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.
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