本研究以真空電弧熔煉法，製備由Al, Co, Cr, Fe, Ni與Ti組成的，多種五元及六元等莫合與非等莫耳高熵合金。所得到的鑄造態合金經過1100℃持溫24小時，均質化水淬，得到均質化態合金。鑄造態合金及均質化態合金，作微結構、熱膨脹係數及磁場大至9 T的霍爾效應量測。微結構主要由FCC, BCC, Full Heusler與Sigma相組成，大部分具有磁性。均質化態合金的磁性Invar效應，使合金熱膨脹係數在居禮溫度處下降，此乃磁矩排列由有序變成無序的結果。等莫耳均質化態合金居禮溫度在480 K到750 K之間，非等莫耳均質化態合金居禮溫度在780 K到870 K之間。
本研究許多合金出現異常霍爾效應，與各約一半合金出現類電子與類電洞行為，5 K及300 K載子濃度在10^22 ~ 10^23 cm^-3之間，與純金屬和傳統合金的相似。載子遷移率為0.21 ~ 2.61 cm^2 V^-1 s^-1之間，比純金屬與傳統合金的低，與多元素混合產生的嚴重晶格缺陷，增加電子散射效應，阻礙電子的移動有關。自由電子模型有效自由電子數即價數，在0.171至4.716之間。等莫耳均質化態合金殘餘電阻率比值(RRR)分布在0.988至1.259之間，在1.032至1.056與1.139至1.207兩區出現正霍爾係數，其餘為負。等莫耳均質化態合金殘餘電阻率比值(RRR)分布在1.114至1.174之間，在1.158至1.173區間出現正霍爾係數，其餘為負。霍爾係數的正負值，被解讀成與聲子和晶格缺陷共同作用，導致Brillouin 區與Fermi區形狀隨溫度變化有關。
高熵合金具有簡單的晶體結構，故滿足Bloch定律，可以平均效應的概念，修正並應用Kronig-Penney模型於高熵合金中。

In this study, selected from Al, Co, Cr, Fe, Ni, and Ti, various quinary equal- and non-equal-mole and senary high-entropy alloys (HEAs) are made by vacuum-arc melting. The as-cast alloys are sequentially homogenized at 1100℃ for 24 h and water-quenched to obtain as-homogenized alloys. Microstructure examination, coefficient of thermal expansion (CTE) measurement, and Hall effect up to 9T are performed. Microstructure is composed of FCC, BCC, full Heusler, and sigma phases. Most alloys are of magnetic. CTE of as-homogenized alloys drops at Curie point due to magnetic order-disorder transition, the so-called magnetic Invar effect. Curie points thus obtained are in the intervals of 480 K to 750 K and 780 K to 870 K for equal- and non-equal-mole as-homogenized alloys, respectively.
Anomalous Hall effect occurs in most alloys, and electron-like and hole-like electrical conduction behaviors emerge in 50 %-50 % ratio of alloys. Similar to their pure elements and conventional alloy counterparts, carrier concentration of HEAs in this study at 5 K and 300 K varies from 10^22 to 10^23 cm^-3, while lower than their pure elements and conventional alloy counterparts, mobility of carriers varies from 0.21 to 2.61 cm^2 V^-1 s^-1, due to multielement mixing-caused heavy imperfection in alloy lattices, making strong electron scattering effect and impeding the electron transport. The effective number of free electrons in the free electron model is the valence number, ranging from 0.171 to 4.716. The residual electrical resistivity ratio (RRR) for equal-mole and non-equal-mole alloys ranges from 0.988 to 1.259, and from 1.114 to 1.174, respectively. The sign of Hall coefficient is positive for RRR intervals of 1.032 to 1.056 and 1.139 to 1.207 for equal-mole alloys, while it is positive for RRR in the interval of 1.158 to 1.173 for non-equal-mole AlxCoCrFeNi alloys. Otherwise, it has negative sign. The sign of Hal coefficient is explained t have something to do with the interaction between phonon and lattice defects that determines the shape of temperature-dependent Brillouin zone and the Fermi surface.
HEAs have simple crystal structure, thus they obey the Bloch theorem. One can apply a modified Kronig-Penney model t them with an idea of “average” effect in potential.

[1] J. W. Yeh, S. Y. Chang, Y. D. Hong, S. K. Chen and S. J. Lin, “Anomalous decrease in x-ray diffraction intensities of Cu-Ni-Al-Co-Cr-Fe-Si alloy systems with multi-principal elements,” Materials Chemistry and Physics 103 (2007) 41.
[2] 謝承安, "微結構對高熵合金的電與磁性質之影響," 國立清華大學材料科學與工程學所碩士論文 (2005)
[3] 陳廷傑, "簡單相高熵合金AlxCoCrFeNi(0≦x≦2)之電性質研究," 國立清華大學材料科學與工程學所碩士論文 (2006)
[4] 林川翔, "從Al, Co, Cr, Fe, Ni, Ti選取五至六元高熵合金之電與磁性質研究," 國立清華大學材料科學與工程學所碩士論文 (2008)
[5] P. Hidnert and W. Souder, Thermal expansion of solids. (NBS Circular, 1950)
[6] R. E. Taylor, “Estimation techniques and selected effects,” Thermal expansion of solids. (ASM International, 1998)
[7] K. Hori, “On the effect of added elements on the volume change due to solidification and on the thermal expansion of aluminum,” Nippon-Kinzoku Gakkai-Si 19 (1955).
[8] W. Lee, “Magnetostriction and magnetomechanical effects,” Reports on Progress in Physics 18 (1955) 184.
[9] M. Shiga, “Invar alloys,” Current Opinion in Solid State & Materials Science 1 (1996) 340.
[10] C. Nix and D. MacNair, “The thermal expansion of pure metals copper, gold, aluminum, nickel, and iron,” Physical Review 60 (1941) 597.
[11] N. W. Ashcroft and N. D. Mermin, Solid state physics. (Saunders College, 1976)
[12] Wei, An Introduction to electronic and ionic materials. (NJ World Scientific, Singapore, 1999)
[13] "Lorentz force." (http://en.wikipedia.org/wiki/Lorentz_force)
[14] Benson, University physics. (John Wiley & Sons, New York, 1995)
[15] C. Kittel, Introduction to solid state physics. (John Wiley & Sons, New York, 2005)
[16] F. Seitz, The modern theory of solids. (McGraw-Hill, New York, 1940)
[17] W. Smith and R. W. Sears, “The hall effect in permalloy,” Physical Review 34 (1929) 1466.
[18] J. Rhyne, “Anomalous and ordinary hall effect in terbium,” Journal of Applied Physics 40 (1969) 1001.
[19] M. Hurd, The hall effect in metals and alloys. (Plenum Press, New York, 1972)
[20] Inoue and H. Ohno, “Taking the hall effect for a spin,” Science 309 (2005) 2004.
[21] G. T. Meaden, Electrical resistance of metals. (Plenum Press, New York, 1965)
[22] W. B. Pearson, A handbook of lattice spacings and strucures of metals and alloys. (Pergamon Press, New York, 1958)
[23] R. K. Kirby, "Thermal expansion," American institute of physics handbook. (American Institute of Physics, 1972)
[24] E. S. Borovik and V. G. Volotska, “Anisotropy of galvanomagetic properties of pure aluminum large effective fields,” Soviet Physics Jetp-Ussr 21 (1965) 1041.
[25] H. Powell and E. J. Evans, “The hall effect and some other physical constants of alloys part vii. The aluminium-silver series of alloys,” Philosophical Magazine Series 7 34 (1943) 145.
[26] N. V. Volkenshtein, G. V. Fedorov and V. P. Shirokovskii, Physics of metals and metallography 11 (1961) 151.
[27] J. Arko, J. A. Marcus and W. A. Reed, “High-field galvanomagnetic effects in antiferromagnetic chromium,” Physical Review 176 (1968) 671.
[28] P. N. Dheer, “Galvanomagnetic effects in iron whiskers,” Physical Review 156 (1967) 637.
[29] G. W. Scovil, “The hall effect in single crystals of titanium,” Applied Physics Letters 9 (1966) 247.
[30] R. Kainuma, K. Urushiyama, K. Ishikawa, C. C. Jia, I. Ohnuma and K. Ishida, “Ordering and phase separation in bcc aluminides of the Ni-Fe-Al-Ti system”, (Elsevier Science, 1997) 235.
[31] H. P. Myers, Introductory solid state physics. (Taylor & Francis, London, 1990)