- Email: [email protected]
The influence of temperature on slip and twinning in uranium
THE
INFLUENCE
OF TEMPERATURE J.
S. DANIEL,t$
B.
ON SLIP AND LESAGEt
and
TWINNING
IN URANIUM*
P. LACOMBET
The deformation modes of a-uranium have been studied by performing mechanical tests on single crystals prepared by both the phase shift and the strain-anneal methods. The critical resolved shear stresses for slip on the systems (OlO)[lOO], (OOl)[lOO], (llO)[liO] and (021)[1i2] have been determined over a wide range of temperatures. The formation of twins on the systems “{197)” and {130) is facilitated by prior slip and the results indicat,e a pole mechanism involving 4 [Ii01 slip dislocat,ions for the lateral growth of (130) twins. Lastly experiments on polyerystalline geranium support the hypot,hesis that the thermally activated deformation mechanism responsible for the temperature dependence of the flow stress is the overcoming of the resistance of the crystal lattice to the passage of disloeat.ions. INFLUENCE
DE LA
TEMPERATURE DANS
SUR LE GLISSEMENT L’URANIUM
ET
LE
MliCLAGE
Les modes de deformation de l’uranium a ont 6te Btudi&s au tours d’essais mecaniques effectuds sur des monocristaux p&pa&s par les m&hodes d%crouissage critique suivi de recuit et de changement dephase. Lesoissions critiques pour le glissement SUP les syst&mcs (OlO)[lOO], (OOl)[lOO], (IlO)(liO] et (OZl)[li2] ont BtB d&ermin&es pour un large domaine de temp6ratures. La formation des m&es sur les systhmes “{197j” et {130} est facilitke par un glissement ant&&r et les r(?sultats mettent en &idence un mbanisme polaire impliquant des dislocations de glissement Y$[liO] pour la croissanoe lat&ale des macles (130). Enfin, les expiriences effectuBes sur de l’uranium polycristallin eonfirment l’hypothhse suivant laquelle la d&formation activee thermiquement responsable de la variation de IR contrainte plastique avec la temperature peut se produire quand la rbsistance du rbseau cristallin au passage des dislocations est surmont6e. DER
EINFLUB
DER
TEMPERATUR
AUF DIE GLEITUNG IN URAN
UND
ZWILLINGSBILDUNG
Die Verformungsmoden von a-Uran wurden untersucht; mit zwei verschiedenen Verfahren hergestellt,e Einkristallo wurden verformt, Die kritischen S~hubspannungen in den Systemen (OlO)[lOO], (OOl)[lOO], (llO)[liO~ und (OZl)[ff2] wurden in einem grol3cn Temperaturbereich bestimmt. Die Bildung von Zwillingen in den Systemen “{197)” und {130) wird durch vorangegangene Gloitungedeichtert. Die Ergebnisse deuten auf einen Polmechanismus hin, bei dem seitliches Wachstum der (130)Experimente an polykristallinom Uran Zwillinge dumb Gleitung von 4 [liO]-Versetzuqgen erfolgt. unterstiitzen die Hypothese, da0 in Uran die Uberwindung des Gitterreibungswiderstandes der die Temperaturabhilngigkeit der Flienspannung bestimmende thormisch-aktivierte Verformungsmechanismus ist. INTRODUCTION
In view of the advances in
our
the of
unde~s~and~~~
common the
of
metallic
crystallography
the
crystal of
made
in
recent
defo~ation structures deformation
years
modes
of
the
study
of
metals
In the case of mercury for example, which has a rhombohedral crystal strncture, Cracker et .Z.(f*2) have shown that both the principal slip direction and the active t~~inning mode violate dell-established criteria for the deformation characteristics of metals. The refinement of existing theory made necessary by the discovery of such anomalies should result in a more general theory of deformation mechanisms. a-uranium, the allotrope of uranium stable at room temperature, (Fig. l), has the orthorhombic crystal structure illust,rated in Kg. 2. Outside the actinide series, gallium is the only pure metal to have an orthorhombic structure, although it should be noted that in the case of gallium the lattice parameters a and h are nearly equal, making the structure almost .* Iteceived May 10, 1970. of lower
symmetry
is of increasing,
interest.
t Laboratoire de MBtallurgie, Faculty des Sciences, $I-Orsay, France. $ Now at.: Ecole Polytechniq~Ie, Mont&al, Canada.
ACTA
METALLURGICA,
VOL.
19, FEBRUARY
1971
tetragonal, a fact which inffuences the twinning behavior of this metal.@) The o~horhombic structure is much more common among polymers and indeed the arrangement of the carbon atoms within the unit cell of polythene closely resembles that of the metal atoms in a-uranium, the principal twinning mode being the same in both cases.c4) The deformation modes of cr-uranium were first investigated nearly twenty years ago by Cahn,@) who studied the slip lines, twins and kinks visible in coarse-grained polyerystalline uranium after deformation or thermal cycling and identified the principal slip and twinning modes. This study revealed the complexity of deformation in M-uranium. Four twinning modes were found, one of which was of type II (irrational composition plane). Cahn also drew attention to the difficulty of preparing good single crystals of uranium of useful size, a di~culty which has hampered all subsequent research and which explains why the experimental work on the deformation of single crystals of uranium has remained largely qualitative. However, Fisher(Q was able to prepare a limited number of good crystals using a grain growth method and this made possible the accurate determination of the elastic constants and
163
ACTA
164
METALLURGICA,
VOL.
19,
1971
orientation are necessary. The phase-shift method remains the only practical means of preparing large numbers of monocrystalline tensile specimens and this method has several disadvantages. Firstly, crystals produced in this way have a rather imperfect, striated structure; secondly, the range of orientations obtained is usually narrow, and lastly the phase transformation often imparts a slight curvature to the tensile specimens, which is obviously inconvenient for the accurate measurement of yield stresses. These difficulties were largely overcome by performing compression tests since such tests can be performed on very small crystals and other methods of crystal preparation, notably the critical strains anneal method, are then available. Crystal preparation
1
0
IO
I
I 30
20
Pressure,
FIG.
1.
The
I 40
I 50
1 60
k bar
temperature/pressure phase uranium.‘35*3B)
diagram
of
expansion coefllcients of cc-uranium over a wide range of temperature. These data provided raw material for several theoretical investigations. Sax1 ef aZ.(7*8)and Yoo(9) attempted to predict the relative importance of the different possible slip modes in a-uranium on the basis of anisotropic elasticity calculations whilst Crockero”) and Leteurtre(ll*ls) have analyzed the possible twinning modes. It being impossible to test many of the predictions arising from this theoretical work on the basis of the existing experimental data, we thought it worthwhile to carry out a systematic investigation of the influence of temperature on the relative importance of the various deformation modes of uranium. The earlier part of the work, in which the slip directions associated with each slip plane were verified, has been described elsewhere(13z14)and the present artiale discusses the variation of the critical resolved shear stress with ~mperature for the different slip systems and reports the results obtained from an investigation of twinning.
Experimental
Whereas the tensile specimens were first stamped out in poleyrystalline sheet and then converted into single crystals, the compression specimens were obtained by spark-cutting from monocrystalline material which was prepared as follows : Strips measuring 50 x 3 x 1.3 mm were spark-cut from 1.3 mm thick cold-rolled sheets of electrolytic uranium (<22 ppm of impurities) and the edges of these strips, which became the eompressed faces in the final specimens, were ground parallel on 600 emery paper. These strips were then converted into single crystals using either the critical strain/ anneal or the phase-shift methods.
t
__I I
‘\
--_
_I
’
,---
---
_-
‘.
_--
--
; ‘\
‘\
--,
I
I
__---_
‘\ I
b
‘\ \ ’
-_
[1001
(010)
~~~
methods
In order to be able to study lattice rotation during deformatiot~ and hence determine slip dire&ions unambiguously tensile testing ‘was employed in the first stage of the investigation.03) However, in order to determine the critical resolved shear stresses for the different deformation modes over a wide range of temperature, large numbers of crystals of various
__ _ __ _ liic
ml
‘
0
___ --__ 4 Dl
7 4 L
’ _____-__ ~:
Y
biOJ
(001)
FOG. 2. The crystal structure of a-uranium and its projection on the principal planes.
DANIEL
et al.:
SLIP
AND
TWIKNIPL’G
IN URANIUM
165
8 FIG. 3. The orientations of the sin&
crvstols studied. A-tensile sDecimens. B--compression specimens. x crystal prepared by the ph&e shift method. 0 crystal prepared by the strain/kneal method.
Critical strain/anneal method In this method the strips were first electropolished and annealed ire,face (< 10e6 torr) at 64O*C for 15 hr in order to obtain a fine and homogeneous grain size. The strips were then critically strained in tension to 0.9 per cent plastic deformation at room temperature before a final recrystallization anneal for 3 hr at 655°C. Best results were obtained if the strained specimens were heated to temperature as rapidly as possible during this final anneal. After recrystallization, electropolishing and etching usually revealed a “bamboo” structure in which grains up to 10 mm length occupied the whole cross section of the strip. Compression specimens of length 2.5-3 mm and approximately square cross section were t)hen obtained by spark-cutting perpendicular to the axis of the strip within these large grains. Some 40 crystals of a wide range of orientations were prepared in this way (Fig. 3). Phase-shift method Although the strain-anneal method yields relatively perfect crystals, the small size of the grains in the bamboo structure limits to two or three the number of compression specimens of identical orientation which can be obtained from a given grain. Since it was often useful, particularly in the study of twinning, to have a large number of specimens of identical orientation, use was also made of the phase shift method. In this case polished strips were sealed in vacua in silica tubes and passed through a furnace having a steep temperature gradient at the fi -+ g transformation temperature (662’C). This treatment converted the greater part of each strip into a single crystal and compression specimens were spark-cut from these as described above.
In all cases the final compression specimens were electropolished to remove all traces of spark-cutting and the orientation of each was determined by the back-reflection Laue method. The sharpness of the spots on the Laue diagram also served as an indication of crystal perfection and crystals prepared by the strain-anneal method gave sharp spots whereas phase-shift crystals usually gave the rather fragmented spots typical of a ~lygonized structure. The length of each specimen was measured and its cross section determined by weighing. Mechanical testing Tests were carried out on an Instron testing machine. The accessories used for low-temperature testing and for high-temperature testing in vacuum of 3 x 10m6 torr have been described elsewhere.‘15*la’ Compression tests were carried out between alumina plates let into R “Nicral” compression jig designed so as to exert a tensile force on the load cell. (A stainless steel compression jig, also with alumina plates, was used for the low temperature work.) Strain rates of 2 x 10T2 per min and a chart speed of 2 cm/min were employed and the load corresponding to the limit of proportionality, as observed on the chart, was used in calculating the critical resolved shear stress (c.r.s.s.) once the operative deformation mode had been identified by the two-surfaces method. ExpeGmental errors The experimental errors involved in this work are likely to be rather greater than is usual in the study of single crystals for three main reasons: 1. As Altshuler and Christian have pointed out, small angular errors in the preparation of compression specimens can give rise to considerable errors in the measured yield stress.
RCTA
166
ioI Q 8
METALLURGICA,
. c
S-
P
6-
4-
I
I
I
200
400
600
800
r;T FIG. 4. The variation with temperature of the shear modulus p& for various combinations of planes and directions in cr-uranium. (1 Pa = 10 dynfcm2.)
2. The fragmented Laue spots characteristic of phase-shift crystals can introduce errors of 2-5’ in orientation determination. 3. All angles were calculated using a stereographic projection based on the values of the lattice parameters at 20°C. The anisotropy of thermal expansion of uranium is such that the position of certain poles on the projection [e.g. (OZl)] would be different by several degrees on a stereographic projection of the actual structure at 6OO’C. Taken together, these factors could result in errors of up to 30 per cent in the measured values of critical resolved shear stresses. Fortunately the differences between the 0.r.s.s. values for the different deformation modes proved to be much greater than this. RESULTS
19,
1971
direction in the a-uranium structure. However, whereas below about 500°C slip in this direction occurs most readily on the (010) plane, at higher temperatures slip is easiest on the (001) plane. Slip on other planes of the [loo] zone, particularly (021) and to a lesser extent (Oil), is also observed at high temperature, always though in a secondary role. (b) Slip on the (110) plane has been observed at temperatures above 15O’C. Studies of lattice rotation and Laue asterism indicated that the slip direction was [lrO] in all cases and above 400% segments of (111) slip are associated with the (110) slip lines. CalaiW and Teeg and Ogilvie(iv) reported slip on the system (llO)[OOl] at room temperature and below. This sysbem was not observed in the present study since all suitably oriented specimens deformed by twinning, kinking, or slip on other systems. (c) Slip can occur on the system (021)[li2] at temperatures above 400% although the stress neeessary to operate this system is several times greater than that required for the system (021)[100].
/
s F
VOL.
AND DISCUSSION: u-URANIUM
SLIP
IN
Sliz, modes
Knowledge of the slip direction associated with each slip plane is a s&e qua mu for the determination of critical resolved shear stresses. The results of this investigation and of previous studies of u-uranium may be summarised as follows: (a) At all temperatures slip occurs most easily in the [IOO] direction, which is the closest packed
Critical resolved shear stresses Once the elements of a slip system have been identified, the crit.ical resolved shear stress TV can be calculat,ed using the Schmid formula. 7, = 0 cos 0 cos a where (T is the applied stress, 0 the angle between the deformation axis and the pole of the slip plane and A the angle bet-rveen the deformation axis and the slip direction. The variation of r with temperature for the slip systems (OlO)[lOO], (OOl)[lOO], (llO)[liO] and (021) [I121 between 20 and 660°U1*) has been studied in the present investigation and Butra(20) has published data showing the variation of ~~ for the system (010) [loo] below 20°C. In comparing slip on different systems and in different metals it is usual to divide the critical resolved shear stress T, by the shear modulus p@ in order to eliminate variations arising from the temperature dependence of interactions between dislocations. In view of the anisotropy of uranium, the shear modulus was calculated as a function of temperature for 14 actual and hypothetical slip systems in cc-uranium using the elastic constant data of Fisher.r2i) The results are shown in Fig. 4. Figure 5 shows the variations of rJpa for the principal slip systems of or-uranium as a function of temperature. This diagram presents two unusual features, the intersections between the curves for different systems and the slight positive slope of the curve for the system (OlO)[lOO] above 300°K. It is int,eresting
DANIEL
et al.:
SLIP
AND
i \(ozu [li2] ‘f
b ;;
.P c”
7; “K
FIG. 5. The varirttion with temperature of T~/,u* (c.r.s.s./ sheer modulus) for the slip modes studied.
to consider to what extent these features are explicable in terms of our present knowledge of t-he ~-uranium structure. Discussion It is well known that the u-uranium structure contains a significant degree of covalent bonding between the atoms in the cor~ga~d (010) planes (which Cahnc5) refers to as giant molecules) and FriedeS2a) has suggested that the changes observed in the variation of certain physical properties of uranium (e.g. electrical resistivity) with increasing temperature are due to the weaking of these covalent bonds as the bonding electrons are excited into the conduction band. The main features of the variation of T~/,I_+, with temperature for different slip systems fit in with this theory. Only on the system (010) (1001 can slip occur without breaking covalent bonds, so that at low temperatures slip will take place most readily on this system. The fact that the value of TJ,u~ for the (OlO)[lOO] system appears to increase slightly with temperature may be related to the contraction of the structure along the b axis which brings the (010) planes closer together as the temperature increases. Slip on the (001) and (021) planes involves rupture of the shortest bond in the structure (d, in Fig. 2) and slip on (110) must break the d, bond. Electrical resistivity measurements indicate that the covalent character of the bonds in a-uranium begins to decrease above about 500°K,(zj) and it can be seen that the critical resolved shear stresses for slip on these planes have high values below this temperature. Rowever, simple considerations of bond lengths, packing densities, interplanar spacings and Burgers vectors
TWINNING
IN
167
URANIUM
are not sufficient to explain the rather complex variations of T,.,+, for the different systems above 500°K. Although it is not yet possible to calculate the stress necessary to move different dislocations in uranium, some progress has been made by applying anisotropic elasticity theory to the problem of slip in uranium. Sax1 and Otruba(‘) began their theoretical study of the mobilities of dislocations in u-uranium by calculating the values of the “ease of Xiding parameter” (EGP) f or d iff erent disIocations in both screw and edge orientations. This parameter, which is simply an approximation for the width of a dislocation divided by its Burgers vector, was proposed by Eshelby(z*) as a measure of dislocation mobility. In a later article Saxl@) also computed the energies of various dislocations in cr-uranium. More detailed comparisons between these theoretical result,s and the experimental data have been given elsewheret7*13*14)and are encouraging. The theory predicts that (~l)[l~] will be the easiest slip system at high temperatures and suggests that, in general, screw dislocations in uranium will be less mobile than edge dislocations, This is borne out by the observation that waviness is a common feature of slip in above about 200°C. TWINNING
IN
a-URANIUM
Introduction The general theory of the crystallography of deformation twinning in lattices has advanced considerably in recent years, particularly as a result of the work of Bilby and Crocker(26) and Bevis and Crocker.(27~28)The elements K,, I”r,, Q, Q, normally used to describe a twin and the twinning shear s, are shown in Fig. 6. Although many different twinning modes of varying complexity are theoretically possible in cubic lattices,(2s) the twins actually observed in the f-cc. and b.c.c. lattices are rather simple. In each case the observed twinning mode involves the smallest possible shear strain which is consistent with the absence of atomic shuffling. For the f.c.c. lattice this means t,he compound twinning mode (lllj, {ill), (%l>, (211), and in the b.c.c. lattice the mode (2111, (211}, (ill), (ill). Although the twinning shear is the same in both cases the b.c.c. mode is not observed in the f.c.c. lattice, where it would necessitate atomic shuf%ng, and vice versa. This suggests that for a fixed value of the twiIlning shear, the mode which necessitates the least shufling will be the easiest to form. Furthermore it was until recently supposed that in the absence of shuffles, or in a hypothetical case where two different twinning
16X
ACTA
METALLURGICA,
VOL.
19,
19’71
in the preparation of good single crystals of c(-uranium most of the recent work on twinning in this metal, which it will be convenient to summarize briefly, has been theoretical rather than experimental. Crockeros) studied the complexity of atomic shuffling for a whole series of theoretically possible twinning modes. He relates the complexity of the shufies to the number of lattice planes (q) parallel to the plane K1 which intersect a unit vector in the yls direction (for type I twins: the plane K2 and the direction vi are used for type II twins) and shows that there are 41 possible twinning modes for which q = 1, 2 or 4 and which have a shear strain less than unity. The first five of these theoretiea~y possible modes (in order of increasing shear) are given in Table 1. Experimental observation has clearly established that some of these twins are in fact operative. These are: No. 1 reciprocal (i.e. K, = “(197}“), No. 2, FTU. 6. The elements of a twin. both twins (K, = (112) and K, = “(172)“) and modes involved identical shuffles, the twinning mode No. 5 reciprocal (K, = (130)). A twin having the requiring the smaller shear strain would be the easier elements: (11211“(141)” “(321)” (311) was reported mode to operate. However, this assumption has by Cahn.(b) This twin has a fairly low shear (s = 0.33) been shaken by the work of Guyoncourt and Crocker(sO) but does not appear in Cracker’s list because of the on twinning in mercury. Although twinning in high value of q (q = 6). The discovery of the “(197)” mercury does not involve shuffling the observed twin (alsoknown as the “(176)” twin) by Lloyd and twinning mode is not the mode with the lowest Chiswik’ss) led to the suggestion that the (121) twin shear, which would be: was really a “(1973” twin, and Crockcr has added that in view of the size of the shear and the complex (011){100}(100)(011) 8 = 0.457 shuffles involved, the formation of a {121f twin is unlikely on theoretical grounds. but a type II mode with a considerably larger shear: Although all authors are agreed that the (130) twin is the most commonly observed mode in a-uranium, “{i%)“(ill)(i21)“{Oil)” .s = 0.633 it is very difficult to class the various twinning These authors thus conclude that, in predicting modes in order of ease of formation on the basis of which twin mode will be operative in a given crystal, previous work since there have been virtually no it is necessary to consider not only the size of the measurements made of the stresses necessary to shear and the complexity of the shuttles, but also induce the various twins. Only limited information whether there exists a convenient dislocation mechcan be obtained by comparing the frequency with anism which would enable the twin to form. which different twins are observed, since the apparA special interest attaches to the study of twinning ition of a given twin will depend not only on it’s own in u-uranium since, at least as far as metallic crystals ease of format,ion but also on t,he ease with which are concerned, it is in this metal that it should be other deformation modes can operate under the easiest to assess the relative importance of the three same conditions. This in turn will depend, for a given factors, shear strain, shuffles and dislocation mechaorientation of the crystal with respect to the applied nisms, in determining the ease of operation of twinstress, on the Schmid factor which relates the resolved ning modes. This is because, since a-uranium has TABLE 1 ~ a double-lattice orthorhombic structure, a variety of twins are observed including a compound twin and twins of types I and II. These twins involve shear strains of different magnitudes and all require varying degrees of atomic shuffling for their formation. Partly, no doubt, because of the difficulties involved
DANIEL
et al.:
SLIP
AND
shear stress on a given deformation mode to the applied stress. Diagrams showing the variation of the Schmid factor with orientation for all deformation modes of a-uranium have been given by Calais(l*) and Daniel et aZ.(lQ and these diagrams highlight the anisotropy of deformation in this metal. To cite one example, the twin”{197)” very rarely forms during tensile tests since in tension the the orientation which produces a high resolved shear stress on this twinning systems also maximizes the shear stress on the principal slip system (010) [loo]. Since twinning is an asymmetric process the orientation which will favor twins of the “(197)” family when a compressive stress is applied will be different. This new orientation corresponds to a low value of the Schmid factor for the principal slip system so that, in compression tests, these twins are common. As well as comparing the stresses necessary to form twins of the various modes at a fixed temperature it would be particularly interesting, in the case of uranium, to know how the different twinning stresses vary with temperature. Leteurtre(12) has proposed dislocation mechanisms for the nucleation and propagation of the various twins observed. Since these mechanisms involve slip dislocations and the previous study of slip has revealed that the mobility of these dislocations varies considerably with temperature in some cases, a study of twinning as a function of temperature should help to assess the validity of these mechanisms. Two twin modes were investigated, the type II mode “(197)” and the compound mode (130). (a) The “(197)”
twin
As noted previously, twins of the “(197)” family form readily for certain crystal orientations during compression tests. These “(197)” twins are usually considerably thicker than the other twins in uranium, probably because, for temperatures below about 2OO”C, the value of the homogeneous twinning shear, s, is smallest for this tuin (Table 1). Compression tests were performed on a series of monocrystals and the stress corresponding to the first “burst” of “{197}” twins was measured. This first burst usually produced twins on both of the most highly stressed “{197)” modes i.e. “(i97)” and “( 197)“, and values of the shear stress 7 were calculated for the “(i97)” mode. The results are shown in Fig 7. At all temperatures studied the minimum value of the shear stress necessary to cause “(197)” twinning falls between 6 and 6 Pa x 10’. However, some crystals required a much higher
TWINNING
IN
URANIUM
100
I50
Temperature,
169
/ 2b*
250
i
3c
‘K
PIG. 7. The shear stress 7 on the “(197)” twin mode corresponding to the first burst of “1197)” twins in crystals of various orientations at different temperatures. (1 Pa x 10’ = 1.0197 kg/mm*)_
shear to initiate twinning and examination revealed that these crystals had orientations such that the Schmid factor (Cos 19Cos 1) for the principal slip system (OlO)[lOO] had a very low value (<0.05). This suggests that the formation of “(197>” twins may be related to the prior movement of slip dislocations. On this hypothesis “(197>” twins will form at a shear stress of 5-6 Pa x 10’ only if some movement of slip dislocations has already occurred. Otherwise the applied stress must be increased until it induces such microscopic slip and twinning will then ensue. (b) The (130) twin. A more detailed study of the variation of twinning stress with temperature was made for the (130) twin. A series of crystals having the same o~entation were deformed in compression at various temperatures in the alpha range. Some typica. stress-strain curves are shown in Fig. 8. At temperatures below about 2OO’C there was a series of isolated twin drops during the elastic part of the stress-strain curve and aubsequent observation revealed that a large number of fine twins had been created. At higher temperatures, e-g_ the stress-strain curves at 404 and 500°C there was a continuous succession of twin drops and in these cases the crystal was found to contain a few very wide twins. Since the number of twins was clearly smaller than the number of twin drops many of the drops must correspond to the widening of existing twins rather than the creation of new ones. Finally, at 6OO”C, the stressstrain curve is characteristic of deformation by
ACTA
METALLURGICA,
VOL.
19,
1971
latter case a large number whereas at
deformation
room
temperature
characteristic 0 Small
twin
drop
of fine twins were formed
of
the
prestrained
produced
the
of high temperature
gradually X 0
lateral
growth
twins
deformation.
in the case of the high temperature mentioned,
specimen
thick
As
tests previously
of the twins
took
place
during the test.
Discussion
hv
The experimental
%3
of
(130)
occurs
twins
results show that lateral growth
is favoured
at high
has been predeformed ature.
20°C
300 *c
404oc
/
I
I
500 oc
I I
0
I
1
1
2
594oc
the
locations.
I
4%
growth
deformation
slip,
the
although
deformed
specimen
contained
The a
slip lines. The
lower,
stress data are collected
continuous
stress corresponding curve,
which
curve
the
shear
to the first twin drop on each
may
be considered
shear stress necessary
as the
minimum
to initiate longitudinal
of a twin nucleus. The upper, dashed, curve indicates at which a continuous This is probably
in Fig. 9.
represents
crystal
the
mechanism
involves
slip will
dislocations, by
4 [liO]
which
slip dis-
The fact that the stress necessary for such
varies with temperature
in roughly
the same
slip (Fig. 9)
also supports this hypothesis.
number of narrow twins in addition to well-developed The twinning
is that
of (110)
way as the stress necessary for (llO)[liO]
FIG. 8. Load/deformation cnrves for a series of crystals of the same orientation which deformed by (130) twinning at different temperatures.
twinning
the
of 4 [liO]
(130) twins grow laterally
I
I
result
the density
implication
when
or when
by (110) slip at high temper-
Since the principal
be to increase
I
either
temperatures
growth
role
of
propagation in detail
dislocations
in the
of the (130)
in the theoretical
who considers
work
that nucleation
by the decomposition
nucleation
of Leteurtre(11*12)
of these twins occurs
under stress of [OOl] dipoles to
give a sessile Frank partial dislocation dislocation. Leteurtre by Cottrell dislocation
For
the
propagation
and a twinning of
applies the pole mechanism, and Bilby, reaction
and
twin has been considered
these
twins,
first proposed
and shows that the simplest
which will furnish the necessary
the shear stress
succession of twin drops begins.
the stress at which significant
lateral
growth of existing twins begins. InJluence of prior strain Whilst the
studying
(130)
twin
the influence
mode
some
on the effect
of prior
many
e.g.
at
metals
room
of temperature
interesting
deformation
iron,
temperature
prior is
were made.
deformation
a means
twinning during a subsequent
of
by
temperature primarily at room
l-2
temperature
slip
quite different
was
slip.
a
exclusively
a(
Crystal by
slip,
subsequent test continuous succession of During
a
from that of other crystals
temperature
*
test at low temperature,
twin drops was observed for a low stress (Fig. 9) and the appearance of the deformed crystal was at room
“, a I.
per cent at 64O”C, at which
deformation
(llO)[liO]
In
c^ e
suppressing
but in uranium the reverse appears to be true. 04 was pre-strained
on
observations
without
prior
strain.
deformed In the
I i
I
200
I
I
400
600 r
/
I
600
OK
FIG. 9. The variation with temperature of: (a) the shear stress 7 corresponding to the first (130) twin drop (lower curve), and (b) the shear stress corresponding to a continuous succession of twin drops (upper, dashed curve). The variation with temperature of the C.P.S.S. on the (110) [ 1lo] slip system is shown for comparison.
dislocations is the reaction -i (1101 +f
+ [loo]*
where f is the Burgers vector of the twinning dislocation and the asterisk implies a dislocation in the twinned lattice. Another possible reaction is : 4 pi01 + 2f + 4.[iio]* Furthermore, Leteurtre has calculated the energies of several reactions between slip dislocations and the (130) twin and shows that : (a) The (130) twin repels the [loo] matrix dislocation. (b) The 4 [lit)] matrix dislocation is also repelled, but by a much smaller force. (c) The 6 [llO] matrix dislocation is attracted by the twin. Thus both theory and experiment indicate that dislocations of the & (1 IO) type play an important role in (130) twinning. We may conclude that predeformation by (1101 slip leaves a high density of + (110) dislocations which can assist (130) twin propagation by the above mechanism. In a freshly grown crystal where the density of $ (110) dislocations is much lower, lateral growth of (130) twins will depend on the ease with which the 4 (110) dislocations present can move towards the twin under the influence of the applied stress. Thus the results obtained during the study of both the “(197)” and {130) twins indicate that slip dislocations play an important role in twin formatio~l. More experiments will be necessary to determine the dislocation reactions involved in “{197>” twinning, but Guyoncourt and Crocker,c30) in their analysis of the “(135)” twin in mercury also postulate the creation of twinning dislocat,ions by the intersection of slip dislocations with the type II twin interface. Furthermore, just as in mercury the (111) slip plane intersects the “‘(135)” twin along the twinning direction, in uranium the (021) slip plane intersects both the “1197)” and “(172)” twinning planes along their respective twinning directions. THE
DEFORMATION
The mechanical properties of polycrystalline ccuranium between 20 and 600°C have been studied in detail by Jean-Louis et aZ.(31*32j and Fig. 10, which shows the variation with temperature of the 0.3 percent proof stress of polycrystalline uranium of various purities and grain sizes, summarizes some of the results. In order to compare the data for single crystals and polycrystals the proof stresses for the latter have been divided by an arbitrarily chosen Taylor factor (o/-r) of 3. According to the von Mises criterion five independent slip systems must be available if a polycrystal is to undergo homogeneous deformation without a change in volume. In ccuranium the principal slip systems studied in this work i.e. (OlO)[lOO], (OOl)[lOO] and (llO)(liO) yield only three independent modes, so that the deformation of polycrystalline uranium must involve slip on other systems, particularly systems having a slip direction whieh does not lie in the (OOl\ plane such as (llO)[OOl] or (02l)[li2]. The ~mperatu~ dependence of slip in polycrystalline uranium will thus reflect the temperature dependence of slip on these systems. Figure 5 shows that for each of the slip systems in or-uranium there exists a critical temperature below which the critical resolved shear stress is strongly temperature-dependent. Bearing in mind the partially covalent nature of the bonding in this metal, it would seem likely that the thermally activated mechanism responsible is the overcoming of the PeierIs forces and experiments were undertaken to test this hypothesis. Although it would have been preferable to measure activation enthalpies
OF POLYCRYSTALLINE URANIUM
To conclude, it is worthwhile to consider to what extent the above results for twinning and slip in single crystals of uranium increase our understanding of the mechanical behavior of polycrystalline uranium. Two aspects of this behavior are of special interest, the ductility and the origin of the temperature dependence of the yield and flow stresses.
I
‘.
\
T “C FIG. 10. Comparison of the temperature variation of the c.r.s.s. for the different slip modes with the temperature variation of the proof stress (oo.spercent/3) for polycrystalline uranium of various purities and grain sizes.
ACTA
172
l
This work
0
Jean-Louis
METALLURGICA,
VOL.
IS,
1971
thermally activated mechanism operative over the temperature range studied. If this is the case then the activation enthalpy at constant strain rate should vary linearly with temperature according to an expression of the type:
(31)
(.EQIP = CT where c is a constant. Figure 11, which includes results obtained from tensile and results from compression tests by Jea~l-Louis tests on polycrystalline uranium performed during the present work, shows that the activation enthalpies for uranium, despite considerable scatter, are not inconsistent with such an expression. The activation volume is defined as the first derivative of the free energy of activation with respect to the st’ress, and its value is given, to a first approximation, b*y the expression:
Tempcrotun,
‘K
FIG. 11. Tho variation with temperature of the activation cnthalpy at constant strain-rate for the deformation of polyery~talline uranium.
and activation volumes and study their variation with temperature for each of the slip systems individually, this approach proved to be rather dii%cult in view of the number and size of the single crystals available, and measurements were made on polycrystalline material instead.
The activation volume can thus be calculated using data obtained by changing the strain rate during a test at constant t’emperature, provided the relationship between the applied stress o and the shear stress r is known. With certain assumptions it can be shown(33) that this volume is equal to the product of the Burgers’ vector of the dislocation and the area swept by the dislocation per successful activation. In Fig. 12 the values of the activation volume for polyoristalline uranium are compared with
l
Results
This work
c Jean-Louis
(31)
The obtention and interpretation of the measurements necessary for the study of thermally activated processes in plastic deformation have been described in detail elsewhere.(17*33*34) From measurements of the stress increments resulting from: (a) a change in temperature at constant strain rate, and (b) a change in strain rate at constant temperature, it is possible to calculate the activation enthalpy at con&ant strain rate using the following expression : I
0
I 40
Fe
fo,
-a,,);
u
(m,
-~873)i
I
I
60
80
12. The activation volume for the deformation of polycrystalline uranium a8 a function of stress. (Activation volumes were calculated assuming 4 = 37 and expressed in terms of the Burgers vector, b, ofthe $ [IlO] dislocation.) Results obtained by Altshuler and Christian(17’ for iron are shown for comparison. Frc.
where P is a frequency factor and the other terms have their usual meaning. The object of this measurement is to determine whether there is a single
I 20
DANIEL
similar
data
for
iron
Christian.(17)
At
the
2O”C, the activation
obtained lowest
volume
by
in uranium
20b3 (where b is the Burgers
4 [110]
dislocation) these they
seem
of the crystal
locations,
is approxi-
vector
for the
to
cannot
be
considered
support
the
hypothesis
of uranium
that the temperature
stress in uranium
and
studied,
at lower temperatures.
results
that the crystal structure namely
AND
and it seems likely that smaller
values would be obtained conclusive,
Altshuler
temperature
mately
Although
SLIP
et al.:
dependence
has its origin
lattice
itself suggests, of the flow
in the
resistance
itself to the movement
of dis-
otherwise known as the Peierl’s force. ACKNOWLEDGEMENTS
One of the authors edges
financial
Council Atomique
(U.K.)
(J. S. D.) gratefully
support
the Science
acknowlResearch
IN URANIUM
173
11.
J. LETEURTRE, Report CEA Paris No. 372 (1964). 12. J. LETEURTRE, Thesis, University of Paris (1968). 13. J. S. DANIEL, F. JEAN-LOUIS and P. LACOMBE, J. nucE. Mater. 26, 319 (1968). 14. J. S. DANIEL, A. LE FLOCH, F. JEAN-LOUIS et P. LACOMBE CT. hebd. S&znc. Acad. Sci., Paris 267, 1103 (1968). 15. J. F. FRIES and G. CIZERON, to be published. 16. F. JEAN-LOUIS, G. CIZERON et J. PERRIER. M&n. s&e&. Revue. M&all. 62, 495 (1965). 17. T. L. ALTSHULER and J. W. CHRISTIAN, Phil. Trans. R. Soe. A261, 253 (1967). 18. D. CALAIS, P. LACOMBE and N. SIMENEL, J. nucl. Mater. 1, 325 (1959). 19. 0. TEEC+and R. E. OQILVIE, J. nucl. Mater. 1, 81 (1961). 20. F. P. BUTRA, Z. F. EVKINA, 0. L. FUFAEVA, I. A. KOROBEINIKOV and L. M. LEBEDEV, Soviet. atom. Energy 19, 1307 (1965). 21. E. S. FISCHER, J. nucl. Mater. 18, 39 (1966). 22. L. T. LLOYD and H. H. CHISWIK, Trans. Am. Inst. Min. Engrs 203, 1206 (1955). 23. J. FRIEDEL, J. phys. Chem. Solids 1, 175 (1956). 24. J. D. ESHELBY. Phil. Maa. 40.903 (19491. The Physical MekzZZurby of Uranium. 25. A. N. HOLD&, Addison-Wesley
(1958).
during the course of this work.
26. B. A. BILBY and A. G. CROCKER, Proc. R. Sot. A228, 240 (1965). 27. A. G. CROCKER, Phil. Mag. 7, 1901 (1962). 28. M. BEVIS and A. G. CROCKER, Proc. R. Sot. A304, 123
REFERENCES
29. M. BEVIS and A. G. CROCKER. Proc. R. Sot. A313. 509
and
(France)
from
TWINNING
the
Commissariat
B 1’Energie
(1968).
1. A. G. CROCKER, F. HECKSHER and M. BEVIS, Phil. Mug. 8, 1863 (1963). 2. A. G. CROCKER, F. HECKSHER, M. BEVIS and D. M. M. GUYONCOURT, Phil. Mag. 13, 1191 (1966). 3. F. J. SPOONER and C. G. WILSON, J. less-common Metals
10, 169 (1966).
4. F. C. FRANK, A. KELLER and A. O’CONNOR, Phil. Mug.
3, 64 (1958).
5. R. W. CAHN, Acta Met. 1, 49 (1953). 6. E. S. FISHER, USAEC ANL Report 5075 (1953). 7. I. SAXL and J. OTRUBA, J. wucl. Mater. 26, 325 (1968). 8. I. SAXL, Czech. J. Phys. B19, 456 (1969). 9. M. H. YOO, J. nucl. Mater. 26, 307 (1968). 10. A. G. CROCKER, J.nucZ. Mater. 16, 366 (1965).
(1969).
30. D. M. GUYONCO~RT and A. G. CROCKER, Acta Met. 16, 523 (1968). 31. F. JEAN-LOUIS, Thesis, Universitv of Paris (1968). 32. F. JEAN-LOUIS and P. LACOXBE,” MBm. scieni. Revue. M&aZZ. 4, 296 (1966). 33. J. W. CHRISTIAN and B. C. MASTERS, Proc. R. Sot. 281 223 (1964). 34. A. G. EVASS and R. D. RAWLINGS. Phvs. Status SoZirZi 34. 9 (1969). 35. L. T. ‘LLOYD, R. G. LIPTAI and R. J. FRIDDLE, J. nucl. Mater. 19, 173 (1966). 36. W. E. GARDNER and T. F. SIITH, Phys. Rev. 154. 309 (1967).
INFLUENCE
OF TEMPERATURE J.
S. DANIEL,t$
B.
ON SLIP AND LESAGEt
and
TWINNING
IN URANIUM*
P. LACOMBET
The deformation modes of a-uranium have been studied by performing mechanical tests on single crystals prepared by both the phase shift and the strain-anneal methods. The critical resolved shear stresses for slip on the systems (OlO)[lOO], (OOl)[lOO], (llO)[liO] and (021)[1i2] have been determined over a wide range of temperatures. The formation of twins on the systems “{197)” and {130) is facilitated by prior slip and the results indicat,e a pole mechanism involving 4 [Ii01 slip dislocat,ions for the lateral growth of (130) twins. Lastly experiments on polyerystalline geranium support the hypot,hesis that the thermally activated deformation mechanism responsible for the temperature dependence of the flow stress is the overcoming of the resistance of the crystal lattice to the passage of disloeat.ions. INFLUENCE
DE LA
TEMPERATURE DANS
SUR LE GLISSEMENT L’URANIUM
ET
LE
MliCLAGE
Les modes de deformation de l’uranium a ont 6te Btudi&s au tours d’essais mecaniques effectuds sur des monocristaux p&pa&s par les m&hodes d%crouissage critique suivi de recuit et de changement dephase. Lesoissions critiques pour le glissement SUP les syst&mcs (OlO)[lOO], (OOl)[lOO], (IlO)(liO] et (OZl)[li2] ont BtB d&ermin&es pour un large domaine de temp6ratures. La formation des m&es sur les systhmes “{197j” et {130} est facilitke par un glissement ant&&r et les r(?sultats mettent en &idence un mbanisme polaire impliquant des dislocations de glissement Y$[liO] pour la croissanoe lat&ale des macles (130). Enfin, les expiriences effectuBes sur de l’uranium polycristallin eonfirment l’hypothhse suivant laquelle la d&formation activee thermiquement responsable de la variation de IR contrainte plastique avec la temperature peut se produire quand la rbsistance du rbseau cristallin au passage des dislocations est surmont6e. DER
EINFLUB
DER
TEMPERATUR
AUF DIE GLEITUNG IN URAN
UND
ZWILLINGSBILDUNG
Die Verformungsmoden von a-Uran wurden untersucht; mit zwei verschiedenen Verfahren hergestellt,e Einkristallo wurden verformt, Die kritischen S~hubspannungen in den Systemen (OlO)[lOO], (OOl)[lOO], (llO)[liO~ und (OZl)[ff2] wurden in einem grol3cn Temperaturbereich bestimmt. Die Bildung von Zwillingen in den Systemen “{197)” und {130) wird durch vorangegangene Gloitungedeichtert. Die Ergebnisse deuten auf einen Polmechanismus hin, bei dem seitliches Wachstum der (130)Experimente an polykristallinom Uran Zwillinge dumb Gleitung von 4 [liO]-Versetzuqgen erfolgt. unterstiitzen die Hypothese, da0 in Uran die Uberwindung des Gitterreibungswiderstandes der die Temperaturabhilngigkeit der Flienspannung bestimmende thormisch-aktivierte Verformungsmechanismus ist. INTRODUCTION
In view of the advances in
our
the of
unde~s~and~~~
common the
of
metallic
crystallography
the
crystal of
made
in
recent
defo~ation structures deformation
years
modes
of
the
study
of
metals
In the case of mercury for example, which has a rhombohedral crystal strncture, Cracker et .Z.(f*2) have shown that both the principal slip direction and the active t~~inning mode violate dell-established criteria for the deformation characteristics of metals. The refinement of existing theory made necessary by the discovery of such anomalies should result in a more general theory of deformation mechanisms. a-uranium, the allotrope of uranium stable at room temperature, (Fig. l), has the orthorhombic crystal structure illust,rated in Kg. 2. Outside the actinide series, gallium is the only pure metal to have an orthorhombic structure, although it should be noted that in the case of gallium the lattice parameters a and h are nearly equal, making the structure almost .* Iteceived May 10, 1970. of lower
symmetry
is of increasing,
interest.
t Laboratoire de MBtallurgie, Faculty des Sciences, $I-Orsay, France. $ Now at.: Ecole Polytechniq~Ie, Mont&al, Canada.
ACTA
METALLURGICA,
VOL.
19, FEBRUARY
1971
tetragonal, a fact which inffuences the twinning behavior of this metal.@) The o~horhombic structure is much more common among polymers and indeed the arrangement of the carbon atoms within the unit cell of polythene closely resembles that of the metal atoms in a-uranium, the principal twinning mode being the same in both cases.c4) The deformation modes of cr-uranium were first investigated nearly twenty years ago by Cahn,@) who studied the slip lines, twins and kinks visible in coarse-grained polyerystalline uranium after deformation or thermal cycling and identified the principal slip and twinning modes. This study revealed the complexity of deformation in M-uranium. Four twinning modes were found, one of which was of type II (irrational composition plane). Cahn also drew attention to the difficulty of preparing good single crystals of uranium of useful size, a di~culty which has hampered all subsequent research and which explains why the experimental work on the deformation of single crystals of uranium has remained largely qualitative. However, Fisher(Q was able to prepare a limited number of good crystals using a grain growth method and this made possible the accurate determination of the elastic constants and
163
ACTA
164
METALLURGICA,
VOL.
19,
1971
orientation are necessary. The phase-shift method remains the only practical means of preparing large numbers of monocrystalline tensile specimens and this method has several disadvantages. Firstly, crystals produced in this way have a rather imperfect, striated structure; secondly, the range of orientations obtained is usually narrow, and lastly the phase transformation often imparts a slight curvature to the tensile specimens, which is obviously inconvenient for the accurate measurement of yield stresses. These difficulties were largely overcome by performing compression tests since such tests can be performed on very small crystals and other methods of crystal preparation, notably the critical strains anneal method, are then available. Crystal preparation
1
0
IO
I
I 30
20
Pressure,
FIG.
1.
The
I 40
I 50
1 60
k bar
temperature/pressure phase uranium.‘35*3B)
diagram
of
expansion coefllcients of cc-uranium over a wide range of temperature. These data provided raw material for several theoretical investigations. Sax1 ef aZ.(7*8)and Yoo(9) attempted to predict the relative importance of the different possible slip modes in a-uranium on the basis of anisotropic elasticity calculations whilst Crockero”) and Leteurtre(ll*ls) have analyzed the possible twinning modes. It being impossible to test many of the predictions arising from this theoretical work on the basis of the existing experimental data, we thought it worthwhile to carry out a systematic investigation of the influence of temperature on the relative importance of the various deformation modes of uranium. The earlier part of the work, in which the slip directions associated with each slip plane were verified, has been described elsewhere(13z14)and the present artiale discusses the variation of the critical resolved shear stress with ~mperature for the different slip systems and reports the results obtained from an investigation of twinning.
Experimental
Whereas the tensile specimens were first stamped out in poleyrystalline sheet and then converted into single crystals, the compression specimens were obtained by spark-cutting from monocrystalline material which was prepared as follows : Strips measuring 50 x 3 x 1.3 mm were spark-cut from 1.3 mm thick cold-rolled sheets of electrolytic uranium (<22 ppm of impurities) and the edges of these strips, which became the eompressed faces in the final specimens, were ground parallel on 600 emery paper. These strips were then converted into single crystals using either the critical strain/ anneal or the phase-shift methods.
t
__I I
‘\
--_
_I
’
,---
---
_-
‘.
_--
--
; ‘\
‘\
--,
I
I
__---_
‘\ I
b
‘\ \ ’
-_
[1001
(010)
~~~
methods
In order to be able to study lattice rotation during deformatiot~ and hence determine slip dire&ions unambiguously tensile testing ‘was employed in the first stage of the investigation.03) However, in order to determine the critical resolved shear stresses for the different deformation modes over a wide range of temperature, large numbers of crystals of various
__ _ __ _ liic
ml
‘
0
___ --__ 4 Dl
7 4 L
’ _____-__ ~:
Y
biOJ
(001)
FOG. 2. The crystal structure of a-uranium and its projection on the principal planes.
DANIEL
et al.:
SLIP
AND
TWIKNIPL’G
IN URANIUM
165
8 FIG. 3. The orientations of the sin&
crvstols studied. A-tensile sDecimens. B--compression specimens. x crystal prepared by the ph&e shift method. 0 crystal prepared by the strain/kneal method.
Critical strain/anneal method In this method the strips were first electropolished and annealed ire,face (< 10e6 torr) at 64O*C for 15 hr in order to obtain a fine and homogeneous grain size. The strips were then critically strained in tension to 0.9 per cent plastic deformation at room temperature before a final recrystallization anneal for 3 hr at 655°C. Best results were obtained if the strained specimens were heated to temperature as rapidly as possible during this final anneal. After recrystallization, electropolishing and etching usually revealed a “bamboo” structure in which grains up to 10 mm length occupied the whole cross section of the strip. Compression specimens of length 2.5-3 mm and approximately square cross section were t)hen obtained by spark-cutting perpendicular to the axis of the strip within these large grains. Some 40 crystals of a wide range of orientations were prepared in this way (Fig. 3). Phase-shift method Although the strain-anneal method yields relatively perfect crystals, the small size of the grains in the bamboo structure limits to two or three the number of compression specimens of identical orientation which can be obtained from a given grain. Since it was often useful, particularly in the study of twinning, to have a large number of specimens of identical orientation, use was also made of the phase shift method. In this case polished strips were sealed in vacua in silica tubes and passed through a furnace having a steep temperature gradient at the fi -+ g transformation temperature (662’C). This treatment converted the greater part of each strip into a single crystal and compression specimens were spark-cut from these as described above.
In all cases the final compression specimens were electropolished to remove all traces of spark-cutting and the orientation of each was determined by the back-reflection Laue method. The sharpness of the spots on the Laue diagram also served as an indication of crystal perfection and crystals prepared by the strain-anneal method gave sharp spots whereas phase-shift crystals usually gave the rather fragmented spots typical of a ~lygonized structure. The length of each specimen was measured and its cross section determined by weighing. Mechanical testing Tests were carried out on an Instron testing machine. The accessories used for low-temperature testing and for high-temperature testing in vacuum of 3 x 10m6 torr have been described elsewhere.‘15*la’ Compression tests were carried out between alumina plates let into R “Nicral” compression jig designed so as to exert a tensile force on the load cell. (A stainless steel compression jig, also with alumina plates, was used for the low temperature work.) Strain rates of 2 x 10T2 per min and a chart speed of 2 cm/min were employed and the load corresponding to the limit of proportionality, as observed on the chart, was used in calculating the critical resolved shear stress (c.r.s.s.) once the operative deformation mode had been identified by the two-surfaces method. ExpeGmental errors The experimental errors involved in this work are likely to be rather greater than is usual in the study of single crystals for three main reasons: 1. As Altshuler and Christian have pointed out, small angular errors in the preparation of compression specimens can give rise to considerable errors in the measured yield stress.
RCTA
166
ioI Q 8
METALLURGICA,
. c
S-
P
6-
4-
I
I
I
200
400
600
800
r;T FIG. 4. The variation with temperature of the shear modulus p& for various combinations of planes and directions in cr-uranium. (1 Pa = 10 dynfcm2.)
2. The fragmented Laue spots characteristic of phase-shift crystals can introduce errors of 2-5’ in orientation determination. 3. All angles were calculated using a stereographic projection based on the values of the lattice parameters at 20°C. The anisotropy of thermal expansion of uranium is such that the position of certain poles on the projection [e.g. (OZl)] would be different by several degrees on a stereographic projection of the actual structure at 6OO’C. Taken together, these factors could result in errors of up to 30 per cent in the measured values of critical resolved shear stresses. Fortunately the differences between the 0.r.s.s. values for the different deformation modes proved to be much greater than this. RESULTS
19,
1971
direction in the a-uranium structure. However, whereas below about 500°C slip in this direction occurs most readily on the (010) plane, at higher temperatures slip is easiest on the (001) plane. Slip on other planes of the [loo] zone, particularly (021) and to a lesser extent (Oil), is also observed at high temperature, always though in a secondary role. (b) Slip on the (110) plane has been observed at temperatures above 15O’C. Studies of lattice rotation and Laue asterism indicated that the slip direction was [lrO] in all cases and above 400% segments of (111) slip are associated with the (110) slip lines. CalaiW and Teeg and Ogilvie(iv) reported slip on the system (llO)[OOl] at room temperature and below. This sysbem was not observed in the present study since all suitably oriented specimens deformed by twinning, kinking, or slip on other systems. (c) Slip can occur on the system (021)[li2] at temperatures above 400% although the stress neeessary to operate this system is several times greater than that required for the system (021)[100].
/
s F
VOL.
AND DISCUSSION: u-URANIUM
SLIP
IN
Sliz, modes
Knowledge of the slip direction associated with each slip plane is a s&e qua mu for the determination of critical resolved shear stresses. The results of this investigation and of previous studies of u-uranium may be summarised as follows: (a) At all temperatures slip occurs most easily in the [IOO] direction, which is the closest packed
Critical resolved shear stresses Once the elements of a slip system have been identified, the crit.ical resolved shear stress TV can be calculat,ed using the Schmid formula. 7, = 0 cos 0 cos a where (T is the applied stress, 0 the angle between the deformation axis and the pole of the slip plane and A the angle bet-rveen the deformation axis and the slip direction. The variation of r with temperature for the slip systems (OlO)[lOO], (OOl)[lOO], (llO)[liO] and (021) [I121 between 20 and 660°U1*) has been studied in the present investigation and Butra(20) has published data showing the variation of ~~ for the system (010) [loo] below 20°C. In comparing slip on different systems and in different metals it is usual to divide the critical resolved shear stress T, by the shear modulus p@ in order to eliminate variations arising from the temperature dependence of interactions between dislocations. In view of the anisotropy of uranium, the shear modulus was calculated as a function of temperature for 14 actual and hypothetical slip systems in cc-uranium using the elastic constant data of Fisher.r2i) The results are shown in Fig. 4. Figure 5 shows the variations of rJpa for the principal slip systems of or-uranium as a function of temperature. This diagram presents two unusual features, the intersections between the curves for different systems and the slight positive slope of the curve for the system (OlO)[lOO] above 300°K. It is int,eresting
DANIEL
et al.:
SLIP
AND
i \(ozu [li2] ‘f
b ;;
.P c”
7; “K
FIG. 5. The varirttion with temperature of T~/,u* (c.r.s.s./ sheer modulus) for the slip modes studied.
to consider to what extent these features are explicable in terms of our present knowledge of t-he ~-uranium structure. Discussion It is well known that the u-uranium structure contains a significant degree of covalent bonding between the atoms in the cor~ga~d (010) planes (which Cahnc5) refers to as giant molecules) and FriedeS2a) has suggested that the changes observed in the variation of certain physical properties of uranium (e.g. electrical resistivity) with increasing temperature are due to the weaking of these covalent bonds as the bonding electrons are excited into the conduction band. The main features of the variation of T~/,I_+, with temperature for different slip systems fit in with this theory. Only on the system (010) (1001 can slip occur without breaking covalent bonds, so that at low temperatures slip will take place most readily on this system. The fact that the value of TJ,u~ for the (OlO)[lOO] system appears to increase slightly with temperature may be related to the contraction of the structure along the b axis which brings the (010) planes closer together as the temperature increases. Slip on the (001) and (021) planes involves rupture of the shortest bond in the structure (d, in Fig. 2) and slip on (110) must break the d, bond. Electrical resistivity measurements indicate that the covalent character of the bonds in a-uranium begins to decrease above about 500°K,(zj) and it can be seen that the critical resolved shear stresses for slip on these planes have high values below this temperature. Rowever, simple considerations of bond lengths, packing densities, interplanar spacings and Burgers vectors
TWINNING
IN
167
URANIUM
are not sufficient to explain the rather complex variations of T,.,+, for the different systems above 500°K. Although it is not yet possible to calculate the stress necessary to move different dislocations in uranium, some progress has been made by applying anisotropic elasticity theory to the problem of slip in uranium. Sax1 and Otruba(‘) began their theoretical study of the mobilities of dislocations in u-uranium by calculating the values of the “ease of Xiding parameter” (EGP) f or d iff erent disIocations in both screw and edge orientations. This parameter, which is simply an approximation for the width of a dislocation divided by its Burgers vector, was proposed by Eshelby(z*) as a measure of dislocation mobility. In a later article Saxl@) also computed the energies of various dislocations in cr-uranium. More detailed comparisons between these theoretical result,s and the experimental data have been given elsewheret7*13*14)and are encouraging. The theory predicts that (~l)[l~] will be the easiest slip system at high temperatures and suggests that, in general, screw dislocations in uranium will be less mobile than edge dislocations, This is borne out by the observation that waviness is a common feature of slip in above about 200°C. TWINNING
IN
a-URANIUM
Introduction The general theory of the crystallography of deformation twinning in lattices has advanced considerably in recent years, particularly as a result of the work of Bilby and Crocker(26) and Bevis and Crocker.(27~28)The elements K,, I”r,, Q, Q, normally used to describe a twin and the twinning shear s, are shown in Fig. 6. Although many different twinning modes of varying complexity are theoretically possible in cubic lattices,(2s) the twins actually observed in the f-cc. and b.c.c. lattices are rather simple. In each case the observed twinning mode involves the smallest possible shear strain which is consistent with the absence of atomic shuffling. For the f.c.c. lattice this means t,he compound twinning mode (lllj, {ill), (%l>, (211), and in the b.c.c. lattice the mode (2111, (211}, (ill), (ill). Although the twinning shear is the same in both cases the b.c.c. mode is not observed in the f.c.c. lattice, where it would necessitate atomic shuf%ng, and vice versa. This suggests that for a fixed value of the twiIlning shear, the mode which necessitates the least shufling will be the easiest to form. Furthermore it was until recently supposed that in the absence of shuffles, or in a hypothetical case where two different twinning
16X
ACTA
METALLURGICA,
VOL.
19,
19’71
in the preparation of good single crystals of c(-uranium most of the recent work on twinning in this metal, which it will be convenient to summarize briefly, has been theoretical rather than experimental. Crockeros) studied the complexity of atomic shuffling for a whole series of theoretically possible twinning modes. He relates the complexity of the shufies to the number of lattice planes (q) parallel to the plane K1 which intersect a unit vector in the yls direction (for type I twins: the plane K2 and the direction vi are used for type II twins) and shows that there are 41 possible twinning modes for which q = 1, 2 or 4 and which have a shear strain less than unity. The first five of these theoretiea~y possible modes (in order of increasing shear) are given in Table 1. Experimental observation has clearly established that some of these twins are in fact operative. These are: No. 1 reciprocal (i.e. K, = “(197}“), No. 2, FTU. 6. The elements of a twin. both twins (K, = (112) and K, = “(172)“) and modes involved identical shuffles, the twinning mode No. 5 reciprocal (K, = (130)). A twin having the requiring the smaller shear strain would be the easier elements: (11211“(141)” “(321)” (311) was reported mode to operate. However, this assumption has by Cahn.(b) This twin has a fairly low shear (s = 0.33) been shaken by the work of Guyoncourt and Crocker(sO) but does not appear in Cracker’s list because of the on twinning in mercury. Although twinning in high value of q (q = 6). The discovery of the “(197)” mercury does not involve shuffling the observed twin (alsoknown as the “(176)” twin) by Lloyd and twinning mode is not the mode with the lowest Chiswik’ss) led to the suggestion that the (121) twin shear, which would be: was really a “(1973” twin, and Crockcr has added that in view of the size of the shear and the complex (011){100}(100)(011) 8 = 0.457 shuffles involved, the formation of a {121f twin is unlikely on theoretical grounds. but a type II mode with a considerably larger shear: Although all authors are agreed that the (130) twin is the most commonly observed mode in a-uranium, “{i%)“(ill)(i21)“{Oil)” .s = 0.633 it is very difficult to class the various twinning These authors thus conclude that, in predicting modes in order of ease of formation on the basis of which twin mode will be operative in a given crystal, previous work since there have been virtually no it is necessary to consider not only the size of the measurements made of the stresses necessary to shear and the complexity of the shuttles, but also induce the various twins. Only limited information whether there exists a convenient dislocation mechcan be obtained by comparing the frequency with anism which would enable the twin to form. which different twins are observed, since the apparA special interest attaches to the study of twinning ition of a given twin will depend not only on it’s own in u-uranium since, at least as far as metallic crystals ease of format,ion but also on t,he ease with which are concerned, it is in this metal that it should be other deformation modes can operate under the easiest to assess the relative importance of the three same conditions. This in turn will depend, for a given factors, shear strain, shuffles and dislocation mechaorientation of the crystal with respect to the applied nisms, in determining the ease of operation of twinstress, on the Schmid factor which relates the resolved ning modes. This is because, since a-uranium has TABLE 1 ~ a double-lattice orthorhombic structure, a variety of twins are observed including a compound twin and twins of types I and II. These twins involve shear strains of different magnitudes and all require varying degrees of atomic shuffling for their formation. Partly, no doubt, because of the difficulties involved
DANIEL
et al.:
SLIP
AND
shear stress on a given deformation mode to the applied stress. Diagrams showing the variation of the Schmid factor with orientation for all deformation modes of a-uranium have been given by Calais(l*) and Daniel et aZ.(lQ and these diagrams highlight the anisotropy of deformation in this metal. To cite one example, the twin”{197)” very rarely forms during tensile tests since in tension the the orientation which produces a high resolved shear stress on this twinning systems also maximizes the shear stress on the principal slip system (010) [loo]. Since twinning is an asymmetric process the orientation which will favor twins of the “(197)” family when a compressive stress is applied will be different. This new orientation corresponds to a low value of the Schmid factor for the principal slip system so that, in compression tests, these twins are common. As well as comparing the stresses necessary to form twins of the various modes at a fixed temperature it would be particularly interesting, in the case of uranium, to know how the different twinning stresses vary with temperature. Leteurtre(12) has proposed dislocation mechanisms for the nucleation and propagation of the various twins observed. Since these mechanisms involve slip dislocations and the previous study of slip has revealed that the mobility of these dislocations varies considerably with temperature in some cases, a study of twinning as a function of temperature should help to assess the validity of these mechanisms. Two twin modes were investigated, the type II mode “(197)” and the compound mode (130). (a) The “(197)”
twin
As noted previously, twins of the “(197)” family form readily for certain crystal orientations during compression tests. These “(197)” twins are usually considerably thicker than the other twins in uranium, probably because, for temperatures below about 2OO”C, the value of the homogeneous twinning shear, s, is smallest for this tuin (Table 1). Compression tests were performed on a series of monocrystals and the stress corresponding to the first “burst” of “{197}” twins was measured. This first burst usually produced twins on both of the most highly stressed “{197)” modes i.e. “(i97)” and “( 197)“, and values of the shear stress 7 were calculated for the “(i97)” mode. The results are shown in Fig 7. At all temperatures studied the minimum value of the shear stress necessary to cause “(197)” twinning falls between 6 and 6 Pa x 10’. However, some crystals required a much higher
TWINNING
IN
URANIUM
100
I50
Temperature,
169
/ 2b*
250
i
3c
‘K
PIG. 7. The shear stress 7 on the “(197)” twin mode corresponding to the first burst of “1197)” twins in crystals of various orientations at different temperatures. (1 Pa x 10’ = 1.0197 kg/mm*)_
shear to initiate twinning and examination revealed that these crystals had orientations such that the Schmid factor (Cos 19Cos 1) for the principal slip system (OlO)[lOO] had a very low value (<0.05). This suggests that the formation of “(197>” twins may be related to the prior movement of slip dislocations. On this hypothesis “(197>” twins will form at a shear stress of 5-6 Pa x 10’ only if some movement of slip dislocations has already occurred. Otherwise the applied stress must be increased until it induces such microscopic slip and twinning will then ensue. (b) The (130) twin. A more detailed study of the variation of twinning stress with temperature was made for the (130) twin. A series of crystals having the same o~entation were deformed in compression at various temperatures in the alpha range. Some typica. stress-strain curves are shown in Fig. 8. At temperatures below about 2OO’C there was a series of isolated twin drops during the elastic part of the stress-strain curve and aubsequent observation revealed that a large number of fine twins had been created. At higher temperatures, e-g_ the stress-strain curves at 404 and 500°C there was a continuous succession of twin drops and in these cases the crystal was found to contain a few very wide twins. Since the number of twins was clearly smaller than the number of twin drops many of the drops must correspond to the widening of existing twins rather than the creation of new ones. Finally, at 6OO”C, the stressstrain curve is characteristic of deformation by
ACTA
METALLURGICA,
VOL.
19,
1971
latter case a large number whereas at
deformation
room
temperature
characteristic 0 Small
twin
drop
of fine twins were formed
of
the
prestrained
produced
the
of high temperature
gradually X 0
lateral
growth
twins
deformation.
in the case of the high temperature mentioned,
specimen
thick
As
tests previously
of the twins
took
place
during the test.
Discussion
hv
The experimental
%3
of
(130)
occurs
twins
results show that lateral growth
is favoured
at high
has been predeformed ature.
20°C
300 *c
404oc
/
I
I
500 oc
I I
0
I
1
1
2
594oc
the
locations.
I
4%
growth
deformation
slip,
the
although
deformed
specimen
contained
The a
slip lines. The
lower,
stress data are collected
continuous
stress corresponding curve,
which
curve
the
shear
to the first twin drop on each
may
be considered
shear stress necessary
as the
minimum
to initiate longitudinal
of a twin nucleus. The upper, dashed, curve indicates at which a continuous This is probably
in Fig. 9.
represents
crystal
the
mechanism
involves
slip will
dislocations, by
4 [liO]
which
slip dis-
The fact that the stress necessary for such
varies with temperature
in roughly
the same
slip (Fig. 9)
also supports this hypothesis.
number of narrow twins in addition to well-developed The twinning
is that
of (110)
way as the stress necessary for (llO)[liO]
FIG. 8. Load/deformation cnrves for a series of crystals of the same orientation which deformed by (130) twinning at different temperatures.
twinning
the
of 4 [liO]
(130) twins grow laterally
I
I
result
the density
implication
when
or when
by (110) slip at high temper-
Since the principal
be to increase
I
either
temperatures
growth
role
of
propagation in detail
dislocations
in the
of the (130)
in the theoretical
who considers
work
that nucleation
by the decomposition
nucleation
of Leteurtre(11*12)
of these twins occurs
under stress of [OOl] dipoles to
give a sessile Frank partial dislocation dislocation. Leteurtre by Cottrell dislocation
For
the
propagation
and a twinning of
applies the pole mechanism, and Bilby, reaction
and
twin has been considered
these
twins,
first proposed
and shows that the simplest
which will furnish the necessary
the shear stress
succession of twin drops begins.
the stress at which significant
lateral
growth of existing twins begins. InJluence of prior strain Whilst the
studying
(130)
twin
the influence
mode
some
on the effect
of prior
many
e.g.
at
metals
room
of temperature
interesting
deformation
iron,
temperature
prior is
were made.
deformation
a means
twinning during a subsequent
of
by
temperature primarily at room
l-2
temperature
slip
quite different
was
slip.
a
exclusively
a(
Crystal by
slip,
subsequent test continuous succession of During
a
from that of other crystals
temperature
*
test at low temperature,
twin drops was observed for a low stress (Fig. 9) and the appearance of the deformed crystal was at room
“, a I.
per cent at 64O”C, at which
deformation
(llO)[liO]
In
c^ e
suppressing
but in uranium the reverse appears to be true. 04 was pre-strained
on
observations
without
prior
strain.
deformed In the
I i
I
200
I
I
400
600 r
/
I
600
OK
FIG. 9. The variation with temperature of: (a) the shear stress 7 corresponding to the first (130) twin drop (lower curve), and (b) the shear stress corresponding to a continuous succession of twin drops (upper, dashed curve). The variation with temperature of the C.P.S.S. on the (110) [ 1lo] slip system is shown for comparison.
dislocations is the reaction -i (1101 +f
+ [loo]*
where f is the Burgers vector of the twinning dislocation and the asterisk implies a dislocation in the twinned lattice. Another possible reaction is : 4 pi01 + 2f + 4.[iio]* Furthermore, Leteurtre has calculated the energies of several reactions between slip dislocations and the (130) twin and shows that : (a) The (130) twin repels the [loo] matrix dislocation. (b) The 4 [lit)] matrix dislocation is also repelled, but by a much smaller force. (c) The 6 [llO] matrix dislocation is attracted by the twin. Thus both theory and experiment indicate that dislocations of the & (1 IO) type play an important role in (130) twinning. We may conclude that predeformation by (1101 slip leaves a high density of + (110) dislocations which can assist (130) twin propagation by the above mechanism. In a freshly grown crystal where the density of $ (110) dislocations is much lower, lateral growth of (130) twins will depend on the ease with which the 4 (110) dislocations present can move towards the twin under the influence of the applied stress. Thus the results obtained during the study of both the “(197)” and {130) twins indicate that slip dislocations play an important role in twin formatio~l. More experiments will be necessary to determine the dislocation reactions involved in “{197>” twinning, but Guyoncourt and Crocker,c30) in their analysis of the “(135)” twin in mercury also postulate the creation of twinning dislocat,ions by the intersection of slip dislocations with the type II twin interface. Furthermore, just as in mercury the (111) slip plane intersects the “‘(135)” twin along the twinning direction, in uranium the (021) slip plane intersects both the “1197)” and “(172)” twinning planes along their respective twinning directions. THE
DEFORMATION
The mechanical properties of polycrystalline ccuranium between 20 and 600°C have been studied in detail by Jean-Louis et aZ.(31*32j and Fig. 10, which shows the variation with temperature of the 0.3 percent proof stress of polycrystalline uranium of various purities and grain sizes, summarizes some of the results. In order to compare the data for single crystals and polycrystals the proof stresses for the latter have been divided by an arbitrarily chosen Taylor factor (o/-r) of 3. According to the von Mises criterion five independent slip systems must be available if a polycrystal is to undergo homogeneous deformation without a change in volume. In ccuranium the principal slip systems studied in this work i.e. (OlO)[lOO], (OOl)[lOO] and (llO)(liO) yield only three independent modes, so that the deformation of polycrystalline uranium must involve slip on other systems, particularly systems having a slip direction whieh does not lie in the (OOl\ plane such as (llO)[OOl] or (02l)[li2]. The ~mperatu~ dependence of slip in polycrystalline uranium will thus reflect the temperature dependence of slip on these systems. Figure 5 shows that for each of the slip systems in or-uranium there exists a critical temperature below which the critical resolved shear stress is strongly temperature-dependent. Bearing in mind the partially covalent nature of the bonding in this metal, it would seem likely that the thermally activated mechanism responsible is the overcoming of the PeierIs forces and experiments were undertaken to test this hypothesis. Although it would have been preferable to measure activation enthalpies
OF POLYCRYSTALLINE URANIUM
To conclude, it is worthwhile to consider to what extent the above results for twinning and slip in single crystals of uranium increase our understanding of the mechanical behavior of polycrystalline uranium. Two aspects of this behavior are of special interest, the ductility and the origin of the temperature dependence of the yield and flow stresses.
I
‘.
\
T “C FIG. 10. Comparison of the temperature variation of the c.r.s.s. for the different slip modes with the temperature variation of the proof stress (oo.spercent/3) for polycrystalline uranium of various purities and grain sizes.
ACTA
172
l
This work
0
Jean-Louis
METALLURGICA,
VOL.
IS,
1971
thermally activated mechanism operative over the temperature range studied. If this is the case then the activation enthalpy at constant strain rate should vary linearly with temperature according to an expression of the type:
(31)
(.EQIP = CT where c is a constant. Figure 11, which includes results obtained from tensile and results from compression tests by Jea~l-Louis tests on polycrystalline uranium performed during the present work, shows that the activation enthalpies for uranium, despite considerable scatter, are not inconsistent with such an expression. The activation volume is defined as the first derivative of the free energy of activation with respect to the st’ress, and its value is given, to a first approximation, b*y the expression:
Tempcrotun,
‘K
FIG. 11. Tho variation with temperature of the activation cnthalpy at constant strain-rate for the deformation of polyery~talline uranium.
and activation volumes and study their variation with temperature for each of the slip systems individually, this approach proved to be rather dii%cult in view of the number and size of the single crystals available, and measurements were made on polycrystalline material instead.
The activation volume can thus be calculated using data obtained by changing the strain rate during a test at constant t’emperature, provided the relationship between the applied stress o and the shear stress r is known. With certain assumptions it can be shown(33) that this volume is equal to the product of the Burgers’ vector of the dislocation and the area swept by the dislocation per successful activation. In Fig. 12 the values of the activation volume for polyoristalline uranium are compared with
l
Results
This work
c Jean-Louis
(31)
The obtention and interpretation of the measurements necessary for the study of thermally activated processes in plastic deformation have been described in detail elsewhere.(17*33*34) From measurements of the stress increments resulting from: (a) a change in temperature at constant strain rate, and (b) a change in strain rate at constant temperature, it is possible to calculate the activation enthalpy at con&ant strain rate using the following expression : I
0
I 40
Fe
fo,
-a,,);
u
(m,
-~873)i
I
I
60
80
12. The activation volume for the deformation of polycrystalline uranium a8 a function of stress. (Activation volumes were calculated assuming 4 = 37 and expressed in terms of the Burgers vector, b, ofthe $ [IlO] dislocation.) Results obtained by Altshuler and Christian(17’ for iron are shown for comparison. Frc.
where P is a frequency factor and the other terms have their usual meaning. The object of this measurement is to determine whether there is a single
I 20
DANIEL
similar
data
for
iron
Christian.(17)
At
the
2O”C, the activation
obtained lowest
volume
by
in uranium
20b3 (where b is the Burgers
4 [110]
dislocation) these they
seem
of the crystal
locations,
is approxi-
vector
for the
to
cannot
be
considered
support
the
hypothesis
of uranium
that the temperature
stress in uranium
and
studied,
at lower temperatures.
results
that the crystal structure namely
AND
and it seems likely that smaller
values would be obtained conclusive,
Altshuler
temperature
mately
Although
SLIP
et al.:
dependence
has its origin
lattice
itself suggests, of the flow
in the
resistance
itself to the movement
of dis-
otherwise known as the Peierl’s force. ACKNOWLEDGEMENTS
One of the authors edges
financial
Council Atomique
(U.K.)
(J. S. D.) gratefully
support
the Science
acknowlResearch
IN URANIUM
173
11.
J. LETEURTRE, Report CEA Paris No. 372 (1964). 12. J. LETEURTRE, Thesis, University of Paris (1968). 13. J. S. DANIEL, F. JEAN-LOUIS and P. LACOMBE, J. nucE. Mater. 26, 319 (1968). 14. J. S. DANIEL, A. LE FLOCH, F. JEAN-LOUIS et P. LACOMBE CT. hebd. S&znc. Acad. Sci., Paris 267, 1103 (1968). 15. J. F. FRIES and G. CIZERON, to be published. 16. F. JEAN-LOUIS, G. CIZERON et J. PERRIER. M&n. s&e&. Revue. M&all. 62, 495 (1965). 17. T. L. ALTSHULER and J. W. CHRISTIAN, Phil. Trans. R. Soe. A261, 253 (1967). 18. D. CALAIS, P. LACOMBE and N. SIMENEL, J. nucl. Mater. 1, 325 (1959). 19. 0. TEEC+and R. E. OQILVIE, J. nucl. Mater. 1, 81 (1961). 20. F. P. BUTRA, Z. F. EVKINA, 0. L. FUFAEVA, I. A. KOROBEINIKOV and L. M. LEBEDEV, Soviet. atom. Energy 19, 1307 (1965). 21. E. S. FISCHER, J. nucl. Mater. 18, 39 (1966). 22. L. T. LLOYD and H. H. CHISWIK, Trans. Am. Inst. Min. Engrs 203, 1206 (1955). 23. J. FRIEDEL, J. phys. Chem. Solids 1, 175 (1956). 24. J. D. ESHELBY. Phil. Maa. 40.903 (19491. The Physical MekzZZurby of Uranium. 25. A. N. HOLD&, Addison-Wesley
(1958).
during the course of this work.
26. B. A. BILBY and A. G. CROCKER, Proc. R. Sot. A228, 240 (1965). 27. A. G. CROCKER, Phil. Mag. 7, 1901 (1962). 28. M. BEVIS and A. G. CROCKER, Proc. R. Sot. A304, 123
REFERENCES
29. M. BEVIS and A. G. CROCKER. Proc. R. Sot. A313. 509
and
(France)
from
TWINNING
the
Commissariat
B 1’Energie
(1968).
1. A. G. CROCKER, F. HECKSHER and M. BEVIS, Phil. Mug. 8, 1863 (1963). 2. A. G. CROCKER, F. HECKSHER, M. BEVIS and D. M. M. GUYONCOURT, Phil. Mag. 13, 1191 (1966). 3. F. J. SPOONER and C. G. WILSON, J. less-common Metals
10, 169 (1966).
4. F. C. FRANK, A. KELLER and A. O’CONNOR, Phil. Mug.
3, 64 (1958).
5. R. W. CAHN, Acta Met. 1, 49 (1953). 6. E. S. FISHER, USAEC ANL Report 5075 (1953). 7. I. SAXL and J. OTRUBA, J. wucl. Mater. 26, 325 (1968). 8. I. SAXL, Czech. J. Phys. B19, 456 (1969). 9. M. H. YOO, J. nucl. Mater. 26, 307 (1968). 10. A. G. CROCKER, J.nucZ. Mater. 16, 366 (1965).
(1969).
30. D. M. GUYONCO~RT and A. G. CROCKER, Acta Met. 16, 523 (1968). 31. F. JEAN-LOUIS, Thesis, Universitv of Paris (1968). 32. F. JEAN-LOUIS and P. LACOXBE,” MBm. scieni. Revue. M&aZZ. 4, 296 (1966). 33. J. W. CHRISTIAN and B. C. MASTERS, Proc. R. Sot. 281 223 (1964). 34. A. G. EVASS and R. D. RAWLINGS. Phvs. Status SoZirZi 34. 9 (1969). 35. L. T. ‘LLOYD, R. G. LIPTAI and R. J. FRIDDLE, J. nucl. Mater. 19, 173 (1966). 36. W. E. GARDNER and T. F. SIITH, Phys. Rev. 154. 309 (1967).