File: SECT02.WM of Tape: Various/ETH/s10-diss
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II.METALLURGICAL PROPERTIES OF ALUMINUM AND INDIUM SYSTEMS.
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1). Properties of aluminum and indium host.
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a). Aluminum.
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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).
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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^).
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FIGURE 1
Figure 2.1:Fermi surface of Al
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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.
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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).
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FIGURE 2
Figure 2.2: (110) section of Al Fermi surface
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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.
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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.
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FIGURE 3
Figure 2.3:Relative length change of a and c
lattice spacings of In.
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b) Indium.
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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.
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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.
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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.
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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)
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FIGURE 4
Figure 2.4: Temperature dependence of c/a ratio
and relative volume change in In.
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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).
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FIGURE 5
Figure 2.5:Fermi surface of In
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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.
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FIGURE 6
Figure 2.6:(110) section of Fermi surface of In
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2) Solutes in Al and In solvents.
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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).
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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.
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The ratios of the atomic masses are the appropriate numbers for
converting weight percents to atomic percents through the
formula:
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(2.1) X = 100 W / [ W + M\sol\/ M\solv\ ( 100 - W ) ]
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here W is the percent in weight and X the atomic percents.
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| | | | |
| I | II | III | IV |
| | | | |
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| |
|1.08 180 |
| |
| Li |
| |
| .257 |
| |
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| | | | |
| |1.12 650 |R\solut\ 660 |.92 1410 |
| | |------ M.P. | |
| | Mg | R\Al\ Al M\solut\ | Si |
| | | ------ | |
| | .901 | M\Al\ | 1.041 |
| | | | |
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| | | | |
|.90 1083 |.97 420 |.99 30 |.96 937 |
| | | | |
| Cu | Zn | Ga | Ge |
| | | | |
| 2.355 | 2.423 | 2.584 | 2.69 |
| | | | |
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| | |
|1.01 961 |1.08 321 |
| | |
| Ag | Cd |
| | |
| 3.998 | 4.166 |
| | |
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| | | | |
| IV | V | VI | VII |
| | | | |
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| | | | |
|1.03 1668 |.94 1900 |.91 1875 |.94 1245 |
| | | | |
| Ti | V | Cr | Mn |
| | | | |
| 1.775 | 1.888 | 1.927 | 2.036 |
| | | | |
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Table 2.1 : Partial periodic table for Al solutes
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| |
| I |
| |
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| |
|.93 180 |
| |
| Li |
| |
| .060 |
| |
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| | | | |
| II | III | IV | V |
| | | | |
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| |
|.96 650 |
| |
| Mg |
| |
| .212 |
| |
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| | |
|.83 420 |.85 30 |
| | |
| Zn | Ga |
| | |
| .569 | .607 |
| | |
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| | | |
|.93 321 |R\solut\ 156 |.98 232 |
| |------ M.P. | |
| Cd | R\In\ In M\solut\ | Sn |
| | ------ | |
| .979 | M\In\ | 1.034 |
| | | |
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| | | | |
|.95 -38 |1.03 303 |1.05 327 |1.02 271 |
| | | | |
| Hg | Tl | Pb | Bi |
| | | | |
| 1.747 | 1.780 | 1.804 | 1.820 |
| | | | |
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Table 2.2: Partial periodic table for In solutes.
R = Ionic Radius
M = Atomic Mass
M.P. = Melting Point
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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.
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a) Aluminum solutes.
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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.
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FIGURE 7
Figure 2.7:Reduced phase-diagrams of eutectic alloys in Al
( Si - Ge - Ga - Mg - Li - Cu - Ag )
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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.
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| | | | | | | |
|Solute | Seg. |Sol.sol|Max.sol|Eutect.|Interm.|1/a * |
| | coeff.|At% 20C|Eut. T.| Temp. | comp. | da/dC |
| | | | | | | |
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| | | | | | | |
| 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) |
| | | | | | | |
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Table 2.3: Properties of Al solutes
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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.).
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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.
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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).
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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.
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b) Indium solutes.
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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.
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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.
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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.
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FIGURE 8
Figure 2.8:In-Hg Phase diagram.
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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.
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| | | | | | |
| Solute|Segreg.|Sol.sol|Max.sol|Eutect.|Interm.|
| | coeff.|At% 20C|Eut. T.| Temp. | comp. |
| | | | | | |
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| | | | | | |
| 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 |
| | | | | | |
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| | | | | | |
| |1/a * |1/c * |dln(&r) | 1/v * | 1/v * |
| Solute| da/dC | dc/dC |-------| dv/dC | dv/dC |
| | | | dC | | Vegard|
| | | | | | |
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| | | | | | |
| 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 |
| | | | | | |
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Table 2.4: Properties of In solutes.
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