NON-RANDOM ORIENTATION DISTRIBUTION FUNCTIONS WITH RANDOM POLE FIGURES

It is shown, theoretically and with numerical examples, that the orientation distribution function may vary between zero and two or even more times random while a corresponding pole figure is completely random. INTRODUCTION In the literature on textures the opinion has been expressed that from a completely random pole figure one must conclude that the orientation distribution is also completely random. It is the aim of the present paper to show that the orientation distribution function may vary between zero and two times random (or even more) while a corresponding pole fi,gure is completely random. i. Non-random ODF with one random pole fiqure In the case of cubic crystal symmetry and orthorhombic sample symmetry the orientation distribution function (ODF) can be expressed by the series expansion,


INTRODUCTION
In the literature on textures the opinion has been ex- pressed that from a completely random pole figure one must conclude that the orientation distribution is also completely random.
It is the aim of the present paper to show that the ori- entation distribution function may vary between zero and two times random (or even more) while a corresponding pole fi,gure is completely random.i.
Non-random ODF with one random pole fiqure In the case of cubic crystal symmetry and ortho- rhombic sample symmetry the orientation distribution function (ODF) can be expressed by the series expansion, =4 =1%=0 (i) where ' (g) are generalized spherical functions of the ori- entation g which may be expressed for example by the Euler angles.The functions are invariant with respect to the ortho- rhombic sample symmetry and the cubic crystal symmetry.We consider the special choice of the coefficients: for a given value of chosen in the set (12,16,18,20,22) and C 9 0 for {% # I , (3)   In this case the function Eq. (i) takes on the following form f' (g) 1 + C n Tn(g) + C n Tn(g) ( 4) The (hkl)-pole figure corresponding to the general case Eq.

+ C .2 (hkl)
If the coefficient C In is not zero this is a non-random ori- entation distribution function with a random (hkl) pole figure according to the assumption Eq. ( 9).If the function f' (g) in Eq. ( 13) is to represent an orientation density then it must not be negative.
f' (g) > 0 (14)   This condition leads to a restriction for the permitted values of the coefficient C n.If we replace the functions n by their upper limit :.
hkl) The values of the cubic spherical harmonics have been tabu- lated (see Ref. i).With these values one easily obtains the following maximum values for the coefficients C n in Eq. (13) (compare  The term in brackes in Eq. ( 13) takes on positive and negative values of about the same magnitude.If the coefficient C n is chosen as the maximum value according to Eq. ( 18) then the second term in Eq. ( 13) varies between plus and minus 1 and the orientation distribution function f' (g) itself varies between 0 < f' (g) < [Of course other pole figures of the same texture, for example the (h'k'l') pole figure, will generally not be random.] 2. Non-random fiber texture with one random pole fiqure A fiber texture (axially symmetric texture) is com- pletely described by the inverse pole figure (of the fiber axis) (see Ref. i).
In the case of cubic crystal symmetry a fiber texture can thus be represented by the series expansion R(8) The (hkl) pole figure of this texture can be written Phkl($) where PI() are the associated normalized Legendre functions with the coefficients M(1) F I (hkl) 2 21' + 'i +/-C K(hkl) We introduce the analogous specialization to Eq. (2)    are the values given in Eq. (18).The same where C max arguments as in the case of general textures then lead to the upper and lower limit for the orientation density while the (hkl) pole figure is random according to Eq. ( 28).
(Other pole figures of the same texture, e.g. the (h'k' i' pole figure, will generally not be random.) 3. Non-random ODF with two random pole figures The considerations of section 1 can be extended to the case that two random pole figures are required.If in this case the ODF is not to be zero then there must be at least three linearly independent coefficients of the order , i.e.M() 3. We thus set cin # 0, C 2n # 0, C 3n # 0 with taken out of the set (24, 28, 30, 32, 34) and (36) The general ODF Eq. (i) thus takes on the following form (37) in :'In(g) + c2n :'2n(g) + c3n :'3n(g) f"(g) The (hkl) and (h'k'l') pole figure of this texture are PkI(@Y) 1 + F(hkl) K(@y) We postulate that both pole figure Eq. ( 39) and (40) be random This leads to an homogeneous system of two equations [the brackets in Eq. ( 41) and (42) with three unknowns which has a solution of the form Thus the ODF Eq. (38) becomes f" (g) The condition (47) f" (g) > 0 (48) leads then to the condition in 1 (49)

IC max
The function f"(g) Eq. ( 47) is a non-random ODF with random (hkl) and (h'k' i') pole figures.If the coefficient C n is chosen according to Eq. ( 49) then the maximum negative value of the second term in Eq. ( 47) does not exceed i; hence Eq. ( 48) is fulfilled.If we may assume that in this case the maximum positive value is approximately of the same magni- tude then f" (g) varies in the limits 0 < f"(g) < 2 (50) 4. Non-random fiber texture with two random pole figures In a similar way one obtains for fiber textures (51 with the values A and B of Eq. ( 45) and (46).The estimation of the maximum value of the symmetric spherical harmonics Eq. ( 33) yields the estimation of the maximum value of the coefficient C in order that the function R" is not negative The assumption that the maximum positive value of the expression in brackets of Eq. ( 51) be approximately the same as the maximum negative value leads thus to the estimation 0 < R"(8) < 2 (53) while the (hkl) pole figure as well as the (h'k'l') pole figure both are random.
It is easily seen that these considerations can be ex- tended to more than two pole figures.Indeed one can postulate that M() 1 pole figures be random and at the same time the texture varies in the range from zero to about two according to Eq. ( 50) or (53).

Generalizations
In the case of M() 2 and one random pole figure the indices and n could be chosen out of a set of values given in Eq. ( 2) and (3).Instead of the orientation distri- bution function f' (g) Eq. ( 13) we can sum up over all possible values of and n and still the (hkl) pole figure will be random. f"'(g) The non-negativity condition f"' (g) > 0 (55) leads to the following condition for the coefficients which defines a permitted range in the space of the coeffi- cients C In Figure 1 shows a two-dimensional section of the multidimensional space defined by Eq. ( 56).The equality sign in Eq. ( 56) defines a straight line or in the multidimensional case an hyperplane in the space of the coefficients which in- tersects the coordinate axes at certain positive values.
Because only the absolute value of C n enters Eq. (56) the corresponding hyperplanes intersecting at negative values represent further boundaries of the permitted region in the space of coefficients.Every choice of coefficients C n within this region represents a non-random texture with a ran- dom (hkl) pole figure.
Thus far we have considered special orientation distri- bution functions with only such values of that M() was con- stant, and we postulated that M-1 pole figures be random.We now look at the specific case M() 3 and we postulate that one pole figure be random.This leaves two degrees of freedom for every choice of and n.We assume a texture with coefficients according to Eq. ( 36) and (37) but we postulate that only F(hkl) of Eq. (41) be zero [and not F(h'k'l') of Eq. ( 42) ].This leads to the condition 3n The function Eq. (38) can then be written f"(g) 1 + C (g) K I (hkl) :'3n K 2 (hkl) :. (58) If C n and C n are not zero at, the same time then Eq. ( 58) represents a non-random texture with a random (hkl) pole figure.The non-negativity condition together with the esti- mation of the maximum values of n according to Eq. ( 15) leads to the condition for the coefficients (59) which is very similar to Eq. (56).If we set Figure i.Two-dimensional section of the multi-dimen- sional coefficient space defined by Eq. (56).6.The general case of a non-random texture with one random pole fiqure Finally we consider an orientation distribution func- tion according to Eq. (i) and we do not introduce the restric- tions Eq. ( 2) and ( 3) to the coefficients.The (hkl) pole figure is postulated to be random according to Eq. (9).Equation (10) introduced in Eq. -(6) thus leads to the condi- tion M(X)-I_ KU (hkl) CX M (hkl) C u (62) with this condition the distribution function Eq. (i) takes on the following form where I runs over all those values for which M(X) > 2 (64) The non-negativity condition together with the estimation Eq. ( 15) leads to the condition for the coefficients (66) where (67) Equation (66) defines a permitted region in the space of the coefficients which is limited by several hyperplanes defined by the equality sign in Eq. (66).All functions in Eq. (I) with coefficients C 9 satisfying Eq. ( 66) are non-random tex- tures with a random (hkl) pole figure.
The permitted region Eq. ( 66) is so chosen that the non- negativity condition is necessarily fulfilled.
There may however be coefficients corresponding to points outside the region Eq. (66)that do not violate the non- negativity condition, that is to say, that the maximum permitted region may be larger than that defined by Eq. (66).Hence Eq. ( 66) is a sufficient but not a necessary condition for non-negativity of the distribution function.
In the special cases with only very few coefficients C different from zero, e.g.Eq. ( 2) and (3), it was assumed that the term in brackets in Eq. ( 13) takes on maximum posi- tive values of the same order as the maximum negative values.This condition holds for every single function T (g).Hence we obtained the upper bound Eq. ( 19) for the function f' (g).If the series expansion of a distribution function contains however a large number of terms the upper bound of the as- sumed positive values may be much higher than that of the negative values Hence a special choice of coefficients C may exist [certainly outside the region defined by Eq. (67) such that for the function Eq. ( 63) the condition holds 0 < f(g) < fmax (68 with fmax > 2. A function of this type can be constructed if in Eq. ( 63) the coefficient C is given the same sign as the function in brackets takes on at a certain orientation go.Thus we may set for example K (hkl) :-M (g0)] Then all the terms in the sum Eq. ( 63) are positive at the orientation go whereas they will take on positive and negative values at other orientations thus leading to smaller (positive or negative) values.If we put C T 9(g)

9=0
K u (hkl) :'M (g)] M T K (hkl) Min (70) then the maximum negative value of the sum in Eq. (63) will remain above-i everywhere so that the non-negativity condi- tion Eq. ( 68) is not violated whereas fmax will be larger than 2 according to Eq. (68).At the same time the (hkl) pole figure is random as was assumed.
In the case of fiber textures, Eq. ( 20), with one random pole figure we obtain similarly In order that the function takes on a high value at the ori- entation 080 we put For this function a variability range will be obtained 0 < R(8) < Rmax (74) with Rmax > 2 (75) These considerations have shown that it is possible to construct orientation distribution functions with a consider- able range of variation according-to Eq. ( 68) or at least ac- cording to Eq. ( 19) which have one or even more random pole figures.
If f(n)(g) is a so constructed non-random function with n random pole figures then is a non-random function with n pole figures which are iden- tically zero.
[Of course, Af(g) will generally not obey the non-negativity condition.]If f(g) is a certain orientation distribution function then the function f' (g) f(g) + Af(g) (77) has exactly the same n pole figures as the functi.onf(g) which we started with.The two functions cannot be distinguished from the n pole figures under consideration.The function Af(g) thus describes the ambiguity of the orientation dis- tribution determined from n pole figures.The variability range of the function Af (g) may be restricted to a certain degree since the non-negativity condition must hold for f' (g).
The restriction is however the smaller the less f(g) deviates from 1 (the random distribution).
These considerations must be kept in mind when an orienta- tion distribution function f(g) is calculated from pole figure data by methods not based on series expansion such as the meth- 2 Ruer3 In these methods ods described by Williams, or Imhof.a solution f(g) is constructed from pole figures.
There is however no way of estimating the ambiguity range of the obtained solution.
The three methods have been described with one or two pole figures as the starting experimental data.
If these pole figures are random the solution according to the methods 2-is a random distribution function.As has been shown however in this paper the solution can equally well be one of the functions f' (g) or f" (g) constructed above which can vary at least in the region from zero to twice the random density.

Numerical examples
In order to get an impression of the shape of non- random orientation distribution functions with random pole figures according to the above formulae some numerical examples have been calculated and have been graphically represented in the following  All the examples correspond to one random pole figure respectively.
The particular case of functions of one rank only has been considered in section i.They consist of two generalized spherical harmonics Eq. ( 4) the coefficients of which are de- termined according to Eq. ( 17) and (12) Table I gives the numerical calculation of these coefficients starting with the values (hkl) and (hkl) of cubic spherical harmonics for the cases (hkl) (iii) where (hkl) are the in- dices of the pole figure which is assumed to be random.The i n max were already given in Eq. so obtained coefficients IC (18).The choice of the index n is deliberate.Figure 2a shows a distribution function of this type of the rank 12 with n 2 the (i00) pole figure of which is random.The function varies between its minimum value 0.57 and the maxi- mum value 1.43 thus showing that the estimation of the maxi- In which does not violate the mum value of the coefficient IC non-negativity condition Eq. ( 14) was very much on the safe side.
Really the coefficients C n and C n could have been chosen considerably larger than the ones given in Table I (about twice as large) and still Eq. ( 14 would not have been violated.Figure 2b shows a function of rank 22 with n 2 and a randon (i00) pole figure .Here the maximum and mini- mum values are 1.33 and 0.67 respectively.
The case of axially symmetric distribution functions (in- verse pole figures) of one rank only has been considered in section 2. They consist of two spherical harmonics Eq. (25) IA3 LEVELS 3-l.OO 4-1.20 5-.4o l (a) Figure 2. Non-random orientation distribution functions having random (100) pole figures according to Eq. ( 4) with coefficients defined by Eq. ( 17) and Eq. ( 12).a) 12, n the coefficients of which are to be chosen according to Eq. ( 34) and (30) The coefficients correspond to the ones of the non-axially symmetric case but for the factor 24 +i These coefficients are also given in Table I. Figure 3 shows four examples of distribution functions of this type correspond- ing to the ranks 16, 18 and 22 respectively.The random pole figures being (i00), (ii0) and (ill) respectively.The maximum and minimum values indicated in the figures show that in all cases the coefficients could have been chosen larger than the ones of Table I without violating the non-negativity condition.
the case of axially symmetric functions) takes on a large value at the point [080] which also can be described by the Miller indices [uvw].Coefficients C which correspond to this assumption may be obtained according to Eq.. (72) where the factor B takes care of the non-negativity condition.
Coefficients obtained this way for I < 22 are given in Table II for random pole figures of the type (i00), (ii0) and (Iii) the orientation of maximum value [uvw] being [i00], [110] and [Iii] respectively.The quantity B was so chosen that the minimum functional value is equal to zero.It is obvious that a function of this type with (hkl) [uvw] does not exist.Figure 4 shows four examples of functions of this type with various combinations of (hkl) and [uvw].According to the different combinations of (hkl) and [uvw] different maximum values were obtained.The largest value occurs in Figure 4a which corresponds to a random (Iii) pole figure.The maximum functional value of 4.17 is being taken on at the orientation [100].
The degree of the series expansion in Figure 4 was re- stricted to < 22.One can easily estimate that distribution functions with one random pole figure (or even several ones) can be obtained with unlimited maximum functional values.As is shown in Figure 5a the value of the (hkl) pole figure at the angle (in the case of axially symmetric textures) can be obtained from the inverse pole figure by taking the mean value of this figure along a small-circle of radius about the point [hkl].In Figure 5a this is shown for the (iii) pole figure and a texture having its maximum value at the orientation [i00] corresponding to the situation of Figure 4a. Figure 5b is a schematic representation of the functional values of the distribution function along the circle .The mean value of this distribution is the value of the (Iii) pole figure at the angle (from the axis of symmetry).If this pole figure is to take on the value 1 at all angles (random pole figure) then the maximum value fm may roughly be estimated by the condition Here the length of the path L is determined by the number of equivalent directions of the type [uvw] (in this case three directions of the type [100] and the angle between these directions and the normal to the reflecting lattice plane of the (hkl) pole figure [in this case (iii)].If the distri- bution function is represented by a series expansion, b may be taken as the half-value-width of a peak represented by a series of maximum degree .This was called the angular re- solving power corresponding to the series degree .As was shown in Ref. i, p. 109, b decreases to zero if is increased to infinity.Hence, it is obvious from Eq. (78) that non- random distribution functions with infinitely high peak values can be constructed which have one (or even more) random pole figures. 169 equal to zero this is a non-random orientation If C distribution function with a random pole figure according to Eq.(28).As in the case of general textures the condition of non-negativity R' (8) >/ 0 (32) imposes a restriction on the possible values of the coeffi- i The upper bound of the cubic spherical harmonics cient C .may be estimated from the following relation IK()I 24 + 1 iTO(=)l < 4 (33) 1 from which one obtains the upper bound of the coefficient C in Eq. (31).
permitted region for the coefficients similar to that one shown in Figurei.
Table I).