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Low-temperature structure of Fe-Ni alloys
Journal of Magnetism and Magnetic Materials 10 (1979) 152-154 © North-HollandPublishing Company
LOW-TEMPERATURE STRUCTURE OF F e - N i ALLOYS M. MATSUI and K. ADACHI Faculty of Engineering, Nagoya University, Nagoya, Japan
The Debye temperature (0M) of Fe-Ni alloys was obtained from measurements of the X-ray integrated intensity and the electrical resistivity at low temperatures from 4 to 300 K. A decrease of 0M, implying a lattice softening effect, was found by lowering the temperature or increasing the iron concentration in the Invar region.
A decrease in the elastic stiffness constants of Fe-Ni Invar alloys below the Curie point [1] can be regarded as being due to lattice softening and should be associated with a reduction of the Debye temperature. On the other hand, the anomalous negative thermal expansion coefficient in the Invar region (aa) has recently been estimated [2] by the subtraction of an anharmonic term of the lattice vibration from the experimental data (a a = aobs - O q a t t i c e ) in which lattice softening is not specifically invoked. By assuming a two-states model of Fe atoms in the alloy a quantitative explanation of eta(T) as well as the anomalous volume change, Cos(T), has been proposed by Matsui and Chikazumi ([2]). This report attempts to clarify the relationship between the lattice softening effect and a a, especially for the low-temperature region using X-ray diffraction (from 4 to 300 K) and electrical resistivity measurements. Polycrystalline samples with 31 to 35 at% Ni and single crystals with 32, 36 and 38 at% Ni as well as pure Ni were prepared. The single crystals were cut with a perpendicular plane to the scattering vector for X-ray diffraction. The maximum error in the temperature measurement was estimated to be +1 K. The integrated intensity of Bragg reflection was measured at low temperatures below 300 K, using Mo and Cu targets. The basic equations of the X-ray scattering intensity are given by the following expressions.
I o = ALp f 2 exp [-2B(sin 0/X) 2 ] Es ,
with
B = (6h2/makOM)[(F(x)/x) + 1] ,
(2)
x
F(x) =X -1 f [(e r -- 1) -1 d~" ; 0
x = OMIT,
(3)
where lo, Lp and A are the observed intensity, the Lorentz polarization factor and a constant respect i v e l y ; fis the mean atomic form factor of Fe and Ni. The secondary extinction coefficient, Es, was estimated at the lowest temperature by means of Hamilton's equation, [3] for a flat plate sample. A correction for the volume expansion effect was also made, utilizing the measured lattice parameters. The temperature dependence of the lattice parameters for the single crystals is shown in fig. 1. The lattice parameters at 300 K are in good agreement with the report of Owen et al. [4]. The lattice parameter of alloys containing 36 and 38% Ni shows a minimum at about 50 and 40 K, respectively, which is in good agreement with the thermal expansion coefficient measured by previous authors [ 5 - 7 ] . The start of the martensitic transformation (Ms) can be observed at 92 K for 32% Ni. Just below 92 K some 17% by volume of the retained fcc phase can be detected from the integrated Bragg intensity relative to the value at 92 K. The lattice parameter of 32% Ni does not show any minimum in the temperature range from 300 K to the M s point. This suggests that the mini-
(1) 152
M. Matsui and K. Adachi / Low.temperature structure of Fe-Ni alloys 3.525
%
259C
153
The X-ray Debye temperature was calculated from the ratio of the integrated intensity of the same (800) reflection plane at different temperatures using the following equation: 2 B ( 7 ) - 2B(To) = ~
lnFI°(T°)Es(7)l
(4)
Lio (7)Es(To ) J • .<
3.520
v
.<
3.585
v
3.515
3.ssof 0
I
160
200
The reference Debye temperature 0M (To) was first determined from the slope of In [Io (To)/Es (To)Lo 72 ] vs. (sin 8IX)2 plot for To = 300 K. The temperature dependences of 0M(7) were then calculated from eq. (4). The maximum error of 0M (To) was estimated to be +10 K due to the uncertainty of the measured intensity and uncompensated errors such as thermal diffuse scattering. The values thus obtained for 8M(7) are shown in fig. 2. For pure Ni, 0M(T0) is determined to be 460 + 5 K, which is consistent with the results by Bower et al. [8]. For 38% Ni, 0M (7) decreases gradually with lowering temperature, while the values for 36% and 32% Ni decrease abruptly. The values of 8M (To) are 415 -+ 10, 411 +--5 and 410 +--5 K for 32,
300
T (K) Fig. 1. Temperature dependence of lattice parameter of F e - N i alloys. The arrows on the 32% Ni alloy curve indicate the sequence of measurements. The vertical axis on the right side should be employed only for pure Ni.
mum of lattice parameter or the negative thermal expansion coefficient [6] below 50 K has no direct connection with the martensitic transformation. It is noteworthy that, as shown for the 32% Ni alloy in fig. 1, the lattice parameter of the residual fcc phase just below 92 K is larger than that of the fcc just above the M s point and that the lattice parameter of the residual fcc phase decreases on raising the temperature. The observed half width of the (800) Bragg reflection profile of the single crystals was found to be 0.278 °, 0.245 °, 0.244 ° and 0.260 ° for the 32%, 36%, 38% alloys and pure Ni, respectively. The half width does not change with temperature within an error of +0.003 ° . Therefore, it is concluded that the samples do not contain any abnormal internal strain as well as any remarkable composition fluctuation, such as a clustering effect, but the atomic arrangement may be regarded as homogeneous.
Ni
4OO
38 */, x/X~ x x/,~ ~/
~
~
~
!
g
,
30
I I I
~ I
J.
25
'II
I I 200 t 0
Ms
, 100
, 200 T
, 300
(K)
Fig. 2. Temperature dependence of the X-ray Debye temperature, 0M, of F e - N i alloys.
154
M. Matsui and K. Adachi / Low-temperature structure o f F e - N i alloys
36 and 38% Ni, respectively. By extrapolation to 0 K the decrease of 0M, defined as A0M = 0M(300 K) 0M(0K),is 131 + 15, 81 +- 10 and 31 + 10 K for the 32, 36 and 38% Ni alloys respectively. Intensity measurements of the (222) reflection for 36% Ni shows a similar behaviour for 0M (7). The anisotropic behaviour of 0M(T) could not be separated within an error of-+10 K. It should be noted that an extraordinarily low value for the Debye temperature [0M --- 220 -+ 10 K] of the residual fcc phase is found at 71.5 K below the Ms temperature. The root mean s ~ a r e displacement of vibrating atoms, x / ~ = x/B/81r~ , can be evaluated from 0M(T). The displacement at low temperatures therefore increases with increasing Fe concentration. The displacement, x / ~ at 4.2 K (extrapolated to 0 K) is 0.049, 0.044, 0.041 and 0.037 A, for the 32, 36, 38% alloys and pure Ni, respectively. For the 32% Ni alloy, the X / ~ for the residual fcc at 71.5 K is 0.071 A, which is much larger than that of the fcc structure just above the Ms temperature (0.053 A). The electrical resistivity, p, of single crystals was also measured in the [ 100] direction. The sample used for the electrical resistivity measurements of a 32% Ni alloy did not transform to martensite even at 4.2 K, although it was cut from the same block as that used for the X-ray diffraction experiment. It seems that the nickel concentration is very critical for martensitic transformation. The residual resistivity, Po, increases abruptly with increase of Fe concentration as reported previously [9,10]. The phonon contribution to electrical resistivity was estimated from the temperature dependence of the slope (o: 096) of (p - p o ) / T 2 vs. T 3 plot at low temperatures (<30 K). The observation shows that the slope increased abruptly with lowering temperature and increasing Fe concentration, which is consistent with X-ray experiments. On the basis of present experimental results, it is concluded that the lattice softening of F e - N i Invar
alloys in the low-temperature range occurs with lowering temperature and increasing Fe concentration. This result is consistent with the Debye temperature calculated from elastic constants by Hausch [12], excluding the case of 32% Ni alloy. The crystal lattice, as shown in present work, contracts with increasing Fe concentration and expands with lowering temperature below 50 and 40 K for the 36 and 38% Ni alloy, respectively, while 0M decreases in all cases. By contrast the anomalous volume contraction, COs,obtained from an integration of aa(T), shows a monotonic increase with raising temperature and increasing Fe concentration [2]. In consequence, it would seem that the volume changes in Invar alloys are not directly related to the low temperature lattice softening found in the present work. Further investigations are required in order to make clear the relationship between the lattice softening effect and the martensitic transformation.
References [1] G. Hausch and H. Warlimont, Acta Met. 21 (1973) 401. [2] M. Matsui and S. Chikazumi, J. Phys. Soc. Japan 45 (1978) 458. [3] W.C. Hamilton, Acta Cryst. 10 (1957) 629. [4] E.A. Owen, E.L. Yates and A.H. Sully, Proc. Phys. Soc. 49 (1937) 315. [5] G.K. White, Proc. Phys. Soc. 86 (1965) 159. [6] A.I. Zakharovand L.N. Fedotov, Phys. Metal. Metallog. 23 (1967) 201. [7] C.D. Kim, M. Matsui and S. Chikazumi, J. Phys. Soc. Japan 44 (1978) 1152. [8] R.H. Wilson, E.F. Skelton and J.L. Katz, Acta Cryst. 21 (1966) 635. [9] D.I. Bower, E. Claridge and I.S.T. Tsong, Phys. Stat. Sol. 29 (1968) 617. [10] E.I. Kondorskii and V.L. Sedov, J. Appl. Phys. 31 (1960) 331S. [11] B.E. Armstrongand R. Fletcher, Can. J. Phys. 50 (1972) 244.
LOW-TEMPERATURE STRUCTURE OF F e - N i ALLOYS M. MATSUI and K. ADACHI Faculty of Engineering, Nagoya University, Nagoya, Japan
The Debye temperature (0M) of Fe-Ni alloys was obtained from measurements of the X-ray integrated intensity and the electrical resistivity at low temperatures from 4 to 300 K. A decrease of 0M, implying a lattice softening effect, was found by lowering the temperature or increasing the iron concentration in the Invar region.
A decrease in the elastic stiffness constants of Fe-Ni Invar alloys below the Curie point [1] can be regarded as being due to lattice softening and should be associated with a reduction of the Debye temperature. On the other hand, the anomalous negative thermal expansion coefficient in the Invar region (aa) has recently been estimated [2] by the subtraction of an anharmonic term of the lattice vibration from the experimental data (a a = aobs - O q a t t i c e ) in which lattice softening is not specifically invoked. By assuming a two-states model of Fe atoms in the alloy a quantitative explanation of eta(T) as well as the anomalous volume change, Cos(T), has been proposed by Matsui and Chikazumi ([2]). This report attempts to clarify the relationship between the lattice softening effect and a a, especially for the low-temperature region using X-ray diffraction (from 4 to 300 K) and electrical resistivity measurements. Polycrystalline samples with 31 to 35 at% Ni and single crystals with 32, 36 and 38 at% Ni as well as pure Ni were prepared. The single crystals were cut with a perpendicular plane to the scattering vector for X-ray diffraction. The maximum error in the temperature measurement was estimated to be +1 K. The integrated intensity of Bragg reflection was measured at low temperatures below 300 K, using Mo and Cu targets. The basic equations of the X-ray scattering intensity are given by the following expressions.
I o = ALp f 2 exp [-2B(sin 0/X) 2 ] Es ,
with
B = (6h2/makOM)[(F(x)/x) + 1] ,
(2)
x
F(x) =X -1 f [(e r -- 1) -1 d~" ; 0
x = OMIT,
(3)
where lo, Lp and A are the observed intensity, the Lorentz polarization factor and a constant respect i v e l y ; fis the mean atomic form factor of Fe and Ni. The secondary extinction coefficient, Es, was estimated at the lowest temperature by means of Hamilton's equation, [3] for a flat plate sample. A correction for the volume expansion effect was also made, utilizing the measured lattice parameters. The temperature dependence of the lattice parameters for the single crystals is shown in fig. 1. The lattice parameters at 300 K are in good agreement with the report of Owen et al. [4]. The lattice parameter of alloys containing 36 and 38% Ni shows a minimum at about 50 and 40 K, respectively, which is in good agreement with the thermal expansion coefficient measured by previous authors [ 5 - 7 ] . The start of the martensitic transformation (Ms) can be observed at 92 K for 32% Ni. Just below 92 K some 17% by volume of the retained fcc phase can be detected from the integrated Bragg intensity relative to the value at 92 K. The lattice parameter of 32% Ni does not show any minimum in the temperature range from 300 K to the M s point. This suggests that the mini-
(1) 152
M. Matsui and K. Adachi / Low.temperature structure of Fe-Ni alloys 3.525
%
259C
153
The X-ray Debye temperature was calculated from the ratio of the integrated intensity of the same (800) reflection plane at different temperatures using the following equation: 2 B ( 7 ) - 2B(To) = ~
lnFI°(T°)Es(7)l
(4)
Lio (7)Es(To ) J • .<
3.520
v
.<
3.585
v
3.515
3.ssof 0
I
160
200
The reference Debye temperature 0M (To) was first determined from the slope of In [Io (To)/Es (To)Lo 72 ] vs. (sin 8IX)2 plot for To = 300 K. The temperature dependences of 0M(7) were then calculated from eq. (4). The maximum error of 0M (To) was estimated to be +10 K due to the uncertainty of the measured intensity and uncompensated errors such as thermal diffuse scattering. The values thus obtained for 8M(7) are shown in fig. 2. For pure Ni, 0M(T0) is determined to be 460 + 5 K, which is consistent with the results by Bower et al. [8]. For 38% Ni, 0M (7) decreases gradually with lowering temperature, while the values for 36% and 32% Ni decrease abruptly. The values of 8M (To) are 415 -+ 10, 411 +--5 and 410 +--5 K for 32,
300
T (K) Fig. 1. Temperature dependence of lattice parameter of F e - N i alloys. The arrows on the 32% Ni alloy curve indicate the sequence of measurements. The vertical axis on the right side should be employed only for pure Ni.
mum of lattice parameter or the negative thermal expansion coefficient [6] below 50 K has no direct connection with the martensitic transformation. It is noteworthy that, as shown for the 32% Ni alloy in fig. 1, the lattice parameter of the residual fcc phase just below 92 K is larger than that of the fcc just above the M s point and that the lattice parameter of the residual fcc phase decreases on raising the temperature. The observed half width of the (800) Bragg reflection profile of the single crystals was found to be 0.278 °, 0.245 °, 0.244 ° and 0.260 ° for the 32%, 36%, 38% alloys and pure Ni, respectively. The half width does not change with temperature within an error of +0.003 ° . Therefore, it is concluded that the samples do not contain any abnormal internal strain as well as any remarkable composition fluctuation, such as a clustering effect, but the atomic arrangement may be regarded as homogeneous.
Ni
4OO
38 */, x/X~ x x/,~ ~/
~
~
~
!
g
,
30
I I I
~ I
J.
25
'II
I I 200 t 0
Ms
, 100
, 200 T
, 300
(K)
Fig. 2. Temperature dependence of the X-ray Debye temperature, 0M, of F e - N i alloys.
154
M. Matsui and K. Adachi / Low-temperature structure o f F e - N i alloys
36 and 38% Ni, respectively. By extrapolation to 0 K the decrease of 0M, defined as A0M = 0M(300 K) 0M(0K),is 131 + 15, 81 +- 10 and 31 + 10 K for the 32, 36 and 38% Ni alloys respectively. Intensity measurements of the (222) reflection for 36% Ni shows a similar behaviour for 0M (7). The anisotropic behaviour of 0M(T) could not be separated within an error of-+10 K. It should be noted that an extraordinarily low value for the Debye temperature [0M --- 220 -+ 10 K] of the residual fcc phase is found at 71.5 K below the Ms temperature. The root mean s ~ a r e displacement of vibrating atoms, x / ~ = x/B/81r~ , can be evaluated from 0M(T). The displacement at low temperatures therefore increases with increasing Fe concentration. The displacement, x / ~ at 4.2 K (extrapolated to 0 K) is 0.049, 0.044, 0.041 and 0.037 A, for the 32, 36, 38% alloys and pure Ni, respectively. For the 32% Ni alloy, the X / ~ for the residual fcc at 71.5 K is 0.071 A, which is much larger than that of the fcc structure just above the Ms temperature (0.053 A). The electrical resistivity, p, of single crystals was also measured in the [ 100] direction. The sample used for the electrical resistivity measurements of a 32% Ni alloy did not transform to martensite even at 4.2 K, although it was cut from the same block as that used for the X-ray diffraction experiment. It seems that the nickel concentration is very critical for martensitic transformation. The residual resistivity, Po, increases abruptly with increase of Fe concentration as reported previously [9,10]. The phonon contribution to electrical resistivity was estimated from the temperature dependence of the slope (o: 096) of (p - p o ) / T 2 vs. T 3 plot at low temperatures (<30 K). The observation shows that the slope increased abruptly with lowering temperature and increasing Fe concentration, which is consistent with X-ray experiments. On the basis of present experimental results, it is concluded that the lattice softening of F e - N i Invar
alloys in the low-temperature range occurs with lowering temperature and increasing Fe concentration. This result is consistent with the Debye temperature calculated from elastic constants by Hausch [12], excluding the case of 32% Ni alloy. The crystal lattice, as shown in present work, contracts with increasing Fe concentration and expands with lowering temperature below 50 and 40 K for the 36 and 38% Ni alloy, respectively, while 0M decreases in all cases. By contrast the anomalous volume contraction, COs,obtained from an integration of aa(T), shows a monotonic increase with raising temperature and increasing Fe concentration [2]. In consequence, it would seem that the volume changes in Invar alloys are not directly related to the low temperature lattice softening found in the present work. Further investigations are required in order to make clear the relationship between the lattice softening effect and the martensitic transformation.
References [1] G. Hausch and H. Warlimont, Acta Met. 21 (1973) 401. [2] M. Matsui and S. Chikazumi, J. Phys. Soc. Japan 45 (1978) 458. [3] W.C. Hamilton, Acta Cryst. 10 (1957) 629. [4] E.A. Owen, E.L. Yates and A.H. Sully, Proc. Phys. Soc. 49 (1937) 315. [5] G.K. White, Proc. Phys. Soc. 86 (1965) 159. [6] A.I. Zakharovand L.N. Fedotov, Phys. Metal. Metallog. 23 (1967) 201. [7] C.D. Kim, M. Matsui and S. Chikazumi, J. Phys. Soc. Japan 44 (1978) 1152. [8] R.H. Wilson, E.F. Skelton and J.L. Katz, Acta Cryst. 21 (1966) 635. [9] D.I. Bower, E. Claridge and I.S.T. Tsong, Phys. Stat. Sol. 29 (1968) 617. [10] E.I. Kondorskii and V.L. Sedov, J. Appl. Phys. 31 (1960) 331S. [11] B.E. Armstrongand R. Fletcher, Can. J. Phys. 50 (1972) 244.