QUANTITATIVE ANALYSIS OF THE MISORIENTATION DISTRIBUTION AFTER THE RECRYSTALLISATION OF TENSILE DEFORMED COPPER SINGLE CRYSTALS

Recrystallisation experiments in tensile deformed 〈100〉- and 〈111〉-oriented single crystals of high 
purity copper yielded very accurate information about the orientations of recrystallised grains and 
deformation microstructure. For a statistical evaluation of the orientation relationships between the 
dominant recrystallised grains and the deformation microstructure the misorientation distribution 
function was calculated. The most frequently occurring orientation relationships can be described by 
coincidence relationships. Always several coincidence relationships are needed to characterise the 
results. The specimen treatment strongly influences the occurrence of the individual coincidence 
orientation relationships. A particularly preferred growth of grains with a 30° or 40° 〈111〉 orientation 
relationship was not observed.

ships with Y values of 13a, 21a, 19b and 25b.In addition, the less deformed (111) samples showed an orientation relationship characterised by a 145.7 (541)   rotation ( value of 23).The results were purposely described by coincidence orientation relationships because in most cases the deviation of those orientation relationships actually found from these in particular was less than 6  A specific orientation relationship can be described in various symmetrically equivalent ways by rotations around mutual crystallographic axes.In an individual orientation relationship it does not matter which of the possible combinations was employed in the characterisation.The situation is quite different, however, for a frequency analysis of orientation relationships.Then the ambiguity of the various descriptions has to be removed.The combination of smallest angle of rotation with rotational axis in the standard (base) triangle 001-101-111 is such a singular reduction (Mackenzie, 1958).The first statistical analysis which takes these considerations into account was performed on misorientations in rolled copper (Pospiech et al., 1986).
In this paper the statistical method is applied to the analysis of the orientation relationships found in the above mentioned recrystallisation experiments by Ernst and Klement.In the discussion we shall particularly focus on the question whether a characterisation of the experimental orientation relationships by coincidence orientation relationships may be meaningful.

CALCULATION AND REPRESENTATION OF MISORIENTATION DISTRIBUTION
A misorientation can mathematically be represented by a multiplicity of mutually symmetric points lying in an orientation space.Of these points each one lies in a different sub-domain (base domain).Their number depends on the lattice symmetry of the material.In individual cases, a misorientation can be described by different orientation parameters.Their choice is made according to the nature of the problem (Ruer, 1976; Bunge, 1982; Haegner et al. 1983; Frank, 1988).If for instance, as here, one is interested in the characterisation of the boundaries between two crystallites, then it is expedient to choose a description of the misorientation using axis and angle of rotation.Nevertheless, this representation of a misorientation is accompanied by strong non-linearities of the base domain.
In general, when evaluating experimentally determined misorientations, one should of course take the limitations of accuracy of the data into account, caused by errors of measurement.Furthermore, as mentioned in the introduction, comparison of misorientations is only possible after they have been reduced to the same base domain of misorientation space.
If these factors are disregarded in an analysis of misorientations, false conclusions can be drawn in the interpretation of the misorientation statistics.These problems do not arise if the so-called Misorientation Distribution Function (MDF) is applied to the analysis of data.dI --f(r) dr  (1) Here dI is the grain boundary area between grains showing a misorientation r within the interval dr, and I is the total grain boundary area.
The function f(F) can be expanded in a series of generalised spherical functions '"(r), which are invariant with respect to the erystallite symmetry (Bunge and   Weiland 1988): m The coefficients C" may be calculated from the experimentally determined misorientations r (i 1,..., N) according to: C'" 1 exp (-12e/4) exp (-1 + 1)2e2o/4 1 exp (-e/4) ,T '(F,) Here e0 characterises the scattering of a gaussian frequency distribution around each single misorientation F (Pospiech and Liicke, 1975).Its value depends on the accuracy of the measurement.The coefficients C were determined up to lmx 34 according to Eq. (3).The function f(F) was then at first represented in Euler-space for reasons of the series technique.In the last step the MDF was then transformed to orientation space with rotational parameters to, O, p. Thereby to symbolizes the angle of rotation, and p are the spherical polar coordinates of the axis of rotation.
The Misorientation Distribution Function will be represented in the next section with the aid of cross sections to const, of the so-called base domain.
After Mackenzie (1958) the base domain is defined as follows" the angle of rotation is positive and the smallest of all possible angles of rotation.It is called the disorientation angle rod.If both crystallites, whose mutual misorientation is to be calculated, exhibit cubic symmetry, then 0 < rod < 62.8 is true.All the axes of rotation [r, r r, r] lie in the standard triangle 001-101-111.The components of the axes of rotation are given by r >-r->ry >-0 with r =cos p sin , ry sin p sin O, r cos O.

EXPERIMENTAL PROCEDURE AND RESULTS
Since the recrystallisation experiments on tension deformed (100) and ( 111) oriented copper single crystals have been comprehensively described elsewhere (Ernst and Wilbrandt, 1984; Ernst, 1984; Klement, 1987; Klement and Wil-  brandt, 1988), essential details only will be briefly summarized here.(100) oriented single crystals were deformed by 10% and 20% respectively in a strain experiment.The extension of the (111) oriented crystals was 4%.Under these deformation conditions the crystals of both orientations show a uniformly oriented microstructure without deformation inhomogeneities.To generate recrystallisation nuclei the deformed samples were abraded at one end.After annealing, the orientations of the largest recrystallised grains were determined by the Laue technique and the orientation relationships to the deformation mierostructure calculated.Tables 1 to 4 collate the results of the individual experiments.The limit of accuracy of the orientation determinations on recrystallised grains is 1.The orientations of the crystal and with it the orientation of the deformation microstructure can be quoted with the same degree of accuracy.As a result, the uncertainty of the orientation relationships is estimated to be 3 This value is also assumed in the calculation of the MDF for the scattering length eo of the gaussian distribution around each single misorientation Fi.
Using the data in Tables 1 to 4 the MDFs shown in the Figures 1 to 4 were calculated.Since every misorientation in each subdomain of the orientation space is represented by a point, only the base domain is shown.It encloses the whole standard triangle 001-101-111 up to rotation angles of 45 and contracts on further increase of the angles of rotation to a strip along the symmetry line ( 101)-( 111).Individual sections through the base region were taken at 5 intervals; the position of the maxima of the MDF is indicated by the contour lines.
tendency to irregular distribution.In every case, at angles of rotation less than 25 , the (100) direction proves to be the preferred axis of rotation.For larger angles of rotation the axis of rotation is displaced in the direction of the symmetry line ( 101)-(111).A prominent maximum is found for all (100) oriented samples at a rotation angle of 50 with the axis of rotation close to the (331) direction.These rotations also occur in the (111) samples.Upon increasing the angle of rotation to 55 the rotational axes near (331) are still preferred, yet one may observe a tendency for the maximum to broaden in the direction (256).Indeed, for the 10% extended (100) samples the maximum itself is displaced in this III 101 ,0, Figare 3 Misorientation distribution function of the largest recrystallised grains in tensile deformed copper single crystals after annealing ((100)-oriented crystals, 20% elongation, annealing tempera- ture 1080 K; MDF calculated for the data in Table 3, contour lines 1, 2, 4, 7, 11, 16, Regions under the level 1. are dotted.Maxima marked by: A, B, etc., axis notation, avlue of the MDF, value of ).
direction.Angles of rotation of 60 only occur in the 20% deformed (100) samples (annealing temperature 750 K) and in the (111) samples.

DISCUSSION
Evaluation of the individual data using the MDF shows that the results of the recrystallisation experiments on deformed copper single crystals can only be Figare 4 Misorientation distribution function of the largest recrystallised grains in tensile deformed copper single crystals after annealing ((111)-oriented crystals, 4% elongation, annealing temperature 750 K; MDF calculated for the data in Table 4, contour lines 1, 2, 4, 7, 11, 16, Regions under the level 1. are dotted.Maxima marked by: A, B, etc., axis notation, value of the MDF, value of ).
correctly described by a larger number of orientation relationships.It is necessary to investigate to what extent the orientation relationships important to the recrystallisations process can be approximated by coincidence orientation re- lationships.Coincidence grain boundaries are noticeably different from usual grain boundaries with respect to grain boundary energy and mobility in the event of impurities (Rutter and Aust, 1965; Maurer, 1987).The significance of coincidence grain boundaries for recrystallisation has been described elsewhere (Berger et al., 1988).
In accordance with the work of Klement et al. ( 1988), an initial attempt to describe the MDF maxima was made using coincidence orientation relationships with maximum X values of 25.However, with these restrictions the characterisa- tion of the results by coincidence orientation relationships proved to be unsatisfactory.For example, the axes of rotation found at angles of rotation of 50 on or near the symmetry line ( 101)-( 111) cannot be adequately approxim- ated, since for Y. 25 the (221) and the (331) directions each can only once act as an axis of rotation.
As a result of these difficulties a new characterisation of the MDF maxima followed, with the aid of coincidence orientation relationships having values up to 101.The results compiled in Tables 5.1 to 5.4 show that, on this basis, all the MDF maxima can be very accurately described by coincidence orientation relationships.The tables only show those MDF maxima to which more than one measured value is assigned.In the majority of cases the axes of rotation are coincident.Moreover, this evaluation also shows that the coincidence orientation relationships Y. 133, Z 19b, Y. 213, Z 23, and Z 25b used by Klement at al. (1988) only in a few cases describe the MDF maxima, i.e. they cannot be universally used to characterise the results, as had originally been supposed.According to  221)), Z37a (18.9 (100)) and Z 673 (24.4 (111)).Nevertheless, these orientation relationships do not occur equally in all three sets of (100) samples.In all three cases the 25b and the Z 37a orientation relationships turned out to be dominant.
The effect of the sample treatment on preferential occurrence of orientations is indicated above all by the results obtained with the (111) samples.The low degree of deformation of 4% chosen for this sample type results, on the one hand, in the uniform distribution of the orientation relationships on the individual MDF maxima as already described, and on the other hand, it suppresses the Z25b orientation relationship favoured by all (100) samples.The general absence of the 23 orientation relationship in the (111) samples observed by Klement et al. ( 1988) can be explained by the altered evaluation method.As mentioned, the idealisation of certain orientation relationships occurred under the assumption of a maximum Z value of 25, where by an error of 10 was tolerated.Under these presuppositions a series of orientation relationships with Z 23 were idealised which were assigned to various maxima of the MDF on repeated evaluation.Furthermore, the improved evaluation of the data shows that recrystallised grains with a 40 (111) orientation relationship to the strained structure in pure copper are not distinguished by a particularly strong growth.The same conclusion holds for grains with a 30 (111) orientation relationship, since the MDF shows a weak maximum at the appropriate site in the misorientation space for a set of (100) samples only.According to investigations by M6hlmann (1966) on silver, recrystallised grains with a 30 (111) orientation relationship to the matrix exhibit preferential growth.
To sum up: the description of the orientation relationships which dominate the recrystallisation process using coincidence relationships is justified, then as now.However a Z value of 25 is not a meaningful upper limit since the characterisation of certain orientation relationships requires higher values.It is not clear what are the physical reasons for the occurrence of orientation relationships with large .
values.Berger et al. ( 1988) were actually able in some cases to identify the grain boundaries of preferentially growing grains as being unequivocally of low energy.The current understanding of grain boundaries does however not permit these sort of conclusions to be drawn for grain boundaries with high Z values.
Figure I Misorientation distribution function of the largest recrystallised grains in tensile deformed

Table 3
Orientation relationships between the recrystallised grains and the deforma-

Table 4
Orientation relationships between the recrystallised grains and the deforma-

Table 5 .
1 Idealisation of the observed orientation relationships by coincidence orientation relationships (MDF calculated for the data in

Table 1 )
Table5.4 Idealisation of the observed orientation relationship by coincidence orientation relationships (MDF calculated for the data in Table4)