CHAPTER VII
A. X-:-.ii:£~VESTIGATION OF Ni-Zr SYSTEJvi
B. THEHJVJAL EX£'At~S_Icm COEFFICIEN'r OB' I1\lTEf1-
JYIETALLIC PHASES IN Ni-Zr SYSTEr1;
VII A
X-RAY INVESTIGATION OF Ni-Zr SYSTEM
VII A. 1 INTRODUCTION
H.apidly cooled alloys ( upto 55 wt. % Zr) were
first investigated50 microscopically and based on these
findings a tentative phase diagram has been presented35 •
Phase diagram study on Zr-rich alloys was done by various
workers 1 • 56 •77. Examination of liquidus, for 0.55 atfoZr
shows75 the occurence of Ni4Zr and Ni3zr phases. Hayes
et a1. 36 , while investigating phase diagram (upto 50 at.
% Ni) reported intermediate phases, NiZr2 (24.34 wt.% Ni)
and NiZr (39.15 wt. % Ni) by microscopic work. The
eutectoid temperature was determined by DTA, and the
P-Zr field was outlined metallographically43, This
system has been investigated43 ,, (practically complete
range of composition) by X-ray, metallography and resis
tometric measurements and liquidus and solidus were deter
mined accurately (f!ig. VII A. 1) • The NiZr2-Zr portion
of the diagram is similar but the J3 -Zr · eutectoid was 36 found to be appreciably higher than that given earlier
and the existence of phases, Ni5Zr, Ni7Zr2 , Ni5zr2 ,
Ni10zr7 , NiZr, Ni11 zr9 and NiZr2 has been shown. Smith
and Guarct79 also reported phase diagram of partial
system Ni-Ni~fd~hich is very similar to that shown in .. : :~:~~- ;:~;;s .
Fig. VII A .1 ei:ee'pt ~at the peri tectically forming _,,.._~ -~-'
WEIGHT P'tFt C'ENT ZIRCONIUM
1900,.-..... 10-J.20_30"-_.40;;...._.;;.50;;_,.:.6.:.,0 _10.:.:..__;80;.:..,_.:i90~--;
tBOO
t700
1600
1500
1400 I .}J I f w 1300 130cr._ .. ~ I . / ~ 1200 17ff
~ ....... ...( 15 ·• ~ HOOf
..... 1000 t
900 f I I I
' I I
' ' I 10 20 30 40 50 60 10
~ATOMIC P&::R CENT Z'fRCONIUM
FIG. W ijittJ- Zr PHASE DIAGRAM , s-;t -· .. ~.~!'- ~· .
I I 1: I f I , • • I
t .. -' ' ·~ .
•.
152
Ni10zr7 and Ni11 zr9 were not separately distinguished
but. called Ni 5Zr4 • Electron beam microprobe analysis88
exhibited the presence of NiZr5 , NiZr2 , NiZr and
Ni10zr7 phases. Hayes et al.36 reported that solid
solubility of Ni inZr is very small, there being no
detectable change in lattice parameter in alloys anneale~
at 950 °C. The solubility of Zr in Ni was found1 •13 •75 .
Phase diagram and solid solubility data have been given
by Kubaschewski and Goldbeck53 also.
Kramer50 indicated homogeneity range of Ni5Zr
as 14.5 - 18.0 at.% Zr at 900 °C and reported AuBe5 type structure of this phase which was also confirmed by
a number of researchers14 •43,79. However, Kripyakevich
et a1. 51 assumed it to be Ni4Zr phase. At goo °C (21.0-
22.5 at.% Zr) a phase Ni7
Zr2 exists5°.Jcnr.\J was tentatively
assumed43 as orthorhombic, later on parameters of ortho
rhombic lattice was also reported91 • However, another
group of workers 27 •30 treated it, as Ni7Ht2 type,
monoclinic structure. X-ray analysis show~·i~at structure of Ni3Zr is similar to Ni3Sn phase. Later on9
it was possible to isolate this phase, at room tempera
ture. The phase decays by a peritectic reaction
Ni3
Zr --1- Ni7Zr2 + Ni5Zr2 at 940 °C. Earlier presented
phase diagram43 has been completed to show that .the ' •'~
homogeneity ~~e o£ this phase lies between 20-30 at.%1v-. :; ;-~'~ , . ..._, ',
153
The 1ntermetallic compound Ni5Zr2 , which forms
peritectically at 1180 °c, has pseudo-orthorhombic unit
ce1143 • The structure of Ni10zr7 phase, which is formed
peritectically at 1160 °C (at stoichiometric composition),
is base-centered orthorhombic43. Dilution with lr (41.1-
at.% Zr) causes distortion to primitive orthorhombic
cell . 45 of Ni 19~r15 phase Very recently, rapidly cooled
(melt spinning method) alloy Ni64zr36 has exhibited amor
phous behaviour81 •82 • Using X-ray and microscopy techni
que Bsenko13 has reported the phases: Ni 21 zr8 , formed
peritectically and stable down to 800 °C, Ni3
Zr is formed
in reaction between Ni 21 zr8 and Ni7Zr2 at 920 ! 10 °C
Peritectically formed (at 117 °C). Ni11 zr9
phase crystal
lizes34•43 into body centered tetragonal structure with
2 molecules per unit cell. Orthorhombic structure of
NiZr compound has been confirmed44 •48 ,55,S9. Single
crystal X-ray analysis44 exhibits tetragonal unit cell
of NiZr2 phase, and confirms the earlier reportect63• 64
results. The alloy Ni24zr76 was found to show the glassy
nature17 •
Table VII A. 1 gives the details of crystallo
graphic data of all the reported phases, along with the
present work.
15~
'fABLE VII A.1
Crystallographic data of binary phases in Ni-Zr system
-----------------------------------------------------o--------Phase Crystal Lattice parameters f3 Ref.
structure Ao or -- c/a a b c
--------------------------------------------------------------(Ni,Zr) Ni Cu 3.525- 1,13,
3.535 53,75.
Ni5Zr(t~Ni4Zr) AuBe5 6.708 - 14,43' 50,79.
Ni23zr6 Co23Ht6 11.532 - Present work,
Ni7Zr2(h) Ortho. 12.209 9.005 9.326 0.764 43,91,
Ni7Zr2(r) Mono. 4.691 8.225 12.192 95.59 27,30,
Ni3
Zr Ni3Sn(Do 19) 5.309 4.303 0.810 8,9, 13.
Ni5zr2 Pseudo 10.100 12.100 6.500 0.644 43.
-·Ortho,
Ni21 zr8 Ni21Ht8type 6.472 8.064 8.588 68.04 13.
Ni1 0zr7 Ortho. 12.386 9.156 9.211 0. 744 43' 44. +0.006 :!:0 • 008 ±0. 005
Ni11 zr5 Hex. 6.277 13.182 - present
work.
Ni13zr6 Tetrasonal 7-593 .9.527 1.255 . , Ni19Zr15 Ortho. 12.497 9.210 9.325 0.746 45.
Ni11 zr9 Tetragonal 9.900 6.620 0.669 34,43
N~11 zr9 Tetra. 10.299 6.936 0.673 Preseni ,. work ..
" NiZr Ortho. 3.268 9-937 4.101 1.255 43,44, •• 55 •
NiZr2 A1Cu2 6.477 5.241 0.809 43,44, 63,64.
--.-----------------------------------~-----------------------., . ,
155
VII A. 2 ~~TERIALS AND METHOD
The alloys were prepared from reacter grade
zirconium and nickel of 99,99% purity (Johnson Matthey
& Co. Ltd., London). The alloys were melted at Bhabha
Atomic kesearch Centre, Bombay, in a non consumable
electrode argon arc furnace. l"or homogeneous mixing
the alloys were inverted several times during melting.
The alloys were further homogenized in evacuated and
sealed quartz tubes. The change in weight after melting
was found to be less than 0.1% of the total weight of
the alloy. Therefore no chemical analysis of the alloys
were considered necessary.
Powder obtained after filing the ingot were
sieved through a 325 mesh screen and were sealed under
vacuum in silica tubes, for desired heat treatments.
The Debye-Scherrer photographs of these alloys
were taken with filtered CuK~ radiation. An exposure
of about 10 hrs. was found sufficient to bring out the
details of the X-ray patterns. Each X-ray photograph
was followed by duplicate runs to show the reproducibi-
11 ty of the data within the range of experimental error.
·:rp.e line positions were measured 2-3 times for achieving
better accuracy and the mean values have been considered
for the computation of lattice constants. -.\o\_.T, __
-:-~\~=-~~',
,
156
VII A. 3. RESULTS AND DISCUSSION
Many binary phases have been reported in this
system. For the present investigation, the composi
tions of the alloys were selected in the range, 55 to
80 at.% nickel. The X-ray analysis shows the presence
of atleast three new phases viz., Ni13zr6, Ni23Zr6 and
Ni11 Lr5• The occurrence of these phases may be further
verified by using other standard techniques.
VII A. 3. 1. Alloy: Ni68~r32
The X-ray photograph of this alloy can be easily
indexed with a tetragonal unit cell. The following
Table VII A.2 gives the details of X-ray analysis.
In the vicinity of this composition, Nevitt65
ha5 shown the existence of Pd2Hf phase, which is MoSi2 type with a= 3.399 A0 and c = 8.658 A0
, having 6 atoms
per unit cell. In Pq-Ti system also a similar phase,
Pd2Ti exists50(a). As Pd and Ni belong to the T10
group of the Periodic ·rable, their alloying behaviour
with Ti, Hf and Zr is.expected to be similar. In
Table VII A.3 mean atomic volume (MAV) and the number
of atoms in unit cell of different binary phases, exist-
.· ing in this system haye been listed. From the curve drawn
~~Fig. VII A.2) between MAV and composition of different
phases, the corresponding point of this phase, shows the
presence of 38 atoms in unit cell, which in turn suggest .. the possible formula for this new phase as ~i13zr6 •
~------------------------------------~--~,z
0 • JZ StN
t.,~zEZtN ZJ\-'tH
IJzlt,' IN e z.~zstN
f s.~zu,N IJztltN .
t!.,~zOitN SIJZ6ltN 1Jzli1N
~
oL-~~o~--~oL---~~----oL---~o~.----~0~--~o~---o-'N u\ ... ..:. - .:.. .;. ... .;. -N -~r¥1 N ..- - - - -
...---. .,w· 3tfft1CM 3fW01Y NW'ilt .£
.i 0 -'· .,.,,
rr \ ' '
.. .
' .~ • f'"j .
;I I. ~ • -,·.· ... a "" -• I ...
'~· ... .. ... I ..
u i • 0 ... ~ 0 ...
0 _, q.
N
~
• . 0 -...
TABLE VII A.2
X-ray analysis of the new phase Ni 1~
Alloy composition:
Heat treatment:
Radiation:
Exposure:
Structure:
Lattice parameters:
Ni68Zr32
Powder annealed at 1150 °C for 20 minutes and quenched in water.
Filtered CuKI(, Debye-Scherrer camera.
10 hrs.
Tetragonal with 38 atoms/cell
a: 7.593 A0, c = 9.527 A0
•
----------------------------------------------------I obs. (hkl)
----------------------------------------------------1 2 3 4
M 58.71 58.92 (003)
M- 68.90 68.92 ( 103) -M 77.08 77.12 (212)
vs 82.35 82.45 (220)
M+ 104.17 104.75 (004)
s 109.58 109.62 (311) ',
vvs 146.10 146.08 (204)
vs 160.5.3 160.18 (322)
w- 170.52 171 .46 (401)
w- 174.38 175.22 (410) : - ~ ...
(411) F ·' "l'B2.15 181.77
vs 192.34 192.07 (311) •
----------------------------------------------------" ~ ',., ,, ...
157
158 TABLE VII A.2 continued ----------------------------------------------------1 2 3 2 ----------------------------------------------------
S(diff.) 216.05 215.21 (215)
F 258.91 257.67 (430)
VF 283.76 283.86 (432)
S(diff.) 298.81 298.90 (520)
M 320.45 320.80 (007)
M(diff.) 329.08 328.45 (306)
VS 357 .t31 357.83 (523)
vs 381.04 381.36 (610)
VS 422.82 422.89 (540)
w- 434.09 435.46 (444)
Mtdiff.) 521 .98 522.74 (633)
w+ ( di.ff.) 544.42 544.28 (641)
M 569.02 568.57 (634)
M 581.70 581.84 ( 219)
M(diff.) 654.88 654.91~ ~g6~U 654.71
M(diff,) 733.47 733.48 ~ ~654p 733.08 627 J VS(diff.) 778.10 778.86 (832)
w ( diff,) 844.51 843.72 (2111) -w- 901.25 901 .36 (609)
M(diff.) 949.63 949.53 (657)
----------------------------------------------------
159 TABLE VII A. 3
Mean atomic volume in Ni-Zr system
---------------------------------------------------------Phase Volume of the
unit cell A0 3 Atoms/ cell
Ref.
----------------------------------------------------------~-Ni
Ni5Zr
Ni23zr6
Ni7Zr2(r)
Ni3
L:r
Ni5zr2
Ni21Zr8
Ni11 zr5
Ni13zr6 Ni10zr7 Ni19Zr15
Ni11 zr9
NiZr
N1Zr2 o(. -Zr
f' -Zr
43.75
301.62
1506.04
466.56
105.03
794.36
416.83
449.79
549.31
1044.58
1073.28
648.82
133.17
219.86
46.60
47.20
4
24
116
36
8
56
29
32
.$8
68
68
40
8
12
2
2
10.93 9(a)
12.56 43
12.98 Present work
12.96 27' 30
13.13 a, 9
14.18 43
13 14.32
14.06 Present work
15.36 43
15.78 44
16.12 43
16.64 43
18.32 43
23.30 93
23.60 78
, '
~ .~ . . ;_ ....
VII A, 3.2 Alloy: Ni80zr20
The X-ray photograph ot this alloy shows the
lines of Ni-solid solution along with those of a new
intermetallic phase Ni23zr6 , which is stable at room
temperature also (as-cast X-ray_pattern also shows this
phase with a= 11·1!53 A0). It is a face centered cubic
phase, with 116 atomsper unit cell. All the lines could
be very well indexed with this structure. The· details
of the X-ray diffraction analysis 1i,.e given in the
following Table VII A.4.
TABLE VII A. 4
X-ray analysis.Ef the ne~hase Ni23zr6 in equilibrium
with nickel solid solution
Alloy composition
Heat treatment:
Radiation:
Exposure:
Structure:
Lattice parameter:
Powder annealed at 1220 °C for 30 minutes and quenched in water.
Filtered CuKI(. , DebyeSciherrer camera.
8 hrs.
Face.p::~Ul~eii ~ie, co23zr5 type
a= 11.532 A0
-------------------------------------------------------3 2 3 2 !obs,' 10 Sin eobs • 10 Sin 9cal. 'hkll -----------------------------------------------~--- ---1 2 3 4 -------------------------------------------------------
53.03 53.64 71.52 84.93
(222) (400)
(331)
161 TABLE VII A.4 continued
-------------------------------------------------------1 2 3 4 -------------------------------------------------------w+ s vvs s www+ M
F
M
liJ w s s F w-M-w-s w-F
W(diff.)
107.15 120.31 143.08 160.81 177.95 197.82 227.34 232.82 251 • 98 286.51 299.90 304.52 322.24 335.63 514.13 537.66 643.71 732.54 768.81 836.80 858.14 944.87
107.28 120.69 143.04 160.92 178.80 196.68 227.97 232.44 250.32 286.08 299.49 303.96 321.04 335.75 514.05 536.40 643.68 733.08 768.84 835.89 858.24 943.17
(422) (511) (440) (600) (620) (622) (711) (640) (642) (800) (733) (820) (822) (751) (953)
( 1 042) -(884) (886)
(1066) -(995) (888) (997)
-------------------------------------------------------Note: Nickel solid solution lines have been omitted,
------------------------------------~-------------------
182
In Co-Zr system Pechine et al.71 have shown the
existence of cubic phase, Co 23zr6 with a= 11.515 A0•
As cobalt and nickel both belong to group VIII of the
Periodic Table, their alloying behaviour with zirconium
will be very similar as discussed by Panda and Bhan67.
Hence, the new phase Ni23zr6 at the Ni80zr20 composition
seems to be isostructural with Co 23zr6 phase and comes
under the category of Mn23Th6 type of phases.
X-ray photograph of this alloy shows the exist
ence of a new phase. The lines can be very well indexed
with a hexagonal structure (Table VII A.5).
The value of MAV from the MAV versus composition
graph (Fig. VII A. 3) and the atomic volume computed
according to the present finding of lattice parameters
of this phase, suggest that this phase contains 32 atoms
in unit cell. Therefore, the molecular formula of this
new intermetallic phase has been proposed as Ni11 Zr5 •
The Debye-Scherrer X-ray pattern of this alloy
at 1220 °C can be easily indexed as bet structure. The !
. following Table VII A.6 gives the details of X-ray data.
-..r:
''
TABLE VII A. 5
Alloy composition:
Heat treatment:
Radiation:
Exposure:
Structure:
Lattice parameters:
Ni70Zr30
Powder annealed at 1100°C for 40 minutes and quenched in water.
Filtered Cul\(_, Debye-Scherrer camera.
8 hrs.
Hexagonal, own type
a= 6.277 A0, c = 13.182 A0
----------------------------------------------------------I
0b 103s1n2e b 103Sin2e 1 (hkl) s. o s. ca . --------------------------------------------------------
1 2 3 4 --------------------------------------------------------F 61 .02 60.33 (110)
w- 74.25 74.01 ( 112)
M 83.70 83.86 (201)
w- 86.15 85.50 (005)
F 93.83 94.16 (202)
M- 111.07 111.22 (203)
W(diff .) 114.89 115.05 (114)
M 123.13 123.12 (006)
w+ 134.96 135'.16 (204)
W(diff .) 139.92 140.77 ~. '
~ t~ifJII!1r'' >
.. ~!+ 1'i~· 21
----':'";_< ~;.
16~
TABLE VII A. 5 continued
--------------------------------------------------------1 2 3 4
--------------------------------------------------------M 145.94 145.83 ( 115)
F 155.90 154.45 (211)
F 168.71 167.58 (007)
F 172.48 171 • 55 ( 213)
F 180.90 180.99 (300)
w 219.14 218.88 (008)
VF 254.87 255.00 ( 222)
F 271.33 272.10 (223)
F 278.05 277.02 (009)
w 300.39 299.32 (208)
F 316.54 316.15 (315)
F 338.06 337.35 ( 119)
w+ 377.09 376.48 (404)
F 386.11 385.51 {321)
w+ 396.18 395.77 (322)
w 421.94 422.31 ~ ?410) ] 422.44 2010)
F 433.74 433.93 (101])
F 445.12 444.88 (406)
F 458.62 458.0'7 . (~Q~) .
F 495.63 494 • .26 (2011)
F 518.73 '·~~/.34 .· (229) :1~-:~~:·-- ·~ ~i·t;· . , ·._ ....
------------------------------------~·.;.;.~-";f.~--------------•
--~ • I>< ')•
"
165
TABLE VII.A.5 continued
--------------------------------------------------------1 2 3 4 --------------------------------------------------------W(diff .) 534.88 533.53 (503)
VF' 545.01 545.43 (416)
F 562.65 563.08 (420)
F 658.06 657.20 j (512) } 658.40 (2013)
W(diff .) 734.79 735.60 (40~)
VF 759.55 758.97 (3013)
W(diff.) 896.13 895.63 ~ ~1016) 1 895.51 611) j "
F 931.80 931.75 (3114) -W(diff .) 943.52 942.84 (608)
-------------------------------------------------------~
166
TABLE VII A. 6
X-ray powder data of Ni 11~9 phase
Alloy composition:
Heat treatment:
Radiation:
Exposure:
Structure:
Lattice parameters:
Powder annealed at 1220 °C for 30 minutes and quenched in watE
F'il tered CuKD(, De bye Scherrer pattern.
10 hrs.
Body centered tetragonal with :: 40 atoms per unit cell, ·"' Pt11 zr
9 type. ;~ •
a = 10.299 A0 and c = 6.936 A0
--------------------------------------------------------(hkl)
--------------------------------------------------------1 2 3 4 --------------------------------------------------------VF 4 r>•2" ... ~ ~ 40.36
F 46.41 44.82
VF 59.98 60.62
vs 62.24 62.77
vs 71.51 71 .82
w+ 90.12 . e9.63
w 95.52 94.23
s 100.56 100.84
VF 112.94 "'it 12.04 ,, ~- .
F 11 8 • 1 3 ;1 {6 '."it~' .~-, "-·;"' - . ' .
( 211)
(220)
( 112)
(301)
(202)
(400)
'(222)
~~\ ',- -:-,'-:?.t{h~'.J
(420)
( 1 03)' Jllllf' . . ' ~·· . '
. ' . --.-----------------------------~-------------------.
'\,.
-~> -"f~·
16"/ TABLE VII A. 6 continued
--------------------------------------------------------1 2 3 4
--------------------------------------------------------M
F
vs
F
M
M
M
VF
\'/
M
M
F
F
w
F
w
F
w
VF
w ,;".F -~ ,~- .·
139,02
144.60
152.31
161 .05
175.12
178.57
183.92
190.66
207.13
220.43
228.01
237.90
252.27
254.20
267.99
275.44
287.90
298.40
324.72
344.99
375.54
~§§:64 ~ ~~6~~} 145.65 (510)
152.40 (501)
161.59 (303)
174.81 (521)
179.26 (440)
184.00 (323)
190.47 (530)
208.85 (114)
220.06 (204)
228.61 (442)
239.88 (532)
242.46 ) (224)) 242.03) (541)j
253.67 (314)
264.44 (631)
273.63 (523)
287.28 (404)
298.48 (~34)
. ;f$i~~f
. .., .... ·~- ( 514) ' ,~-,
TABLE VII A. 6 continued
-------------------------------------------------------' 1 2 3 4
-------------------------------------------------------· F 409.40 408.08 (723)
w 448.17 448.16 ~ ~~6gn 448.37
F 469.65 467.12 (206)
w 490.13 488.52 (921)
Vl" 502.80 500.73 (316)
F 537.19 538.41 (545)
w 563.78 564.94 (903)
F 581 .87 583.32 (705)
F 600.32 600.99 (664)
F 632.41 633.31 (217)
VF 699.49 700.53 (417)
w 761.93 762.59 (905)
VF 812.87 813.00 ~ ( 208~] 812~57 (617
1({ 825.73 825.64 (826)
F 847.06 848.05 ~ ~666~ I_ 846.61 318 J
VF 877.46 879.79 (707)
w 903.94 904.07 (916) . ~ '
-----------------------------------------------~1~~---
.. : i .;~~~ ···=' · ....
169
Kirkpatric and Larsen43 have shown the exi~tenc.e
of this phase as body centered
parameters, a = 9.90 + 0.02 A0
with 2 molecules per unit cell.
tetragonal with lattice 0 and c = 6.62 + 0.04 A
The mean atomic volume
calculated by these values of lattice.constants £its
nearly in the mean atomic volume (~~V) versus composi
tion curve. The point on the curve corresponding to
the unit cell found by us, will lie slightly above the
curve, since this phase is related to b.c.c. structure
and such types of structures are known to have lattice
vacancies with correspondingly higher MAV. Such is also
the case with pure zirconium. Therefore, the correctness
of the structure has to be decided with the comparison
of the isostructural phase Pt11 Zr9 reported by Panda
arid Bhan68 (having 40 atoms per unit cell and is bet
with parametric values a = 10.259 A0 and c = 6.888 A0,
at 1200 °C), Gomparison of lattice constants and number
of atoms in the unit cell of Ni11 Zr9 with Pt11 Zr9 phase
indicates that both the phases are isostructural. This
is expected as Ni and Pt both are the members o£ the 10 T group with ground state electron configuration of
their free atoms as 3d84s 2 anq,_.5d86s 2 respectively and
'VII A. 4
••• -~~
... .. The investigation on Ni-Zz;ibinary system in the
composition range. 65 to · 80 at.% nickel reveals· the exist-'. ~.
170
-ence of at least three new intermetallic phases besides
the known phases. The alloy Ni80zr20 exhibits the
presence of nickel solid solution along with Ni23~r6 phase which seems to be isostructural with the phase
Co 23~r6 • The cubic lattice constant of this phase has
been found as, a= 11.532 A0• The phase Ni13zr6 (at
68 at.% nickel) crystallizes in tetragonal structure
with lattice parameters, a= 7.593 A0 and c = 9.527 A0•
In Ni70zr30 alloy a new phase Ni11 ~r5 (h) having hexa
gonal structure with parametric values a = 6.277 Ao
and c = 13.182 A0 has been observed. The mean atomic
volume (MAV) values of these new phases also fit well
in the MAV versus composition curve. At 55 at.% nickel,
the phase Ni 11 Zr9 occurs. Comparison of crystal
structure of this phase with the phase Pt11 ~r9 in Pt-Zr
system shows that they are isomorphous.
VII B
THERMAL EXPANSION COEFFICIENT OF INTERMETALLIC PHASES
IN Ni - Zr SYSTEM
VII B.1. INTRODUCTION
In any crystal the atoms, that go in its consti- ,,,
tution are arranged in a regular pattern in three dimension
The lattice parameters of the crystals are determined by '
the spacing between these atoms{ or groups of atoms. Wi~ I
increasing temperature, as the amplitude of thermal
vibration • of the constituent atoms increases, the average,
interatomic distance also increases, giving rise to the
phenomenon of thermal expansion. Studies of the dimen
sional changes that result from a given temperature cha~ge
can give some insight into the nature and directional , ·
character of the interatomic forces that hold the solid
together37 . Thermal expansion is considered to be a
measure of the anharrnonicity of the lattice vibrations.
Thermal expansion plaY1il an important role in .£:~,', ;,·~ '~- ' -" t,' -.· -
crystal physics, in view of ftill applications,, &'me of ·
'~~~-;, which are mentioned below: '.·, L_ '~--
(a). In case of crystals ti!4d.~r;g0i~g »hase tran~ltions, \ .. _'~-':,,.,.---, ··~-' 1t"·' '
;he .thermal expansion coffffiq·i=~~tt·"· varies anomal-
l:;.y near the transition tejii~t • : • .. • · .·
17 ~
(b) Several attempts31 ,54,60,76 have been made to
use thermal expansion data as means of investija-t'iYl~ tile
nature of defects in crystals.
(c)
on the
The thermal expansion data can be
theories of lattice dynamios5,6.
used as a check< ' ~.
,;} '
(d) It is useful in estimating the thermal conductiv~ty
of insulators26.
(e) Resistance to thermal spalling and thermal shoe~
and the design of casting moulds are all dependent on the
. h t . t. f th t . 1 16 expans~on c arac er~s ~cs o e rna er~a s •
(f) In the calculations of some of the physical prGper-
ties, viz., refractive index, the elastic and photoelastic
constant at different temperatures, the knowledge of thermal
expansion data is essential.
(g) The-knowledge of thermal expansion gives an indi
cation of interatomic and intermolecular forces and their
variations with direction, temperature, impurity, vacant
sites, spin orientations, electronic transitions .. etc.
VII B. 2 THERNAL EXPANSION IN RELATION TO CRYSTAL
STRUCTURE AND OTHER PROPERTIES:
The ever increasing knowledge about the atomic .. ~
arrangements in cry?tals make it ~ossi'Q;La~~to obtain a ·<·. ,, ' •' II> ·'- ' • •
be'tter understanding of the s , e.:..expansi~ relation-
ships. Thus crystals could ed together according
to the general nature of es. In isosthenio
..
173
structures, where the atoms or ions are linked together
to their neighbours with bonds of the same strength, the
anisotropy of thermal expansion is small. In the laye.r
structures in which the atoms in the_layers are more
tightly bound than the atoms in the perpendicular directi,n
the coefficient of expansion perpendicular to the layers i•
is larger than that in the layers. In antilayer structures
the coefficient of expansion along the basal plane will be
more than that in the perpendicular direction as the at~s .,,,.,
in the basal planes are comparatively 'weakly bound than'
those in the perpendicular direction97 • A few more
structure-expansion relations.hips . which include·. the - 58
following facts. have been mentioned by Lonsdale •
1. ~xpansion is greater for crystals with small
molecules than for their larger homologues.
2. The expansion along the direction of H-bonds is
relatively small.
3. Expansion is relatively large when only Vander
Waal's forces are present or along directions where these
forces form the links between molecules.
4. Covalent bonds perm! t only a very small .• f/;Xpansion • ..
may
.. 6. The expansion of may be greater ._ ·
than that of either pure
17~
7. The thermal-expansion coefficient of an alioy
can not be predicted from its constitution.
Several attempts have been made in the past to
relate the thermal expansion to the lattice energy, the
melting point, the heat of formation, the, bond strength
and the index of refraction. For many f.c.c., b.c.c.
elements and binary compounds there is an empirical ''
relationship between the thermal expansion'coefficient
and the melting point18 •85 •95, The spacings and expans~Gl
coefficient are both, of course, seriously affected by
strain in a metal or alloy and near the melting point
a rise in the number of vacant sites may cause a change in 46 .
slope of the expansion coefficient curve. Klemm found
that for ionic crystals product of thermal expansion
coefficient and melting point is a constant quantity.,
The thermal expansion is found to be large for crystals of
low hardness. The value of expansion coefficient a~.lso
22 . 15 86 depends largely on texture , impur~ty ' and on second
order transformation94 within the crystals.
VII B. 3 BRIEF SUMMARY OF THE PAST WORK:
A comprehensive bibliography69 •83 on thetb.ermal
expansion data is available. The data on thermal expansion
of metals have been. compi:l~d ·~y·.P~~so~7<t- · The thermal
expansion data can also be 23 29 42 -66 sources ' ' •,... · • To .~.v'~"J"'
.;.:' :
~ -~~
the.major other
beautifully edited a
hand book series, consisting of.several volumes which
include thermal expansion data reported on elements,
intermetallic compounds and alloys.
17~
i 1 Survey of the literature shows that thermal expan• l.
sion coefficient of intermetallic phases in Ni-Zr system r has not been investigated in past. The aim of the present
study is an attempt in this direction.
VII B. 4 METHODS OF DETERMINING THERMAL EXPANSION
OF SOLIDS:
Many different methods for measuring the thermal
expansion of solids have been developed during the past
50 years. The choice of method may depend upon the material
to be measured, the quantity of material, the temperature
range of the measurement and the type of information
(accuracy, sensitivity and time factor etc.) required,
There are two main classes of experimental methods
of determining thermal expansion. They are (i) Macroscopic
methods and (ii) Lattice expansion method. The two methods · 38 D are not always in agreement · • isagreement have been
attributed to a difference between the lattice' expansion,
which measures correc.tly overCall expansion in single
crystals or in separate crystallites, and overall expansion - ·, . :< t ·•'·
of poly crystalline solids, may iiav1hve reorientation
G:J; crystal;Li tes: ·or other changes that affect the . . ··. ' ..
spacing 'betw.een, them~
A ~~·, ~escription ,, of .the macroscopic methods is
given below:
176
VII B. 4.1 MACROSCOPIC METHODS:
' VII B. 4.1.1 Push-rod dilatometer methods: The push-rod,
dilatometer method7• 20 •74 is experimentally simple, sensi~ tive, reliable and easy to automate. It is useful for J
i quality control measurements and for studying phase tran~ ,,
sitions. With this method the expansion of the specimen( ,,
is transferred to one of the heated zone to an extenso--
meter by means of rods (or tubes) of some stable material.
VII B. 4.1 • 2 Twin-Telemicroscope method: This method33; is
most useful for measuring the absolute expansion of large
specimens at high temperatures. The best results are
obtained when the two telemicroscopes (microscopes with
a relay lens) are rigidly mounted to a bar of low expansion
material and the length change· is measured with filar
micrometer eyepieces. The sensitivity of this method
isrv 10-6 m.
VII B. 4. 1.3 Interferometric methods: These methods40
are based on the interference of monochromatic light
reflected from two surfaces that are separated by a speci
men or by the combination of a specimen and a 'reference
material. 2 ....
The fizeau interferometer can be used to
measure either the absolute or relative expansion of a
specimen. When the optical f+~t$ '9f a F.~.wY-Perot inter
fer,ter are mad~ ~hl~.:r~fe-~;ting the multiply reflected
''beam.S ~-cause a great i:ilcreaa_.,,,lf.sl:ilil.rpness of .the fringes
which in turn g~:<fes :h:igher sensitivity in the expansion
measurements.
A technique which utilizes the frequency shift
of a .to'abry-Perot interferometer has been developed by
Foster32 and by Jacobs39, The latter has used his system
to measure the low
with a sensitivity
expansion materials
of 10-9 m,
from 273 to
The polarizing interferometer28 is also a
600 K ~ ,,
sensi- i -:c
tive apparatus in which laser is used to emit polarised :
beam.
VII B. 4.1.4 High-sensitivity methods: At low tempera-
tures (< 30 K) expansivities are very small, and hence,·
high sensitivity (0J10-11 m) methods are used, In three
terminal capacitor technique49 •96 variation of either
capacitance or mutual inductance of the specimen in a
balancing bridge circuit is measured. In variable diffe
rential transformer technique80 the relative displacement
between the primary and secondary windings of a tr~for
mer is detected with a mutual inductance bridge. In
mechanical-optical system72 •73 the expansion of a 6 em.
specimen causes the rotation of a mirror which is attached
to the centre of a doubly twisted strip, The .;rptation of
the mirror is detected with a sensitive photo~lectric
device.
VII B. 4.1 .5 . c ' 19 Hig!l-sP:!el!i:· methads: . , . e~~1r,liyan has ....
described a· system in which t,Q..~.· elX:pansion of a specimen is
m'Msured by detecting 1}~ ~h·;t:tl•radiance from a
constan:t radiatillbn ·soi.u;~/;~':Cl'fi:bb.ange in radiance is '1/ ~· -~~: . .,. . ·:" ~' ,'·- <" '
c caused by the expansion .. pf the specimen which partially
blocks the radiation.
~
178
The use of stabilized lasers39, the use of
computer controlled data acquisition system to measure
a specimen during a rapid change20 and the use
diffraction method12 are the outcome of modern
of neutron ~
technology o{
VII B. 4.2 X-RAY DIFfRACTION METHOD:
The macroscopic methods, have some inhe.rent dis-·
advantages. The specimens required for these methods are
of fairly large size and for non cubic crystals differen~
k
specimens, oriented in appropriate directions are necessary
to determine the thermal expansion along different crysta
llographic axes. For the optical methods, crystals with
polished faces and fine finish are needed. Moreover,
hygroscopic and volatile substances can not be studied,
In the X-ray method, the changes in the lattice•
spacings produced by the thermal expansion cause shift-s
in the Bragg angles of X-ray reflections and these shifts 0.
in the angles, measured a:>,/unction of temperature, give
the lattice thermal expansion of crystal. The X-ray method
provides some unique advantages.in the measureiJ!eht of i(<
thermal expansion of solids. These advantages .:may be
roughly classified into~,two groups. ·~)~~
( 1) Technical advanta'gesl_ . • . . ... ~
(a) The posslbilj.':z of th~ _u,se . .:. . --~·.
spe'eime.ri. · . . """" · · ' - • > ~
:;·. ~-.·. '
-,"-::· of· a vecy.i. tiny
.. . ·, ,,,. _ _f....,. --~: . (b) All the linear lm,d"anj;ill~ar dimension (of the
! '\~~:;;,.
179. unit cell) of an anisotropic material are obtained simul
taneously.
(c) The possibilities of using specimens of
irregular shape, without special surface preparation.
(d) Convenient examination of mechanically weak
specimens, such as molecular crystals.
(e) The possibility of studying hygroscopic and
volatile materials.
(f) The ability of studying conveniently substances
which condense or solidify only at cryogenic temperatures,.
2. Scientific advantages:
(a) There is unique possibility of determining
j
and relating the atomic structure of a crystal to the
thermal expansion57• 97 which in general is quite complieated
(b) Evidence on the extent and nature of crystal
line perfection in solids can often be obtained simultane
ously from the X-ray method.
(c) Information of intrinsic defect structure and
its possible contribution to thermal expansion can be
obtained25 •76 •84•
(d) In the case of simple structures, knowledge
of the precise values of the.lattice parameters provides
a direct measure oi inter nuclear distances. ' ' - ._ ' ', "~· 11': .
( e) Lattice parameters:· are .requi~ed: :for- the calcu-JIIIIIf .. . 411 lation of other physic~l prop,~'f!P:~~· ~uch. as couipressibili ty'
lattice energy, refractive index and photo elastic ;·, .t t 4,11,61 ,_ons an s .•
1~0
(f) For the construction of phase equilibrium
diagram the lattice parameter data are found to be nece
ssary87.
VII B. 5 DETERMINATION OF LATTICE PARAMETERS:
For the calculations of thermal expansion coeffi
cient, by X-ray method, knowledge of lattice parameters
at different known temperatures is essential. The
precision in the measurement of lattice parameters and
the specimen temperature are the important factors, which
affects the accuracy of thermal expansion measurement.
Excellent reviews on different methods of evaluating
accurate cell parameters are available3• 10 • 24 •47. The
two types of major sources of errors, the random errors
and the systematic errors, are to be considered in the
determination of accurate values of lattice constants.
By making careful and repeated experimental observations
on the same sample, the random errors of observations
can be minimi~ed. In the present work, extrapolation
method and least$quares method have been used for the
determination of accurate values of lattice coqstants,
which account for the corrections due to systematic errors.
Brief descriptions of these two methods are given below:
VII B. 9.1 Extrapo~tion metho4~,.;· '~, ~'{- ., .
~, ~ the Debye-Scherrer .. ~ay ph:Otographs the signi-
fic~f':$urces of systematic errors ar~ as follows:
181
1. Shrinkage of the film.
2. Lack of knowledge of the exact film radius.
3. Eccentricity of the specimen.
4. Absorption of X-rays by the specimen.
On id N 1 d R'l 6; cons ering the above errors,. e son an ~ e~
and Taylor and Sinclair9~ after rigorous analysis, proposE
the use of error function -
1 2 +
for getting the accurate values of cell parameters. The
systematic error approaches zero as e approaches 90°,
as can be seen from above expression. Precise values of
cell parameters can be obtained by plotting (large scale)
the calculated values of lattice parameters for different
(Cos 2e/Sin9 + reflections, against the
Cos2eje) values and then
corresponding
extrapolatingithe'lin~ar curve
to the zero value of the error functi0ri. for all practical
purposes lines at all anglesmay be employed in making the
extrapolation.
obtained, is as
better resu;L ts the foll0Wing ppin-ts should be.;i•~bserved in
practice:
(i) In
relativ~ly
18;
(ii) itEcdiation may
one good line in the
be cllosen so as to yield at least
far back-reflection re~:>ion (0) <30°)
in order to fix with highest accuracy, the point or intersection or the plot with the axis of ordinates.
(iii) Cnly lines between e = 30 and 90° should be used.
VII 3. 5.2 Least-sguares method:
The principle oi' this method is tllat, if the
angular dependence of the combined systematic errors is
expressible as a simple mathematical function, the effect
of these errors can be eliminated by Least-squares analy-
tical treatment of the data. The procedure suggested by
Cohen21
is in principle the analytical equivalent of a
graphical extrapolation, but it is superior to later
method in that it is applicable to noncubic crystals
also (if strong reflections (boo), (oko) and(ool) are
absent). However, the disadvantages of this method are
(i) labosrious calculations and (ii) assigning equal
weightage to all reflections, regardle_ss of 1rrhether they
can be measured with good or poor accuracy.
The method is as follows 1 ·
Square of the Bragg's equation gives
-~<,. ' ... -l:, "'f;,
-- (7.1)
183
- - (7 .2)
where ,6 Sin2e is the error in the observed Sin2e value
and ~d is the corresponding error in the 1 d 1 spacing.
If f(S) is the linear extrapolation function for
any camera, then the fractional error in the d-spacing
can be expressed by:
= D'f(e) -- (.7·3)
where D' is a constant of proportionality.
Combining equation (7.2) and (7.3) we have,
(7.4)
which can be written as
= -Db -- (7.5)
where D, is proportionality constant, called the drift
constant, which has the same value. for all the lines
recorded on one film and 6 is a t~igonometric expression
different for different lines recorded on a film •..
Equation (7 .5) indicaj:;e~ that t:[(Jlc·o values
of Sin2e will have/~ .error b;· ~ a~o~~ ' - Db,
and that this term has to be ~€ldeac tb:"~tJi?e quadratic ·-_,:;.. ' __ "
18~
A I C '( 0 C><.i + 0 i (7.6)
for a tetragonal crystal
.. 2 2 J_ 2 2 )\/4c
0 1 i =(h +k )i'
~ = [i and D bi = appropriate form of drift
error -
LlSin2ei
for a hexagonal crystal,
A0 = >f/3a~ , C0 =
and Y'1 = L/.
Equation (7 .6) is called the observational equation for
each line of the diffraction pattern and the number of'
such equations will be equal to the number of lines us~d
in evaluating the lattice parameters.
The effect of random errors, which is reduced by
the averaging procedure used, during measurements can be
J
minimized further by applying the principle
If such error do exist, the equation (7.6)
small residue in R.H.S., representing the
tional error. According to the principle
squaref
a
·observa-
the .. best value of the quanti ties A0
, ·t:0
and D are those . !~4:.~- ..... ~.
whi!~·afE! obtained by minimizing of the squares
observational
.filli!Uations o:! the ~ .,; ' ·-;/t"~· ; - . ,_ ~ -•
,.,~~·~·mbined to give a set e!
ines can be
equations.
185
These are as follows:
A f_J. 2 + co lii Y'i + D I)..i si --
=t/·fin2e1 0 i
Ao[_J.i Yi + coLYi2 + nLJ'd'"i = _LYisin2-ei (7.7).
Aof ~i~ + coi'{1ri +D,l-~L r- 2 =_oisin ei J
When these equations are solved simultaneously
for A0
, C0
and D, then, a0
and c0
are calculated from th• .·
2 '1.2 2 \2 relationships ao = " I 4Ao and co· = (I I 4C 0 for tetra-
gonal crystals or a0
2 = A213A0
and c0
2 = ~214C0 for
hexagonal crystals,
The precision of the results is greatly improved
by using a comparatively large number of lines. Again,
all the values of the observed Sin2e, are normalised to
a common wavelength, preferably to theJ~ wavelength. So
if a number of~ 1 ,(2 pairs are to be used, the Sin2e value
of theo( 2 reflections mus~be converted to the equivale
Sin2e values, corresponding to the o(1 wavelen
plying them with ( A.c1 / 'A .~, 2 ) 2 • The factor
for CuK~ radiation is 0.99503. It is a~so i1Portant to
observe that the value of b for -~ny ~iveh po;der line depen ~~~.,..... . ' ' ,·
_.upon the .{'bserved Bragg angle of th~t line and hence to ... . ' •- .
compute the c; value for.that'parj;icula flection, the ,'+. ' "' < •• ·~ ,: ""' ..,_ -
.obf'~v~d':;,a:i,;t~t··e·:~ -must "De used, and • ·' . "h,
value
af' ~. normalized to a d"iffe~~wa,y,elength. -.-.&.-. . ·<l:J.~.-'"1-"-~: •• -···
186
Orthorhombic crystals can be evaluated using the
Least-squares method . by a logical extension of the mathe
matical theory given above. However, the opportunities
for successfully utilizing powder data from orthorhombic
crystals are rare and the calculations become very tedious:.
' VII B. 6 STANDARD ERRORS IN THE VALUES OF LATTICE CONSTENTf
--------~~~~~~~~~~~~-=~~~~~~~~
Jette and E'oote41 have suggested a statistical m~ttho<
for the estimation of the errors in the cell parameters . f.
determined by Cohen's method21 from the data of an indivfdua:
film. Using this method, the limits of errors in the para
met~ic values have been evaluated from the variance, s2
of Sin2e given by :
= L_L~-~~!!~~l~ N- W
(7 .8)
where ~Sin2e = Sin2eobs.- Sin
2ecal.
N = number of reflections employed and W = number
of constants to be determined (for a tetragonal crystal,
W = 3). In the case of a tetragonal system,
of the constants A ( = \2/4a 2) and B ( = 0 I' 0 0
the,.·variance 2
0 ) are
187
The standard errors in the lattice parameters
I a I 0
and 1 c 1 0
are then
aoSA -----2 A
0
given by 1
• - - (7.10)
VII B. 7 EVALUATION OF THERMAL EXPANSION COEFFICIENT:
The linear thermal expansion coefficient is defi~a
as the change
range of 1°C.
in length per unit length for a temperatur•
For a crystal, if at' bt and ct are the
lattice parameters 0 ~ at t C along the three crystallograp~c
'0~
axes and if their dependence with temperature is expressed 2 by a quadratic relation of type at = a
0 + a1t + a 2t or
by a linear relation of at = a0
+ a1
t type, then the t~.ermaJ ,;"
expansion coefficients along the three axes are given lily
the following relations.
=
and =
(1/a0
) (dat/dt)
(1/b0
) (dbt/dt)
(1/c0
) (dct/dt)
VII B. 8 EXPERIMENTAL:
.. .•. For the determination of
cient, . three binary alloys of ::j:;he
exact compositions Ni55_zr 45 ,, .~l.-,tl'"~.~:<::>llf"!
':,
(7.11) '
coeffi-
the
The detail1
1&8
employed have already been mentioned in chapter VII A.
The powders of alloys Ni55zr45 and Ni80Zr20 were
annealed at 660°C for 3 hours in pyrex glass capsules,
sealed under vacuum and coo.led within the furnace. These II I ' strain-free' powders were used-for the room temperature
X-ray photographs. However, for obtaining the as-cast
X-ray photograph of Ni75zr25 alloy; the method suggested
by Becle et a19 has been followed, ~; '
''·t-' s4eved powders sealed under vacuum in silica tubes
were heat treated at different temperatures (alloy Ni 55zr45 0 .
and Ni80zr20 , from 730 to 1220 C and alloy Ni75zr25 , from
380 to 820°C) and quenched rapidly in water. These pow~er
samples were utilized for obtaining Debye-Scherrer X-ray
photographs. It was observed that ail the X-ray patterns c
of a particular alloy were similar thereby showing thitno "i
phase transformation occurs withill the entire range fff
temperature, studied,
VII B. 9 RESULTS AND DISCUSSION
VII B-. 9. 1 THERMAL EXPANSION OF ALLOY:
All the X-ray photographs o!- :lthiil'
presenc.e_ of Ni11 zr9 phase having-b¢.t:' .
.._ of'_,~~~ X-~, diffraction;,;ata of ~.~r; alri,ady- given (chapter._V~~-~ ~l·-~
the
The detailE
189
Accurate unit cell dimensions of Ni11 zr9 phase at
each temperature were determined {Table VII B,1) using
Least-squares treatment21 • The standard errors in the
lattice parameters so obtained, were calculated by the
method suggested by Jette and Foote41 • The s.n. of the
lattice parameters for all the x-ray films taken at diffej" t rent temperatures were found to be of the same order of / ; ' magnitude (equal to : 0.0058 A0 and + 0.0042 A0 for 1a 1
and 'c' parameters respectivel~.
Refraction correction has not been computed, bec:~us1 ,,·
it has a very small value in comparison to experimental ··
errors. .'-'
Graphs between lattice parameters and temperatur'e
were drawn in a large scale. Figure VII B,ishows that the i lattice parameters vary linearly with temperature and~.the
dependence may be analytical11 expressed as
at = 9.9591(11) + 2.795(13) X 1o-4t ·-~,
--(7.12)
and ct = 6.6894(27) + 1.989(31) X
\!>lhere, the numerals within tb.e '
S.D. of the respective constant_s. ·The ;u.ne:
of · . .thermal expansion for both .the were
... calculated using relation ( 7.11) cl : •io:; . . ..• ' ' . .. ' ,·
~;;,iii.£ '. '~·e'~~~~~·,~ at ~:~ ~ :· ~~~t-v ·a c
have: oeen dete~i~ee.·-~~-\h'e u~~al method59. .-,;~~~:.:: ·tr;:. • • ''•~: • ·-~~~
values of
I0·34r-----------------... 7·04
1 . "<
IX UJ ... UJ ~
~ -~-
... . ·'
10·10
. ,, ....._.,. ·.·
!i" .,. '" FIG."fl!l..,$.1
~ • a· PARAMETER
e e •c' PARAMETER
8.7Z
••••
. . ,
'"t, ···._, TABLE VII B .1 '.t ~Lattice parameters and thermal expansion
~:· '
data of Ni11 zr9
phase at
~ltifferent temperatures ' '"
--~--~~~~--~-~--~---------------------------------------------------------------------------·. ·He~t :.fef:~~tme* .. and 1 a 1 parameter 1 c 1 parameter r:f. x106 /°C o( x106 f 0 c · water q~!9jlching A 0 A o a c
-------~-------------------------------------------------------------------------------------Temp. 0 c \ Time Observed Calculated Observed Calculated
_.. ·;. minutes -~ .,
--------~------------------------------------------------------------------------------------. . 11 1'-.::,.-"' : .:~Jtc '
'!5(1\,s..:;~ast)' - 9.9646(58) 9.9661
1'5~ ··.1 70 tlO .1626 (58) 10.1631
~ -~ 60 10.1892(58) 10.1883
)~o~~ ··· . 'd:·, '·'_so 10.225o(ss) 10.2246
:f<l20' ' .~ .''40 10.2558(58) 10.2442 ' l,l.\<~ 111. ~. 'f'
'< 1150 . 1.:~ ;;;.f,,~.~ ' ;. 10.2703 (58) 10.2805
1220 :'::'" ~:0. 2996 (58) • . . 10. 3001
---.--____ &_ .. _.-...
6.6940(42) 6.6944
6.8357(42) 6.8346
6.8538(42) 6.8525
6.8758(42) 6.8783
6.8950(42) 6.8923
6.9125(42) 6.9181
6.9359(42) 6.9320
28.05(1)
27.50(1)
27.43(1)
27.33(1)
27.25(1)
27.21(1)
27.14(1)
29.71(4)
29.10(4)
29.02(4)
28.93(4)
28.85(4)
28.77{4)
28.68(4)
,.._ = C)
191
standard deviations of thermal expansion coefficient at
different temperatures are found (Table VII B.1) to be
nearly same (equal to + 0.0122 for rJ. and + 0.044 ford ) • ~ - a - c 1
VII B. 9.2 THERMAL EXPANSION OF ALLOY: Ni80
Zr20 J The X-ray photographs of this alloy at different ,
"''' temperatures showed similar pattern, nickel solid soluti4:' ,,
lines along with a new intermetallic phase Ni23zr6 , which'' '
has a f.c.c. structure. The details of the X-ray analys~j ,,,,
of this new phase have been already given (chapter VII A ,,,,
page iC:O).
Values of lattice parameter were calculated f~m
the line positions of the intense reflections (640), (8Q,O),
(733), (822), (751), (884) and (1Q66). Using the meth~dJ,
suggested by Taylor and Sinclair9° and by Nelson and ~. ·. '""''
Riley62" 'the accurate value of lattice constant at ~ch L·'
temperature , was determined by extra]Hiillating the plft~ of · · 2 · :r2
lattice parameter values against ~2 (cos 9/sine + Ccis e/e)
function to 8 = 90°. One such plot, at 122o°C h~s been
shown in Fig. VII B.2.
The standard error of so
obtained, have been calculate<Ltl.Qi·: ·, :·~·.;_/.
·of the
'
··'-<·-·.;· ' ...•
'
-~
"' I ' ... • :-_ ~ " -
0 ,.l· ~ ;.(
. '
l 19 -.........
o.1JG ~ v
~ o.,.
"' 1')1 0
·<-
+ .. ·r 'jc .u ·;
........ . _,,.,.
• • 'I
I
·~ II! c -... ~ ;::1 II.
Cl: 0 Cl: a: Lll . • > Cl: LiJ ...
. W
·~ .a: ·~ ·a.;
Lll u
t: • _, .u 0 N N
N
CD
192
TABLE VII B. 2
Lattice constant and thermal expansion data of Ni2~
phase. I
1 ------------------------------------------------------------! Heat treatment
and water quenching
Temp. °C Time minutes
'a' parameter Ao
Observed Calculated
----------------------------------------------------------f/;i.-•.1} ', ,.
25(As-..cast) - 11.3534(37) 11.3547 13.03(1)
730 70 11.4603(37) 11.4590 12 .900)
820 60 11.4702(37) 11.4723 'k
12.89(~) /
950 50 11.4910(37) 11.4915 12.87(~)
1020 40 11.5027(37) ;\
11.5018 12 .86(:¥1) 1':
1150 60 11.5201(37) 11 .5211 ~:l:~~ 1220 30 11.5324(37) 11.5314 t~·--
-l.: ~:.h
----------------------------------------------------~-------
A graph between lattice parameter
was drawn preferably .with a large ~Q~J:;~,. :.
represents a .linear variation of
temperature.
• 1.479(12) xto-4t
B. 3
-- .. "(7.13) ~
I a""'"' the S.D. of the
of linear
u.ss , _______ .-.. _______ _,
u.s,
I f.49
1 11.47
•.c ex 11.45 w .... w ::E .( ex n-43 ~ • " •
. "·" ,,··.,
' ',,t/
. '.
11.39
11-31
WIT I
193 The thermal expansion coefficients at each temper
ature were evaluated using equation (7.11) along with their
S.D. values. The numerals within the brackets o£ columns :~
3 and 5 of Table VII B. 2 represent S,D. values of lattice -; {~
parameter and thermal expansion coefficient respectively. ~·
VII B. 9.3 THERMAL EXPANSION OF ALLOY: Ni75zr25 Jr.,· ~;~~' ~
All the high temperatures as well as as-cast X-rfY' •
photograph exhibited similar pattern. ·The hexagonal unit
cell
ment
parameters of Ni3
Zr phase were
with the reported values8•9.
t found to be in agre1:
i'~
Accurate cell parameters of this phase at each .. -:· . "'''' / "~ . 21 W·
temperature were computed by Least-squares method • Jhe , ..
standard errors of the parametric values were calculat~d 41 ;·
using the procedure suggested by Jette and Foote • ~e
accuracy in the lattice parameter measurements were
as ± 0.0006 A0 and + 0.0004 A0 for 'a' and· 'c'
respect! vely. ~- -·~ '
The observed values of lattice
listed in Table VII B. 3 and the n.ature
temperature is shown in Fig •. VII~&"':'*''
analysis of the experimental . : ' '
analytical expressions .• · "' .~.-· . . :~' . . ... i,, ~-··-. V.iP
.. are
ers
-~ "' 0
3 0
·~· "" "' 31 ..; • .,
I ~'I , ,
r i! J
I l i ·"' 1:.
-~ .., ; ~'ii! ~ ·,~ In o:r 0: w
~·-a l i' ... • 0: • I "" w .... ... ·:· ·, -·~ '- " _, ·w I . : ~· ~-·. 2 , w ~ I
'·:,., u I= t. '!('
3 IlL
• • • u . • 1&. 0
I 1 z 0 I= ·.t
:! a: J ,., .,
~ ,•,. • ·" ID ""
, , -· ,::i: •!._ ,, .;. .;. !'!" , ' ·.~
':·· ,'f -
•"
.~ ' \
':t:. ,, .
' ~~-, . , . '
~
TABLE VII B. 3
.,Lattic-e parameters and thermal expansion coefficients at various temperatures of the compound Ni
3Zr
'a' parameter 'c' parameter
Observed Calculated Observed Calculated from least- from least-squares squares analysis analysis
Ao Ao Ao Ao
5.2529 4.2454
Coefficient of thermal expansion
tXa ~
x106/°C x106/°C
44.58 34.35 as t) .. :-. . .... ~'"~~~
5.2530(6) 4.2452(4)
5.3186(6) 5. 31.89 .:;3'61 ( 6) 5.3290
352(6) 5.3356 .3416(6) 5.3426 ,,
.3482(6). 5.34~0, ' -· ' ~
) 5.3517 5.3534
------',.-,,._~!.!,.
4.2885(4) 4.2882 4.2961(4) 4.2953 4.3002(4) 4.3003 4.3053(4) 4.3057 4.3100(4) 4.3103 4.31-41(4) 4.3142 4.3163(4) 4.3164
26.30 22.16 19.08 14.96 10.82
6.71 3.62
22.47 19.80 17.80 15.13 12.45 9.78 7.78
..... ~ ...-..
195
It may be seen from the Fig, VII B. 4 that the
lattice parameters increase in a non-linear manner over the
whole range of temperature considered which is also impli
from the above analytical expressions. The non-linearity
with the increase of temperature is the manifestation of
the anharmonic contribution and the vacancy concentrati
It is, however, interesting to see that, linear thermal
expansion coefficient values decrease. with increasing
temperature. This suggests that although there is uni
cell expansion with the increase in temperature, the
of change of this expansion with the rate of change
temperature is not influenced accordingly. This
in a net decrease in the value of the expansion coeff
This behaviour can only be predicted f~Qm the nature
binding forces between the constituent
temperature variation of the Grl:l!De:is~ paJ•ameter,
which is a weighted average of th~ G~ef'sen nmram
of various lattice vibrational I~oetes.
of thermal expansion coefficient, vri th,
1 a' parameter is noted to be l~rg~,, 'c' parameter, showing that at()m$
loosely bou:ii,§ than those in i;h~: ·
the
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APPENDIX
List of Published Work of the Author
1. X-ray Studies of Phase Transitions in Ni-Zr Alloys,
XI National Conference on Crystallography, -~adavpur, Calcutta (17-19 January 1980), c4-4
2, Crystal Structure of Two New Intermetallic Compounds: Pt-Zr System,
XII National Conference on Crystallography, Hyderabad (25-28 Pebruary 1981), 8.
3. New Intermetallic Phases in Ni-Zr System.
XIII National Seminar on Crystallography, Nagpur (15-18 March 1982), 169.
4. Structure of the Ni11 zr9 Phase and its Thermal Expansion-Coefficient.
(accepted for publication in Acta Cryst. B).
5. Thermal-Expansion-Coefficient of Ni3Zr PhasE
(Communi __ ..__,.,,
6. Studies on Ni-Sn-In Binary Systems. . . i,
",;~ ;,
and Ni-Sn""'Sb Quas