File: SECT02.WM of Tape: Various/ETH/s10-diss
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- 7 - @ka II.METALLURGICAL PROPERTIES OF ALUMINUM AND INDIUM SYSTEMS. ___________________________________________________________ 1). Properties of aluminum and indium host. __________________________________________ a). Aluminum. ____________ @ke @ba Sir Humphry Davy first showed in 1809 that potash alum has a metallic base. He proposed the name "aluminum". This spelling has been retained in North America and we shall also use it here. In 1825 , Hans Christian Oersted was able to separate the metal as such and we think it is quite fitting to investigate the electronic properties of this metal, which has been discovered by the discoverer of electromagnetism and which also is the third most abundant element on earth (after oxygen and silicon). @be @ba Aluminum crystallizes in the face centered cubic system (A1). The unit cell contains 4 atoms at the positions (0 0 0 , 0 1/2 1/2 , 1/2 0 1/2 , 1/2 1/2 0). The lattice constant of the FCC cell at 25C is 4.0497 A^o^ngstroem and the lattice constant at 4K is 4.03185 A^o^ngstroem. We obtained this lattice constant by extrapolation of the low temperature lattice constants measured by Figgins et al.(1956). There seems to have been some confusion about the low temperature lattice constants in aluminum: Ashcroft(1963) used the room temperature lattice constant in kX units (4.04 kX), while Anderson and Lane(1970) only quote the free-electron Fermi energy, but this value corresponds to still another lattice constant (4.019A^o^). @be - 8 - @fa FIGURE 1 Figure 2.1:Fermi surface of Al @fe @ba The electronic configuration of Al is (Ne) 3s2 3p1. Thus Al has a full rare gas shell, no d states and 3 valence electrons. These 3 valence are responsible for the conduction band electron states and will fill exactly 1.5 Brillouin zones; therefore Al is a so-called "uncompensated" metal. @be @ba The dHvA results on the third band 'monster' of Al of Gunnersen(1957) and Larson and Gordon(1967) have been incorporated into a model for the real Fermi surface (FS) by Ashcroft(1963). @be - 9 - @fa FIGURE 2 Figure 2.2: (110) section of Al Fermi surface @fe @ba This fitting was done in the pseudopotential spirit and he obtained the values of two matrix elements: V(111)=.0179Ryd, V(200)=.0562Ryd. (where V(111) are the matrix-elements connecting plane-waves in the {111} directions and V(200) those in the {100} directions.) Anderson and Lane(1970) made supplementary dHvA measurements on the second zone of Al and they made some minor modifications to the matrix elements: V(111)=.018, V(200)=.062Ryd. @be - 10 - @ba A reconsideration of these results and the use of magneto- striction data by Griessen and Sorbello(1972) led to the result of: v(111)=.0171, V(200)=.0562Ryd. A perspective view of the real Fermi surface corresponding to this model is shown in Fig. 2.1 and a (110) section of the FS in Fig. 2.2. @be @fa FIGURE 3 Figure 2.3:Relative length change of a and c lattice spacings of In. @fe - 11 - @ka b) Indium. __________ @ke @ba Indium was discovered in 1863 by F. Reich and T. Richter. The spectral lines they discovered were of indigo-blue colour and, thus, they gave the name indium to this element. @be @ba The lattice structure of indium can be described by an FCC cubic structure which is slightly distorted in the (001) direction (The third index is in the so-called c direction). The lattice constants at 25C are: a=4.5984A^o^, c=4.9468A^o^ and the axial ratio c/a= 1.0758. The lattice constants have been measured at 4K by Barret(1962) and the values a=4.5557A^o^, c=4.9343A^o^ and c/a=1.0831 were given. @be @ba We extrapolated the room temperature values with the help of the integrated thermal expansion data of Collins et al.(1967). This gives the values of a=4.5529A^o^, c=4.9347A^o^ and c/a= 1.0839. We prefer these values (which are almost equal to the quoted 4K values) because this give us a continuous transition over the temperature range 4K to 300K, of the a, c and c/a values. @be @ba The temperature dependence of these parameters is shown in Figs.2.3 and 2.4. The fact that the c/a ratio has a maximum at ~70K and drops down to the room temperature value, while going through the same value as the 4K c/a ratio at the temperature of 115K, will be of importance in discussing effects which are measured over a large temperature range. (see Sect.V) @be - 12 - @fa FIGURE 4 Figure 2.4: Temperature dependence of c/a ratio and relative volume change in In. @fe @ba The electronic structure of In is rather similar to that of Al. It is in the same column of the periodic table but in the next heavier row and has a (Kr) 4d10 5s2 5p1 shell. The d states are all occupied and rather far away (in energy) from the 3 valence states. The departure from cubicity, on the other hand, has a strong influence on the topology of the real Fermi surface. The 'monster' in the third zone of Al is reduced into disconnected rings located in a plane perpendicular to the c axis. The data which has helped in determining the Fermi surface comes from Gantmakher and Krylov(1966), Hughes and Shepherd(1969) and van Weeren and Anderson(1973). @be - 13 - @fa FIGURE 5 Figure 2.5:Fermi surface of In @fe @ba These results have been incorporated into pseudopotential models of the FS by the last two authors and by Ashcroft and Lawrence(1968). The model of Ashcroft and Lawrence is a good average between the different results and the approximate matrix elements are: V(111)=-.05, V(200)=+.003, V(002)=-.02 Ryd.. A perspective view is shown in Fig.2.5 and a (110) section of the FS in Fig.2.6. @be - 14 - @fa FIGURE 6 Figure 2.6:(110) section of Fermi surface of In @fe - 15 - @ka 2) Solutes in Al and In solvents. _________________________________ @ke @ba In investigations of alloys it will be important to know which alloys are suitable for dHvA or conductivity measurements. For this purpose we will give a review of the possible solutes in Aluminum and in Indium. What we mean by possible is the existence of a solid solution phase. Our criterion is, first of all, that phase diagrams exist with the appropriate solid solution phase and, secondly, the availability of X-ray lattice spacings for these solutions. There will be some borderline cases with either peculiar phase diagrams or a very restricted solubility range, which will be duly discussed. The data for the phase diagrams was critically evaluated from Hansen(1958), Elliot(1965) and Shunk(1969). @be @ba The partial periodic tables in Table.2.1 for Al and in Table.2.2 for In show the elements which we selected as possible candidates for solid solutions. These tables also show the ratios of atomic radii and the ratios of the atomic masses with respect to each host and, furthermore, the melting points of these elements. @be @ba The ratios of the atomic masses are the appropriate numbers for converting weight percents to atomic percents through the formula: @be @ea (2.1) X = 100 W / [ W + M\sol\/ M\solv\ ( 100 - W ) ] @ee @ba here W is the percent in weight and X the atomic percents. @be - 16 - @ka _________________________________________________________________ | | | | | | I | II | III | IV | | | | | | _________________________________________________________________ _________________ | | |1.08 180 | | | | Li | | | | .257 | | | _________________________________________________________________ | | | | | | |1.12 650 |R\solut\ 660 |.92 1410 | | | |------ M.P. | | | | Mg | R\Al\ Al M\solut\ | Si | | | | ------ | | | | .901 | M\Al\ | 1.041 | | | | | | _________________________________________________________________ | | | | | |.90 1083 |.97 420 |.99 30 |.96 937 | | | | | | | Cu | Zn | Ga | Ge | | | | | | | 2.355 | 2.423 | 2.584 | 2.69 | | | | | | _________________________________________________________________ | | | |1.01 961 |1.08 321 | | | | | Ag | Cd | | | | | 3.998 | 4.166 | | | | _________________________________ _________________________________________________________________ | | | | | | IV | V | VI | VII | | | | | | _________________________________________________________________ _________________________________________________________________ | | | | | |1.03 1668 |.94 1900 |.91 1875 |.94 1245 | | | | | | | Ti | V | Cr | Mn | | | | | | | 1.775 | 1.888 | 1.927 | 2.036 | | | | | | _________________________________________________________________ Table 2.1 : Partial periodic table for Al solutes @ke - 17 - @ka _________________ | | | I | | | _________________ _________________ | | |.93 180 | | | | Li | | | | .060 | | | _________________ _________________________________________________________________ | | | | | | II | III | IV | V | | | | | | _________________________________________________________________ _________________ | | |.96 650 | | | | Mg | | | | .212 | | | _________________________________ | | | |.83 420 |.85 30 | | | | | Zn | Ga | | | | | .569 | .607 | | | | _________________________________________________ | | | | |.93 321 |R\solut\ 156 |.98 232 | | |------ M.P. | | | Cd | R\In\ In M\solut\ | Sn | | | ------ | | | .979 | M\In\ | 1.034 | | | | | _________________________________________________________________ | | | | | |.95 -38 |1.03 303 |1.05 327 |1.02 271 | | | | | | | Hg | Tl | Pb | Bi | | | | | | | 1.747 | 1.780 | 1.804 | 1.820 | | | | | | _________________________________________________________________ Table 2.2: Partial periodic table for In solutes. R = Ionic Radius M = Atomic Mass M.P. = Melting Point @ke - 18 - @ba The ratios of the atomic radii are a first indication on the existance of a possible solid solution phase. An empirical rule given by Hume-Rothery states that unless the solute and solvent radii lie within about 15% of each other, solid solutions cannot be formed even though all other factors are favourable. It can be seen from Table 2.1 and Table 2.2 that the only alloy which does not satisfy this rule is In-Zn; this means we should be sceptical about the quality of a solution of Zn in In. @be @ka a) Aluminum solutes. ____________________ @ke @ba The phase diagrams of the solutes Si, Ge, Ga, Mg, Li, Cu and Ag are all of a very similar type, i.e. a well defined eutectic temperature and a maximum solid solubility at this same temperature. In order to see the similarity more easily we have replotted these phase diagrams in a reduced manner: the 3-phase equilibrium point between (Al), (Al)Liquid, (Al)Eutect. has been made a fixed point of Fig. 2.7 and the scaling factors corresponding to each solute can be found in Table 2.3 under Max.sol.Eut.Temp.[At%] and under Eutectic Temp. This figure shows that all the relative solvus curves tend to have a vertical tangent at low temperatures and this should imply that some upper concentration limit of solid solution exists, even at low temperatures. In the upper part of the diagram, the double lines for each solute corresponding to the solidus- liquidus curves are also quite similar and will give segregation coefficients of around 0.1 - 0.5. @be - 19 - @fa FIGURE 7 Figure 2.7:Reduced phase-diagrams of eutectic alloys in Al ( Si - Ge - Ga - Mg - Li - Cu - Ag ) @fe @ba The solutes Si and Ge in this group have a doubtful solubility and alloys of this type will have to be analysed carefully if they are to be used for scattering measurements. @be - 20 - @ka _________________________________________________________ | | | | | | | | |Solute | Seg. |Sol.sol|Max.sol|Eutect.|Interm.|1/a * | | | coeff.|At% 20C|Eut. T.| Temp. | comp. | da/dC | | | | | | | | | _________________________________________________________ | | | | | | | | | Si | .12 | <.01 | 1.6 | 577 | Si |(-.047)| | | | | | | | | | Ge | .08 | .05 | 2.8 | 424 | Ge | .039 | | | | | | | | | | Ga | .07 | 8.0 | 9.5 | 26 | Ga | .031 | | | | | | | | | | Mg | .30 | 1.5 | 18.9 | 450 |Mg2Al3 | .072 | | | | | | | | | | Li | .50 | 5.0 | 22.0 | 601 | LiAl | -.011 | | | | | | | | | | Cu | .13 | .05 | 2.5 | 548 | CuAl2 | (-.12)| | | | | | | | | | Ag | .30 | .10 | 23.8 | 566 | Ag3Al | .006 | | | | | | | | | | Zn | .35 | .70 | 16.0 | 275 | Zn | -.016 | | | | | | | | | | Cd | .35 | .01? | 0.14 | 649 | Cd | (-.1) | | | | | | | | | | Mn | ~1 | .01? | 0.90 | 658 | MnAl6 |(-.148)| | | | | | | | | | Cr | ~1 | .01? | 0.37 | 661 | CrAl7 |(-.23) | | | | | | | | | | Ti | 1.4 | ? | 0.20 | 665 | TiAl3 |(-.23) | | | | | | | | | _________________________________________________________ Table 2.3: Properties of Al solutes @ka @ba The remaining candidates for Al are Zn, Cd, and Mn, Cr, Ti. The phase diagram for Zn is peculiar; there is a miscibility gap at higher Zn concentrations and the solidus and solvus are disconnected. This should not have any effect on the properies of the solid solution phase. Cd is a very marginal case and it's phase diagram is badly known (very restricted sol. sol.). @be - 21 - @ba The phase diagrams for Mn, Cr and Ti are of the peritectic type and are not very well established. The sol. sol. are quite restricted, but considering that the scattering properties of transition metal solutes are usually higher, they could still be interesting candidates; their segregation coefficients are >=1. @be @ba The data connected with the phase diagrams of the different solutes is collected in Table 2.3, i.e. segregation coefficient, solid solubility[At%] at room temperature, maximum solubility[At%] at the eutectic or peritectic temperature and the eutectic or peritectic temperature itself. Two other quantities are also given in this table: The relative change in lattice constant for the solid solution phase (will be needed in calculations of scattering properties) and the nearest intermediate compound the solute forms with Al (This compound will be in equilibrium with the sol.sol. if the concentration is higher than the solvus line and will show the most likely pseudo-molecule the solute will form with Al). @be @ba We obtained the data for the alloy lattice spacings by fitting low concentration tangents to values taken from Pearson(1964) and Pearson(1967). In some cases data was only available at concentrations near or above the solvus curves and represents lattice constants of metastable solutions. These values are marked by being enclosed in parenthesis in the table. @be - 22 - @ka b) Indium solutes. _________________ @ke @ba The melting points of the In solutes are, with the exeption of Mg, much closer to the M.P. of In than those of the Al solutes to the M.P. of Al and the phase diagrams are simpler, with fewer intermediate compounds. It is to be noted that there seem to be no transition metal solutes with a non-vanishing solubility in Indium. @be @ba The fact that In has a tetragonally distorted face-centered lattice presents one extra feature which affects the phase diagrams. It is quite certain that this tetragonal distortion is due to electronic effects which tend to minimize the cohesive energy of the crystal; adding solutes with other valencies will change the Fermi energy and this change will force the crystal to seek a new equilibrium c/a ratio. @be @ba Typical second-order phase transitions corresponding to this effect are to be seen in the alloys with Sn, Pb, Tl, Cd and Hg. As a typical example we give the phase diagram of Hg in Fig.2.8. The characteristic parameters of the solutes in this group can be taken from Table 2.4. The beta phases adjacent to the solid solution are FCC for Tl, Cd and Hg, FCT for Sn and Pb (c/a=1.3 and .93 resp.). The phase diagrams for Bi,Ga, Mg and Zn are of the normal eutectic type (see Fig.2.7 ). As already mentioned the Zn solid solution is not very certain: we have tried to make an In-Zn alloy, but it showed no increase in residual resistivity. The alloy In - Li is of the peritectic type and the (Li) sol.sol. phase is not very certain either. @be - 23 - @fa FIGURE 8 Figure 2.8:In-Hg Phase diagram. @fe @ba All of the parameters for these alloys have been collected in Table 2.4. The tetragonal structure of In makes it necessary to indicate both the a and c relative changes in lattice con- stants with concentration C (C in units of 100At.%). We also give the the relative changes for &r=c/a and v=a^2^c. @be - 24 - @ka _________________________________________________ | | | | | | | | Solute|Segreg.|Sol.sol|Max.sol|Eutect.|Interm.| | | coeff.|At% 20C|Eut. T.| Temp. | comp. | | | | | | | | _________________________________________________ | | | | | | | | Bi | 0.38 | 2.50 | 12 | 74 | In2Bi | | | | | | | | | Sn | 0.30 | 11.5 | 10 | 144 | Sn | | | | | | | | | Pb | 2.50 | 13.0 | 10 | 160 | Pb | | | | | | | | | Ga | 0.65 | 15.0 | 18 | 16 | Ga | | | | | | | | | Tl | ~1 | 22.0 | 17 | 156 | Tl | | | | | | | | | Mg | 1.60 | 37.0 | 43 | 330 | ? | | | | | | | | | Zn | 0.36 | .05? | 2 | 143 | Zn | | | | | | | | | Cd | 0.46 | 4.50 | 3 | 148 | Cd | | | | | | | | | Hg | 0.38 | 6.00 | 6 | 108 | HgIn11| | | | | | | | | Li | >1 | 2.00 | 10 | 159 | LiIn | | | | | | | | _________________________________________________ _________________________________________________ | | | | | | | | |1/a * |1/c * |dln(&r) | 1/v * | 1/v * | | Solute| da/dC | dc/dC |-------| dv/dC | dv/dC | | | | | dC | | Vegard| | | | | | | | _________________________________________________ | | | | | | | | Bi | -.022 | .34 | .35 | .30 | .37 | | | | | | | | | Sn | -.092 | .23 | .33 | .045 | .042 | | | | | | | | | Pb | .02 | .154 | .13 | .195 | .16 | | | | | | | | | Ga | - | - | - | - | -.24 | | | | | | | | | Tl | .065 | -.054 | -.12 | .075 | .08 | | | | | | | | | Mg | 0 | -.003 | -.003 | -.003 | -.11 | | | | | | | | | Zn | - | - | - | - | -.42 | | | | | | | | | Cd | .13 | -.38 | -.52 | -.12 | -.17 | | | | | | | | | Hg | .046 | -.22 | -.25 | -.13 | -.10 | | | | | | | | | Li | .004 | -.017 | -.023 | -.01 | -.16 | | | | | | | | _________________________________________________ Table 2.4: Properties of In solutes. @ke