The Atomic Defect Relaxation Processes in the Ti–Mo Alloys

Abstract

The relaxation processes correlated to atomic defects were investigated by the internal friction method using a multifunctional internal friction apparatus for Ti–Mo alloys. The microstructures of the Ti–Mo alloys with different Mo content and heat treatments were observed using an optic microscopy and a scanning electronic microscopy. Their phase constitutions were detected by X-ray diffraction (XRD). The β phase increases and α phase decreases in volume with increasing Mo content for the furnace-cooled alloys. Similarly, the β_M phase volume increases when Mo content is increased. The two relaxational internal friction peaks, named as P1 for the low temperature peak and P2 for the high temperature peak, respectively, are found on the internal friction temperature dependent curves in the alloys. The two peaks always appear in the present experimental specimens with different Mo content and different heat treatments except for the water quenched Ti–24 Mo alloys. The P1 and P2 peaks are all influenced by Mo content and heat treatments and related to phase constitutions and microstructures. The P1 peak height increases with increasing Mo content for the water quenched alloys and the P1 peak temperature is not changed with Mo content. The P2 peak height increases also with increasing Mo content and the P2 peak temperature is raised when Mo content is increased but for the water quenched Ti–3Mo and Ti–5Mo alloys. Relaxation parameters of P1 peak are H_wq1 = 1.56 ± 0.1 eV and τ_0wq1 = 2.5 × 10^−16±0.1 s and those of P2 peak are H_wq2 = 2.3 ± 0.1 eV and τ_0wq2 = 2.1 × 10^−17±0.1 s for the water-quenched Ti–5Mo alloy, respectively. The P1 peak is attributed to the stress-induced short-range ordering of oxygen complexes around Ti atoms in the Ti–Mo alloys. The increase of the P1 peak height with increasing Mo content is attributed to the increase of β phase in volume. The P2 peak is resulted from the interaction of Mo–O atoms or the reorientation of Mo–Mo atomic pairs by means of vacancies. The Mo–O interaction is strengthened when Mo content is increased, which results in the increase of both the peak temperature and the activation energy of the P2 peak.

The P1 and P2 peak-temperatures in water quenched Ti–5Mo alloy increase with increasing the vibration frequency.

1. Introduction

Recently, low-modulus, non-toxic β-type titanium alloys have been paid much attention due to their containing nontoxic elements, low elastic modulus, high strength, good corrosion resistance and excellent biocompatibility. The alloys have many important applications as biomedical materials in orthopedic implant.^1–4) Mo is the most effective β stabilizer, among the nontoxic β-type alloying elements (e.g., Mo, Nb, Ta, Zr, etc.).⁵⁾ Therefore, the Ti–Mo alloys have been also extensively studied as one of the β-type Ti alloys for biomedical applications.^6–10) The reported results in above these references are helpful for our understanding the mechanism of high performance of Ti–Mo alloys.^9–11)

The atomic defects in metals or crystalline materials have important effects on their physical, mechanical properties and anelasticity, which is generally characterized by internal friction. However, the emphasis of the above-mentioned studies on the Ti–Mo alloys is focused on their microstructures and mechanical properties.⁹⁾ The relaxation processes correlated to atomic defects, such as interstitial oxygen, substitutional Mo atoms vacancies and so on in Ti–Mo alloys has not systematically been investigated although it has been studied in Ti–Nb alloys.^12–14) Therefore, it is necessary to investigate the diffusion movements of the interstitial and substantial atoms (e.g., O, N, Mo and so on) in Ti-based alloys and to give a clear physical image of relaxation processes.

In this work, the relaxation processes and mechanism of atomic defects in Ti–Mo alloys was investigated though examining the internal friction behavior of the Ti–Mo alloys and correlating this behavior to the occurrence and movement of defects.

2. Experimental Procedures

Ti–Mo alloys with different Mo content (nominal Mo content 3, 5, 10, 18, 24 (mass) %) were melted in a vacuum arc furnace under argon atmosphere using commercial pure Ti (99.99% purity) and Mo (99.9% purity) particles, which are named as Ti–3Mo, Ti–5Mo, Ti–10Mo, Ti–18Mo, Ti–24Mo alloys, respectively. The ingots were hot forged at about 1100°C and changed into disc samples. All specimens to be detected were spark cut from the dics samples. The specimen dimensions are 1 × 2 × 50 (mm) for the internal friction tests and 5 × 5 × 10 (mm) for the microstructure detect. They were heated and homogenized using a GSL-1600X tube furnace in an argon atmosphere at 950°C for 20 mins., and then quenched into water at room temperature and furnace-cooled to room temperature, respectively.

The internal friction (Q⁻¹) of the specimens with different Mo contents and different heat treatments were measured using a multifunctional internal friction apparatus during heating and cooling. Four forced vibration frequencies (0.5, 1, 2, 4 Hz) were swept when the specimens were heated and cooled with a changing temperature rate of 3°C/min since sufficient data can be obtained for the above four frequencies at the rate, with applied strain amplitude of 3 × 10⁻⁵.

X-ray diffraction (XRD) experiments were carried out in order to detect the differences in the phase constitutes of the Ti–Mo alloy specimens with different heat treatments and different Mo contents. The microstructures of Ti–Mo alloy specimens are also observed using an optical microscopy. The specimens to be detected are polished and etched in a solution (5 vol.-%HNO3 + 10 vol.-%HF + water (bal.)). An oxygen nitrogen analyzer is used to detect the oxygen content of the Ti–Mo alloys.

3. Results and Discussion

3.1 The phase constitutions and microstructures of the Ti–Mo alloys

Pure Ti possesses α phase with hexagonal close-packed (hcp) structure and pure Mo is body-centered cubic (bcc) structure at low temperature. Ti–Mo alloys possess body-centered cubic (bcc) structure at high temperature, which is called as β-Ti–Mo alloys. If the β-Ti–Mo alloys were furnace-cooled from high temperature, the stable α or α+β_S (stable β) or whole β_S phases would be formed and their relative volume depends on Mo content. Whole β_S phase can be obtained when Mo content exceeds about 12.6 (mass)% according to Ref. 15). In contrast, the β-Ti–Mo alloys at high temperatures will form either α′ (with a hcp structure) or α′′ (with an orthorhombic structure) or β_M (metastable β) or their combination, which is dependent on Mo content when they are rapidly cooled.^3,10,11,15) Figure 1 shows the XRD results of the furnace-cooled Ti–Mo alloys. It can be seen that the furnace-cooled Ti–Mo alloys consist of α and β phases. The volume of α phase decreases and the volume of β phase increases with increasing Mo content, which are in accordance with those in Ref. 4). From Fig. 2, it can be known that the water quenched Ti–Mo alloys show different phase constitutes compared with the furnace-cooled alloys with the identical Mo content. The water quenched Ti–Mo alloys are composed of α′, α′′, β_M or their combination. α′ phase exists in the water quenched Ti–Mo alloys with low Mo content and β_M exists in those with high Mo content. The water quenched Ti–Mo alloys with medium Mo content possess either α′ + α′′ or α′′ + β_M. The whole β_M is found in the water quenched Ti–18Mo and Ti–24Mo alloys, in agreement with Ref. 10). From Figs. 3 and 4, it can be seen that the water quenched Ti–12Mo alloy possesses α′′ + β_M phases, in accordance with the XRD results. The present observed phase constitutions are also similar to those of TI–Mo–Ag (or Sn, Ga) in Refs. 16, 17).

Fig. 1

The XRD results of the furnace-cooled Ti–Mo alloys with different Mo content.

Fig. 2

The XRD results of the water quenched Ti–Mo alloys with different Mo content.

Fig. 3

Microstructures of the water quenched Ti–12Mo alloy observed using an optical microscope (red line length in the image denotes 100 µm).

Fig. 4

Microstructures of the furnace-cooled Ti–12Mo alloy observed using an optical microscope (red line length in the image denotes 100 µm).

3.2 The internal friction of the Ti–Mo alloys

Figure 5 shows the internal friction (Q⁻¹) and relative dynamic modulus vs. Temperature during heating for the water-quenched Ti–5Mo alloy at different vibration frequencies. It can be seen that two internal friction peaks appears at around 260°C (named as P1) and 460°C (named as P2) (f = 1 Hz) and the relative dynamic modulus declines rapidly in the vicinity of the peak position for the low temperature peak, which is similar to Kronig-Kramers relation.^18,19) Their peak temperature increases with increasing the vibration frequency for the two internal friction peaks, indicating that the two peaks are a thermally activated relaxation process, being a linear anelastic relaxation.

Fig. 5

The internal friction and relative dynamic modulus vs. temperature during heating for the water-quenched Ti–5Mo alloy from 950°C with different vibration frequency.

For a thermally activated relaxation process with a single relaxation time, the relaxation time τ follows the Arrhenius law:¹⁹⁾   

\begin{equation} \tau = \tau_{o}\exp(H/kT) \end{equation} (1)

Where τ₀ is the pre-exponential factor and H is the activation energy of the relaxation process. At the peak position, ωτ_p = 1 is satisfied, where ω = 2πf is the angular frequency and τ_p is the relaxation time at the peak temperature.¹⁹⁾ Relaxation parameters of P1 peak can be calculated from Fig. 3 to be H_wq1 = 1.56 ± 0.1 eV and τ_0wq1 = 2.5 × 10^−16±0.1 s and those of P2 peak can be calculated to be H_wq2 = 2.3 ± 0.1 eV and τ_0wq2 = 2.1 × 10^−17±0.1 s for the water-quenched Ti–5Mo alloy, respectively. Martins J.R. et al.¹¹⁾ found the two internal friction peaks at 460 K and 525 K in Ti–Mo alloys, respectively. The 525 K peak temperature is similar to that of the present P1 peak, and its activation energy is different from that of the present P1 peak. However, the present P1 peak is very similar to the internal friction peak found in the water-quenched Ti–Nb alloys from the peak-temperature and the activation energy, whose peak-temperature is about 250°C and activation energy is H = 1.67 ± 0.1 eV,¹²⁾ which is correlated to oxygen.

From Fig. 5, it can be also seen that there are evidently the internal friction background (IFB) on the internal friction-temperature curves. The importance of the IFB (also called as high temperature background (HTB)) have two aspects. On one hand, it could provide information about microscopic mechanisms involved in high-temperature creep behavior,²⁰⁾ and on the other hand, because once the HTB is characterized, it is possible to remove it from the internal friction spectra, and then the internal friction peak can be better analyzed.¹⁰⁾ Figure 6 shows the net peak after being subtracted IFB for the water quenched Ti–5Mo alloy at f = 1 Hz. It can be seen that the net peak-heights are reduced and the peak temperatures is very similar to the original internal friction peaks.

Fig. 6

The net peak subtracted internal friction background for the water quenched Ti–5Mo alloy at different frequencies.

Figure 7 shows the internal friction-temperature curve of the water quenched Ti–5Mo alloy during heating and cooling. Obviously, the P1 peak-height is seriously reduced and the peak-temperature is little changed during cooling. For the furnace-cooled Ti–5Mo alloy, the internal friction-temperature curves during heating are the same as those during cooling except a little difference of their IFB, as shown in Fig. 8. The difference from the water quenched Ti–Mo alloys to the furnace-cooled Ti–Mo alloys in their peak height is also seen in Fig. 9 and Fig. 10.

Fig. 7

The internal friction-temperature curves of the water-quenched Ti–5Mo alloy during heating and cooling at f = 1 Hz.

Fig. 8

The internal friction-temperature curves of the furnace-cooled Ti–5Mo alloy during heating and cooling at f = 1 Hz.

Fig. 9

The internal friction-temperature curves of the furnace-cooled and water-quenched Ti–5Mo specimens during heating at f = 1 Hz.

Fig. 10

The internal friction-temperature curves of the furnace-cooled and water-quenched Ti–24Mo specimens during heating at f = 1 Hz.

Figure 11 shows the internal friction-temperature curves of the water-quenched Ti–Mo alloys with different Mo content at f = 1 Hz. It can be seen that Mo contents have great effects on the two peaks. In order to clarify accurately the influence of Mo content on the two peaks. Their net peaks are obtained by subtracting the IFB. Generally, IFB, $Q_{b}^{ - 1}$, are expressed as:²¹⁾   

\begin{equation} Q_{b}^{-1} = A + B\exp(C/kT) \end{equation} (2)

where A, B and C are constants, k is the Boltzamann constant and T is the absolute temperature. The net peak heights vary with Mo content in Fig. 12. It can be seen that the two peak-heights increase when Mo content is increased. The P1 peak temperature is 260°C and is little changed with Mo content. The P2 peak temperature is varied with Mo content and is raised with increasing Mo content but a little exception for the water quenched Ti–3Mo and Ti–5Mo alloys. It should be pointed out that the P2 peak cannot be observed in the water quenched Ti–24Mo alloy, which can be resulted from the higher P2 peak temperature in the alloy and cannot be detected within the present measuring temperature limit. It can be deduced that the characteristics of the P1 peak show a obvious relation to β phase. For the furnace-cooled Ti–Mo alloys, Mo content also has influence on the internal friction behavior, as shown in Fig. 13. Similarly, the internal friction-temperature curves after being subtracted IFB are shown in Fig. 14 for the furnace-cooled Ti–Mo alloys with different Mo content. Obviously, the P1 and P2 peak-heights increase with increasing Mo content. The P1 peak temperature is little changed with Mo content and P2 peak temperature is increased with increasing Mo content, which is similar to that of the water quenched alloys.

Fig. 11

The internal friction-temperature curves of the water-quenched Ti–Mo alloys with different Mo contents at f = 1 Hz.

Fig. 12

The variation of The P1 and P2 net peaks with Mo content for the water quenched Ti–Mo alloys.

Fig. 13

The internal friction-temperature curves of the furnace-cooled Ti–Mo alloys with different Mo content during heating at f = 1 Hz.

Fig. 14

The variation of The P1 and P2 net peaks with Mo content for the furnace-cooled Ti–Mo alloys.

Therefore, it can be simply deduced that the P1 and P2 peak heights increase with increasing the volume of β_M or β phase for the water quenched and furnace-cooled Ti–Mo alloys from the microstructures, which depends on Mo content.

3.3 The relaxation mechanisms of atomic defects of P1 and P2 peaks

Atomic defects in metals or crystalline materials can produce a time-dependent strain under the application of an external cycle stress due to the reorientation of the point defects, which is anelastic relaxation when defect symmetry is smaller than that of host metals or crystalline materials.^12,19) Oxygen is common impurity in Ti-based alloys and generally is located in the interstitials of metals. In Ti–Mo alloys, Mo atoms are substitutional atomic defects and occupy Ti positions. In addition, the Mo or Ti vacancies are also major atomic defects in Ti–Mo alloys, especially in the water quenched Ti–Mo alloys. The Snoek relaxation resulted from interstitial impurities of carbon or nitrogen in α-Fe and was experimentally investigated by Snoek et al.¹⁹⁾ Recently, Saitoh et al. investigated the influences of substitutional atoms on the Snoek peak of carbon in bcc iron.^22–24) In β-type Ti-based alloys, the Snoek-type relaxation peak is caused by oxygen and/or nitrogen.^25–29)

The present P1 peak temperature (260°C, f = 1 Hz) is similar to the peak found by Martins JR in the T–Mo alloys at 525 K (f = 1 Hz) reported in Ref. 11), in which the peak is attributed to the stress-induced short-range ordering of oxygen complexes around Ti atoms in the metal matrix. Oxygen in Ti-based alloys is difficult to remove even though the heat treatment is conducted at high vacuum. Therefore, the specimens contain oxygen under the condition of non-vacuum heat treatment. Oxygen contents are measured to be about 0.05% (mass%) for the four specimens, which are the furnace cooled Ti–3Mo, Ti–12Mo, Ti–24Mo alloys and the water quenched Ti–12Mo alloys. If the oxygen atoms are located in the octahedral positions of the β phase in Ti-based alloys with bcc structure, they will produce relaxation process under the application of cycle stress. An internal friction peak known as the Snoek peak will appear on the internal friction-temperature curve. Oxygen jumps or some atomic interactions among O and substitute atoms in bcc solution are the main physical mechanism of the peak.¹²⁾ Therefore, the present P1 peak is also related to oxygen. The Mo is an effective β stabilizer. When Mo content is increased, the volume of β phase is also raised. Correspondingly, the oxygen numbers producing relaxation in β phase can be increased, resulting in the increase of the P1 peak height since only the oxygen atoms in β phase can contribute relaxation. From Fig. 5, it can be seen that the P1 peak height is severely reduced during cooling, as compared with that during heating. On one hand, the metastable β_M is transformed into the stable β or α or ω, causing β_M to reduce in volume. On the other hand, oxygen amount solid-dissolved in β solid solution will be decreased, resulting in the reduction of P1 peak height during cooling.

The present P2 peak appears at about 446°C (f = 1 Hz) on the internal friction-temperature curves for the water quenched Ti–5Mo alloy. The activation energy of P2 peak is H_wq2 = 2.3 ± 0.1 eV, which is similar to the mutual diffusion energy value (210 kJ/mol (2.18 eV)) of Ti and Mo in Ti–5Mo of the reported in Ref. 30). In addition, the P2 heights increases with increasing Mo content. Thus, the present P2 relaxation process can be related to not only oxygen but also the substitutional atoms of Mo. The interaction of Mo–O may be involved in the P2 peak. The Mo–O interaction in β phase with low Mo content is weak and the P2 peak height is small. Correspondingly, the Mo–O interaction in Ti–Mo alloys with high Mo concentration will be strengthened. That the peak temperature is increased with increasing Mo content indicates the relaxation activation energy of the Mo–O interaction is also increased according to Ref. 31).

The present results show the P2 peak height in the furnace-cooled Ti–Mo alloys is smaller than that in the water quenched alloys for any specimen with identical Mo content, which should be resulted from different microstructures, as shown in Figs. 7, 12, 14, 15, either is P1 peak. Nevertheless, the furnace-cooled Ti–Mo alloys is greatly different from the water quenched Ti–Mo alloys in vacancy concentration in addition to their difference in microstructures. The quenched vacancies play an important role in relaxational processes produced by atomic defects, especially substitutional atomic defects since the reorientation or movements of the substitutional atomic pairs depends on the vacancies in the relaxational processes, e.g., Zener relaxation.^10,19) According to Ref. 11, 18), the specimens containing more substitutional impurities may produce the internal friction peaks around 697–754 K for frequency around 1 Hz and some complexes might have been involved, which is a little similar to the present P2 peak from the peak temperature, indicating that the relaxational mechanisms of the P2 peak is complicated. L. Usategui et al.¹⁰⁾ found an relaxation internal friction peak in Mo-rich Ti–44Al–7Mo alloy at high temperature, which is attributed to the reorientation of Mo–Mo atomic pairs by means of vacancies. It can be deduced that vacancies in Ti–Mo alloys influences the relaxational processes.

Fig. 15

The XRD results of the water quenched and furnace-cooled Ti–5Mo alloys.

The metastable β_M is obtained by the quenching and the stable β is formed during furnace-cooling. The quenched Ti–Mo alloys not only possess different the microstructures from the furnace-cooled Ti–Mo alloys but also contain a large number of the quenched vacancies.³²⁾ The vacancies may strengthen the interaction between the interstitial atoms (O, N and so on) and the substitute atoms (Mo, Ti) and increase the internal friction peak-height. When the internal friction of the quenched Ti–Mo alloys is measured during the heating, the vacancies are annihilated and thereby the peaks height is reduced during cooling. It should be pointed out that there is no great difference in their oxygen contents between the water quenched and the furnace-cooled Ti–Mo alloys. The peak-height difference between the water quenched and the furnace-cooled Ti–Mo alloys for an identical Mo content can be resulted from the difference of their vacancy numbers.

4. Conclusion

The β phase increases and α phase decreases in volume with increasing Mo content for the furnace-cooled alloys. Similarly, the β_M phase increases when Mo content is increased. The P1 and P2 peaks always appear in the present experimental specimens with different Mo content and different heat treatments except for the water quenched Ti–24Mo alloys. The P1 and P2 peaks are all influenced by Mo content and heat treatments and related to phase constitutions and microstructures. The P1 peak height increases with increasing Mo content for the water quenched alloys and the P1 peak temperature is not changed with Mo content. The P2 peak height increases also with increasing Mo content and the P2 peak temperature is raised when Mo content is increased but a little exception for the water quenched Ti–3Mo and Ti–5Mo alloys. Relaxation parameters of P1 peak are H_wq1 = 1.56 ± 0.1 eV and τ_0wq1 = 2.5 × 10^−16±0.1 s and those of P2 peak are H_wq2 = 2.3 ± 0.1 eV and τ_0wq2 = 2.1 × 10^−17±0.1 s for the water-quenched Ti–5Mo alloy, respectively. The P1 peak is attributed to the stress-induced short-range ordering of oxygen complexes around Ti atoms in the Ti–Mo alloys. The increase of the P1 peak height with increasing Mo content is attributed to the increase of β phase in volume. The P2 peak is resulted from the interaction of Mo–O atoms or the reorientation of Mo–Mo atomic pairs by means of vacancies. The Mo–O interaction is strengthened when Mo content is increased, which results in the increase of both the peak temperature and the activation energy of the P2 peak.

Acknowledgments

This work is supported by Research Center of New Functional Materials and Intelligent Testing Instruments in Suzhou Vocational University. This work is also supported by Natural Science Foundation of the Jiangsu Higher Education Institutions of China under Project (19KJD470005) and Suzhou Key Industry Technology Innovation Project (SYG201939).

REFERENCES

• 1)   Q.  Chen and  G.A.  Thouas: Mater. Sci. Eng. R 87 (2015) 1–57.
• 2)   M.  Geetha,  A.K.  Singh,  R.  Asokamani and  A.K.  Gogia: Prog. Mater. Sci. 54 (2009) 397–425.
• 3)   X.  Zhao,  M.  Niinomi,  M.  Nakai and  J.  Hieda: Acta Biomater. 8 (2012) 1990–1997.
• 4)   J.L.  Xu,  S.C.  Tao,  L.Z.  Bao,  J.M.  Luo and  Y.F.  Zheng: Mater. Sci. Eng. C 97 (2019) 156–165.
• 5)   Y.L.  Zhou and  D.M.  Luo: J. Alloys Compd. 509 (2011) 6267–6272.
• 6)   R.P.  Kolli,  W.J.  Joost and  S.  Ankem: JOM 67 (2015) 1273–1280.
• 7)   A.  Almeida,  D.  Gupta,  C.  Loable and  R.  Vilar: Mater. Sci. Eng. C 32 (2012) 1190–1195.
• 8)   M.  Sabeena,  S.  Murugesan,  P.  Anees,  E.  Mohandas and  M.  Vijayalakshmi: J. Alloys Compd. 705 (2017) 769–781.
• 9)   N.T.C.  Oliveira,  G.  Aleixo,  R.  Caram and  A.C.  Guastaldi: Mater. Sci. Eng. A 452–453 (2007) 727–731.
• 10)   L.  Usategui,  M.L.  Nó,  S.  Mayer,  H.  Clemens and  J.  San Juan: Mater. Sci. Eng. A 700 (2017) 495–502.
• 11)   J.R.S.  Martins, Jr.,  R.O.  Araújo,  R.A.  Nogueira and  C.R.  Grandini: Arch. Metall. Mater. 61 (2016) 25–30.
• 12)   Z.C.  Zhou,  J.Y.  Xiong,  S.Y.  Gu,  D.K.  Yang,  Y.J.  Yan and  J.  Du: J. Alloys Compd. 509 (2011) 7356–7360.
• 13)   Z.C.  Zhou,  Y.K.  Zhang,  J.  Du,  Y.J.  Yan and  S.Y.  Gu: Appl. Mech. Mater. 487 (2014) 3–6.
• 14)   Z.-C.  Zhou,  J.  Du,  S.-Y.  Gu and  J.Y.  Yong: Solid State Phenomena 298 (2019) 59–63.
• 15)  J.L. Murray: Phase Diagram of Binary Titanium Alloys, (ASM, Materials Park, Ohio, 1987).
• 16)   T.  Maeshima and  M.  Nishida: Mater. Trans. 45 (2004) 1096–1100.
• 17)   H.Y.  Kim,  Y.  Ohmatsu,  J.I.  Kim,  H.  Hosoda and  S.  Miyazaki: Mater. Trans. 45 (2004) 1090–1095.
• 18)  M.S. Blanter, I.S. Golovin, H. Neuhäuser and H.-R. Sinning: Internal Friction in Metallic Materials: A Handbook (Springer Series in Materials Science), Vol. 90, (Springer Berlin Heidelberg, Berlin, Germany, 2007).
• 19)  A.S. Nowick and B.S. Berry: Anelastic Relaxation in Crystalline Solids, (Academic Press, Now York and London, 1972).
• 20)   P.  Simas,  J.  San Juan and  M.L.  Nó: Intermetallics 18 (2010) 1348–1352.
• 21)  P.R. Bevington: Data Reduction and Error Analysis for the Physical Science, (McGraw-Hill, New York, 1969) p. 235.
• 22)   H.  Saitoh,  N.  Yoshinaga and  K.  Ushioda: Acta Mater. 52 (2004) 1255–1261.
• 23)   I.S.  Golovin and  S.B.  Golovina: Phys. Met. Metallogr. 102 (2006) 593–603.
• 24)   I.S.  Golovin,  S.B.  Golovina and  O.A.  Sokolova: Phys. Met. Metallogr. 105 (2008) 193–201.
• 25)   F.  Yin,  S.  Iwasaki,  D.  Ping and  K.  Nagai: Adv. Mater. 18 (2006) 1541–1544.
• 26)   F.  Yin,  L.  Yu,  D.  Ping and  S.  Iwasaki: Mater. Sci. Forum 614 (2009) 175–180.
• 27)   O.  Florêncio,  F.W.J.  Botta,  C.R.  Grandini,  H.  Tejima and  J.A.R.  Jodāo: J. Alloys Compd. 211–212 (1994) 37–40.
• 28)   L.H.  Almeida,  C.R.  Grandini and  R.  Caram: Mater. Sci. Eng. A 521–522 (2009) 59–62.
• 29)   F.  Yin,  L.  Yu and  D.  Ping: Mater. Sci. Eng. A 521–522 (2009) 372–375.
• 30)   L.  Feng,  J.-S.  Li,  L.  Huang,  H.  Chang,  Y.-W.  Cui and  L.  Zhou: Chin. J. Nonferrous Met. 19 (2009) 1766–1771.
• 31)   M.  Weller: Mater. Sci. Eng. A 442 (2006) 21–30.
• 32)   Y.  Nagai,  T.  Toyama,  Z.  Tang,  M.  Hasegawa,  H.  Matsumoto and  S.  Hanada: Scr. Mater. 54 (2006) 1751–1753.