalar product of two spherical tensors of rank l. The geometric factor ( )� in Eq. (2) is determined by the geometry and intensity registration method. In our case [12], this factor is proportional to the specific inter-plane separation squared: ( ) / sin � � � � � �d a Hhkl 2 2 2 2 24 , (7) where H h k l2 2 2 2� � � and� is the electron wavelength. The temperature factor fT ( )q in Eq. (2) is chosen in the standard Debye approximation [12], ideally applica- ble for our case with the Debye temperature � equal [13] to ( . . )151 8 1 5� K. The main reason why the approach under discussion was efficient and simple to apply for all diatomic crystals such as nitrogen [2], deuterium [3], or oxygen [5,11] is the small value of the parameter as defined in Eq. (4), which permitted us to leave only the first two terms in the expansion in Eq. (5) to stay within an accuracy better than 2–3%. In a triatomic crystal such as CO2 the parameter is roughly twice as large, which does not allow so short a truncation. Therefore, we have to consider higher terms, the thermodynamic interpretation of which is not so sim- ple as that of the rank-2 term, which yields the standard orientational order parameter �. Let us start our analysis of Eq. (5) with the first two terms of the series in l. It can be shown [2] (see Appendix A) that the l � 2 term after averagingr yields precisely the orientational order parameter as defined in Eq. (1): � � �C Cm c m c2 2( ) ( )w m� , (8) where mc is the unit vectors along the preferable orientation of the molecule in sublattice c, i.e., along the corresponding cube diagonal in the Pa3 structure. Moreover [10], ( ( ) ( )) ( )C n C m n m2 2 2� � �c cP , (9) where P2 is again the Legendre polynomial. Finally, the first two terms yield [2]: F Fs s ( ) ( )0 2� � � � � 4 8 10 0f f j q f j C O O regular reflections,( ) ( ) ( ) ( ) q q q � � 2 2( ) ( )�q Q q superstructure reflections, � � � (10) where we define Q PN i c N c c( ) ( )q n m qR� � e 2� . (11) The term with l � 4 in Eq. (5) differ from the previous that with l � 2 in that not only the m � 0 component in the intrinsic reference frame will survive (see Appendix A). making use of Eq. (A.1) we can write the third ( )l � 4 term of the scattering amplitude in the form F f j q Qs ( ) ( ) ( ) { ( )4 4 4 418� �O q q� � 784 Fizika Nizkikh Temperatur, 2007, v. 33, Nos. 6/7 V.V. Danchuk, A.A. Solodovnik, and M.A. Strzhemechny � � � � 43 2 43 0 43 0e i c c C C qR n n[ ( ) ( )]} (12) with the anisotropy parameter � 43 defined in Eq. (A.5). The corresponding expression for the l � 6 contribu- tion to the scattering amplitude can be obtained in a simi- lar way with that difference that the final formula analo- gous to Eq. (12) will contain also terms with m � � 6: F f j q Qs ( ) ( ) ( ){ ( )6 6 6 626� � �O q q� � � � � � � 63 2 63 0 63 0e i c c C C qR n n[ ( ) ( )] � � � � 66 2 66 0 66 0e qR n nc c C C[ ( ) ( )]} , (13) where �6 is the rank-6 order parameter, defined similarly as �4 , and the two possible rank-6 anisotropy parameters are � 63 63 0 63 0 1 2 � � � � � �[ ( ) ( ) ]C Cc cw w , (14) � 66 66 0 66 0 1 2 � � � � � �[ ( ) ( ) ]C Cc cw w . (15) The presence of the anisotropy parameters in addition to a few orientational order parameters seems to consider- ably complicate the task of reconstructing all these quan- tities from diffraction intensities. However, we will fur- ther show that the anisotropy parameters in the triatomic cryocrystal CO2 are negligible compared to the respective order parameters and can be thus discarded (see also Ap- pendix B). This assertion will be verified when making the relevant calculations. 3. Experimental The structure of thin CO2 films was studied by trans- mission high-energy electron diffraction (THEED) with the aid of an EG-100A Electronograph equipped with a special liquid helium cryostat. The film samples were pre- pared in situ inside the column of the instrument by de- positing the CO2 gas at room temperature onto a compos- ite substrate (cooled down to 45 K), which consisted of two parallel strips made of different materials, amorphous carbon and fine grained polycrystalline aluminum. This double-strip substrate was placed onto the sample holder. In order to ensure a better thermal contact between holder and substrate we used indium spacers. The ( )115 5� � thick Al film served as an internal reference for the accu- rate determination of inter-plane distances. The carbon film was utilized to obtain accurate integrated intensities, avoiding the overlap of strong CO2 lines with reflections from aluminum. The purity of the source co gas was better than 99.9%. Before deposition the substrate was kept for 10 min at 45 K, whereupon then a sample was grown at that temper- ature. Then the temperature was raised to 55 K at a rate of 1 K/min and the sample was annealed for 5 min. After that the temperature was brought down to 5.6 K at a rate of 2 K/min and the sample was kept at that temperature for 10 min. The diffraction patterns were recorded at 45 K right after annealing and then during warm up at nine dif- ferent temperatures from 6 to 70 K. The time needed for equilibration in every temperature point was at all times about 3 min. In order to reduce effects of the electron beam, we employed the highest accelerating voltage pos- sible of 80 kV, thereby reducing the duration of the time an electron interacts with the sample. For the same pur- pose we used a Faraday cylinder, which shut the beam off during temperature stabilization processes and ensured extra control of the beam stability. The photo takings were analyzed on an IZA-2 optical comparator; the reflec- tion intensities were determined after densitometric runs using an IFO-451 photometric device. The sample temperature was varied with the aid of a dc heater at a fixed helium pumping rate. Temperature mea- surements were performed with a semiconductor thermome- ter, which was in good thermal contact with the substrate. The temperature measurement accuracy was not worse than � 0 25. K over the temperature range 5 to 75 K. The CO2 film thickness, which was typically about ( )450 50� �, was first controlled and subsequently evaluated from the follow- ing data: (i) the pressure of the gas in the filling container (close to 200 Torr as measured by an oil manometer); (ii) de- position time (about 30 s); (iii) the amount of the material deposited (an equivalent of about 1 Torr pressure drop); (iv) the orientation of the gas jet respective to substrate (usually, at an angle of ( )45 5� �); (v) the distance between the filling pipe opening and the substrate (typically about 18 mm). Such a regime allowed us to grow polycrystalline CO2 films with the characteristics that were close to those of «bulk» carbon dioxide crystals. During experiments, when the helium vessel of the cryostat was filled to a sufficient level, the pressure in the column did not exceed 3 10 7� � Torr. To prevent the con- densation of residue gases, the working chamber was pro- tected by two screens, one at the liquid-nitrogen and the other, at liquid-helium temperature, which actually were additional cryopumps. During the whole diffraction ex- periment both intensity and focusing of the electron beam did not change. 4. Calculations In this Section we describe some details of our method employed to reconstruct the absolute values of the orientational order parameters. This is necessary for the assessment of the validity of the method. Orientational order parameter in CO2-based alloys with rare gases from THEED data: pure CO2 Fizika Nizkikh Temperatur, 2007, v. 33, Nos. 6/7 785 4.1. Truncation accuracy evaluation Exactly as done for diatomic cryocrystals [2,3,5], for the first step we do as follows. We set all the molecules rigidly oriented along the respective axes in the sublattices, which corresponds to the case � � �� 4 6 1 and all the off-diagonal orientational order parameters � ik are zero. The relevant exact expressions as functions of the pertaining crystallographic and diffraction parameters for this case are well known [15]. The corresponding values, designated for particular reflections j as F j ex ( ), are used for comparison with the respective approximations resulting from a particular truncation of the series in Eq. (5). These evaluations are summarized in Table 1. The notation is as follows. The scattering amplitude terms Fs n( ) are defined in Eqs. (10) through (13); � N j s n n N F ( ) ( )� � 0 (16) and � � N j N j jF ( ) ( ) ( )/� �ex 1 is the mismatch for a particular approximation of rank N for reflection j. As one can see from Table, the approximation that ac- counts for the terms up to N � 4 is acceptable within a 2% error only for the four reflections with the least indices, viz., 111, 200, 210, and 211. To correctly calculate the scattering amplitudes for the other five reflections men- tioned in Table, account of the term with N � 6 is neces- sary. Considering also a larger uncertainties in experi- mentally determined intensities for those five reflections, we did not use them in the subsequent reconstruction of absolute order parameter values. 4.2. Reconstruction of order parameters Thus, the reconstruction procedure dealt with the four small-angle reflections. We sought two unknown values, the orientational order parameters �2 and �4 , treating �6 as known and equal to either 1 or, within a more precise approach, to a value found as explained in Appendix B. These two values were found as solutions to the following two equations, which are intensity ratios of reflections i and j represented by the truncated expansions of the scat- tering amplitudes with explicit unknowns �2 and �4 . These theoretical ratios were equated to the values R i j( / ) known from experimental data: I I i j ( ) ( ) � � f T P F F F F f T P i i i i i i i j j ( ) \| \| ( ) ( ) ( ) ( ) ( ) 0 2 2 4 4 6 6 2� � �� j j j j j F F F F R i j \| \| ( / ) ( ) ( ) ( ) ( ) 0 2 2 4 4 6 6 2� � � � � � � (17) Here all quantities , , ,P F fn were defined above in Eqs. (2), (10)–(13). Since �6 is treated as known, a set of two equations like Eq. (17) involving at least three differ- ent reflections are needed to find �2 and �4 . 5. Results and discussion We have carried out six series of experiments, produc- ing over 100 photo plates. In Fig. 1 we present a typical electron diffraction pattern. We used amorphous carbon for the substrate material and made it as thin as possible in order to reduce the incoherent scattering background, which is quite insignificant in the pattern depicted. Thus, 786 Fizika Nizkikh Temperatur, 2007, v. 33, Nos. 6/7 V.V. Danchuk, A.A. Solodovnik, and M.A. Strzhemechny Table 1. Assessment of the accuracy of the approximating truncations for particular reflections observed in diffraction experiment. The lat- tice parameter a is that at T � 0. Reflection j Exact value [15] F j ex ( ) � 4 ( )j � 4 ( )j (%) � 6 ( )j � 6 ( )j (%) 111 4.3392708 4.3626179 0.5380426 4.3392043 – 0.0015334 200 2.5809769 2.6102143 1.1328012 2.5801121 – 0.0335058 210 – 3.084531 – 3.1331715 1.57692 – 3.0862019 0.0541703 211 –2.106178 – 2.0634765 – 2.0274837 – 2.1032798 –0.1376497 220 1.9100776 1.6408049 – 14.097477 1.9047396 – 0.2794693 311 0.3772583 0.431169 14.2901061 0.3907192 3.5680792 222 1.5078859 2.1396547 41.8976518 1.5049553 –-0.194354 320 – 2.2477249 – 2.4510779 9.0470569 – 2.2344827 – 0.5891404 321 – 1.5319902 – 1.4051779 – 8.2776198 – 1.5263462 – 0.3684124 the inevitable small-angle halo is not seen. Here we also mark the absence of any appreciable effects of texture, multiple scattering. The rings are perfectly uniform and nicely circularly shaped, which is evidence of a high sam- ple quality. Diffraction patterns like that in Fig. 1 were next run through a standard densitometric procedure. The result- ing patterns in electronic form were reduced to leave only fractions that contained the four reflections of in- terest, namely, 111, 200, 210, and 211. These shortened patterns were further processed to resolve them into in- dividual reflections in order to evaluate their intensities. The fitting was performed using the Lorentzian line shape, which yielded a much better result compared to the Gaussian one, which also suggests a good sample quality. The final fitting is shown in Fig. 2. The experi- mental data run nicely along the fitting curve, the devia- tion not exceeding 1%. Ratios of these intensity values were further utilized to reconstruct the absolute values of the orientational order parameters �2 and �4 . Since we dealt only with four reflections, by combining three of them in the respective set of two equations like Eq. (17), we could obtain six independent pairs of �2 and �4 values. The average of those four values was taken to be its true value, reconstructed from diffraction experi- ments. The temperature dependence of the orientational order parameter �2, as determined according to the procedure just described, is shown in Fig. 3 as empty squares. Almost all the values exceed unity, which is physically impossible. This fact suggests that we miss some factor that could bring the values to a physically reasonable level. One of such factors could be the fact that we use an isotropic Debye–Waller temperature factor f Ti ( ), although in fact it might be anisotropic. However, variation of f Ti ( ) did not lead to appreciable changes in the resulting �2 values. There is another factor of importance. As mentioned above, for the half of the internuclear spacing in the mole- cule we employed the value d �11599. � derived from op- tical measurements [9]. We argue that instead we should use an effective value deff which is by a few percent larger than d. The argument leans on the speculation that elec- trons are scattered on the electron cloud, the size of which exceeds the internuclear spacing. If we put d deff �1 04. , we arrive at the �2 values shown as solid circles in the same Fig. 3. Such a modification does not affect the general be- havior of the �2( )T dependence, which is nonmonotone with a distinct dip at about T � 20 K. We are confident that this behavior reflects some changes that take place in the rotational and vibrational dynamics of the co crystal. Figure 4 represents the temperature dependence of the rank-4 orientational order parameter �4 . As with �2 in Fig. 3, the estimated values are too high; in order to bring them down we used the same renormalized value d deff � 1 04. for the effective internuclear distance in the molecule. Again, the temperature dependence of �4 ex- hibits a dip close to 20 K. Compared to �2 as a function of temperature, the rank-4 order parameter shows a much steeper decrease, which is in accord with our reasonings in Appendix B. We should stress that the dip minimum at around 20 K and the maximum at 60 K in both �2 and �4 as a function of temperature were persis- tent for all series. Orientational order parameter in CO2-based alloys with rare gases from THEED data: pure CO2 Fizika Nizkikh Temperatur, 2007, v. 33, Nos. 6/7 787 Fig. 1. THEED pattern from solid CO2 on an amorphous car- bon substrate; T � 60 K. In te n si ty , ar b . u n it s 2000 1500 1000 500 0 1.8 2.1 2.4 2.7 3.0 Fig. 2. An example of the intensity distributions from a solid CO2 sample at T � 60 K. long 0 10 20 30 40 50 60 70 80 T, K 1.10 1.05 1.00 0.95 0.90 0.85 0.80 � 2 � 2 �� Fig. 3. Temperature dependence of the orientational order pa- rameter �2 of solid CO2. The effective 2d values: 2.3198 � (squares) and 2.4198 � (circles). It should be noted that the anomalies observed in this study correlate with the anomalies in the dilatometric expansivity data by Manzhelii et al. [13]. The heat capaci- ty versus T curve also has an inflection, which might re- flect a nonmonotone variation of �2 with T . The rank-4 orientational order parameter �4 is an important para- meter, which determines the correlative and rotational- unharmonic effects in the librational subsystem, which are known to be large in triatomic molecular cryocrystals [16]. Detailed comparison of experimental �4 values with theoretical estimates is needed. We must also call attention to the temperature depend- ence of the cubic lattice parameter, which have been ob- tained in these measurements and depicted in Fig. 5. The values obtained by THEED are somewhat larger than those measured by powder x-ray diffraction, which is quite common, especially at low temperatures. But there is a kink at the lowest temperature, which is within the er- ror bars shown but which was persistently observed for all runs. 6. Conclusions The lattice parameter as well as the integrated reflec- tion intensities of solid CO2 have been determined by transmissive high-energy electron diffraction at tempera- tures from 6 to 70 K. The method of reconstruction of absolute orientational order parameter values has been modified for the case of the triatomic CO2 cryocrystal. It is shown that, because of the comparatively large length of the CO2 molecule, a consistent description of the integrated intensity data can be obtained only if an orientational order parameter of a higher (fourth) rank, �4 is taken into consideration to- gether with the standard (rank-2) order parameter �2. Both orientational order parameters �2 and �4 in pure CO2 have been reconstructed for temperatures from 6 to 70 K. A self-consistent reconstruction of these values was possible only if the internuclear distance in the CO2 mole- cule, 2d, is taken to be slightly (by about 4%) longer than determined spectroscopically. The order parameter �4 , determined for the first time in this work, is responsible for correlative effects in the thermodynamics and rota- tional dynamics of solid CO2. The authors are grateful to N.N. Galtsov for helping with data processing. Appendix A: Transformations of spherical harmonics To average A Cm m c2 2� � �( ) ,w it is convenient to rotate the reference frame so that its axis z would be directed along the 3-fold axis (one of the cube diagonals) of the Pa3 structure, which is the natural frame for sublattice c. Then the value in the old frame is expressed via the value in the new, natural frame as � � � � �� C D Cm c m m m m c2 2 2 0( ) ( ) ( ), w w� . (A.1) Here � is the set of Eulerian angles that effectuate the rotation; Dm m, * ( )� 2 � is the Wigner functions [8]; wc0 is the unit vector along the molecular axis in the new reference frame. All components of � ��C m c2 0( )w except that with m � 0 will average to zero from symmetry considerations, namely, because the maximum value of m, which is 2, is less than the order of the symmetry axis. The m � 0 aver- age is exactly �as defined in Eq. (1). Now, since [8] D Cm m c, * ( ) ( )0 2 2� � m , (A.2) where mc is the unit vector along the cube diagonal chosen in the «old» reference frame, which brings us to Eq. (8). On the example of the term with l � 4 we show the emerging distinctions in the terms of higher orders. Now, in the expression � ��C m c4 0( )w not only the � �m 0 term is nonzero. The presence of the 3-fold axis allows terms with m � � 3 to survive as well. Since the scalar 788 Fizika Nizkikh Temperatur, 2007, v. 33, Nos. 6/7 V.V. Danchuk, A.A. Solodovnik, and M.A. Strzhemechny 0 10 20 30 40 50 70 8060 1.2 0.8 0.6 0.4 T, K �� 1.0 Fig. 4. Temperature dependence of the orientational order pa- rameter �4 of solid CO2. The effective 2d values: 2.3198 � (squares) and 2.4198 � (circles). L at ti ce p ar am et er 5.59 5.58 5.57 5.56 5.55 0 15 30 45 60 75 T, K our data x-ray data Fig. 5. Temperature dependence of the lattice parameter. The powder x-ray data are from Krupskii et al. [10]. product ( ( ) ( ))C n C w4 4� c is invariant under rotation, we may consider this product in the intrinsic frame for the respective sublattice. Finally, this scalar can be recast to (we provide vectors in the intrinsic reference frame with subscript 0): C Cm m m c4 4 * ( ) ( ) � � �n w � � � � � � C C Pm m m c c4 0 4 0 4 4 * ( ) ( ) ( )n w n m� � � � � � � � � � [ ( ) ( ) ] [ ( ) ( )]* * C C C C c c43 0 43 0 43 0 43 0 2 w w n n � � � � � � � � [ ( ) ( ) ] [ ( ) ( )]* * C C C C c c43 0 43 0 43 0 43 0 2 w w n n . (A.3) By suitably choosing the origin of the variable we are able to nullify the last term in Eq. (A.3) and bring it to the form C Cm m m c4 4 * ( ) ( ) � � �n w � � � �� 4 4 43 43 0 43 0P C Cc( ) [ ( ) ( )]n m n n . (A.4) Here �4 is a rank-4 generalization of the orientational or- der parameter, Eq. (1), and we introduced a real-valued quantity with the proper subscripts, � 43 43 0 43 0 2 � � � � � �C Cc c( ) ( ) , w w (A.5) which characterizes the anisotropy of librations respec- tive to the intrinsic frame. Appendix B: Evaluation of order anisotropy parameters Let us evaluate a typical anisotropy parameter, for example, � 43. The distribution function !( , )x of the orien- tational variables x � � and , which in the intrinsic refer- ence frame is centered around � � 0, can be expanded as ! ! !( , ) ( ) ( ) cos ( )x x x nn n n � � � 0 , (B.1) where, by symmetry, any positive nonzero n is a multiple of 3. It is self-evident that the amplitudes !n x( ) do not ex- ceed in magnitude the main isotropic amplitude !0( )x . In classical cryocrystals made up of linear molecules, in which the distribution is sharp around the value x �1, !0( )x can be approximated as ! "� " 0 3 2 2 2 1 1 ( ) exp ( ) / x x � � �# $ % % & ' ( ( , (B.2) where " is the rms libration amplitude, which is small even in the ordered phases of nitrogen and carbon monoxide, not to mention co under discussion, i.e., the characteristic devia- tion � � �1 � is small compared to unity. If � is sufficiently small, then � )4 1 10 3� � ( / ) . Here we make a general re- mark that the higher order parameters are more sensitive to deterioration of the orientational order. The quantity � is easily related with ": � � �3 1 2"� / . Averaging of C 43( )w � � P x i 43 3( )e , where P x43( ) is the respective associated Legendre polynomial, gives � �C x43 2146( ) .� � . In N2 close to the orientational transition point [14], � � 0 27. so that � �C x43 0106( ) .� is substantially smaller than � � 0 73. or smaller even than �4 0 39� . . There are solid grounds to ex- pect that in co � �C x43( ) will be more negligible compared to the rank-4 isotropic order parameter �4 . 1. M.A. Strzhemechny, A.A. Solodovnik, and S.I. Kovalenko, Fiz. Nizk. Temp. 24, 889 (1998) [Low Temp. Phys. 24, 669 (1998)]. 2. N.N. Galtsov, O.A. Klenova, and M.A. Strzhemechny, Fiz. Nizk. Temp. 28, 517 (2002) [Low Temp. Phys. 28, 365 (2002)]. 3. V.V. Danchuk, N.N. Galtsov, M.A. Strzhemechny, and A.I. Prokhvatilov, Fiz. Nizk. Temp. 30, 163 (2004) [Low Temp. Phys. 30, 118 (2004)]. 4. J.L. Yarnell, R.L. Mills, and A.F. Schuch, Fiz. Nizk. Temp. 1, 760 (1975) [Sov. J. Low Temp. Phys. 1, 366 (1975)]. 5. A.A. Solodovnik, V.V. Danchuk, and M.A. Strzhemechny, J. Non-Cryst. Solids 352, 4331 (2006). 6. M.A. Krivoglaz, Theory of Scattering of x-rays and Ther- mal Neutrons by Real Crystals, Nauka, Moscow (1967) [in Russian]. 7. G.B. Bokii and M.A. Porai-Koshitz, X-ray Structure Anal- ysis, Moscow University (1964) [in Russian]. 8. L.V. Gurvich et al., Thermodynamic Properties of Individ- ual Substances, Vol. 2, Book 1, Nauka, Moscow (1979). 9. I.N. Krupskii, A.I. Prokhvatilov, A.I. Erenburg, and A.S. Barylnik, Fiz. Nizk. Temp. 8, 533 (1982) [Sov. J. Low Temp. Phys. 8, 263 (1982)]. 10. D.A. Varshalovitch, A.N. Moskalev, and V.K. Khersonskii, Theory of Angular Momentum, World Scientific, Singapore (1988). 11. M.A. Strzhemechny, J. Low Temp. Phys. 139, 581 (2005). 12. B.K. Vainshtein, Structural Analysis by Electron Diffrac- tion, Pergamon, Oxford (1964). 13. V.G. Manzhelii, A.M. Tolkachev, M.I. Bagatskii, and E.I. Voitovich, Phys. Status Solidi B44, 39 (1971). 14. T.A. Scott, Phys. Rep. C27, 89 (1976). 15. International Tables for x-ray Crystallography, vol. 1, Sym- metry Groups, N.F.M. Henry and K. Lonsdale (eds.), Kynoch Press, Birmingham (1969). 16. The Physics of Cryocrystals, V.G. Manzhelii, Yu.A. Frei- man, M.L. Klein, and A.A. Maradudin (eds.), AIP Press, Woodbury (1997). Orientational order parameter in CO2-based alloys with rare gases from THEED data: pure CO2 Fizika Nizkikh Temperatur, 2007, v. 33, Nos. 6/7 789

Orientational order parameter in CO₂-based alloys with rare gases from THEED data: pure CO₂

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