Designing High-Strength Copper Alloys Based on the
Crystallographic Structure of Precipitates
by Hidemichi Fujiwara *We have developed a technique for advancing the design of precipitation
hardened copper alloys using calculations to extract, from a crystallographic
data base, those elements and compounds capable of acting as the precipitate phase, and wehave verified experimentally the results of those calculations. We have found that in those alloys
in which the compounds that were identified by calculation as those in which precipitation would
occur during the aging process, precipitation did occur on the habit plane that was predicted by
calculation, and precipitation hardening behavior was clearly exhibited. By also taking accountof considerations of solid mechanics it has been possible to predict strength after aging, raising
the expectation of integrating precipitate (i.e., the alloy system) selection with design for afteraging strength.

ABSTRACT1.

INTRODUCTION

In recent years precipitation-hardened copper alloys havecome into use in applications requiring high-strength
electrical conductors, creating an active movement forthe development of new alloys. One of the most obvious
needs is for high-strength copper alloys, and a large
number of precipitation-hardened alloys have beendeveloped 1).
In the conventional approach to alloy development,
an alloy system was generally identified on the basisof the temperature dependence of the solid solution
limit of the precipitate component, as obtained from aphase diagram for the alloy, and alloy composition wasestablished by evaluating its properties experimentally.
At the present time it is rare for alloy design to focus on
the parameters that determine alloy strength, i.e., thecrystallographic structure of the precipitate phase and theprecipitation state itself.
What is more, the actual identification of an alloy using
an experimental approach would, in the case of ternary
alloy systems, involve upwards of 2400 combinations.
When we also consider methods of optimizing thecomposition, identifying the optimum alloy becomes apractical impossibility.
We accordingly propose, as a means of identifying
new alloy systems from among the huge number ofexisting alloy system combinations, to focus on the
crystallographic structure of the precipitate phase, which
contributes to precipitation hardening, and are working
to identify those alloys that could exist as a precipitation
hardened phase, and to predict the conditions ofexistence of the precipitate phase and the precipitation* Ecology and Energy Lab., R&D Div.

hardenability 2).
We believe that the method proposed here makes it
possible to design alloys by bench-top investigation,using crystallographic structures that form a hardened
phase together with a combination of phase diagrams
and calculations capable of predicting its volume fraction.
In the following we discuss the methodology used,
together with the analytical results obtained.

2.

EXPERIMENTAL AND ANALYTICALMETHODOLOGY

2.1

Calculation Method for Predicting the Conditionsof Existence of Precipitates in a Matrix
The state of precipitates in a matrix may be characterizedwith reference to precipitate phase size, distribution
density, and the conditions at the interface between the
precipitate and matrix phases. Since precipitate size anddistribution density are determined by the concentrationof the alloy components and the aging conditions, themost important parameter in identifying an alloy system
for alloy design is the conditions at the interface between
the precipitate and matrix phases, so that the focus must
be on the habit plane between the precipitate and thematrix.
Among the conditions for calculating the habit
plane, the matching of the atomic alignment period at
the interface between the precipitate and the matrix
is a necessary condition but when one extends thediscussion to interface stability it becomes necessary totake into account the conditions of atomic bonding--thatis to say, the magnitude of chemical interfacial energy
and the strain energy caused by misfit between the matrix
and precipitate phases.
However, in order to keep the interfacial energy value

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Designing High-Strength Copper Alloys Based on the Crystallographic Structure of Precipitates

low, a minimal match of the atomic alignment periods is anecessary condition. Therefore in selecting compounds
for which a habit plane can exist between the precipitate
and matrix phases, we may say the necessary conditionis satisfied by matching atomic alignments.
Turning our attention next to the crystallographic
structure on the compound side, and considering thesymmetricality of the copper crystals that form thematrix, we find ourselves limited to those crystallographic
systems having comparatively high symmetry: 1) cubic;
2) tetragonal; 3) orthorhombic; and 4) hexagonal. Since
we consider that matching can be easily obtained inthese systems we select from these four crystallographic
systems those compounds in which a precipitationhardened phase can occur.
2.1.1 Cubic Crystals
Considering precipitates in cubic systems, they are of thesame crystallographic system as the matrix, so that theorientation relationship between the fundamental planesof the two may be said to be:(001)Cu//(001)p. t., [100]Cu//[100]p.t.
In this case two things are necessary:1) that the fundamental structure of the precipitate be aface-centered structure; and2) that the lattice constant of the precipitate be close to
an integer multiple of that of the matrix (misfit is lessthan 4 %).
We may express this asap.t. = {aCu (1 + δ )} · n(1)where: ap.t. is the lattice constant of the precipitate;aCu is the lattice constant of the copper matrix; andn is any integerof the method. Following is a detailed description of thecalculating procedure:(1) It is essential that at least one lattice constant of
the chemical compound be close to an integer
multiple of a certain plane spacing of the matrix(misfit less than 4 %), and that a directional vector
perpendicular to the plane satisfying this conditionand one lattice axis of the compound be therotational pole. At this point one of the matrix-side
vectors included in the habit plane is established.
(2) If, when a unit cell of the compound is rotatedabout an axis (e.g., the a axis of the compound),the origin of which is one of the atoms locatedon the matrix plane, the other lattice points of thecompound (e.g., atomic points on the b or c axis)
match or are extremely close to atomic points on
the low-index plane of the matrix (i.e., misfit is lessthan 4 %), the planes formed by two axes of theabove-mentioned compound become the habit
planes of the matrix and precipitate respectively,
and the orientation relationship can be established.
In terms of the specific method of calculation, it is firstnecessary to establish the rotational axis of the matrix. In
such a case there are few bonding defects in the interface
at the low-index plane, so that in terms of interfacial
energy it is stable. There are an infinite number of matrix
directional vectors for the rotational axis, but if zero isincluded among the indices, there will of necessity be
low-index vectors in the perpendicular direction. These,
therefore, must be the basis of our discussion here.
If the relationships described above be represented
with reference to rotational axes [001] and [011], we havethe following:I) Considering a rotational axis [001] perpendicular tothe (001) plane:
the directional vectors may be represented by the
relationship[a b 0] = (n1[0.5 0.5 0] + n2[0.5 –0.5 0]
where n1 + n2 must be an even number;the atomic spacing may be calculated by
dcu = d100 {1/2 (n12 + n22)} 1/2(2)

the fitting conditions may be represented by
ap.t. = dCu (1 + δ ) · n

Figure 1

Schematic of the method used to establish the habit
plane. (atomic positions of the matrix and precipitate
are shown by blank and solid circles respectively)

2.1.2 Tetragonal and Orthorhombic Crystals
To find the habit plane it is necessary to establish two
directional vectors of the matrix within that plane, andthis involves calculating the fit at the unit vectors for the
spacing between the fundamental plane of the precipitate
and the azimuthal vectors included within the interfaceof the matrix side. Figure 1 is a schematic representation(3)and the plane index for the predicted plane will be(b a 0).
II) Considering a rotational axis [011] perpendicular tothe (011) plane:
the directional vectors may be represented by the
relationship[a b b] = (n1[1 0 0] + n2[0 0.5 0.5]
where n1 must be an integer and n2 must be an evennumber;the atomic spacing may be calculated by
dCu = d100 {(n12 + 1/2n22)} 1/2(4)

the fitting conditions may be represented by

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Designing High-Strength Copper Alloys Based on the Crystallographic Structure of Precipitates

ap.t. = dCu (1 + δ ) · n(5)and the plane index for the predicted plane will be(2b a a).
By means of the two relationships described above,it is possible to identify phases having a relationshipmatching the matrix planes indexed by the relationships(b a 0) and (2b a a).
Further, as stated above, it is also possible toextrapolate the above relationships for higher-index
planes, but since the atomic plane density that could
make a habit plane would be lower, we confined theanalysis presented in this paper to two directional
vectors.
2.1.3 Hexagonal Crystals
In the case of a hexagonal crystallographic system, theazimuthal relationship, considered in terms of the atomic
surface layer period, would be (111)Cu//(0001)hex, and acondition of partial matching can be obtained in cases
in which the (110) plane spacing in the matrix (0.256
nm for copper) and the length of the a and b axes of thecompound are close to the same.
The fitting conditions are represented by
ap.t. = {aCu /2 [110] (1 + δ )} · n(6)Analyzing for Precipitation Hardenability fromState of Precipitation
When the precipitate is minutely dispersed and has notreached the over-aging stage, precipitation hardening
most often consists of strengthening due to the elasticstrain caused by precipitation. The crystallographic
structure of the precipitate differs from that of thematrix, and when the interface between the precipitate
and matrix phases is coherent, the precipitate forms
a coherent dislocation loop and passes through theprecipitate without shearing, and in the case of a cubic
system that is matched with the precipitate, is thought tobe sheared due to dislocation. In either case it is thoughtthat the strengthening due to the elastic strain existing
between the matrix and precipitate phases is what
determines the degree of precipitation hardening. Using
the equations evaluated by Gerold and Haberkom 3), thisdegree of precipitation hardening may be represented as

due to precipitation, which may be found using, inter alia,
Linde’s Rule 4), we find that δ C ∝ δρ , and considering
the above-mentioned conditions we may represent thedegree of precipitation hardening by∆τ ∞ | ε |3/2 {δ C · ra (a/b)} 1/2/sin θ(8)where: a is the minimum length of the precipitatephase; b is the maximum length of the precipitatephase; and ra is the average precipitate diameter;
and where θ is the angle formed by the habit plane
and the slip plane; and the slip system having the
maximum value for sin θ is used.
The precipitation hardenability per unit of precipitate
can then be found by the partial differential equation∂∆τ /∂ C ∞ | ε | 3/2 · δ C –1/2 · {ra (a/b)} 1/2/sin θ(9)On the other hand it is thought that the change
in hardness δ H due to age hardening, as foundexperimentally, is proportional to ∆τ , and that the valueof δ H/δ C, obtained by dividing the change in hardness
δ H by the change in solute concentration δ C contributing
to precipitation, found from the change in the value forresistivity, is proportional to ∂∆τ /∂ C in Equation (9), sothat it becomes possible to compare the calculated
and experimental values for the degree of precipitation
hardening and the precipitation hardenability.

2.2∆τ ∞ | ε |3/2 ( f · r) 1/2(7)

where: ε is the degree of misfit; f is the volume
fraction of precipitate; and r is precipitate radius.
With reference to the relationship between the habit
plane and the slip plane, priority should be given to
adopting a slip system having a directional vector
such that the angle formed between the direction
perpendicular to the habit plane at which the precipitate
size factor is minimized and the slip plane is the smallest.
As for the precipitate volume fraction f, if we let δρ be
the change in resistivity during the aging process andδ C be the change in solute concentration in the matrix

3.

RESULTS AND DISCUSSION

3.1

Results of Compound Selection Based on Fitting
Calculations
As a database for compound selection we performed
fitting calculations with reference to compounds inthe cubic, tetragonal, orthorhombic and hexagonal
crystallographic systems contained in Peason’s
Handbook 5).
Table 1Results calculated for crystal lattice fitting using dataon chemical compounds in Peason's Handbook.

Cubic

Tetragonal

Orthorhombic

Hexagonal

101

16

107

12

As shown in Table 1, the compounds selectedcomprise 101 of the cubic crystallographic system, 16of the tetragonal, 107 of the orthorhombic and 12 of
the hexagonal, and it may be considered that when
compounds were present as a precipitate, there wasa habit plane between it and the matrix. It is worthy ofnote that this occurred more frequently in tetragonal andorthorhombic systems.

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Designing High-Strength Copper Alloys Based on the Crystallographic Structure of Precipitates

Table 2

Results calculated for habit plane selected from data
shown in Table 1. (shading shows data that match
fitting conditions)

Alloy

Precip.

CuNiTiCuTiSi

Crystal a axis b axis c axis
sys.
(nm) (nm) (nm)

Rot.
Pole

Fitting

Habit P.

Ni3Ti

Hex. 0.5019 0.5019 0.8299 <111>

(110)

(111)

CuTiSi0rtho. 0.6193 3.746 0.713 <100>

(100)

(100)

(110)

(110)

TiSi

0rtho. 0.6544 0.3638 0.499 <100>

CuCoSiCo2Si

0rtho. 0.408 0.373 0.7097 <100> (110, 100) (110, 100)

CuCoP

Co2P

0rtho. 0.565 0.3508 0.6604 <100>

CuNiSiNi2Si

0rtho. 0.5

CuNiMnP
CuNiAl

(130)

(130)

0.372 0.706 <100> (110, 100) (110, 100)

NiMnP 0rtho. 0.595 0.3557 0.6828 <100>

(130)

(130)

NiMn3P2 0rtho. 0.5949 0.3486 0.6865 <100>

(130)

(130)

(100)

(100)

Ni3Al

Cubic 0.357 0.357 0.357 <100>

Striped area's data matched to fitting condition.

Table 2 shows a single result selected from thecalculated results referred to above. Depending on
the compound there are cases in which a plurality ofhabit plane were selected by calculation, but since,as was described above, interface stability changes inaccordance with the atomic bonding condition at the
habit plane interface, interfaces with lower interfacial
energy were the ones selected 1).
Figure 2

3.2

Comparison Between Habit Plane According toCalculation and Experimental Habit Plane
In order to check the match between the habit planeobtained by calculation and the actual habit plane,comparisons are being performed with the aged phasesof actual alloys for the alloy systems shown in Table 2.
Figure 2 shows typical high-resolution observations
made using a JEOL JEM3010 transmission electron
microscope at 300 kV with reference to Cu-Ti-Si, Cu-NiSi, Cu-Ni-Mn-P alloy systems, in which orthorhombic
p r e c i p i t a t e s m i g h t b e e x p e c t e d t o o c c u r. T h eobservations were made after aging at temperaturesfrom 723 to 773 K for from 2 to 24 hr. Since the predicted
habit plane index was {0 k l}, we calculated the angles
formed between the predicted habit plane and the (010)
and (001) planes using an electron beam impingement
direction of [100]Cu, and those compared with thehigh-resolution images observed are also shown. The
calculated plane angle and the angle between the(010) plane and the habit plane in the high-resolution
images showed good agreement, demonstrating thatthe precipitation concentration was in accordance withthe predictive calculations. Good agreement with thepredicted results has been obtained for all of the cubic,
hexagonal and orthorhombic crystallographic systems 2).
The agreement of atomic alignment on the habit plane
is, as noted in Section 2.1, a necessary condition forstructuring the habit plane, and are thought to be usefulas conditions for selecting compounds.

High-resolution TEM images of orthorhombic system
precipitates in selected alloys after aging.

3.3

Evaluation of Precipitation Hardenability of
Selected Compounds using Calculation andExperiment
In evaluating precipitation hardenability by means ofEquation (9) in Section 2.2, we carried out an analysis
of precipitation hardenability δ H/δ C obtained from aging
experiments with reference to the contribution of thevarious parameters in Equation (9), among which we may
mention: a) the change in solute concentration: δ C; b)
the boundary misfit between matrix and precipitate: ε ; c)the precipitate morphology factor: a/b; and d) the angle
between the habit plane and the slip plane in the systemthat moves most easily: θ .
a) Change in solute concentration: δ Cb) Boundary misfit between matrix and precipitate: εc) Precipitate morphology factor: a/bd) Angle between slip plane and habit plane: θLet it be noted that precipitate size is not considered
because, as shown in Table 3, the spherical equivalentdiameter of precipitates in the condition where maximum
hardness had been reached were virtually identical,
irrespective of the alloy system.

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Designing High-Strength Copper Alloys Based on the Crystallographic Structure of Precipitates

Experimental results for selected specimens from
Table 2 after aging.

Crystal Experimental

Alloy

Aging

Hardness

Precipitate

Solutesystem habit plane condition change δ H change δ Cdiameter/mm

Morphology
factor a/b

CuNiTi

Hex.

(111)

723K-5h96.6

2.3806

6.2

0.075

CuTiSiOrtho.

(100)

773K-5h54.6

1.5064

7.2

0.013

CuCoSiOrtho.

(110)

748K-5h149

2.4142

5.8

0.273

CuCoP

Ortho.

(130)723K-2h

89.3

0.387

5.2

0.875

CuNiSiOrtho.

(110)

773K-2h1302.4055

6.2

0.288

CuNiMnP Ortho.

(130)

723K-2h850.9227

5.6

0.72

CuNiAl

(100)723K-5h23.1

0.58215.8

0.5

Hardening factor

Hardening factor

Cubic

Figure 5

1/sin θEffect of various parameters in Equation (9) on the
precipitation hardening factor.

Morphology factor a/b

Figure 3 shows the relationship between precipitation
hardenability δ H/δ C and the various parameters. The
highest correlation was between experimental δ H/δ C andprecipitate morphology factor a/b, and it can also be seenthat a/b is large, i.e., the closer it is to granular the greater
the precipitation hardenability.

baMorphology factor a/b

Atomic packaged density, ρ dInterface bonding ratio, Zi

Figure 4

Effect of atomic arrangement parameters on
precipitate morphology factor.

Relationship between experimental and calculated
values for precipitation hardening factor.
Hardness change, H(Hv)

Hardening factor
Boundary misfit, εFigure 3

dH/dC∞d ∆τ /dC

Calculated hardening factor, d ∆τ /dC
Precipitates morphology fact or a/b

Solution conc entration change, δ C(%)
Hardening factor

As can be seen from Figure 4, the precipitate
morphology factor has a correlation with the atomic
density of the matrix in the precipitation plane--i.e., thelower the atomic density in the precipitation plane thegreater the morphology factor--indicating a tendency
to granularity. If this relationship is plotted against
Z i which is the bonding number at the interface withrespect to atomic bonding number of matrix (fcc is 12), a
substantially linear relationship is obtained.

Hardening factor

Table 3H∞ ∆τCalculated hardening content, ∆τFigure 6

Relationship between experimental and calculated
values for parameters of precipitation hardness
change.

Figure 5 shows the results of a comparison between
experimental values for precipitation hardenability andcalculated values for the parameters, and Figure 6 shows
the results of a comparison between experimental values
for the degree of precipitation hardening and calculated
values for a parameter that is proportional to the degreeof precipitation hardening. For both precipitation
hardenability and degree of precipitation hardening, agood correlation can be observed between the calculated
and experimental values, suggesting the validity ofthe prediction of degree of precipitation hardening
using Equation (8) and the evaluation of precipitation
hardenability using Equation (9).
In the case of the Ni3Ti phase, which is a hexagonal
system, there is a slight deviation from the distributionof calculated and experimental values shown in Figures
5 and 6 for each of the precipitates. As a cause for
this, it may be that, unlike in the precipitate phases ofan orthorhombic system, there may be fewer bonding
defects on the atomic alignment on the habit line, sothat different precipitation hardening mechanisms, suchas ordered strengthening may be at work, and it ispossible that comparisons with reference to precipitation
hardenability must be carried out within hexagonal
crystallographic systems.

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Designing High-Strength Copper Alloys Based on the Crystallographic Structure of Precipitates∂∆τ /∂ C ∞ | ε | 3/2 · δ C –1/2 · Zi 1/2/sin θ

σ e = µ a (1 + φ – (1 + φ 2) 1/2) /2 π 2

3.4

Factors Establishing the Habit Plane ofPrecipitate and Matrix
As we may see from the calculated results shown in Table
2, cases may exist in which a plurality of habit planes
may be inferred for a single precipitate. This indicatesthat it is not possible to fully characterize the habit planes
merely with reference to atomic alignment alone. Here, asa method for characterizing habit planes from calculated
results, let us proceed with reference to interfacial energy.
Interfacial energy understood in the broad sense may be
represented as

This shows that if the misfit is large interfacial energy
increases, making for instability. Figure 3, however,showed no correlation between the degree of misfit andthe morphology parameter, so that it can be inferredthat its influence is less than that of chemical interfacial
energy, and this has in part been experimentally verified 2).
Crystallographic database on chemical compounds

Temperature dependence of solid solution limit

Cartesian coordinate system

Hexagonal system

Plane spacing calculation; fitting calculation(11)

Secondary phase?

αPrediction of habit planewhere: σ c is chemical energy and σ e is elastic strainenergy, and σ c may be represented in accordancewith Servi and Turnbull 7) by

σ c = EONSZS (C – Ca)2/NOZL(13)

where: φ is 2 πε / (1 – ν ); a is the lattice constant;
ε is misfit strain; ν is Poisson's ratio; and µ is theshear modulus.

(10)

The relationship in Equation (10) shows that if forthe habit plane, θ , Zi and ε are determined based on
the calculated results about habit plane, precipitation
hardenability ∂∆τ /∂ C can be found by calculation basedon the amount of alloy elements.

σ = σc + σeplane Zi, may be considered to have been brought about
by the magnitude of chemical interfacial energy.
Then again the elastic strain component of interfacial
energy produced by misfit may be represented as 8)

Temperature

To summarize the foregoing discussion, when it isconsidered that as an equation for predicting precipitation
hardening parameter from crystallographic data, theprecipitate morphology parameter a/b at the habit line andthe atomic bonding ratio Zi are in a substantially linear
relationship, we may derive

α

XaYb

+ XaYbComposition

Determination of precipitation hardness parameter(12)

where: N s is the number of atoms per unit ofinterface area; Z L is the number contributing tobonding (12 for fcc); ZS is the number of bonds
per atom passing through the interface, which issmaller for planes with a higher atomic density ofthe matrix; E0 is a value equivalent to the amountof increase of energy due to the average atomic
bonds of different kinds; N 0 is the Avogadro
number; C is the solute concentration in the
precipitate; and Ca is the solute concentration in thematrix.
Provided different kind of atom bonding energy is
not negative, the larger the number of atomic bonding
defects at the interface the larger E0 will be and the higher
the interfacial energy itself, so that habit plane instability
will result. That is to say from the standpoint of chemicalenergy, the lower the atomic density and the greater
the number of bonding defects at the habit plane, themore unstable the habit plane will be. Figure 4 showed
the relationship of the precipitate morphology factor a/b
and atomic plane density with atomic bonding ratio at
the habit plane Zi, but when it is considered that Zs/ZL in
Equation (12) is equivalent to the atomic bonding ratioat the habit plane, the result of a greater tendency togranularity the lower the atomic density of a plane, thatis to say the higher the atomic bonding ratio at the habit

Selection of alloy system

Figure 7

Flowchart showing design procedure for
precipitation hardening copper alloys.

3.5

Methodology of Alloy Design Based on
Crystallographic Structure of Precipitates
Figure 7 shows a flowchart summarizing the procedure
for the design of copper alloys based on atomicalignment and solid mechanics. These are:1) To extract from a database data and lattice constants
pertaining to chemical compounds belonging the
cubic, tetragonal, orthorhombic and hexagonal
crystallographic systems, while excluding thosecompounds formed from elements for which there
is no solid solution limit or solid-solution limit
temperature dependence with reference to copper inthe solid-state temperature range.
2) For each crystallographic system and based on
the principles described in Section 2.1, to performfitting calculations, select compounds that satisfy theconditions, and predict the habit plane, and when,in the above calculations, a plurality of habit planes
are identified for the same compound, to narrow thefocus based on the number of bonding defects as
judged from the habit plane atomic density (atomic
bonding ratio) at the habit plane.
3) To measure experimentally and estimate using

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Designing High-Strength Copper Alloys Based on the Crystallographic Structure of Precipitatesa calculation phase diagram the degree ofprecipitation δ C of the selected compound attemperatures around 723~773K, and to estimate,using Equation (10) the precipitation hardening
parameter ∂∆τ /∂ C.

∂∆τ /∂ C ∞ | ε | 3/2 · δ C -1/2 · Zi 1/2/sin δ(14)where: ε is the degree of lattice misfit; δ C is thechange in solute concentration in the matrix due
to precipitation; θ is the angle formed by the habit
plane and slip plane; and Zi is the atomic bonding
ratio4) Since the relationship between an alloy content andthe degree of precipitation strengthening can be
estimated using precipitation hardening parameter∂∆τ /∂ C, the focus on alloy systems can be further
narrowed, taking account of the amount of admixture
in the alloy.
5) Finally, at the point at which the alloy systemand the amount of alloy is designed, to verify bymeans of prototype production, and to design the
manufacturing process.
In accordance with the alloy design process describedabove, it is necessary to know the temperaturedependence of the solid-solution limit of the precipitatephase, and if a calculation phase diagram program 9) canbe put into practice, the time required to narrow the focus
to specific alloy systems will be significantly shortened.
An approach to the structuring of a phase diagram bymeans of first principle calculations has also been made,and it is necessary to press forward with the creation ofa more complete compound database, as well as theapplication of phase diagrams.

4.

CONCLUSIONS

1) In crystallographic systems using Cartesian
coordinates--cubic, tetragonal and orthorhombic,
as well as in hexagonal systems, we selectedalloy systems using Peason's Handbook on
Crystallographic Data on chemical compounds, as aresult of which we identified 101 cubic alloy systems,
16 tetragonal systems, 107 orthorhombic systems
and 12 hexagonal systems.
2) We produced prototypes of some of the alloys
selected, and for alloys that exhibited the property
of precipitation hardening, we compared theresults of orientation prediction and the results ofobservation by transmission electron microscope,and satisfactory agreement was obtained.
3) As a result of investigations of the relationshipbetween precipitation hardenability ∂∆τ /∂ C andprecipitate parameters, it was found that, morethan the degree of lattice misfit or the angle theprecipitate formed to the slip plane, influence wasexerted by precipitate morphology, determined by

the atomic plane density plane of the habit plane(atomic bonding ratio), and it was found that this
could be expressed by∂∆τ /∂ C ∞ | ε | 3/2 · δ C -1/2 · Zi 1/2/sin δ
where: ε is the degree of lattice misfit; δ C is thechange in solute concentration in the matrix due
to precipitation; θ is the angle formed by the habit
plane and slip plane; and Zi is the atomic bonding
ratio4) From crystallographic data on the chemicalcompound forming the precipitate and data on
the solid solution limit change for precipitate
constituents, it becomes possible to narrow
the focus to alloy systems capable of reachinghigh strength, and to predict the precipitation
hardenability. By proceeding with calculations inaccordance with the procedures presented in thispaper and by some of the experiments, it is believed
possible to design new high-strength alloys with
far fewer process steps than with the conventional
method.

REFERENCES
1) Japan Copper and Brass Association: Basis and Engineering
Technology of Copper and Copper Alloys (Revised edition), 236. (inJapanese)
2) Fujiwara et al.: Transactions of Copper and Brass Workshop, 37,
106 (1998). (in Japanese)
3) V.Gerold and H.Haberkom: Phy. Status Solidi, 16, 675 (1966).
4) J.O.Linde: Ann. d. Phy., 15, 219 (1932).
5) P.Villars and L.D.Calvert: Peason's Handbook of Crystallographic
Data, ASM, 1~46) P.Guyot: Phil. Mag., 24, 987 (1971).
7) I.S.Servi and D.Turnbull: Acta Met., 14, 161 (1966).
8) J.H.van der Merwe: J. Appl. Phys., 34, 117 (1963).
9) The Japan Institute of Metals: Basis and Application of Phase
Diagram in Material Development and Design, 93 (1994). (inJapanese)

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