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The crystallography of the δ to α martensitic transformation in plutonium alloys
Journal of Nuclear North-Holland,
Materials Amsterdam
THE CRYSTALLOGRAPHY PLUTONIUM ALLOYS M.A. CHOUDHRY
Received
10 August
119
127 (1985) 119-124
OF THE 6 to a MARTENSITIC
* and A.G.
1984; accepted
TRANSFORMATION
IN
CROCKER
6 October
1984
The CRAB formulation of the theory of martensite crystallography has been applied to the 6 to a phase transformation in plutonium alloys. The low symmetry of the monoclinic a-phase results in 41 lattice invariant transformation shear modes being considered possible. The predictions, when these shears are used with the three available correspondences have been analysed and the 20 most likely mechanisms selected for presentation. Because of the lack of experimental information with which to compare the results it is difficult to select the best mechanism. However twinning of the product phase on (001) using the third correspondence is suggested. This has a habit plane near {258)s which is the crucial piece of information which needs to be checked experimentally. Further theoretical work on the atomic shuffles associated with' the correspondence also needs to be carried out
habit planes, shape deformations and orientation relationships for the transformation, which may in turn be compared directly with experimental observations. The CRAB formulation of the theory of martensite crystallography [6,7], which has been successfully applied recently to the tetragonal to monoclinic transformation in zirconia [8] is equally suitable for investigating the characteristics of the 6 to (Y transformation in plutonium alloys. Using the three lattice correspondences suggested by Olson and Adler [l], together with a number of lattice invariant shear modes, the possible crystallographic features for this transformation have been studied. In this paper, following a description in section 2 of the data required in the analysis, the results of this theoretical investigation are presented in section 3. The predictions are discussed in section 4 and compared with some recent experimental observations of Olsen [9] regarding the orientation relation between the 6 and a phases. A suggestion is also made about the most likely mechanism.
1. Introduction The low temperature S to a phase change which occurs on cooling or deformation of plutonium alloys exhibits many of the characteristics of martensitic transformations [l]. The crystallography of this transformation, which involves a large volume contraction of about 20%, has not been elucidated. This is because of the complexity of the crystal structure of the product phase, which contains 16 atoms in its monoclinic unit cell and the difficulties of performing detailed experimental work on this material. The transformation therefore provides a challenging application for theories of martensite crystallography [2]. These theories require a correspondence relating the parent 6 face centred cubic and the product a monoclinic structures. One such correspondence was proposed by Lomer [3] and the associated lattice deformation deduced by Spriet [4]. However this is inconsistent with experimental evidence that the transformation involves a contraction normal to the (020), mirror plane. This result is based on the crystallographic texture of the a phase produced from 6 by uniaxial compressive deformation [5]. Recently Olson and Adler [l] have proposed three possible lattice correspondences consistent with this contraction. These can now be used for systematic calculations of possible * On leave from: Islamia University,
Bahawalpur,
2. Data required in the analysis The theory of martensite crystallography [6,7] requires as data the metrics c,, and pi, (i, j = 1, 2, 3) of the parent and product phases, a correspondence matrix relating these phases and a lattice invariant shear mode
Pakistan.
0022-3115/85/$03.30 0 Elsevier Science Publishers (North-Holland Physics Publishing Division)
B.V.
M.A. Choudhry, A.G. Cracker / 8 IOa martensitic transformation
120
in plutonium
Table 1
The principal distortions
VI
d, 112 d, q3 d, zsf
I
II
111
0.938 0.833, -0.546, 0.084 1.114 0.552, 0.818, - 0.156 0.767 0.016, 0.177, 0.984 2.711
1.132 0.632, -0.187, 0.752 0.912 0.771, 0.058, -0.633 0.776 0.074, 0.980, 0.181 2.718
0.758 0.994, 0.098, 0.037 1.083 -0.100. 0.791, 0.603 0.976 0.029, - 0.603. 0.796 2.702
The magnitudes q, and directions d, (i = 1, 2, 3) of the principal distortions are given for correspondences
1, II and III together with
the quantity &I~.
of either the parent or the product structure. For the transformation in plutonium the metrics of the fee 6 and monoclinic a: phases are given by
C'J =
ao’
0
0
0
002
0
0
u; I
i 0
P,, =
and
a2
0
0
b20
Lac cos p
nc cos p
0
1 3
c2
whereao=4.183A,a=6.183A,b=4.822A,c=10.963 A and /3 = 101.79” [lo]. The three correspondences duced by Olson and Adler [1] are n=Q[
-i
i
I;],
&+$[
-I
a
de-
criterion for deciding which is the most likely mechanism to be active in practice [Ill. Unfortunately, as shown in table 1, the three plutonium correspondences have almost identical magnitudes of X$. It is therefore not possible to select a correspondence at this stage and all three have been used in the calculations. The lattice invariant shear modes which may be associated with the transformation are considered to be twinning or slip of the (Yphase and slip of the 6 phase. The likely twinning modes in o-plutonium have been studied by Cracker [12]. Seven of these which have small twinning shears and relatively simple atomic shuffle mechanisms, have been selected as possible shears for the transformation. They are given in table 2 toTable 2 Possible lattice invariant shear modes
J. a-twinning
a-slip
I (ool)[ioo];
0.159 , 1 (olo)]loo]
3 (001)]100]; 0.417
3 (loi)[oTo] 15 (lGi)]llo]
[ u J of the a and 6
4 (100)[001]; 0.417
4 (iOi)(OlO] 16 (114)(110]
4(1li)(liO]
(h),
5 (401)[104]; 0.129
5 (io2)]010] 17 (114)[lio]
5 (lli)]lol]
6 (lOl)[lOi];
6 (102)[010] 18 (114)[1iO]
6 (lli)[Oll]
7 (16?)[301]; 0.167
7 (lil)[llO]
19 (liS)[llO]
7 (lil)[llO]
_
8 (ill)~llo]
20 (i18)[110]
8 (lil)[loi]
_ _
9 (lllNlio]
21 (llq[lio]
9 (liT)[olT]
lo (ilT)[lio]
22 (li8)[lio]
_
11 (liZ)[llO]
-
11 (ill)[lol]
_
12 (il2)]110] -
12 (ill)[oli]
= (h),
,Cs,
(h),=(h), & [#lb =&[~I,, 1~1, =&[u18, where sCa = (,Cs)-’ [2]. The magnitudes 9, and directions d, (i = 1, 2, 3) of the principal distortions associated with these correspondences [ll] have been determined and are given in table 1. Note that in each case one of the distortions is greater than unity and two are less than unity. Hence they all satisfy the primary criterion that correspondences must have principal strains ( TJ,- 1) with mixed signs if an invariant plane is to be predicted for the habit plane of the transformation [2]. The quantity &~f - 3 is a measure of the strain energy associated with a correspondence and can therefore be used as a
0.167
2 (100)lo1ol14 (iiZ)[lio]
1 (iTl)]lio] 2 (lll)]loi] 3 (lll)]oli]
2 (201)[102]; 0.159 These relate planes (h ) and directions phases by means of the equations
S-slip
13 (112)]lio]
lo (iil)[llo]
The indices are given relative to the a-basis in columns and the d-basis in column 3; in the theory all indices are relative to the &basis. In column 1 the magnitudes twinning shears are also included; note that modes 5-7 more complex atomic shuffles than modes 1-4.
1 and 2 needed of the involve
M.A.
Choudhry,
A.G. Cracker
/ 6 to a markwsitic
gether with the magnitudes of the twinning shear strains. in the theory [6,7] the indices of the shear mode are needed relative to the parent crystal basis so that each of these seven a-modes has to be converted to S-indices. using the equations given above, for each of the three correspondences. Hence 21 distinct modes are obtained. Experimental information on the active slip planes of a-plutonium has been reported by Bronisz et al. [13] and Liptai et al. 1141. Most of these planes are consistent with the only feasible Burgers vectors of slip dislocations in the monoclinic crystal structure, which the two most closely packed are [OlO], and [loo],, directions, and (110), arising from stable interactions between these 1121. The resulting 22 a-slip modes are again presented in table 2 and give 66 lattice invariant shear modes when converted to S-indices using the correspondences. Finally slip in the d-phase should occur on { 111) planes in (110) directions. There are 12 crystallographically equivalent variants of this mode and these are given in the third column of table 2. Each of these can be used with the three correspondences providing a further 36 mechanisms. Thus there are in all 123 possible lattice invariant shear modes which can be used as data for the CRAB Tahle 3 Predictions
of the crystallographic Mode
1
2 3
a-T; I: 7 a-T; III; 3 a-T; III; 4
121
in plutonium
theory. Also, each set of data gives two values for the magnitude of the lattice invariant shear (g) and of the shape defo~ation (I). Then each pair of values for g and f gives two habit planes (h) and directions (u) of the invariant plane strain [7]. Potentially therefore 492 distinct habit plane predictions might be obtained, together with the associated shape deformations and orientation relationships.
3. Results All 123 possible transformation mechanisms which arise from the twinning and slip modes summarised in table 2 have been investigated using the CRAB formulation of the theory of martensite crystallography [6,7]. However many of these mechanisms give imaginary solutions of the equations and others can be eliminated because they involve large values of the shear g and shape deformation f. In particular for a given shear mode only results associated with the smaller of the two values of g and / have been considered and, even for these, upper limits of g = 0.20 and f = 0.42 were imposed. For the twinning mechanisms only positive val-
features f
g
transformation
h
u
0.129 0.065 0.053
0.316 0.323 0.318
0.328. 0.456. 0.826 0.824. 0.529, 0.199 0.831, 0.482. 0.275
0.174, - ,950. - .951,
9
#
0.8s 1.94 1.66
2.34 2.40 2.44
-.986 - ,969 -.988 - ,970 -.964 0.088 0.164 0.362 0.201 - .224 0.292
5.55 4.06 3.49 2.57 3.89 2.40 2.17 4.18 2.17 2.29 2.91
4.98 1.37 5.09 2.23 5.15 2.69 2.11 I .66 1.71 1.48 1.03
0.185. -.929 0.123, -.969 0.322, 0.127 0.178. 0.213 0.264, 0.091 0.210, 0.252
6.01 4.46 1.77 2.57 2.06 1.77
1.77 4.12 2.69 1.60 4.29 2.00
0.240, - ,956 0.262. 0.164 0.245, 0.185
4 5 6 7 8 9 10 I1 12 13 14
a-s: a-s: a-s; n-S: a-s; a-s; a-s; a-s; a-s: a-s; (Y-s;
I; 8 I; 10 I; 12 I: 14 I; 17 III; 9 III; 13 III: 17 III; 18 III; 21 III; 22
- .I35 0.155 - .156 0.135 0.178 0.157 0.097 -.I32 0.072 0.067 - ,085
0.340 0.386 0.316 0.389 0.429 0.396 0.363 0.330 0.349 0.343 0.332
0.344, 0.626, 0.699 0.351, 0.637. 0.686 0.424, 0.544.0.722 0.448.0.564, 0.692 0.588, 0.447, 0.673 0.689.0.671.0.270 0.745, 0.610. 0.265 0.863,0.383.0.327 0.774.0.568, 0.277 0.790.0.542. 0.284 0.831, 0.463, 0.305
0.125, 0.105, - .005, 0.244. 0.137. 0.056, 0.076. 0.230, -.149, 0.217, -.974, 0.222. -.962, 0.213, -.912, 0.189. -.956, 0.210. - ,952, - .207, -.935, 0.199,
15 16 17 18 19 20
s-s; 6-S; 6-S; 6-S; 6-S; 6-S;
I; 8 I; 11 III; 4 III: 5 III; 1 III; 8
-.155 -.135 - .083 - .052 - .082 - .055
0.376 0.394 0.359 0.341 0.362 0.336
0.424, 0.431, 0.796 0.431, 0.585, 0.686 0.805, 0.550, 0.219 0.780,0.481,0.398 0.757,0.587.0.285 0.819,0.427. 0.382
0.318, 0.211, - .938, -.960, -.959, -.944,
The first column gives the lattice invariant shear mode, either a-twinning (a-T). a-slip (a-S) or S-slip (S-S). the correspondence, I. II or III, and the mode number as defined in table 2. The remaining columns give g and f. the magnitudes of the lattice invariant shear and the total shape deformation respectively. k the habit plane, n the direction of the shape deformation. and 6 and C$the angles between [liO]s and (loOI, and between (Ill), and (020), respectively.
122
M.A. Choudhry, A.G. Cracker / S to a mnrrensiric transformation
in plutonium
from ill, 011 and particularly 001. The directions of the shape deformation all lie between 8” and 25” from 001. In 15 of the mechanisms the angle 0 between [IlO], and [lOOI, is less than 4’ and similarly 15 have the angle + between (ill), and (020),i less than 4”. In 12 cases both angles are less than this value. The larger angles tend to arise for correspondence I mechanisms. i22
4. Discussion
Fig. 1. Predicted habit plane poles for mechanisms 1 to 20 of table 3 plotted in a standard parent stereographic triangle. Correspondence I and III poles are indicated by closed and open circles respectively.
ues of g less than the twinning shear were allowed. Also, recent experiments by Olsen [9] indicate an orientation relation involving parallelism of the (111) 6 and (020) o( planes and of the [ 1101s and [ 1001 u directions to within 7”. Hence theoretical predictions not lying in this range were also eliminated. On applying the above selection criteria the 492 possible sets of solutions reduced to 20 and these are given, relative to the parent cubic basis, in table 3. Three of these arise from o-twinning, eleven from a-slip and six from &slip. Eight are associated with correspondence I, twelve with correspondence III but none with correspondence II. The magnitudes of the lattice invariant shears g range from 0.053 to 0.178 and are thus acceptably small in all cases. The ten smallest shears are associated with correspondence III modes. Note that when referred to the a basis the magnitudes of g for the three twinning mechanisms become 0.144, 0.063 and 0.053 respectively, all less than the corresponding twinning shears given in table 2. The shape deformation f ranges from 0.316 to 0.425 being larger than in many other transformations. The lowest values are for two correspondence I mechanisms but these are closely followed by four correspondence III results. The habit plane results (/I) are plotted on a standard parent stereographic triangle in fig. 1. They lie in a broad band across the centre of the triangle between 112 and 122. The correspondence I and III solutions are distinguished by different symbols and the former tend to lie closer to ill. However all of the poles lie well away
This application of the theory of martensite crystallography, based on the invariant plane strain criterion, has again demonstrated the benefits of the CRAB formulation [6,7]. Through by-passing intermediate steps in the computation and concentrating directly on observable crystallographic features, this analysis avoids many of the hazards associated with alternative versions of the theory. It is particularly useful when treating transformations involving low-symmetry phases with complex structures. The 6 to Q transformation in plutonium alloys therefore provides an ideal challenging but straightforward application. In this case the analogy between the monoclinic unit cell, and a distorted hexagonal close packed structure is invaluable. This was employed by Cracker [7] when analysing the possible deformation twinning modes of a-plutonium. In particular he compared the relative complexities of the atomic shuffling mechanisms associated with the different twinning modes by applying shears to a pseudostructure with only two atoms in its unit cell. This analogy has been extended in a very imaginative way by Olson and Adler [l] in order to examine the S to (Y phase transformation in plutonium. The s-phase is fee and therefore by using the distorted hcp pseudo-structure for the a-phase they liken the transformation to the phase change in cobalt [2]. In this case the - ABCABC - stacking of close packed (111) planes of the parent fee phase is transformed to the - ABABAB stacking of basal planes of the product hcp phase, by the passage of i(112) Shockley partial dislocations on every second plane. There are three variants of this dislocation on each {ill} plane and these may occur in any sequence so that no unique mechanism arises. In particular the total shape strain can take any value up to 0.35. The three correspondences of Olson and Adler [l] which are used in the present paper arise from the three Shockley partial variants. However, because of the distortion of the hcp structure to produce the (Yplutonium pseudo-structure, they are no longer equivalent. Indeed not only are the pure strains of the correspondences different from each other but so are the
M.A.
Choudhry, A.G. Cracker / S to a mariensitic transformation
complex atomic shuffles which must accompany the lattice deformation in order to locate the atoms on their correct product sites [l]. No attempt has yet been made to analyse these shuffles quantitatively but they must to a large extent control which of the correspondences is favoured. It is possible of course that all three occur together. However Olson and Adler [l] have shown that roughly equal proportions of the three correspondences which as in cobalt, would minimise the shape change, does not give an invariant plane of the transformation. This arises because of the large volume change of almost 20% associated with the transformation. Because of its low crystal symmetry cY-plutonium has a large number of independent deformation modes [12]. In addition the large number of crystallographically equivalent deformation modes of the high symmetry d-phase all have a distinct relationship with the a-structure. Thus there are many potential lattice invariant shear mechanisms for the transformation. Some are of course more favoured than others. Thus a-twinning modes 1 and 2 of table 2 have a smaller shear than modes 3 and 4 and simpler shuffle mecahnisms than modes 5-7. Also a-slip modes l-6 have more acceptable Burgers vectors than modes 7-22. Many of the slip planes of this latter group have been reported experimentally [13,14] but a far more detailed study of these deformation mechanisms is desirable. Slip on { 114) and (118) planes is not easy to accept. The 12 variants of the { 111 } (110) S-slip mode must of course be considered although the classical mechanisms of martensite crystallography involve shears in the product structure. The possible lattice invariant shears of table 2 produced a very large number of real solutions of the theory. These included a-twinning, a-slip and d-slip for all three correspondences. However in restricting the number of solutions to the 20 presented in table 3, by imposing limits on the magnitudes of the shear and shape deformations and using the available very rough experimental information on the orientation relationship [9], many cases have been eliminated. In particular no correspondence II mecahnisms survive and no mechanisms with the favoured a-twinning modes 1 and 2 and a-slip modes 1 to 6. All the remaining modes have acceptably small values of g but it is noticeable that the lowest values tend to be for correspondence III. The magnitudes of f are all close to the value of 0.35 associated with the cobalt transformation. It is therefore difficult to select a particular mechanism by using the criterion of small values off. Transformation twinning modes may of course be distinct from those occurring during deformation and in particular modes in which the twinning plane corre-
in plutonium
123
sponds to a mirror plane of the parent phase, or the twinning direction to a 2-fold axis might be favoured [ 151. These conditions are necessary if the two correspondences between the parent phase and the two components of the twinned product are to be variants of each other [2]. Of the three twinning results in table 3 this is only valid for mechanism 2. Thus with correspondence III, twinning mode 3 arises from (113) [ilO] in the a-phase, having a 2-fold shear direction. This particular mechanism has a low value for g corresponding to less than one-sixth of the product plate being twinned. The magnitude of f is relatively small and the habit plane is “{258}“, which as shown in fig. 1 is near the centre of the stereographic triangle. The direction of the total shape deformation is about 18” from (100). The two angles 0 and $J are about 2”, far less than the reported experimental limit of 7”. Note that these results are identical to those for a-twinning mode 1 of table 2 except that the sign of g is reversed and hence becomes unacceptable as the twinning shear must be positive. Mechanism 2 is therefore the one which, very tentatively, the results of the present investigation suggest may be responsible for the 8 to a transformation in plutonium alloys. Clearly more experimental work on the transformation is needed to see whether this suggestion is acceptable. In particular determination of the habit plane would be invaluable especially as pole 2 in fig. 1 is fortuitously on the edge of the scatter of predictions. More refined measurements of the orientation relationship and the magnitude of the shape deformation would probably not help as many of the predictions for the former are about 2” and for the latter about 0.35. Clearly the key experiment would be to determine the lattice invariant shear mode using electron microscopy but on this particular material this may be beyond available experimental techniques. In conclusion therefore many mechanisms seem to be acceptable for the 6 to a martensitic phase transformation in plutonium alloys. In practice several of these may occur but at present the most likely is twinning of the product phase on the (001) plane, with the parent and product structures related by correspondence III. Further theoretical work on the atomic shuffle mechanisms associated with the correspondences and further experimental work on the crystallograhic features of the transformation is needed. Acknowledgements
The authors are indebted to P.H. Adler, G.B. Olson and C. Olsen for stimulating discussions and to the Government of Pakistan for financial support.
124
M.A.
Choudhry,
A.G. Cracker
/ 6 to a murtensitic
References
VI G.B. Olson and P.H. Adler, Scripta Met. 18 (1984) 401. in Metals PI J.W. Christian, The Theory of Transformations and Alloys, Part I (Pergamon Press, Oxford, 1975). [31 W.M. Lomer. Solid State Commun. 1 (1963) 96. 1965. Eds. A.E. Kay and M.B. [41 B. Spriet, in: Plutonium Waldron (Chapman and Hall, London, 1967) p. 88. IS?A. Goldberg, R.L. Rose and J. Shyne, J. Nucl. Mater. 55 (1975) 33. 161A.F. Acton, M. Bevis, A.G. Cracker and N.D.H. Ross. Proc. Roy. Sot. (Lond.) A320 (1970) 101. f71A.G. Crwker, J. de Physique 43 (1982) C4-209.
transform&on
m plutonium
[8] M.A. Choudhry and A.G. Cracker, Adv. in Ceramics, in press (91 C. Olsen. private communication. [IOj W.H. Zachariasen, Acta Cryst. 5 (1952) 660. [Ill A.G. Cracker and N.D.H. Ross, in: The Mech. of Phase Trans. in Cryst. Solids (Inst. of Metals, London. 1969) p. 176. [12] A.G. Cracker, J. Nucl. Mater. 41 (1971) 167. [13] SE. Bronisz and R.E. Tam, in: Plutonium 1965. Eds. A.E. Kay and M.B. Waldron (Chapman and Hall. London. 1967) p. 558. [14] R.G. Liptai and R.J. Friddle, in: Plutonium 1970 and the Other Actinides (Met. Sot. AIME. 1970) p. 406. [15] N.D.H. Ross and A.G. Cracker. Scripta Met. 3 (1969) 37.
Materials Amsterdam
THE CRYSTALLOGRAPHY PLUTONIUM ALLOYS M.A. CHOUDHRY
Received
10 August
119
127 (1985) 119-124
OF THE 6 to a MARTENSITIC
* and A.G.
1984; accepted
TRANSFORMATION
IN
CROCKER
6 October
1984
The CRAB formulation of the theory of martensite crystallography has been applied to the 6 to a phase transformation in plutonium alloys. The low symmetry of the monoclinic a-phase results in 41 lattice invariant transformation shear modes being considered possible. The predictions, when these shears are used with the three available correspondences have been analysed and the 20 most likely mechanisms selected for presentation. Because of the lack of experimental information with which to compare the results it is difficult to select the best mechanism. However twinning of the product phase on (001) using the third correspondence is suggested. This has a habit plane near {258)s which is the crucial piece of information which needs to be checked experimentally. Further theoretical work on the atomic shuffles associated with' the correspondence also needs to be carried out
habit planes, shape deformations and orientation relationships for the transformation, which may in turn be compared directly with experimental observations. The CRAB formulation of the theory of martensite crystallography [6,7], which has been successfully applied recently to the tetragonal to monoclinic transformation in zirconia [8] is equally suitable for investigating the characteristics of the 6 to (Y transformation in plutonium alloys. Using the three lattice correspondences suggested by Olson and Adler [l], together with a number of lattice invariant shear modes, the possible crystallographic features for this transformation have been studied. In this paper, following a description in section 2 of the data required in the analysis, the results of this theoretical investigation are presented in section 3. The predictions are discussed in section 4 and compared with some recent experimental observations of Olsen [9] regarding the orientation relation between the 6 and a phases. A suggestion is also made about the most likely mechanism.
1. Introduction The low temperature S to a phase change which occurs on cooling or deformation of plutonium alloys exhibits many of the characteristics of martensitic transformations [l]. The crystallography of this transformation, which involves a large volume contraction of about 20%, has not been elucidated. This is because of the complexity of the crystal structure of the product phase, which contains 16 atoms in its monoclinic unit cell and the difficulties of performing detailed experimental work on this material. The transformation therefore provides a challenging application for theories of martensite crystallography [2]. These theories require a correspondence relating the parent 6 face centred cubic and the product a monoclinic structures. One such correspondence was proposed by Lomer [3] and the associated lattice deformation deduced by Spriet [4]. However this is inconsistent with experimental evidence that the transformation involves a contraction normal to the (020), mirror plane. This result is based on the crystallographic texture of the a phase produced from 6 by uniaxial compressive deformation [5]. Recently Olson and Adler [l] have proposed three possible lattice correspondences consistent with this contraction. These can now be used for systematic calculations of possible * On leave from: Islamia University,
Bahawalpur,
2. Data required in the analysis The theory of martensite crystallography [6,7] requires as data the metrics c,, and pi, (i, j = 1, 2, 3) of the parent and product phases, a correspondence matrix relating these phases and a lattice invariant shear mode
Pakistan.
0022-3115/85/$03.30 0 Elsevier Science Publishers (North-Holland Physics Publishing Division)
B.V.
M.A. Choudhry, A.G. Cracker / 8 IOa martensitic transformation
120
in plutonium
Table 1
The principal distortions
VI
d, 112 d, q3 d, zsf
I
II
111
0.938 0.833, -0.546, 0.084 1.114 0.552, 0.818, - 0.156 0.767 0.016, 0.177, 0.984 2.711
1.132 0.632, -0.187, 0.752 0.912 0.771, 0.058, -0.633 0.776 0.074, 0.980, 0.181 2.718
0.758 0.994, 0.098, 0.037 1.083 -0.100. 0.791, 0.603 0.976 0.029, - 0.603. 0.796 2.702
The magnitudes q, and directions d, (i = 1, 2, 3) of the principal distortions are given for correspondences
1, II and III together with
the quantity &I~.
of either the parent or the product structure. For the transformation in plutonium the metrics of the fee 6 and monoclinic a: phases are given by
C'J =
ao’
0
0
0
002
0
0
u; I
i 0
P,, =
and
a2
0
0
b20
Lac cos p
nc cos p
0
1 3
c2
whereao=4.183A,a=6.183A,b=4.822A,c=10.963 A and /3 = 101.79” [lo]. The three correspondences duced by Olson and Adler [1] are n=Q[
-i
i
I;],
&+$[
-I
a
de-
criterion for deciding which is the most likely mechanism to be active in practice [Ill. Unfortunately, as shown in table 1, the three plutonium correspondences have almost identical magnitudes of X$. It is therefore not possible to select a correspondence at this stage and all three have been used in the calculations. The lattice invariant shear modes which may be associated with the transformation are considered to be twinning or slip of the (Yphase and slip of the 6 phase. The likely twinning modes in o-plutonium have been studied by Cracker [12]. Seven of these which have small twinning shears and relatively simple atomic shuffle mechanisms, have been selected as possible shears for the transformation. They are given in table 2 toTable 2 Possible lattice invariant shear modes
J. a-twinning
a-slip
I (ool)[ioo];
0.159 , 1 (olo)]loo]
3 (001)]100]; 0.417
3 (loi)[oTo] 15 (lGi)]llo]
[ u J of the a and 6
4 (100)[001]; 0.417
4 (iOi)(OlO] 16 (114)(110]
4(1li)(liO]
(h),
5 (401)[104]; 0.129
5 (io2)]010] 17 (114)[lio]
5 (lli)]lol]
6 (lOl)[lOi];
6 (102)[010] 18 (114)[1iO]
6 (lli)[Oll]
7 (16?)[301]; 0.167
7 (lil)[llO]
19 (liS)[llO]
7 (lil)[llO]
_
8 (ill)~llo]
20 (i18)[110]
8 (lil)[loi]
_ _
9 (lllNlio]
21 (llq[lio]
9 (liT)[olT]
lo (ilT)[lio]
22 (li8)[lio]
_
11 (liZ)[llO]
-
11 (ill)[lol]
_
12 (il2)]110] -
12 (ill)[oli]
= (h),
,Cs,
(h),=(h), & [#lb =&[~I,, 1~1, =&[u18, where sCa = (,Cs)-’ [2]. The magnitudes 9, and directions d, (i = 1, 2, 3) of the principal distortions associated with these correspondences [ll] have been determined and are given in table 1. Note that in each case one of the distortions is greater than unity and two are less than unity. Hence they all satisfy the primary criterion that correspondences must have principal strains ( TJ,- 1) with mixed signs if an invariant plane is to be predicted for the habit plane of the transformation [2]. The quantity &~f - 3 is a measure of the strain energy associated with a correspondence and can therefore be used as a
0.167
2 (100)lo1ol14 (iiZ)[lio]
1 (iTl)]lio] 2 (lll)]loi] 3 (lll)]oli]
2 (201)[102]; 0.159 These relate planes (h ) and directions phases by means of the equations
S-slip
13 (112)]lio]
lo (iil)[llo]
The indices are given relative to the a-basis in columns and the d-basis in column 3; in the theory all indices are relative to the &basis. In column 1 the magnitudes twinning shears are also included; note that modes 5-7 more complex atomic shuffles than modes 1-4.
1 and 2 needed of the involve
M.A.
Choudhry,
A.G. Cracker
/ 6 to a markwsitic
gether with the magnitudes of the twinning shear strains. in the theory [6,7] the indices of the shear mode are needed relative to the parent crystal basis so that each of these seven a-modes has to be converted to S-indices. using the equations given above, for each of the three correspondences. Hence 21 distinct modes are obtained. Experimental information on the active slip planes of a-plutonium has been reported by Bronisz et al. [13] and Liptai et al. 1141. Most of these planes are consistent with the only feasible Burgers vectors of slip dislocations in the monoclinic crystal structure, which the two most closely packed are [OlO], and [loo],, directions, and (110), arising from stable interactions between these 1121. The resulting 22 a-slip modes are again presented in table 2 and give 66 lattice invariant shear modes when converted to S-indices using the correspondences. Finally slip in the d-phase should occur on { 111) planes in (110) directions. There are 12 crystallographically equivalent variants of this mode and these are given in the third column of table 2. Each of these can be used with the three correspondences providing a further 36 mechanisms. Thus there are in all 123 possible lattice invariant shear modes which can be used as data for the CRAB Tahle 3 Predictions
of the crystallographic Mode
1
2 3
a-T; I: 7 a-T; III; 3 a-T; III; 4
121
in plutonium
theory. Also, each set of data gives two values for the magnitude of the lattice invariant shear (g) and of the shape defo~ation (I). Then each pair of values for g and f gives two habit planes (h) and directions (u) of the invariant plane strain [7]. Potentially therefore 492 distinct habit plane predictions might be obtained, together with the associated shape deformations and orientation relationships.
3. Results All 123 possible transformation mechanisms which arise from the twinning and slip modes summarised in table 2 have been investigated using the CRAB formulation of the theory of martensite crystallography [6,7]. However many of these mechanisms give imaginary solutions of the equations and others can be eliminated because they involve large values of the shear g and shape deformation f. In particular for a given shear mode only results associated with the smaller of the two values of g and / have been considered and, even for these, upper limits of g = 0.20 and f = 0.42 were imposed. For the twinning mechanisms only positive val-
features f
g
transformation
h
u
0.129 0.065 0.053
0.316 0.323 0.318
0.328. 0.456. 0.826 0.824. 0.529, 0.199 0.831, 0.482. 0.275
0.174, - ,950. - .951,
9
#
0.8s 1.94 1.66
2.34 2.40 2.44
-.986 - ,969 -.988 - ,970 -.964 0.088 0.164 0.362 0.201 - .224 0.292
5.55 4.06 3.49 2.57 3.89 2.40 2.17 4.18 2.17 2.29 2.91
4.98 1.37 5.09 2.23 5.15 2.69 2.11 I .66 1.71 1.48 1.03
0.185. -.929 0.123, -.969 0.322, 0.127 0.178. 0.213 0.264, 0.091 0.210, 0.252
6.01 4.46 1.77 2.57 2.06 1.77
1.77 4.12 2.69 1.60 4.29 2.00
0.240, - ,956 0.262. 0.164 0.245, 0.185
4 5 6 7 8 9 10 I1 12 13 14
a-s: a-s: a-s; n-S: a-s; a-s; a-s; a-s; a-s: a-s; (Y-s;
I; 8 I; 10 I; 12 I: 14 I; 17 III; 9 III; 13 III: 17 III; 18 III; 21 III; 22
- .I35 0.155 - .156 0.135 0.178 0.157 0.097 -.I32 0.072 0.067 - ,085
0.340 0.386 0.316 0.389 0.429 0.396 0.363 0.330 0.349 0.343 0.332
0.344, 0.626, 0.699 0.351, 0.637. 0.686 0.424, 0.544.0.722 0.448.0.564, 0.692 0.588, 0.447, 0.673 0.689.0.671.0.270 0.745, 0.610. 0.265 0.863,0.383.0.327 0.774.0.568, 0.277 0.790.0.542. 0.284 0.831, 0.463, 0.305
0.125, 0.105, - .005, 0.244. 0.137. 0.056, 0.076. 0.230, -.149, 0.217, -.974, 0.222. -.962, 0.213, -.912, 0.189. -.956, 0.210. - ,952, - .207, -.935, 0.199,
15 16 17 18 19 20
s-s; 6-S; 6-S; 6-S; 6-S; 6-S;
I; 8 I; 11 III; 4 III: 5 III; 1 III; 8
-.155 -.135 - .083 - .052 - .082 - .055
0.376 0.394 0.359 0.341 0.362 0.336
0.424, 0.431, 0.796 0.431, 0.585, 0.686 0.805, 0.550, 0.219 0.780,0.481,0.398 0.757,0.587.0.285 0.819,0.427. 0.382
0.318, 0.211, - .938, -.960, -.959, -.944,
The first column gives the lattice invariant shear mode, either a-twinning (a-T). a-slip (a-S) or S-slip (S-S). the correspondence, I. II or III, and the mode number as defined in table 2. The remaining columns give g and f. the magnitudes of the lattice invariant shear and the total shape deformation respectively. k the habit plane, n the direction of the shape deformation. and 6 and C$the angles between [liO]s and (loOI, and between (Ill), and (020), respectively.
122
M.A. Choudhry, A.G. Cracker / S to a mnrrensiric transformation
in plutonium
from ill, 011 and particularly 001. The directions of the shape deformation all lie between 8” and 25” from 001. In 15 of the mechanisms the angle 0 between [IlO], and [lOOI, is less than 4’ and similarly 15 have the angle + between (ill), and (020),i less than 4”. In 12 cases both angles are less than this value. The larger angles tend to arise for correspondence I mechanisms. i22
4. Discussion
Fig. 1. Predicted habit plane poles for mechanisms 1 to 20 of table 3 plotted in a standard parent stereographic triangle. Correspondence I and III poles are indicated by closed and open circles respectively.
ues of g less than the twinning shear were allowed. Also, recent experiments by Olsen [9] indicate an orientation relation involving parallelism of the (111) 6 and (020) o( planes and of the [ 1101s and [ 1001 u directions to within 7”. Hence theoretical predictions not lying in this range were also eliminated. On applying the above selection criteria the 492 possible sets of solutions reduced to 20 and these are given, relative to the parent cubic basis, in table 3. Three of these arise from o-twinning, eleven from a-slip and six from &slip. Eight are associated with correspondence I, twelve with correspondence III but none with correspondence II. The magnitudes of the lattice invariant shears g range from 0.053 to 0.178 and are thus acceptably small in all cases. The ten smallest shears are associated with correspondence III modes. Note that when referred to the a basis the magnitudes of g for the three twinning mechanisms become 0.144, 0.063 and 0.053 respectively, all less than the corresponding twinning shears given in table 2. The shape deformation f ranges from 0.316 to 0.425 being larger than in many other transformations. The lowest values are for two correspondence I mechanisms but these are closely followed by four correspondence III results. The habit plane results (/I) are plotted on a standard parent stereographic triangle in fig. 1. They lie in a broad band across the centre of the triangle between 112 and 122. The correspondence I and III solutions are distinguished by different symbols and the former tend to lie closer to ill. However all of the poles lie well away
This application of the theory of martensite crystallography, based on the invariant plane strain criterion, has again demonstrated the benefits of the CRAB formulation [6,7]. Through by-passing intermediate steps in the computation and concentrating directly on observable crystallographic features, this analysis avoids many of the hazards associated with alternative versions of the theory. It is particularly useful when treating transformations involving low-symmetry phases with complex structures. The 6 to Q transformation in plutonium alloys therefore provides an ideal challenging but straightforward application. In this case the analogy between the monoclinic unit cell, and a distorted hexagonal close packed structure is invaluable. This was employed by Cracker [7] when analysing the possible deformation twinning modes of a-plutonium. In particular he compared the relative complexities of the atomic shuffling mechanisms associated with the different twinning modes by applying shears to a pseudostructure with only two atoms in its unit cell. This analogy has been extended in a very imaginative way by Olson and Adler [l] in order to examine the S to (Y phase transformation in plutonium. The s-phase is fee and therefore by using the distorted hcp pseudo-structure for the a-phase they liken the transformation to the phase change in cobalt [2]. In this case the - ABCABC - stacking of close packed (111) planes of the parent fee phase is transformed to the - ABABAB stacking of basal planes of the product hcp phase, by the passage of i(112) Shockley partial dislocations on every second plane. There are three variants of this dislocation on each {ill} plane and these may occur in any sequence so that no unique mechanism arises. In particular the total shape strain can take any value up to 0.35. The three correspondences of Olson and Adler [l] which are used in the present paper arise from the three Shockley partial variants. However, because of the distortion of the hcp structure to produce the (Yplutonium pseudo-structure, they are no longer equivalent. Indeed not only are the pure strains of the correspondences different from each other but so are the
M.A.
Choudhry, A.G. Cracker / S to a mariensitic transformation
complex atomic shuffles which must accompany the lattice deformation in order to locate the atoms on their correct product sites [l]. No attempt has yet been made to analyse these shuffles quantitatively but they must to a large extent control which of the correspondences is favoured. It is possible of course that all three occur together. However Olson and Adler [l] have shown that roughly equal proportions of the three correspondences which as in cobalt, would minimise the shape change, does not give an invariant plane of the transformation. This arises because of the large volume change of almost 20% associated with the transformation. Because of its low crystal symmetry cY-plutonium has a large number of independent deformation modes [12]. In addition the large number of crystallographically equivalent deformation modes of the high symmetry d-phase all have a distinct relationship with the a-structure. Thus there are many potential lattice invariant shear mechanisms for the transformation. Some are of course more favoured than others. Thus a-twinning modes 1 and 2 of table 2 have a smaller shear than modes 3 and 4 and simpler shuffle mecahnisms than modes 5-7. Also a-slip modes l-6 have more acceptable Burgers vectors than modes 7-22. Many of the slip planes of this latter group have been reported experimentally [13,14] but a far more detailed study of these deformation mechanisms is desirable. Slip on { 114) and (118) planes is not easy to accept. The 12 variants of the { 111 } (110) S-slip mode must of course be considered although the classical mechanisms of martensite crystallography involve shears in the product structure. The possible lattice invariant shears of table 2 produced a very large number of real solutions of the theory. These included a-twinning, a-slip and d-slip for all three correspondences. However in restricting the number of solutions to the 20 presented in table 3, by imposing limits on the magnitudes of the shear and shape deformations and using the available very rough experimental information on the orientation relationship [9], many cases have been eliminated. In particular no correspondence II mecahnisms survive and no mechanisms with the favoured a-twinning modes 1 and 2 and a-slip modes 1 to 6. All the remaining modes have acceptably small values of g but it is noticeable that the lowest values tend to be for correspondence III. The magnitudes of f are all close to the value of 0.35 associated with the cobalt transformation. It is therefore difficult to select a particular mechanism by using the criterion of small values off. Transformation twinning modes may of course be distinct from those occurring during deformation and in particular modes in which the twinning plane corre-
in plutonium
123
sponds to a mirror plane of the parent phase, or the twinning direction to a 2-fold axis might be favoured [ 151. These conditions are necessary if the two correspondences between the parent phase and the two components of the twinned product are to be variants of each other [2]. Of the three twinning results in table 3 this is only valid for mechanism 2. Thus with correspondence III, twinning mode 3 arises from (113) [ilO] in the a-phase, having a 2-fold shear direction. This particular mechanism has a low value for g corresponding to less than one-sixth of the product plate being twinned. The magnitude of f is relatively small and the habit plane is “{258}“, which as shown in fig. 1 is near the centre of the stereographic triangle. The direction of the total shape deformation is about 18” from (100). The two angles 0 and $J are about 2”, far less than the reported experimental limit of 7”. Note that these results are identical to those for a-twinning mode 1 of table 2 except that the sign of g is reversed and hence becomes unacceptable as the twinning shear must be positive. Mechanism 2 is therefore the one which, very tentatively, the results of the present investigation suggest may be responsible for the 8 to a transformation in plutonium alloys. Clearly more experimental work on the transformation is needed to see whether this suggestion is acceptable. In particular determination of the habit plane would be invaluable especially as pole 2 in fig. 1 is fortuitously on the edge of the scatter of predictions. More refined measurements of the orientation relationship and the magnitude of the shape deformation would probably not help as many of the predictions for the former are about 2” and for the latter about 0.35. Clearly the key experiment would be to determine the lattice invariant shear mode using electron microscopy but on this particular material this may be beyond available experimental techniques. In conclusion therefore many mechanisms seem to be acceptable for the 6 to a martensitic phase transformation in plutonium alloys. In practice several of these may occur but at present the most likely is twinning of the product phase on the (001) plane, with the parent and product structures related by correspondence III. Further theoretical work on the atomic shuffle mechanisms associated with the correspondences and further experimental work on the crystallograhic features of the transformation is needed. Acknowledgements
The authors are indebted to P.H. Adler, G.B. Olson and C. Olsen for stimulating discussions and to the Government of Pakistan for financial support.
124
M.A.
Choudhry,
A.G. Cracker
/ 6 to a murtensitic
References
VI G.B. Olson and P.H. Adler, Scripta Met. 18 (1984) 401. in Metals PI J.W. Christian, The Theory of Transformations and Alloys, Part I (Pergamon Press, Oxford, 1975). [31 W.M. Lomer. Solid State Commun. 1 (1963) 96. 1965. Eds. A.E. Kay and M.B. [41 B. Spriet, in: Plutonium Waldron (Chapman and Hall, London, 1967) p. 88. IS?A. Goldberg, R.L. Rose and J. Shyne, J. Nucl. Mater. 55 (1975) 33. 161A.F. Acton, M. Bevis, A.G. Cracker and N.D.H. Ross. Proc. Roy. Sot. (Lond.) A320 (1970) 101. f71A.G. Crwker, J. de Physique 43 (1982) C4-209.
transform&on
m plutonium
[8] M.A. Choudhry and A.G. Cracker, Adv. in Ceramics, in press (91 C. Olsen. private communication. [IOj W.H. Zachariasen, Acta Cryst. 5 (1952) 660. [Ill A.G. Cracker and N.D.H. Ross, in: The Mech. of Phase Trans. in Cryst. Solids (Inst. of Metals, London. 1969) p. 176. [12] A.G. Cracker, J. Nucl. Mater. 41 (1971) 167. [13] SE. Bronisz and R.E. Tam, in: Plutonium 1965. Eds. A.E. Kay and M.B. Waldron (Chapman and Hall. London. 1967) p. 558. [14] R.G. Liptai and R.J. Friddle, in: Plutonium 1970 and the Other Actinides (Met. Sot. AIME. 1970) p. 406. [15] N.D.H. Ross and A.G. Cracker. Scripta Met. 3 (1969) 37.