Materials Transactions, Vol. 47, No. 3 (2006) pp. 677 to 681 Special Issue on Shape Memory Alloys and Their Applications #2006 The Japan Institute of Metals Root-Mean-Square Displacements of Atoms in the B2-Phase of Titanium Nickelide Vyacheslav M. Gundyrev and Vitaly I. Zel’dovich Institute of Metal Physics, Ural Division, Russian Academy of Science, S. Kovalevskaya Str. 18, Ekaterinburg, 620041 Russia Using of MoK radiation, on a single crystal of a titanium nickelide intensities of 15 structural and 11 superstructural reﬂections of the B2phase are measured. Structural factors of scattering for these reﬂections are calculated, and root-mean-square displacements of atoms of nickel and atoms of titanium from positions of equilibrium are determined. The mean square of displacements of atoms of nickel is equal hu2 iNi ¼ ð8:7 0:6Þ 104 nm2 , atoms of titanium hu2 iTi ¼ ð3:9 0:3Þ 104 nm2 . (Received September 20, 2005; Accepted November 28, 2005; Published March 15, 2006) Keywords: titanium nickelide, single crystal, X-ray study, intensity reﬂections, structural and superstructural reﬂections, root-mean-square displacements of atoms 1. Introduction The high-temperature B2-phase in alloys of a titanium nickelide (structure CsCl) undergoes martensitic transformations at cooling. The structural state of the B2-phase is characterised by the big value of root-mean-square displacements of atoms concerning position of equilibrium hu2 i. In diﬀerent works diﬀerent values of mean squares of displacements hus 2 i concerning a crystallographic plane are resulted. These values change over a wide range: from 1 104 till 7 104 nm2 .1–3) The diﬀerence can be connected both to a measurement technique, and with characteristics of an alloy (diﬀerence in a chemical compound, heat treatment, a diﬀerent position of temperature of measurement in relation to a martensitic point). Values hus i determine from measurements of intensity of X-ray lines on polycrystalline samples. Such measurements usually have a signiﬁcant error, and this circumstance is aﬀected on accuracy of the obtained results. Neutron diﬀraction research, at which measurements of intensity of reﬂections have much higher accuracy, has given value hus 2 i ¼ 3:1 104 nm2 .4) In work4) the conclusion is made, that the mean square of atomic displacement in the B2phase of a titanium nickelide at room temperature approximately ﬁve times exceeds this value for normal metals (copper, iron). At determination hus 2 i the assumption is made, that atoms of nickel and atoms of titanium are displaced equally about the positions of equilibrium. The purpose of the present work was separate determination of root-mean-square displacements of Ni atoms and Ti atoms. 2. Experimental Procedure The work was carried out on a single crystal of a titanium nickelide (Ti–50.8 at%Ni), prepared in the Siberian Physicotechnical Institute (Tomsk). The part of single crystal was cut on the plates parallel to (110) plane of the B2-phase. From the plates X-ray diﬀraction topograms were obtained by method of angular scanning.5) Samples for X-ray researches were cut out from the most perfect sections of the plates. In the given work the electropolished sample in the sizes 4 5 mm2 , having the disorientation on all surface no more than 1 degree, was used. The surface of plate had a deviation from (110) plane of the B2-phase about 1.5 degrees. The plate was aged at 450 C for 6.5 h. At cooling, according to the X-ray data, phase transition B2 ! R, which began at temperature 45 C and came to the end at 30 C, was observed. The integrated intensity of lines of the B2-phase was measured at (90 2) C, and this completely excluded a possibility of formation of R-phase. Measurement of integrated intensities of various reﬂections was carried out on a diﬀractometer on which the X-ray tube with a dot projection of focus was established. Height of focus was 0.2 mm, its width was 0.4 mm. The primary beam which have been cut out by collimator, at crossing an axis of a goniometer had height of 0.67 mm and width of 2.87 mm. The beams reﬂected from a sample were registered by the scintillation counter before which the ﬁlter from zirconium by thickness of 0.18 mm was installed. For removing a researched crystal in various reﬂecting positions a goniometer attachment was used, allowing to incline a sample and to turn it in the plane. The attachment is equipped with the device for heating a sample. Measurement of integrated intensity of reﬂection from the chosen crystallographic plane was carried out at rotation of a crystal with constant speed in the given range of angles and at the motionless position of the counter installed under an angle 2, where —the Bragg angle. The range of rotation of a crystal was given such that the crystal from not reﬂecting position has come in reﬂecting position and has completely left it. For exact measurements of integrated intensity of reﬂections from surface of a single crystal on method Bragg6) it is necessary, that all reﬂected beam passed completely through a slit of the counter. On height of slit of the counter this condition is carried out in all range of angles of reﬂections. On width of the counter this condition to fulﬁl more diﬃculty at the big angles of reﬂection as lines extend. The width of slit of the counter should be increased at increase of the Bragg angle proportionally tg , where —the Bragg angle, —spectral width of the K doublet. The maximal width of slit was 4 mm. To register completely wider reﬂections, measurements of these reﬂections were carried out in parts 678 V. M. Gundyrev and V. I. Zel’dovich with step of moving of the counter of 1.27 degrees as 4 mm of slit of the counter cover an angle of 1.27 degrees. We made corrections to the measured values of integrated intensity of reﬂections on a background, on the contribution of the next reﬂection of other order, on the width of a sample in the event that the width of sample was less, than 2:87= sin mm. Mean squares of displacements of atoms in a direction perpendicular to reﬂecting planes hus 2 iNi and hus 2 iTi are determined from temperature factors which take into account total value of thermal (dynamic) and static displacements of atoms. The temperature factor is included into the structural factor jFj2 ,7,8) which for the B2-phase can be presented as follows: ( sin2 2 2 2 jFj ¼ fNi exp 8 hus iNi 2 þ fTi sin2 exp 82 hus 2 iTi 2 )2 cos½nðh þ k þ lÞ ; ð1Þ where fNi and fTi —atomic factors of scattering of atoms Ni and Ti, —length of a wave of X-ray radiation, hkl—indexes of a reﬂecting plane, n—the order of reﬂection. The structural factor for imperfect crystals is determined from the formula for integrated reﬂective ability from a mosaic crystal: I=I0 ¼ E!=I0 ¼ ðe4 3 =4m2 c4 v2 Þ jFj2 2 ½ð1 þ cos 2Þ= sin 2; ð2Þ where I ¼ E!—integrated intensity of the reﬂected beam, !—speed of rotation of a crystal, E—the reﬂected radiation registered during rotation of a sample in which the abovementioned corrections are made, Io —intensity of the used part of a spectrum of a primary beam, e—charge of electron, —factor of absorption, m—electron mass, c—speed of light, v—volume of a unit cell. For registration of several orders of reﬂections the molybdenum radiation was used. Intensity of K -doublet of a primary beam was equal 23:3 106 pulse/s. Calculating under the formula ¼ ð=Þ1 P1 A1 þ ð=Þ2 P2 A2 ; P1 A1 þ P 2 A2 where ð=Þ1 and ð=Þ2 -mass factors of absorption Ti and Ni, P1 and P2 -atomic concentrations Ti and Ni, A1 and A2 atomic weight Ti and Ni, -density of titanium nickelide, we obtain ¼ 239 cm1 for sample from titanium nickelide in the B2-phase. Taking into account, that ¼ 0:7107 108 cm, e4 =m2 c4 ¼ 7:91 1026 cm2 , v ¼ 3:0163 1024 cm3 , we ﬁnd 4m2 c4 v2 =I0 e4 3 ¼ 1:09 s/pulse: Substituting the calculated value in the eq. (2), we ﬁnd jFj2 ¼ 1:09 E!=Lp ; where Lp ¼ ð1 þ cos2 2Þ= sin 2. ð3Þ Table 1 Values of obtained from experiment structural factors jFj2 and also calculated under tables of handbook9Þ atomic functions of scattering of X-rays. sin2 2 2 hkl 2 (degree) Lp E! (pulse/s) jFj2 fNi fTi 100 13.53 8.313 246.5 32.3 23.09 16.23 200 27.26 3.909 17.43 11.85 11.0 300 41.40 2.363 13.63 9.08 24.7 400 56.24 1.577 11.03 7.65 44.0 500 72.19 1.149 2.97 2.82 9.15 6.65 68.7 600 700 89.97 111.13 1.000 1.211 8.22 1.59 8.96 1.43 7.74 6.69 5.99 5.22 98.9 134.7 800 140.98 2.547 110 220 19.18 38.93 5.758 2.554 330 59.98 1.444 79.3 440 83.60 1.019 12.2 550 112.84 1.249 111 23.55 4.606 222 48.18 1.939 2110 1.05 114 1.88 5240 594 3.28 65.5 257 588 0.48 78.9 0.81 (nm ) 2.75 5.85 4.56 175.9 20.45 14.21 14.24 9.49 5.5 22.0 59.9 10.50 7.37 49.5 13.0 8.18 6.19 88.0 6.63 5.17 137.4 18.75 12.83 12.32 8.33 992 254 2.86 15.5 145 8.24 33.0 333 75.50 1.098 1.49 1.48 8.85 6.50 74.2 444 109.43 1.178 3.11 2.88 6.76 5.26 131.9 311 46.00 2.061 0.47 0.25 12.73 8.56 30.2 622 102.80 1.076 4.17 4.23 7.03 5.47 120.9 221 41.40 2.363 1.60 0.81 13.63 9.08 24.7 442 89.97 1.000 8.28 9.02 7.74 6.00 98.9 320 50.28 1.831 0.66 0.40 11.91 8.10 35.7 640 116.34 1.336 2.96 2.41 6.50 5.07 142.9 210 420 30.55 63.59 3.426 1.337 7.09 53.0 16.39 10.07 11.08 7.14 13.7 55.0 630 104.44 1.097 1.48 6.96 5.47 123.7 3. 22.3 65.1 1.49 Experimental Results and Discussion Integrated intensities of reﬂections h00, hh0 and hhh, and also some other was measured by the method stated above. The structural factor was determined under the eq. (3). The obtained results are submitted in Table 1. Atomic factors of scattering fNi and fTi are calculated under the tables, taken of Mirkin’s handbook.9) 2 The temperature factor expð82 hus 2 i sin 2 Þ, included in the eq. (1) for calculations of inﬂuence of displacements of atoms from sites of a lattice on integrated intensity of reﬂection, is found from a condition, that displacements of atoms are small in comparison with the interplane distance, divided on the order of reﬂection.7) However, this temperature factor is true as well at the big displacements of atoms if Gaussian distribution of density of probabilities of a ﬁnding of atoms to distance u from a reﬂecting plane only takes place.8) Assuming, that displacements of atoms concerning a reﬂecting plane correspond to Gaussian distribution, it is possible to substitute values jFj2 , fNi , fTi and ðsin2 Þ= 2 found for diﬀerent reﬂections, in the eq. (1) and to result system of the equations from which it is possible to ﬁnd mean squares of displacements of atoms Ni and Ti concerning a reﬂecting plane. For cubic crystals the mean square of displacements of the same atoms concerning a crystallographic plane (hus 2 i) is isotropic magnitude. It has allowed to Root-Mean-Square Displacements of Atoms in the B2-Phase of Titanium Nickelide Values of D and values of D (see text). D hus 2 iNi ¼ 2:9 104 nm2 hus 2 iTi ¼ 1:3 104 nm2 D hus 2 iNi ¼ hus 2 iTi ¼ ¼ 1:9 104 nm2 100 1.09 1.37 200 1.01 1.07 300 0.83 20.93 400 1.03 1.21 500 0.77 0.29 600 1.04 1.11 700 0.79 0.03 800 0.99 0.70 110 1.02 1.05 220 1.05 1.17 330 440 1.05 1.04 1.24 1.15 550 0.90 0.80 111 222 0.90 0.97 1.80 1.18 333 1.52 0.43 444 1.06 0.98 hkl 3 Table 3 Values of R-factors. 29.03 311 3 10 622 1.03 1.00 221 442 0.50 1.04 12.48 1.10 320 0.48 12.80 640 0.90 0.78 210 0.76 2.69 420 0.93 1.09 630 0.97 0.04 solve all systems of the equations written for diﬀerent reﬂecting planes, in common. At the joint solution of all equations using program Mathcad we have obtained hus 2 iNi ¼ ð2:9 0:2Þ 104 nm2 and hus 2 iTi ¼ ð1:3 0:1Þ 104 nm2 . From here it is visible, that mean square of displacements of atoms of nickel twice is more, than atoms of the titanium. In Table 2 values D ¼ jF0 j2 =jF1 j2 , determining coincidence theoretically calculated on the found values hus 2 iNi and hus 2 iTi structural factors jF0 j2 with factors jF1 j2 , calculated of integrated intensities of reﬂections under the eq. (3), are given for each reﬂection. For comparison values D ¼ jF0 j2 =jF1 j2 where jF0 j2 -theoretically calculated structural factor at the assumption, that root-mean-square displacement of atoms Ni and Ti are identical, are given. In this case at the joint solution of all equations it was received hus 2 iNi ¼ hus 2 iTi ¼ 1:9 104 nm2 . For a quantitative estimation of coincidence of calculated values of structural factors with experimentally received, R-factors for all investigated reﬂections, and also separately for structural and superstructural reﬂections under the eq. were calculated: X jjF0 j jF1 jj X R¼ : jF0 j The obtained results are submitted in Table 3. As can be seen from obtained values R, the structural factors calculated at All reﬂections Structural reﬂections Superstructural reﬂections hus 2 iNi ¼ 2:9 104 nm2 hus 2 iTi ¼ 1:3 104 nm2 2.8% 1.2% 13.5% hus 2 iNi ¼ hus 2 iTi ¼ ¼ 1:9 104 nm2 13.2% 4.4% 82.6% hus 2 iNi ¼ 2:9 104 nm2 and hus 2 iTi ¼ 1:3 104 nm2 , have come out much more precisely, than calculated at hus 2 iNi ¼ hus 2 iTi ¼ 1:9 104 nm2 . Especially, it concerns superstructural reﬂections. From the obtained values hus 2 iNi and hus 2 iTi have found root-mean-square displacement ofpatoms ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃNi and Ti concerning a crystallographic plane: hus 2 iNi ¼ 0:017 nm and pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ hus 2 iTi ¼ 0:011 nm. For cubic crystals the mean square of displacements of atoms concerning their position of equilibrium hu2 i in 3 times is more, than a mean square of displacements of atoms concerning a plane hus 2 i. Proceeding from this, have found hu2 iNi ¼ ð8:7 0:6Þ 104 nm2 and hu2 iTi ¼ ð3:9 0:3Þ 104 nm2 , and also have calculated the appropriate root-mean-square displacement of atoms Ni and Ti concerning their position of equilibrium: (0:0295 0:0010) nm and (0:0197 0:0007) nm. On Fig. 1 calculated dependencies jFj on parameter ðsin Þ= and experimentally obtained points are given. Lines correspond to calculated values F under the formula (1) at hus 2 iNi ¼ 2:9 104 nm2 and hus 2 iTi ¼ 1:3 104 nm2 . The points show the experimental values appropriate to concrete reﬂections, submitted in Table 1. As can be seen from Fig. 1, for structural reﬂections good coincidence of experimental points to a calculated curve in all range of changes of values ðsin Þ= is observed. From here it is possible to draw the following conclusions. (1) Accepted for a single crystal of a titanium nickelide the model of ideally mosaic single crystal was correct. Primary and secondary 30 20 IFI Table 2 679 10 0 2 4 6 8 (sinθ ) λ 10 nm 12 14 -1 Fig. 1 Dependence of structural amplitude jFj on parameter ðsin Þ=. Lines correspond to calculated values jFj under the eq. (1) at hus 2 iNi ¼ 2:9 104 nm2 and hus 2 iTi ¼ 1:3 104 nm2 . Points show experimental values: –superstructural reﬂections, –structural reﬂections. 680 V. M. Gundyrev and V. I. Zel’dovich 25 10 f 'Ni , IFI f 'Ti 20 15 10 5 0 2 4 6 (sin θ ) λ 8 10 12 -1 nm Fig. 2 Dependence of structural amplitude jFj of superstructural reﬂections on parameter ðsin Þ=. The continuous line corresponds to calculated values jFj under the eq. (1) at hus 2 iNi ¼ 2:9 104 nm2 and hus 2 iTi ¼ 1:3 104 nm2 , the dashed line corresponds to calculated values jFj under the eq. (1) at hus 2 iNi ¼ hus 2 iTi ¼ 1:9 104 nm2 . The experimental values jFj are shown by circles. extinctions were insigniﬁcant. Otherwise the ﬁrst experimental points belonging to reﬂections (110) and (200), because of a primary and secondary extinction would be much below a calculated curve. However it is not observed. (2) The carried out measurements and calculations are performed truly, including is correctly measured Io . We shall notice what to measure intensity of a primary beam there is no necessity as it is possible to ﬁnd Io in the calculated way from measured intensities of reﬂections. However comparison calculated Io with measured allows to check all experiment. In this case calculated Io has coincided with measured accurate to one percent. (3) In spite of the big displacement of atoms in the B2-phase of titanium nickelide concerning their position of equilibrium, good coincidence of experimental points to calculated curve is observed nevertheless even at the big values ðsin Þ= for diﬀerent reﬂections. It shows that distribution of density of probabilities of a ﬁnding of atoms concerning any plane is close to Gaussian. On Fig. 2 for superstructural reﬂections, dependence jFj from ðsin Þ=, calculated under condition of hus 2 iNi ¼ hus 2 iTi is shown by a dashed line. As can be seen, the found experimental points for superstructural reﬂections do not coincide with this curve, at the same time these points lay close to the continuous curve calculated under condition of hus 2 iNi ¼ 2:9 104 nm2 and hus 2 iTi ¼ 1:3 104 nm2 . On Fig. 3, the calculated functions f 0 Ti ¼ fTi exp½1:0 ðsin2 Þ= 2 and f 0 Ni ¼ fNi exp½2:3 2 2 ðsin Þ= are shown depending on ðsin Þ=. As can be seen from ﬁgure, curves are crossed, while atomic functions of scattering fTi and fNi are not crossed (see Table 1). If rootmean-square displacements would be equal or root-meansquare displacement of atoms of nickel would be less than root-mean-square displacement of atoms of titanium, curves in Fig. 3 would not be crossed and, hence, there would be no minimum near to a point ðsin Þ= ¼ 0:55 in Figs. 1 and 2. 0 0 2 4 6 8 10 12 14 (sin θ ) nm -1 λ Fig. 3 Dependencies of coherent scattering by atoms Ni and Ti ( fNi and fTi ) in view of temperature factors from ðsin Þ=. The continuous curve relates to f 0 Ti , the dotted one relates to f 0 Ni . However such minimum is experimentally observed. All superstructural reﬂections, namely (311), (300), (221) and (320), located near to this point, have is anomalous low intensity. Only provided that hus 2 iNi it is much more hus 2 iTi , this fact has an explanation. To this condition there correspond the found values hus 2 iNi ¼ 2:9 104 nm2 and hus 2 iTi ¼ 1:3 104 nm2 mean squares of displacement of atoms of nickel and titanium. As is known,10) in the B2-phase of a titanium nickelide oscillation of planes (110) in a direction ½11 0 takes place. The plane (110) consists of alternating rows of ‘‘large’’ atoms Ti and rows of ‘‘ﬁneer’’ atoms Ni. The rows are parallel to a direction ½11 0. Above titanium rows in the next planes the same titanium rows, above nickel–nickel settle down. Because of a diﬀerence at a size of atoms, the dominant counteraction to movement of a plane from the direction of the next planes occurs through atoms of titanium. Because of this circumstance, the amplitude of oscillations of rows of atoms Ni should be more, than the amplitude of oscillations of rows of atoms Ti if only to not consider oscillating planes, as absolutely rigid. Thus, from the crystallographic analysis of collective oscillations of atoms in a titanium nickelide, having structure B2, we come to the same conclusion, that oscillations of atoms Ni are more, than oscillations of atoms Ti. In work11) the data under anisotropic temperature factors for the B190 -martensite of alloy Ti–49.2 at%Ni are obtained at room temperature, and also their equivalent isotropic values are given. From these data we have determined mean squares of displacement of atoms Ni and Ti, which are given in Table 4, concerning crystallographic planes in the B190 phase, and also values hus 2 iNi and hus 2 iTi from equivalent isotropic factors are calculated. From the table it is visible, that the average and maximal values of a mean square of atomic displacements for atoms of nickel are more, than for Root-Mean-Square Displacements of Atoms in the B2-Phase of Titanium Nickelide Table 4 Mean squares of displacements hus 2 i 104 nm2 atoms Ti and Ni concerning crystallographic planes in the B190 -martensite of Ti–49.2 at%Ni alloy. Mean squares of displacements obtain from equivalent isotropic temperature factors.11Þ (100) (010) (001) ð101 Þ (103) Average values Ti 0.96 0.77 1.43 1.30 1.14 1.06 Ni 1.66 0.40 1.57 2.28 1.11 1.25 The mean squares of displacement obtained from equivalent isotropic temperature factors.11Þ atoms of titanium. Further, if to compare the average values, given in Table 4, with the values obtained in the present work hus 2 iNi ¼ 2:9 104 nm2 and hus 2 iTi ¼ 1:3 104 nm2 for the B2-phase, it is visible, that displacements of atoms in a martensite are less, than in the B2-phase. It corresponds to representation about ‘‘freezing’’ a part of cooperative thermal oscillations at martensitic transformation B2 ! B190 .10) 4. Conclusions In work results of measurements of integrated intensity of 15 structural and 11 superstructural reﬂections of the B2phase of single crystal of titanium nickelide from which structural factors for these reﬂections are determined are given. It is shown, that the single crystal of titanium nickelide can be considered as ideally mosaic concerning scattering X-rays. Computer data processing under structural factors has allowed to determine mean squares of displacements of atoms of nickel and atoms of titanium from sites of crystal lattice. Values hu2 iNi ¼ 8:7 104 nm2 and hu2 iTi ¼ 3:9 104 nm2 were found. 681 Acknowledgment Authors are grateful to Prof. Ju. I. Chumljakov for the given single crystal. REFERENCES 1) V. G. Pushin, S. A. Muslov and V. N. Khachin: Fiz. Metal. Metalloved. 64 (1987) 802–808. 2) A. A. Klopotov: Zakonomernosti fazovykh perekhodov v splavakh TiNi–TiMe i CuPd s B2 sverkhstrukturoi (Laws of Phase Transitions in Alloys TiNi–TiMe and CuPd with B2 superstructure) (Author’s Abstract of Dissertation. of Doctor of Physical and Mathematical Sciences, Tomsk, 2002) pp. 3–32. 3) Ju. P. Mironov, P. G. Erokhin and S. N. Kul’kov: Izvestiya VUZov. Fizika. (1997) 100–104. 4) E. Z. Valiev, V. I. Zel’dovich, A. E. 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