Specification [Title of the Invention] PLATED STEEL SHEET WITH QUASI CRYSTAL [Technical Field of the Invention] [0001] The present invention relates to a surface-treated steel sheet which is excellent in corrosion resistance. Particularly, the present invention relates to a plated steel sheet containing a quasi crystal. [Related Art] [0002] The quasicrystal is the crystal structure which was firstly discovered in 1982 by Dr. Daniel Shechtman, and has an atomic arrangement with a polyhedron with 20 faces (icosahedron). The quasicrystal is known as the crystal structure which has unique rotational symmetry not to be obtained by general metals and alloys, is a non-periodic crystal structure having fivefold symmetry for example, and is equivalent to a non-periodic structure represented by a three-dimensional Penrose Pattern. [0003] After the discovery of the new arrangement of metallic atoms, that is new crystal structure, the quasicrystal having the quasi-periodic structure and having the unique rotational symmetry has received a lot of attention. The quasicrystal has been generally obtained by a liquid quenching method in the past, although it is found that the quasicrystal can be obtained by a crystal growth method in recent years. The shape thereof has been restricted to powder, foil, and chip, and thus, it has been very rare to apply the quasi crystal to a product. [0004] Patent documents I and 2 disclose high strength Mg based alloys and - 1 - producing methods thereof. The Mg based alloys have a mctallographic structure in which the hard quasi crystal phase having a grain size of tens to hundreds of nm is dispersedly precipitated, and thus, the Mg based alloys are excellent in strength and elongation. The patent documents 1 and 2 utilize the properties such that the quasi crystal is hard. [0005] Patent document 3 discloses a thermoelectric material using AI based quasicrystal. The patent document 3 utilizes the properties such that the quasi crystal is excellent in thermoelectric property. Patent document 4 discloses a heat-resistant catalyst whose precursor is a quasicrystalline AI alloy (AI based quasicrystal) and a producing method thereof. The patent document 4 utilizes the properties such that the quasicrystal without a periodic crystal structure is brittle and fracturable. As described above, in the prior inventions, the fine particles of the quasi crystal may be dispersed or consolidated. [0006] As another application different from the above inventions, patent document 8 discloses a metallic coating for cookware containing the quasicrystal. In the patent document 8, the coating excellent in wear resistance and corrosion resistance to salt is applied to the cookware by plasma-spraying the alloy powder containing the quasicrystal which consists of AI, Fe, and Cr and which is excellent in corrosion resistance. [0007] As described above, the Mg based quasicrystal is utilized as the materials excellent in strength, and the AI based quasicrystal is utilized as members which is excellent in strength, thermoelectric materials, coatings for cookwares. However, the - 2 - utilization is limited, and the quasicrystal is not always utilized in many fields. [0008) The quasicrystal has excellent characteristics derived from the unique crystal structure. However, the characteristics thereof are only partially investigated, and the quasicrystal is not widely applied to industrial fields at the moment. The present inventors have tried to improve the corrosion resistance by applying the quasicrystal which is hardly utilized in the industrial field to a plated-metal-layer of a surface-treated steel sheet. [0009) In general, in order to prolong a useful life of steel sheet, the steel sheet is subjected to surface treatment such as metallic plating, paint coating, conversion coating, or organic film laminating in order to ensure an anticorrosive function to a certain extent. In the many steel materials used in fields of automobiles, consumer electronics, building materials or the like, the metallic plating is mainly applied. The plated-metal-layer provides, at a low cost, both barrier protection in which a base metal (steel substrate) is shielded from outside environment and sacrificial protection in which the layer is preferentially corroded as compared with the base metal. [0010) There are various methods to industrially form the plated-metal-layer. In order to make the plated-metal-layer thick, spraying, hot-dip plating or the like is preferable. In order to uniformly form the plated-metal-layer, sputtering, ion plating, evaporating, electro plating or the like is preferable. Among these methods, the hot-dip plating is widely applied, because it is possible to massively and economically produce the steel materials with the plated-metal-layer. [0011) - 3 - In the electro plating, deposited metals are limited, and thus, the elements included in the plated-metal-layer are limited in general. In the methods such as the spraying and the evaporating in which the plated-metal-layer is formed by using reactions such as melting, evaporation, deposition, and solidification of metals, it is possible to form the plated-metal-layer as with that formed by the hot-dip plating in theory. However, each metal has specific melting point and boiling point, and thus, the difference between chemical compositions of the used alloy and the formed plated-metal-layer tends to occur in the spraying and the evaporating. [0012] Since it is possible for the hot-dip plating to form the plated-metal-layer whose chemical composition is about the same as that of the used alloy for the hot-dip bath, the hot-dip plating is well suitable for forming the plated-metal-layer which has predetermined chemical composition as compared with other forming methods. [0013] At present, conventional surface-treated steel sheets which are industrially available are mainly those with the plated-metal-layer of Zn-based alloy orAl-based alloy. The plated-metal-layer of Zn-based alloy includes Zn as main element and a small amount of AI, Mg, or the like, and the metallographic structure thereof includes Zn phase, AI phase, Mg2Zn phase, or the like. The platedcmetal-layer of Al-based alloy includes AI as main element and a small amount of Si, Fe, or the like, and the metallographic structure thereof includes AI phase, Si phase, Fe2Al5 phase, or the like. [0014] As the plated steel materials in which the chemical composition of plated alloy is quite different from that of the conventional surface-treated steel sheets, the present inventors disclosed the steel sheets with the plated layer containing Mg-based - 4 - alloy in patent documents 5 to 7. Based on the above plated steel materials, the present inventors have tried to further improve the corrosion resistance by focusing the quasicrystal which has hardly been considered for the improvement of the corrosion resistance of the plated layer (plated-metal-layer). [Prior Art Document] [Patent Document] [0015] [Patent Document 1] Japanese Unexamined Patent Application, First Publication No. 2005-113235 [Patent Document 2] Japanese Unexamined Patent Application, First Publication No. 2008-69438 [Patent Document 3] Japanese Unexamined Patent Application, First Publication No. H08-176762 [Patent Document 4] Japanese Unexamined Patent Application, First Publication No. 2004-267878 [Patent Document 5] Japanese Unexamined Patent Application, First Publication No. 2008-255464 [Patent Document 6] Japanese Unexamined Patent Application, First Publication No. 2010-248541 [Patent Document 7] Japanese Unexamined Patent Application, First Publication No. 2011-219823 [Patent Document 8] Published Japanese Translation No. 2007-525596 of the PCT International Publication [Disclosure of the Invention] [Problems to be solved by the Invention] - 5 - __ . __ -[_ [0016] An object of the present invention is to provide the plated steel sheet which is further excellent in the corrosion resistance requested for applying building materials, automobiles, consumer electronics or the like. [0017] In particular, an object of the present invention is to provide the plated steel sheet having both excellent corrosion resistance and excellent sacrificial protection by focusing the quasicrystal which has hardly been considered for the improvement of the corrosion resistance of the plated layer and by clarifYing the morphology of the metallographic structure which maximally improve the corrosion resistance. Specifically, the corrosion resistance and the sacrificial protection of the plated steel sheet is to be improved by clarifying the preferable morphology of the quasi crystal in plated-metal-layer (the plated layer) which has hardly been considered but which is expected to improve the corrosion resistance and by clarifying the processes to preferably form the quasicrystal in the plated-metal-layer. [Means for Solving the Problem] [0018] An aspect of the present invention employs the following. (1) A plated steel sheet with a quasicrystal according to an aspect of the present invention includes a steel sheet and a plated-metal-layer arranged on a surface of the steel sheet, wherein: the plated-metal-layer includes, as a chemical composition, by atomic%, 20% to 60% of Zn, 0.3% to 15% of AI, - 6 - O%to 3.5% ofCa, 0% to 3.5% ofY, 0% to 3.5% of La, O%to3.5%ofCe, 0% to 0.5% of Si, 0% to 0.5% ofTi, 0% to 0.5% ofCr, 0% to 2% of Fe, 0% to 0.5% of Co, 0% to 0.5% ofNi, 0% to 0.5% ofV, 0% to 0.5% ofNb, 0% to 0.5% ofCu, 0% to 0.5% ofSn, 0% to 0.2% ofMn, 0% to 0.5% of Sr, 0% to 0.5% of Sb, 0% to 0.5% ofPb, and a balance ofMg and impurities; a zinc content and an aluminum content expressed in atomic% in the chemical composition of the plated-metal-layer satisfY 25% <:: Zn + Al; the plated-metal-layer includes, as a metallographic structure, a quasicrystal phase; a magnesium content, a zinc content, and an aluminum content expressed in atomic% in the quasicrystal phase satisfy 0.5 <:: Mg I (Zn + Al) :S 0.83; and - 7 - ,- an average equivalent circle diameter of the quasi crystal phase is larger than I f.!ID and equal to or smaller than 200 f.!ID. (2) In the plated steel sheet with the quasi crystal according to (1 ), a calcium content, an yttrium content, a lanthanum content, and a cerium content expressed in atomic% in the chemical composition of the plated-metal-layer may satisfy 0.3% :S Ca + Y +La+ Ce :S 3.5%. (3) In the plated steel sheet with the quasicrystal according to (1) or (2), a silicon content, a titanium content, and a chromium content expressed in atomic% in the chemical composition of the plated-metal-layer may satisfy 0.005% :S Si + Ti + Cr :S 0.5%. ( 4) In the plated steel sheet with the quasi crystal according to any one of (1) to (3), a zinc content and an aluminum content expressed in atomic% in the chemical composition of the plated-metal-layer may satisfy 30% :S Zn + Al :S 50% and 3 :S Zn I AI :S 12. ( 5) In the plated steel sheet with the quasi crystal according to any one of (1) to ( 4), when viewed in a cross section, whose cutting direction is parallel to a thickness direction of the plated-metal-layer, the rnetallographic structure of the plated-metal-layer may be a bimodal structure which comprises a fine domain composed of a grain having an equivalent circle diameter of 1 f.!ID or smaller and a coarse domain composed of a grain having an equivalent circle diameter of larger than 1 f.!ID, - 8 - the coarse domain may include the quasi crystal phase, and the fine domain may include at least one selected from a Mg51Zn20 phase, a Mg32(Zn, Al)49 phase, a MgZn phase, a MgZnz phase, and a Zn phase. (6) In the plated steel sheet with the quasicrystal according to any one of (1) to (5), an area fraction of the coarse domain in the metallographic structure may be equal to or more than 5% and equal to or less than 80%, and an area fraction of the fine domain in the metallographic structure may be equal to or more than 20% and equal to or less than 95%. (7) In the plated steel sheet with the quasicrystal according to any one of (l) to (6), an area fraction of the quasi crystal phase included in the coarse domain may be equal to or more than 80% and less than 100% in the coarse domain, and an area fraction in total of the Mg51Zn2o phase, the Mg32(Zn, Al)49 phase, the MgZn phase, the MgZn2 phase, and the Zn phase included in the fine domain may be equal to or more than 80% and less than l 00% in the fine domain. (8) In the plated steel sheet with the quasi crystal according to any one of (l) to (7), when viewed in the cross section and when a thickness of the plated-metal-layer is regarded as D, an area from a surface of the plated-metal-layer toward the steel sheet in the thickness direction to 0.3 x D is regarded as a surface area of the plated-metal-layer, and an area from an interface between the steel sheet and the plated-metal-layer toward the plated-metal-layer in the thickness direction to 0.3 x Dis regarded as a deep area of the plated-metal-layer, an area fraction oft he coarse domain in the surface area of the - 9 - plated-metal-layer may be equal to or more than 10% and less than 100% and an area fraction of the coarse domain in the deep area of the plated-metal-layer may be equal to or more than 10% and less than 100%, and when an area except for the surface area and the deep area in the plated-metal-layer is regarded as a center area of the plated-metal-layer, an area fraction of the fine domain in the center area of the plated-metal-layer may be equal to or more than 50% and less than 100%. (9) In the plated steel sheet with the quasi crystal according to any one of ( 1) to (8), a Mg phase may be absent in the metallographic structure ofthe plated-metal-layer. (1 0) The plated steel sheet with the quasi crystal according to any one of ( 1) to (9) may further include a Fe-Al containing alloy layer, wherein: the Fe-Al containing alloy layer is arranged between the steel sheet and the plated-metal-layer; the Fe-Al containing alloy layer includes at least one selected from Fe5Ab and Al3.2F e; and a thickness of the Fe-Al containing alloy layer is equal to or more than 10 nm and equal to or less than 1000 nm. (11) A method of producing a plated steel sheet with a quasicrystal according to an aspect of the present invention, which is the method of producing the plated steel sheet with the quasicrystal according to any one of (1) to (10), includes: a hot-dip-plating process of dipping a steel sheet into a hot-dip-plating bath having an adjusted composition in order to form a plated-metal-layer on a surface of the steel sheet; - 10 - a first cooling process of cooling the steel sheet after the hot-dip-plating process under conditions such that an average cooling rate of the plated-metal-layer is equal to or faster than 15 °C/sec and equal to or slower than 50 °C/sec in a temperature range where a temperature of the plated-metal-layer is from Tmclt to Tsolid-liquid in unit of °C, when the Tmelt is regarded as a liquidus temperature of the plated-metal-layer and when the Tsolid-Iiquid is a temperature range where the plated-metal-layer is in a coexistence state of a solid phase and a liquid phase and where a volume ratio of the solid phase to the plated-metal-layer is equal to or more than 0.3 and equal to or less than 0.8; and a second cooling process of cooling the steel sheet after the first cooling process under conditions such that an average cooling rate of the plated-metal-layer is equal to or faster than 100 "C/sec and equal to or slower than 3000 °C/sec in a temperature range where a temperature of the plated-metal-layer is from a temperature at finishing the first cooling process to 250°C. (12) In the method of producing the plated steel sheet with the quasicrystal according to (11), in the hot-dip-plating process: an oxide in the hot-dip-plating bath may be 1 g/1 or less; an oxygen concentration of an atmosphere at dipping the steel sheet may be I 00 ppm or less in volume ratio; a plating tub to hold the hot-dip-plating bath may be a steel tub; a dross in the hot-dip-plating bath may be removed by a metal pump; Tbath which is a temperature of the hot-dip-plating bath may be equal to or higher than I 0°C and equal to or lower than 1 00°C higher than the T melt; and a time for dipping the steel sheet into the hot-dip-plating bath may be equal to or longer than 1 sec and equal to or shorter than I 0 sec. - 11 - L-_ [Effects of the Invention] [0019] According to the above aspects of the present invention, it is possible to provide the plated steel sheet which is further excellent in the con-osion resistance requested for applying building materials, automobiles, consumer electronics or the like. Therefore, it is possible to prolong the useful life of the materials as compared with the conventional surface-treated steel sheets. [Brief Description of the Drawings] [0020] FIG. 1 is a SEM micrograph of a plated steel sheet according to an embodiment of the present invention and a metallographic micrograph obtained by observing a cross section whose cutting direction is parallel to a thickness direction of the plated steel sheet. FIG. 2 is a TEM micrograph of a plated-metal-layer of the plated steel sheet according to the embodiment and a metallographic micrograph obtained by observing the cross section whose cutting direction is parallel to the thickness direction of the plated steel sheet. FIG. 3A is an electron diffraction pattern obtained from a local area 2al in a coarse domain 2a shown in FIG. 2. FIG. 3B is an electron diffraction pattern obtained from a local area 2b 1 in a fine domain 2b shown in FIG. 2. '' r' FIG. 4 is a SEM micrograph of the plated steel sheet according to the embodiment and a metallographic micrograph obtained by observing the cross section whose cutting direction is parallel to the thiclmess direction of the plated steel sheet. FIG. 5 is a liquidus surface phase diagram of a ternary Zn-Al-Mg system. - 12 - fi [Embodiments of the Invention] [0021] Hereinafter, a preferable embodiment of the present invention will be described in detail. However, the present invention is not limited only to the configuration which is disclosed in the embodiment, and various modifications are possible without departing from the aspect of the present invention. [0022] The plated steel sheet according to the embodiment includes a steel sheet (base metal) and a plated-metal-layer (plated layer) arranged on a surface of the steel sheet. The plated-metal-layer whose shape is thin film is the alloy which has adhesion to the base metal, equips the steel sheet with a function such as anticorrosion, and does not harm the properties of the base metal such as strength or rigidity. Specifically, the plated steel sheet according to the embodiment is the composite material in which two types of metal-alloy- materials that are the steel sheet and the plated-metal-layer are layered. In the interface between the steel sheet and the plated-metal-layer, an interface alloy layer (Fe-Al containing alloy layer) or a diffused area formed by mutual diffusion of metal atoms may exist as a result of the composition, and thereby, the interface adherence may be increased due to atomic bonding of metal. First, the properties requested to the plated-metal-layer of the plated steel sheet according to the embodiment will be described. [0023] The plated-metal-layer of the plated steel sheet is required to be excellent in anticorrosion performance. The anticorrosion performance is classified into the corrosion resistance and the sacrificial protection. In general, the corrosion resistance of the plated-metal-layer corresponds to the corrosive resistivity of the - 13 - plated-metal-layer itself, and is usually evaluated by the corrosion loss of the plated-metal-layer after a predetermined time in various corrosion tests. [0024] When the corrosion loss is small, the plated-metal-layer remains for a long time as a protective layer of the steel sheet (base metal), and thus, the corrosion resistance is excellent. When the corrosion loss is evaluated by using pure metals, the corrosion resistance of Zn tends to be better than that ofMg, and the corrosion resistance of AI tends to be better than that of Zn in general. [0025] On the other hand, the sacrificial protection of the plated-metal-layer corresponds to the protective function for the steel sheet in which the plated-metal-layer is preferentially corroded instead of the steel sheet when the steel sheet is accidentally exposed in corrosive environment. When the sacrificial protection is evaluated by using pure metals, the metal which is electrochemically less-noble and which tends to be corroded is excellent in the sacrificial protection. Thus, the sacrificial protection of Zn tends to be better than that of AI, and the sacrificial protection of Mg tends to be better than that of Zn in general. [0026] The steel sheet plated the Zn-Mg alloy according to the embodiment includes a large amount ofMg in the plated-metal-layer, and thus, is excellent in the sacrificial protection. On the other hand, the point to be improved is to reduce the corrosion loss of the plated-metal-layer, which is to improve the corrosion resistance of the plated-metal-layer. [0027] The present inventors have investigated the constituent phase of the - 14 - metallographic structure of the plated-metal-layer in order to preferably reduce the corrosion loss of the plated-metal-layer in the steel sheet plated the Zn-Mg alloy. As a result, it is found that the corrosion resistance is drastically improved by including the quasicrystal phase in the plated-metal-layer. [0028] The metallographic structure of the plated-metal-layer is the main characteristic of the plated steel sheet according to the embodiment. In a case where the plated steel sheet is produced based on a chemical composition within a specific range, which will be described later, under specific production conditions, a quasicrystal phase is formed in the plated-metal-layer, and corrosion resistance can be significantly improved. In the embodiment, an average equivalent circle diameter (diameter) of the quasicrystal phase formed in the plated-metal-layer is larger than 1 J..tm and equal to or smaller than 200 J..tm. [0029] The plated-metal-layer of the plated steel sheet according to the embodiment contains the aforementioned quasicrystal phase. Therefore, the corrosion resistance thereof is further improved compared to the corrosion resistance of a plated-metal-layer not containing a quasicrystal phase. Furthermore, the plated-metal-layer of the plated steel sheet according to the embodiment contains a large amount of Mg. Therefore, the plated-metal-layer also exhibits excellent sacrificial protection with respect to the steel sheet. That is, the plated steel sheet according to the embodiment includes an ideal plated-metal-layer excellent in both of the corrosion resistance and the sacrificial protection. [0030] Hereinafter, regarding the plated steel sheet according to the embodiment, the - 15 - chemical composition of the plated-metal-layer, the metallographic structure of the plated-metal-layer, and the production conditions will be specifically described in this order. [0031] Generally, when constitutive equations of metallic phases or intermetallic compounds such as Zn, AI, Mg2Zn, and Fe2Als are described, an atomic ratio is used instead of a mass ratio. The embodiment will be described using an atomic ratio because the embodiment is focused on a quasicrystal phase. That is, unless otherwise specified, "%" showing a chemical composition in the following description means atomic%. [0032] First, regarding the chemical composition of the plated-metal-layer, the way the numerical ranges are limited and why the numerical ranges are limited will be described. [0033] The plated-metal-layer of the plated steel sheet according to the embodiment contains Zn and Al as basic components, optional components as necessary, and Mg and impurities as a balance. [0034] Zn (Zinc): 20% to 60% In order to obtain a quasi crystal phase as a metallographic structure of the plated-metal-layer, the plated-metal-layer must contain Zn within the above range. Therefore, a Zn content in the plated-metal-layer needs to be 20% to 60%. In a case where the Zn content is less than 20%, the quasicrystal phase cannot be formed in the plated-metal-layer. Likewise, in a case where the Zn content is greater than 60%, the - 16 - quasicrystal phase cannot be formed in the plated-metal-layer. Furthermore, in order to preferably control the formation of the quasi crystal phase and the formation of an intermetallic compound, which will be described later, the lower limit and the upper limit of the Zn content may be set to be 25% and 52% respectively. More preferably, the lower limit and the upper limit of the Zn content may be set to be 30% and 45% respectively. [0035] In order to further improve the corrosion resistance by preferably forming the quasicrystal, the Zn content is preferably set to be equal to or greater than 33%. If the Zn content is equal to or greater than 33%, a compositional range is established in which the quasicrystal phase easily grows as a primary phase, and a Mg phase does not easily grow. That is, an amount (area fraction) of the quasicrystal phase in the plated-metal-layer can be increased, and an amount of the Mg phase deteriorating corrosion resistance can be reduced as much as possible. More preferably, the Zn content is set to be equal to or greater than 3 5%. Generally, if the plated steel sheet is produced within the above compositional range by the production method according to the embodiment, the Mg phase practically does not exist. (0036] AI (aluminum): 0.3% to 15% AI is an element improving the corrosion resistance of a planar portion of the plated-metal-layer. Furthermore, AI is an element accelerating the formation of the quasicrystal phase. In order to obtain these effects, an AI content in the plated-metal-layer is set to be equal to or greater than 0.3%. In order to preferably control an average equivalent circle diameter of the quasi crystal phase, the Al content in the plated-metal-layer may be set to be equal to or greater than 5%. When the AI - 17 - content is equal to or greater than 5%, the average equivalent circle diameter of the quasicrystal phase easily becomes larger than 1 ftm, and when the AI content is equal to or greater than 10%, the average equivalent circle diameter of the quasi crystal phase easily becomes larger than 2 ftm. If the average equivalent circle diameter of the quasicrystal phase is controlled and becomes larger than 2 ftm, the corrosion resistance of the planar portion is further improved. In a case where the Zn content is smaller than the values within the above range, in order to preferably form the quasicrystal phase in the plated-metal-layer, it is preferable to control the Zn content and the AI content in combination. Specifically, the Zn content and the AI content, expressed in atomic%, in the chemical composition of the plated-metal-layer preferably satisfy 25% <:: Zn + Al, and more preferably satisfy 28.5% <:: Zn +AI. The upper limit of Zn +AI is not particularly limited and is preferably 50%. In contrast, if the plated-metal-layer contains a large amount of AI, red rust easily occurs, the quasicrystal phase is not easily formed, and thus the corrosion resistance deteriorates. Therefore, the upper limit of the Al content in the plated-metal-layer needs to be set to be 15%. In addition, it is preferable that the elementAl is contained in the plated-metal-layer by forming a Fe-Al interface alloy layer which will be described later. [0037] In order to more preferably form the quasicrystal phase in the plated-metal-layer, it is preferable to control the Zn content and the AI content as below. That is, the Zn content and the Al content, expressed in atomic%, in the chemical composition of the plated-metal-layer preferably satisfy 30% <:: Zn +AI<:: 50% and 3 <:: Zn/ AI<:: 12. When the Zn content and the AI content satisfy the above conditions, the quasicrystal phase is formed in the plated-metal-layer at a preferred area fraction. It is preferable that the Zn content and the AI content satisfy the above - 18 - --------- ·--- conditions, because then the quasi crystal phase is formed in the plated-metal-layer at an area fraction of about 30% to 80% with respect to the total area of the plated-metal-layer. The technical reason is unclear. However, the formation of the quasicrystal phase at the aforementioned area fraction is considered to be related to the facts that the quasicrystal phase in the embodiment has a crystal structure mainly composed of Zn and Mg, the formation of the quasi crystal phase is accelerated by the substitution of A! with Zn, and there is an optimal value of the amount of A! substituted. Because the quasicrystal phase is preferably formed in the plated-metal-layer, the corrosion resistance is improved particularly in a processed portion, and it takes a long time until red rust starts to occur in a base metal. Presumably, these effects result from a fact that the quasicrystal phase is preferably dispersed in the plated-metal-layer because the contents of Zn and A! are precisely controlled. [0038] Mg (magnesium) is a main element which constitutes the plated-metal-layer similarly to Zn and A! and improves the sacrificial protection. Furthermore, Mg is an important element accelerating the formation of the quasi crystal phase. In the embodiment, a content of Mg in the plated-metal-layer does not need to be particularly specified and is equal to the aforementioned balance minus a content of impurities. That is, the Mg content may be greater than 25% and less than 79.7%. However, the Mg content in the balance is preferably equal to or greater than 50%, and more preferably equal to or greater than 55%. In the embodiment, although the plated-metal-layer must contain Mg, in order to improve the corrosion resistance, it is preferable to inhibit Mg contained in the plated-metal-layer from being precipitated as a Mg phase in the plated-metal-layer. That is, because the Mg phase deteriorates the - 19 - corrosion resistance, it is preferable that Mg is contained in the plated-metal-layer is in the form of a quasicrystal phase or a constituent of other intermetallic compounds. [0039] The plated-metal-layer of the plated steel sheet according to the embodiment contains impurities in addition to the aforementioned basic components. Herein, the impurities mean elements such as C, N, 0, P, S, and Cd that are mixed in from raw materials of steel and plated alloys, the production environment, or the like when the plated steel sheet is industrially produced. Even if these elements are contained as impurities in an amount of about 0.1% respectively, the aforementioned effects are not impaired. [0040] The plated-metal-layer of the plated steel sheet according to the embodiment may further contain, instead of a portion of Mg described above as a balance, at least one or more optional components selected from Ca, Y, La, Ce, Si, Ti, Cr, Fe, Co, Ni, V, Nb, Cu, Sn, Mn, Sr, Sb, and Pb. The plated-metal-layer may contain these optional components according to the purpose. Therefore, the lower limit of the content of these optional components does not need to be limited and may be 0%. Even if these optional components are contained as impurities, the aforementioned effects are not impaired. [0041] Ca (calcium): 0% to 3.5% Y (yittrium): 0% to 3.5% La (lanthanum): 0% to 3.5% Ce (cerium): 0% to 3.5% In order to improve workability of hot-dip plating, Ca, Y, La, and Ce may be - 20 - contained in the plated-metal-layer as necessary. In a case where the plated steel sheet according to the embodiment is produced, a highly oxidative hot-dip Mg alloy is held in the atmosphere as a plating bath. Therefore, it is preferable to take a certain measure to prevent the oxidation of Mg. Ca, Y, La, and Ce are more easily oxidized compared to Mg and prevent the oxidation of Mg in the bath by forming a stable oxide layer on the surface of the plating bath in a molten state. Accordingly, in the plated-metal-layer, a Ca content may be set to be 0% to 3.5%, a Y content may be set to be 0% to 3.5%, a La content may be set to be 0% to 3.5%, and aCe content may be set to be 0% to 3.5%. More preferably, the lower limit and the upper limit of each of the Ca content, theY content, the La content, and the Ce content may be set to be 0.3% and 2.0% respectively. [0042] It is preferable that the plated-metal-layer contain at least one element selected from Ca, Y, La, and Ce in an amount of equal to or greater than 0.3% in total, because then the plating bath with a high Mg content can be held in the atmosphere without being oxidized. In contrast, Ca, Y, La, and Ce are easily oxidized and negatively affect the corrosion resistance in some cases. Therefore, the upper limit of the tot the Mg32(Zn, Al)49 phase> the Mg51Znzo phase> the MgZn phase= the MgZn2 phase> the Zn phase>> the Mg phase. In a case.where these constituent phase are mixed together, increasing a fraction of the phase having high conosion resistance favors the conosion resistance of the plated-metal-layer. That is, in the plated steel sheet according to the embodiment, - 35 - it is preferable that the area fraction of the quasicrystal phase is the highest among all of the constituent phases included.in the metallographic structure of the plated-metal-layer. In other words, it is preferable that the quasicrystal phase is a main phase in the metallographic structure of the plated-metal-layer of the plated steel sheet according to the embodiment. [0073] Here, in a case where various metallic phases or intermetallic compounds coexist in the plated-metal-layer, due to the formation of a coupling cell, the corrosion resistance further deteriorates than in a case where a single phase exists in the plated-metal-layer. Generally, if a plurality of phases is mixed into the plated-metal-layer, portions that are noble and less noble in terms of electric energy formed in the plated-metal-layer, and hence a coupling cell reaction occurs. The less noble portions corrode first, and hence the corrosion resistance deteriorates. Here, in the plated steel sheet according to the embodiment, in a case where the plated-metal-layer has the aforementioned biomodal structure, the deterioration of corrosion resistance resulting from the formation of the coupling cell is practically not observed and negligible, and rather, the corrosion resistance is markedly improved · because the plated-metal-layer contains the quasicrystal. [0074] Generally, an intermetallic compound has poor plastic deformation properties. If a fraction of a coarse intermetallic compound having poor plastic workability is reduced, only fine cracks occur in the plated-metal-layer at the time of processing the plated steel sheet. Accordingly, an exposed area of the steel sheet (base metal) is reduced, and the corrosion resistance is preferably improved. Fmihermore, because the peeling of the plated-metal-layer is inhibited, it takes a long time until red rust - 36 - occurs in the processed portion, and hence the corrosion resistance is preferably improved. [0075] The quasicrystal phase is a non-equilibrium phase and thermally unstable. Therefore, if exposed to a high temperature environment with a temperature of around 250°C to 330°C for a long period of time, the quasicrystal phase undergoes phase decomposition, and hence the Mg phase having poor corrosion resistance is formed in addition to the Mg51Zn20 phase in some cases. Consequently, the corrosion resistance as the overall plated steel sheet is likely to deteriorate. Care is required in a case where the plated steel sheet is used in a high temperature environment. [0076] In the plated-metal-layer of the plated steel sheet according to the embodiment, an area fraction of the coarse domain in the metallographic structure of the entirety ofthe plated-metal-layer (area of coarse domain/area of plated-metal-layer) is preferably 5% to 80%, and an area fraction of the fine domain in the metallographic structure of the entirety of the plated-metal-layer (area of fine domain/area of plated-metal-layer) is preferably 20% to 95%. If the above conditions are satisfied, the corrosion resistance of the plated-metal-layer is further improved. FIG. 4 is a SEM micrograph of the plated steel sheet according to the embodiment, which is a metallographic micrograph obtained by observing the cross section whose cutting direction is parallel to the thickness direction of the plated steel sheet. FIG. 4 shows a plated-metal-layer in which an area fraction of the coarse domain is 63% and an area fraction of the fine domain is 3 7%. In was confirmed that the corrosion resistance of the plated-metal-layer is further improved in the plated steel sheet. [0077] - 37 - In a case where further improvement of the corrosion resistance of the plated-metal-layer is prioritized, the lower limit of the area fraction of the coarse domain may be set to be I 0%, 15%, or 25%, and the upper limit of the area fraction of the fine domain may be set to be 90%, 85%, or 75%. In contrast, in a case where the inhibition of peeling at the time of bending is prioritized more than the corrosion resistance of the plated-metal-layer, the upper limit of the area fraction of the coarse domain may be set to be 50%, 35%, or 25%, and the lower limit of the area fraction of the fine domain may be set to be 50%, 65%, or 75%. [0078] In the plated-metal-layer of the plated steel sheet according to the embodiment, an area fraction of the quasi crystal phase included in the coarse domain is preferably 80% to less than 100% as compared with the coarse domain (area of quasicrystal phase in coarse domain/area of coarse domain), and an area fraction in total of the Mg51Zn20 phase, the Mg32(Zn, Al)49 phase, the MgZn phase, the MgZn2 phase, and the Zn phase included in the fine domain is preferably 80% to less than 100% as compared with the fine domain (total area of respective constituent phases in fine domain/area of fine domain). When the above conditions are satisfied, the corrosion resistance of the plated-metal-layer is further improved. Presumably, there may be a certain correlation between the fraction of the quasi crystal phase or the size ;: of the coarse domain in the plating structure (plated-metal-layer) and the II electrochemical properties. For example, as the fraction of the quasicrystal phase !I '! increases, a corrosion potential of the plated-metal-layer shifts to a noble potential (-1.0 V to -0.8 V vs. Ag/AgCl reference electrode) from a less noble potential (-1.3 V to -1.1 V vs. Ag/ AgCl reference electrode), a cathode current value and an anode current value at the corrosion potential decrease, and hence a corrosion current density - 38 - decreases. Presumably, this is because the quasicrystal phase has a unique potential or properties close to those of a passive state. It is considered that, as a result, the corrosion resistance of the plated-metal-layer is improved. The balance of the coarse domain and the balance of the fine domain include an intermetallic compound or a metallic phase other than the above in some cases, but even in these cases, the effects of the embodiment are not impaired. The potential of the coarse domain can be measured using, for example, a scanning Kelvin probe method, and the mapping of the structure can be measured. Generally, as the fraction of the quasi crystal and Mg32(Zn, Al)49 increases, the potential is closer to a value of around -0.8 V which is noble. In contrast, Mg51Zn20 has a potential of about -1.1 V. The potential or the corrosion current density varies with the amount of these phases and is generally within a range of -1.3 V to -0.8 V. Usually, the closer the potential to -0.8 V, the further the corrosion current density tends to be reduced. [0079] It is preferable that the metallographic structrrre of the plated-metal-layer in the plated steel sheet according to the embodiment does not contain the Mg phase. The Mg phase contained in the plated-metal-layer deteriorates the corrosion rcsistar{ce in both the coarse domain and the fine domain. Therefore, it is preferable to suppress the precipitation of the Mg phase as much as possible. Whether or not the Mg phase exists may be determined and confirmed through TEM-EDX, SEM-EDX, XRD, or the like. For example, in a case where a diffraction intensity from a (11 0) surface of the Mg phase is equal to or less than 1% of a diffraction intensity at a diffraction angle (28 = 36.496°) of the Mg51Zn20 phase (or Mg7Zn3 phase) in an XRD diffi·action pattern, it can be said that the metallographic structure of the plated-metal-layer does not contain the Mg phase. Likewise, in a case where a number fraction of grains of the Mg phase - 39 - ------- is equal to or less than 3% when 100 or more grains are randomly sampled in a TEM diffraction pattern, it can be said that the metallographic structure of the plated-metal-layer does not contain the Mg phase. The number fraction of grains of the Mg phase is more preferably less than 2%, and most preferably less than 1%. [0080] In the plated-metal-layer, the Mg phase is easily formed as a primary phase at a temperature immediately below the melting point. Whether the Mg phase will be formed as a primary phase generally depends on the chemical composition of the plated-metal-layer and the production conditions. In a case where the Mg content is higher than in a eutectic composition (Mg 72%-Zn 28%) of an equilibrium state diagram of a binary Mg-Zn system, the Mg phase is likely to be crystallized as a primary phase. In contrast, in a case where the Mg content is lower than the above, in principle, the Mg phase is less likely to be crystallized as a primary phase. The production process according to the embodiment is a process for forming a quasi crystal as a primary phase. Therefore, if the Mg content is higher than in the eutectic composition, it is extremely difficult for the Mg phase to be formed, and even if the formation of the Mg phase could be confirmed, the Mg phase is less likely to present as a main phase. The grain of the Mg phase is present at a number fraction of about up to 3%. The present inventors confirmed that when the Zn content is 28.5% or greater, a proportion of the grain of the Mg phase in grains contained in the metallographic structure of the plated-metal-layer tends to be less than 2% in terms of a number fraction. Furthermore, when the Zn content is 33% or greater, a proportion of the grain of the Mg phase in the grains contained in the metallographic structure of the plated-metal-layer tends to be less than 1% in terms of a number fraction. If the Mg phase is present in the plated-metal-layer, the surface of the plated-metal-layer - 40 - turns black with the passage of time particularly in a humid environment, and hence the appearance of the plating becomes defective in some cases. In this respect, it is preferable to avoid mixing of the Mg phase into the surface layer of the plated-metal-layer in particular. By storing the plated steel sheet in a thermohygrostat tank for a certain period of time, the occurrence of appearance defectiveness, a phenomenon in which the surface of the plated-metal-layer turns black, can be determined. [0081] In the plated-metal-layer of the plated steel sheet according to the embodiment, an area fraction of the quasicrystal phase include in the coarse domain is preferably 80% to less than 100% as compared with the coarse domain (area of quasicrystal phase in coarse domain/area of coarse domain), and an area fraction of the Mg51Zn20 phase included in the fine domain is preferably 80% to less than 100% as compared with the fine domain (area ofMg51Zn2o phase in fine domain/area of fine domain). When the above conditions are satisfied, a fraction of the Mg51Zn20 phase having excellent corrosion resistance is increased, and hence the corrosion resistance of the plated-metal-layer is further improved. [0082) Regarding the plated-metal-layer of the plated steel sheet according to the embodiment, when a cross section whose cutting direction is parallel to a thickness direction of the plated-metal-layer is viewed and when a thickness of the plated-metal-layer in the thickness direction is regarded as D in a unit of f!m, an area from a surface of the plated-metal-layer toward the steel sheet in the thickness direction to 0.3 x Dis regarded as a surface area of the plated-metal-layer, and an area from an interface between the steel sheet and the plated-metal-layer toward the - 41 - plated-metal-layer in the thickness direction to 0.3 x Dis regarded as a deep area of the plated-metal-layer, an area fraction of the coarse domain in the surface area of the plated-metal-layer (area of coarse domain in surface area of plated-metal-layer/area of surface area of plated-metal-layer) is preferably I 0% to less than I 00% and an area fraction of the coarse domain in the deep area of the plated-metal-layer (area of coarse domain in deep area of plated-metal-layer/area of deep area of plated-metal-layer) is preferably I 0% to less than I 00%. Furthermore, when an area except for the surface area and the deep area in the plated-metal-layer is regarded as a center area of the plated-metal-layer, an area fraction of the fine domain in the center area of the plated-metal-layer (area of fine domain in center area of plated-metal-layer/area of center area of plated-metal-layer) is preferably 50% to less than 100%. If the above conditions are satisfied, the constituent phases contained in the plated-metal-layer are preferably arranged, and hence the corrosion resistance of the plated-metal-layer is further improved. In addition, the adherence of the plated-metal-layer tends to be improved. In a case where the grains in the coarse domain are present in a position that extends across the surface area and deep area of the plated-metal-layer, or in a case where the grains in the coarse domain are present in a position that extends across the deep area and center area of the plated-metal-layer, the aforementioned area fraction may be calculated using the area of the grains included in the surface area or deep area of the plated-metal-layer. Likewise, in a case where the grains in the fine domain are present in a position that extends across the surface area and deep area of the plated-metal-layer, or in a case where the grains in the fine domain are present in a position that extends across the deep area and center area of the plated-metal-layer, the aforementioned area fraction may be calculated using the area of grains included in the center area of the plated-metal-layer. - 42 - '_, ·,._,_._;_ ··- -, : __ -_\__._ [0083] The plated steel sheet according to the embodiment preferably further has a Fe-Al containing alloy layer. The Fe-Al containing alloy layer is preferably arranged between the steel sheet and the plated-metal-layer and preferably contains at least one or more kinds of compound between Fe5Ah and Al32Fe, and a thickness of the Fe-Al containing alloy layer in a thickness direction thereof is preferably 10 nm to 1,000 nm. If the Fe-Al containing alloy layer satisfying the above conditions is arranged in an interface between the steel sheet and the plated-metal-layer, peeling of the plated-metal-layer is preferably inhibited. Furthermore, if the Fe-Al containing alloy layer is formed, the adherence of the plated-metal-layer tends to be improved. [0084] The thickness D of the plated-metal-layer of the plated steel sheet according to the embodiment is not particularly limited, and may be controlled as necessary. Generally, the thickness D is set to be 35 f!m smaller in many cases. [0085] The metallographic structure of the plated-metal-layer is observed as below. A sample is collected by cutting the plated steel sheet such that the cross section whose cutting direction is parallel to the thickness direction of the plated steel sheet is observed. The cross section is polished or processed by using a Cross Section Polisher (CP). In a case where the cross section is polished, the cross section is etched with nita!. The cross section is then observed using an optical microscope or SEM, and a metallographic micrograph thereof is captured. If the cross section observed with SEMis a COMPO image as shown in FIG. 1, due to a difference in a chemical composition between the coarse domain and the fine domain, a sharp contrast is made, and hence the boundary between the coarse domain and the fine domain can - 43 - be easily discerned. The chemical composition of the constituent phases can be measured by analysis based on EDX or EPMA. From the result of the chemical analysis, the constituent phases can be simply identified. The metallographic micrograph is binarized through, for example, image analysis; an area ratio of a white portion or black portion of the plated-metal-layer is measured; and in this way, area fractions of the constituent phases can be measured. Furthermore, from the determined area of each coarse domain, an average equivalent circle diameter can be determined by calculation. Alternatively, by observing the metallographic structure of the plated-metal-layer by an Electron Back Scattering Diffraction Pattern (EBSD) method, the constituent phases may be identified, and the area fraction and the average equivalent circle diameter of the constituent phases may be determined. [0086] In order to more specifically identify the constituent phases, the metallographic structure of the plated-metal-layer is observed as below. A thin sample is collected by cutting the plated steel sheet such that the cross section whose cutting direction is parallel to the thickness direction of the plated steel sheet is observed. The thin sample is subjected to ion milling. Alternatively, a thin sample is collected by processing the plated steel sheet with a Focused Ion Beam (FIB) such that the cross section whose cutting direction is parallel to the thickness direction of the plated steel sheet is observed. These thin samples are observed with TEM, and the metallographic micrograph thereof is captured. The constituent phases can be accurately identified using an electron diffraction pattern. By performing image analysis on the metallographic micrograph, the area fractions and the average equivalent circle diameters of the constituent phases can be determined. [0087] - 44 - From XRD diffraction peaks of the plated-metal-layer, the existence ofthe constituent phases can be confirmed in the simplest way, although how the constituent phases exist in a space cannot be ascertained in this way. Here, because the diffraction peak positions of the quasicrystal phase, Mgs1Zn2o, and Mg32(Zn, A1)49 overlap each other, the existence of theses phases can be confirmed, but it is difficult to distinguish them from each other. [0088] The steel sheet as a base metal of the plated steel sheet according to the embodiment is not particularly limited. As the steel sheet, it is possible to use AI killed steel, ultra low carbon steel, high carbon steel, various high tensile strength steel, steel containing Ni or Cr, and the like. [0089] Next, a method of producing a plated steel sheet according to the embodiment will be described. [0090] The method of producing a plated steel sheet according to the embodiment includes a hot-dip-plating process of dipping the steel sheet into a hot-dip-plating bath having an adjusted composition in order to form a plated-metal-layer on a surface of the steel sheet; a first cooling process of cooling the steel sheet after the hot-dip-plating process under conditions such that an average cooling rate of the plated-metal-layer is 15 °C/sec to 50 °C/sec in a temperature range where a temperature of the plated-metal-layer is from Tmelt to Tsolid-liquid in a nnit of°C, when Tmelt is regarded as a liquidus temperature of the plated-metal-layer and when Tsolid-Jiquid is a temperature range where the plated-metal-layer is in a coexistence state of a solid phase and a liquid phase and where a volume ratio of the solid phase to the plated-metal-layer - 45 - ~~------- (volume of solid phase/volume of plated-metal-layer) is 0.3 to 0.8; and a second cooling process of cooling the steel sheet after the first cooling process under conditions such that an average cooling rate of the plated-metal-layer is I 00 oC/sec to 3000 °C/sec in a temperature range where a temperature of the plated-metal-layer is from a temperature at finishing the first cooling process to 250°C. [0091] A value ofT melt which is a liquidus temperature of the plated-metal-layer can be determined using, for example, liquidus temperatures (liquidus surface temperatures) disclosed in a non-patent document (Liang, P., Tarfa, T., Robinson, J. A., Wagner, S., Ochin, P., Harmelin, M. G., Seifert, H. J., Lukas, H. L., Aldinger, F., "Experimental Investigation and Thermodynamic Calculation of the Al-Mg-Zn system", Thermochim. Acta, 314, 87-110 (1998)) written by Liang eta!, as shown in FIG. 5. In this way, a value ofT melt substantially can be estimated by using the fraction of Zn, AI, and Mg contained in the plated-metal-layer. [0092] A value ofT solid-liquid can be accurately determined from an alloy phase diagram. Specifically, by using the chemical composition of the plated-melal-laye~ and the corresponding alloy phase diagram, a volume ratio (volume fi·action) between a plurality of coexisting phases can be determined based on lever rule. That is, by using the alloy phase diagram, a temperature at which a volume ratio of a solid phase becomes 0.3 and a temperature at which a volume ratio of the solid phase becomes 0.8 may be determined. In the method of producing a plated steel sheet according to the embodiment, the value ofT solid-liquid may be determined using the alloy phase diagram. At this time, as the alloy phase diagram, a calculated phase diagram based on a thermodynamic calculation system may be used. Here, because the alloy phase - 46 - diagram merely shows an equilibrium phase, the ratio between the constituent phases determined from the alloy phase diagram does not necessarily totally agree with an actual ratio between the constituent phases in the plated-metal-layer which is being cooled. Regarding Tsolid-liquid as a temperature range where the plated-metal-layer that is being cooled is in a coexistence state of a solid phase and a liquid phase and a volume ratio of the solid phase to the plated-metal-layer is 0.3 to 0.8, the inventors of the present invention conducted intensive investigation. As a result, they found that, by the following expression, {345 + 0.35 x (T melt- 345)} - 5 :S Tsolid-liquid :S {345 + 0.35 x (T melt- 345)} + 5, Tsolid-liquid can be empirically determined. Therefore, in the method of producing a plated steel sheet according to the embodiment, a value of Tsolid-liquid may be determined by the above expression. [0093) In the hot -dip-plating process, the chemical composition of the plating bath is adjusted such that the chemical composition, by atomic%, of the plated-metal-layer formed on the surface of the steel sheet contains Zn: 20% to 60%, AI: 0.3% to 15%, Ca: 0% to 3.5%, Y: 0% to 3.5%, La: 0% to 3.5%, Ce: 0% to 3.5%, Si: 0% to 0.5%, Ti: 0% to 0.5%, Cr: 0% to 0.5%,Fe: 0% to 2%, Co: 0% to 0.5%, Ni: 0% to 0.5%, V: 0% to 0.5%, Nb: 0% to 0.5%, Cu: 0% to 0.5%, Sn: 0% to 0.5%, Mn: 0% to 0.2%, Sr: 0% to 0.5%, Sb: 0% to0.5'Yo, and Pb: 0% to 0.5%, the balance consists of Mg and impurities, and a Zn content and an AI content expressed in atomic% in the chemical composition of the plated-metal-layer satisfy 25% :S Zn +AI. [0094] In the embodiment, the hot-dip-plating process is selected for example. However, the method of forming the plated-metal-layer on the surface of the steel sheet is not limited as long as the plated-metal-layer having the aforementioned chemical - 47 - composition can be formed on the surface of the steel sheet. In addition to the hot-dip-plating, spraying, sputtering, ion plating, evaporating, or electroplating may be applied. [0095] Immediately after being pulled up out of the plating bath, the plated-metal-layer formed on the surface of the steel sheet by the hot-dip-plating process is in a molten state (liquid phase). By cooling the plated-metal-layer in the molten state by a first cooling process and a second cooling process unique to the embodiment, the plated-metal-layer can be controlled to have the aforementioned metallographic structure containing a quasicrystal. [0096] In a case where a plated-metal-layer forming method other than the hot-dip-plating process is selected, by reheating the plated steel sheet, on which the plated-metal-layer is formed, by using a heating furnace so as to melt only the plated-metal-layer, and then cooling the plated-metal-layer by the first cooling process and the second cooling process unique to the embodiment, the plated-metal-layer can be controlled to have the aforementioned metallographic structure containing a quasi crystal. [0097] A melting point of the plated-metal-layer containing Mg and Zn as main components is totally different from a melting point of the steel sheet as a base metal. Therefore, those skilled in the related art can easily determine an optimized temperature at which only the plated-metal-layer is melted and an optimized melting time. CLAIMS 1. A plated steel sheet with a quasicrystal comprising a steel sheet and a plated-metal-layer arranged on a surface of the steel sheet, wherein: the plated-metal-layer comprises, as a chemical composition, by atomic%, 20% to 60% ofZn, 0.3% to 15% of AI, 0% to 3.5% ofCa, 0% to 3.5% ofY, 0% to 3.5% of La, 0% to 3.5% ofCe, 0% to 0.5% of Si, 0% to 0.5% ofTi, 0% to 0.5% of Cr, 0% to 2% of Fe, 0% to 0.5% of Co, 0% to 0.5% ofNi, 0% to 0.5% ofV, 0% to 0.5% ofNb, 0% to 0.5% ofCu, 0% to 0.5% of Sn, 0% to 0.2% ofMn, 0% to 0.5% of Sr, 0% to 0.5% of Sb, 0% to 0.5% ofPb, and - 115 - a balance of Mg and impurities; a zinc content and an aluminum content expressed in atomic% in the chemical composition of the plated-metal-layer satisfy 25% <:; Zn +AI; the plated-metal-layer includes, as a metallographic structure, a quasicrystal phase; a magnesium content, a zinc content, and an aluminum content expressed in atomic% in the quasicrystal phase satisfy 0.5 <:; Mg I (Zn +AI)<:; 0.83; and an average equivalent circle diameter of the quasi crystal phase is larger than 1 f.Lm and equal to or smaller than 200 f.lm. 2. The plated steel sheet with a quasi crystal according to claim 1, wherein a calcium content, an yttrium content, a lanthanum content, and a cerium content expressed in atomic% in the chemical composition of the plated-metal-layer satisfy 0.3% <:; Ca + Y +La+ Ce <:; 3.5%. 3. The plated steel sheet with a quasi crystal according to claim 1, wherein a silicon content, a titanium content, and a chromium content expressed in atomic% in the chemical composition of the plated-metal-layer satisfy 0.005% <:; Si + Ti + Cr <:; 0.5%. 4. The plated steel sheet with a quasi crystal according to claim 1, wherein a zinc content and an aluminum content expressed in atomic% in the chemical composition of the plated-metal-layer satisfy 30% <:; Zn +AI <:; 50%, and 3 <:; Zn/ AI<:; 12. 5. The plated steel sheet with a quasicrystal according to claim I, - 116 - wherein: when viewed in a cross section, whose cutting direction is parallel to a thickness direction of the plated-metal-layer, the metallographic structure of the plated-metal-layer is a bimodal structure which comprises a fine domain composed of a grain having an equivalent circle diameter of 1 rtm or smaller and a coarse domain composed of a grain having an equivalent circle diameter oflarger than 1 fLID; the coarse domain comprises the quasi crystal phase; and the fine domain comprises at least one selected from a Mg51Zn20 phase, a MgJz(Zn, Al)49 phase, a MgZn phase, a MgZnz phase, and a Zn phase. 6. The plated steel sheet with a quasicrystal according to claim 5, wherein: an area fraction of the coarse domain in the metallographic structure is equal to or more than 5% and equal to or less than 80%; and an area fraction of the fine domain in the metallographic structure is equal to or more than 20% and equal to or less than 95%. 7. The plated steel sheet with a quasicrystal according to claim 5, wherein: an area fraction of the quasi crystal phase included in the coarse domain is equal to or more than 80% and less than 100% in the coarse domain; and an area fraction in total of the Mg51Znzo phase, the Mg32(Zn, Al)49 phase, the MgZn phase, the MgZn2 phase, and the Zn phase included in the fine domain is equal to or more than 80% and less than 100% in the fine domain. 8. The plated steel sheet with a quasicrystal according to claim 5, wherein, when viewed in the cross section and when a thickness of the - 117 - plated-metal-layer is regarded as D, an area from a surface of the plated-metal-layer toward the steel sheet in the thickness direction to 0.3 x D is regarded as a surface area of the plated-metal-layer, and an area from an interface between the steel sheet and the plated-metal-layer toward the plated-metal-layer in the thickness direction to 0.3 x Dis regarded as a deep area of the plated-metal-layer, an area fraction of the coarse domain in the surface area of the plated-metal-layer is equal to or more than 10% and less than 100% and an area fraction of the coarse domain in the deep area of the plated-metal-layer is equal to or more than 10% and less than 100%, and wherein, when an area except for the surface area and the deep area in the plated-metal-layer is regarded as a center area of the plated-metal-layer, an area fraction of the fine domain in the center area of the plated-metal-layer is equal to or more than 50% and less than 100%. 9. The plated steel sheet with a quasi crystal according to claim I, wherein a Mg phase is absent in the metallographic structure of the plated-metal-layer. I 0. The plated steel sheet with a quasi crystal according to claim I further comprising a Fe-AI containing alloy layer, wherein: the Fe-AI containing alloy layer is arranged between the steel sheet and the plated-metal-layer; the Fe-Al containing alloy layer comprises at least one selected from Fe5Ah and Ah2 Fe; and a thickness of the Fe-Al containing alloy layer is equal to or more than 10 nm - 118 - and equal to or less than 1000 nm. 11. A method of producing a plated steel sheet with the quasi crystal according to any one of claims 1 to 10, the method comprising: a hot-dip-plating process of dipping a steel sheet into a hot-dip-plating bath having an adjusted composition in order to form a plated-metal-layer on a surface of the steel sheet; a first cooling process of cooling the steel sheet after the hot-dip-plating process under conditions such that an average cooling rate of the plated-metal-layer is equal to or faster than 15 °C/sec and equal to or slower than 50 °C/sec in a temperature range where a temperature of the plated-metal-layer is from Tmelt to Tsolid-l;qu;d in unit of °C, when the Tmelt is regarded as a liquidus temperature of the plated-metal-layer and when the Tsolid-l;qu;d is a temperature range where the plated-metal-layer is in a coexistence state of a solid phase and a liquid phase and where a volume ratio of the solid phase to the plated-metal-layer is equal to or more than 0.3 and equal to or less than 0.8; and a second cooling process of cooling the steel sheet after the first cooling process under conditions such that an average cooling rate of the plated-metal-layer is equal to or faster than 100 °C/sec and equal to or slower than 3000 °C/sec in a ,(,~ ll II temperature range where a temperature of the plated-metal-layer is from a temperature at finishing the first cooling process to 250°C. il !i 12. The method of producing a plated steel sheet with the quasicrystal according to claim 11, wherein: in the hot-dip-plating process, an oxide in the hot-dip-plating bath is - 119 - ' !i il ?: 1 g/1 or less; an oxygen concentration of an atmosphere at dipping the steel sheet is 100 ppm or less in volume ratio; a plating tub to hold the hot-dip-plating bath is a steel tub; a dross in the hot-dip-plating bath is removed by a metal pump; Tbath which is a temperature of the hot-dip-plating bath is equal to or higher than 1 0°C and equal to or lower than 1 00°C higher than the T melt; and a time for dipping the steel sheet into the hot-dip-plating bath is equal to or longer than 1 sec and equal to or shorter than 10 sec.