Characterizing dislocation loops in irradiated polycrystalline Zr alloys by X-ray line profile analysis of powder diffraction patterns with satellites

Satellites generated by small dislocation loops in polycrystalline proton- or neutron-irradiated Zr alloys are characterized by X-ray line profile analysis. The convolutional multiple whole-profile procedure has been extended to characterize irradiation-induced satellites around fundamental Bragg peaks.


Size broadening of diffraction profiles
With decreasing scattering volume the diffraction profiles broaden. This is called size broadening. X-ray or neutron diffraction profiles provide the coherently scattering domain size or crystallite size (Bertaut, 1950). This is often smaller than the grain size obtained by transmission electron microscopy (TEM). In Ungár et al. (S2005) it was shown that the coherently scattering domain size given by X-ray or neutron line profile analysis provides the subgrain or dislocation cell size bounded by small angle grain boundaries or dipolar walls. In the present section the effect of the coherently scattering domain size on the diffraction profiles is discussed.

Size parameters
The definition of the commonly used different size parameters, given in eqs. (12), (13) and (16) below, is presented through a simple case study (Warren, 1969). Let's assume an infinite crystallite plane with the thickness of N atoms. According to the theory of kinematical X-ray scattering, the line profile of this special crystallite is: where x = Ga, G = g+g, g is the diffraction vector, g is a small vector, and a is the unit cell vector chosen to be perpendicular to the plane of the crystallite. The function sin 2 (Nx)/sin 2 (x) describes the shape and position of the peaks in this special case. This function is plotted for different values of N in Fig. 1. It has a maximum value at positions: x = n, n∈ℤ. This condition Fig. 1. The function sin 2 (Nx)/sin 2 (x) plotted close to its first maximum for different values of N. As N tends to infinity, the curve becomes a delta function.
is equivalent to the Laue equations. The maximum value of this function is lim →0 2 2 =N 2 . For large values of N this profile function can be approximated by the following simple function: The FWHM value of this function is given by: This transcendent equation can be numerically solved for Nx: the solution is Nx = 1.39. Consequently, the FWHM, i.e. 2x, is reciprocally proportional to the number of lattice points perpendicular to the diffracting plane: This means that the profile function becomes narrower as the crystallite becomes thicker. The integral breadth of this curve is: A conventional  -2 diffractometer measures the intensity parallel to the direction of the g diffraction vector as a function of s = |g|. The FWHM and integral breadth values of the I(s) intensity function as a function of s can be expressed as: , , where LG,a = Nacos((G,a)∡) is the projection of the crystallite width in the direction of the diffraction vector. This means that by measuring the widths of the peak profiles, only the crystallite size parallel to the diffraction vector can be determined. For the reflection h00, LG,a is equal to the thickness of the crystal, i.e. either to 0.9/FWHM or to 1/, respectively. The following two size parameters can be defined generally for an arbitrary I(s) intensity profile: Eq. (12) is the Scherrer (S1918) equation. The more rigorous theoretical description is done by the Fourier transform of the intensity profile Warren (1959): Eq. (15) shows that that LG,a can be determined from the initial slope of the Fourier transform. The size parameter L0 is generally defined for an arbitrary I(s) intensity profile as the initial slope of the A S (L) Fourier transform (Warren, 1959): .
The definition of L0 is illustrated in Fig. 6 below for the case of spherical crystallites with lognormal size distribution. The different size parameters usually satisfy the following relation (Langford & Wilson, S1978): D ≥ d ≥ L0. For spherical crystallites d and L0 are proportional to the volume-and area-weighted average crystallite sizes, respectively (Langford & Wilson, S1978): , . (18)

Size distribution functions
In the previous section the size broadening of a single crystallite has been discussed. A polycrystalline or fine powder sample consists of many crystallites with different sizes where the size distribution function has to be implemented. By selecting the proper size distribution and assuming a realistic crystallite shape, the size broadened profile can be calculated on a theoretical basis. Several distribution functions can be used to describe the size distribution of crystallites (Langford et al., S2000;Scardi & Leoni, S2002). Among these, one of the most flexible one is the lognormal size distribution (Aitchison & Brown, S1957). The validity of this function was confirmed by several observations and successful applications (Valiev et al., S1994;Terwilliger & Chiang, S1995;Krill & Birringer, S1998, Ungár et al., S1999;Langford et al., S2000). The Gamma distribution (Arley & Buch, S1950) is also suitable to describe the experimental size distributions. York (S1999) proposed another distribution. These distributions are discussed below.
(i) The lognormal distribution. According to TEM observations (Aitchison & Brown, S1957) this is the most commonly used size distribution of particle size in a fine powder. It can be shown that a milling procedure leads to a lognormal size distribution (Hinds, S1982). Therefore, the lognormal distribution is widely used in microstructural investigations. It is obtained by substituting the variable of a normal distribution with its logarithm. When one applies the lognormal distribution to describe the size distribution of crystallites, this means that the logarithm of the crystallite size follows a normal distribution. The density function of the lognormal size distribution has the following form: . ( where m and  are the parameters of the distribution, logm is the median and  the variance of the normal distribution. The parameters m and  are called "median" and "variance" of the lognormal size distribution. In the CMWP procedures ( where a and b are the parameters of the distribution and (x) is the Gamma function.
(iii) York's distribution. The York distribution (York, S1999) was obtained by assuming a normal growth phenomena. The density function of the York distribution has a form similar to the Gamma distribution: , where a and b are the parameters of the distribution and (x) is the Gamma function.
We note that Leoni & Scardi (S2004) proposed a bar-diagram for the size distribution density function and each individual column height of this diagram is fitted independently in their pattern-refining procedure. Since this is an ad-hoc, experimental distribution its discussion is beyond the scope of this theoretical section. Langford et al. (S2000) have shown that the above discussed size distribution functions correlate well with experimental X-ray diffraction profiles and, that it is difficult, if not impossible, to distinguish between size distribution functions on an experimental basis.
2.3. Determination of the size profile (Ribárik et al. S2001;Ribárik et al. S2004;Ungár et al. S2001) Assuming a particular crystallite shape and crystallite size distribution, one can determine the theoretical size profile. Bertaut (S1949) and Guinier (S1963) have shown, that the size profile of a powder specimen consisting of crystallites with arbitrary size and shape can be determined as follows: (a) the crystallites should be divided into columns parallel to the diffraction vector g, (b) the "size profile" is obtained as the volume-weighted sum of the intensity profiles normalized by their integral intensities corresponding to each column. The size profile of a column with area Ai and height Mi, normalized by its integral intensity: .
By summing up the contributions from all columns of all crystallites using the volume of the column as weight, the intensity distribution becomes: .
Introducing g(M)dM as the sum of the volumes of the columns with height between M and M+dM of all crystallites: .
Using this quantity, the intensity distribution can be written as: , Eq. (25) shows that the size profile can be obtained by determining g(M)dM, which depends on the crystallite shape and the size distribution of the crystallites. Now we calculate the size profile according to the lognormal size distribution for (i) spherical and (ii) ellipsoidal crystallite shapes (Ribárik et al. S2001;Ungár et al. S2001). Latter is a simple and general description of deviations from spherical shape. It also accounts for anisotropic size broadening as a function of hkl. For a particular crystallite shape g(M)dM is determined first for one crystallite. The calculation is based on the geometrical properties of the crystallite shape. The size profile is obtained by summing up for all crystallites using the crystallite size distribution function. In the case of spherical crystallites and lognormal size distribution, g(M)dM is obtained as: , where we used the notations of Fig. 3.
Differentiating eq. (26) we obtain: 2ydy =−MdM/2. Therefore for one crystallite: Since f (x)dx is proportional to the number of the crystallites with diameter between x and x+dx and all the crystallites with diameter x ≥ M contain the column length M: .
Using the distribution density function in eq. (19), this integral can be expressed as: , where erfc is the complementary error function: .
For all crystallites in the specimen g(M)dM can be written as: .
Using eq. (25)   Using eq. (34) the Fourier transform of the size profile-function in eq. (33) can be given as: .
Using substitutions and partial integration the above integral can be simplified: Dividing A S (L) by its maximum value, the normalized size Fourier transform is obtained. The maximum value of A S (L) is: .
An example for the plot of the size Fourier transform is shown in Fig. 2.6. Figure 6. The theoretical Fourier transform of a size profile for spherical crystallites with lognormal size distribution. The initial slope defining L0 is also indicated in the figure.
We note that the Fourier transform of the size profile can also be obtained from the Patterson function of the scattering object (Guinier, S1963; Zilahi et al., S2016): , where (r)=1 if r ∈"scattering object" otherwise (r)=0. The two different calculations provide exactly the same result for A S (L) (Ribárik et al., S2001). The different size parameters can be obtained as functions of m and  using the explicit forms of I S (s) and A S (L). The derivation of A S (L) gives the L0 size parameter: . (39) The maximum value of the size profile is: .
The integral of the normalized size profile gives the size parameter d: .
The arithmetic-, area-and volume-weighted mean crystallite size values are: 2.4. Anisotropic size broadening (Ribárik et al. S2001;Ribárik et al. S2004;Ungár et al. S2001) If the crystallite shape is spherical, the size function is isotropic and thus independent of the hkl indices. However, in the case of anisotropic crystallite shape the size function becomes anisotropic and hkl dependent, cf. (Langford et al., S1993). Now we calculate the size profilefunction for spherically ellipsoidal shape and lognormal size distribution (Ribárik et al., S2001;Ribárik et al., S2004). The determination of the g(M)dM expression is done in the following steps taking into account the notations shown in Fig. 7.
1. Construction of the equation of the ellipsoid with ellipticity =c/a in the eigensystem denoted by x′ , y′, z′. 2. Transformation by rotation of the angle  around the axis x into the system x, y, z, where  is the angle between the diffraction vector and the revolution axis of the ellipsoid and z is parallel to g.
where h(,) is given by the equation of M(x, y). We note that in the case of a sphere: h(1,)=1. 6. Calculation of the maximum column length Mmax: .
7. Summing up for all crystallites with the column length M and using the lognormal size distribution density function f (x): .
The size profile-function obtained in this way is identical to the one corresponding to spherical crystallites with the only difference that the m parameter becomes hkl-dependent: , where mA is the now the m parameter of the size distribution. If the relative orientation between the unit cell vectors and the axis of revolution of the ellipsoid are known, coshkl can be expressed with the hkl indices. For cubic crystals, if the axis of revolution is parallel to the unit cell vector a: .
For hexagonal crystals, if the axis of revolution is parallel to the unit cell vector c: . We note that Scardi & Leoni (S2002) calculated the hkl dependent size broadening for different polyhedra, which is another possibility to introduce size anisotropy.

Strain broadening
In a real crystal, due to the lattice defects, the atoms are displaced relative to their ideal position. For this reason the reciprocal space may also be distorted and the condition of diffraction is satisfied not only in the ideal positions of the reciprocal lattice points, but in a finite volume in their proximity. This effect is called strain broadening. Warren (1959) gave the Fourier transform of the X-ray line profile if size and strain effects are present simultaneously: where the strain Fourier coefficients can be expressed in the following form: where g is the absolute value of the diffraction vector,  , 2  is the mean square strain (mss) depending on the displacement of the atoms relative to their ideal position and the brackets indicate spatial averaging. According to the continuum theory of elasticity, the longitudinal strain parallel to the direction of the g diffraction vector is defined as (Wilkens, 1970): where u is the displacement field and L is the distance of the lattice points r+ 2 | | and r-2 | | . With this the mss is obtained as: .
For discrete atoms, the strain, g,L is defined as: , where j and j′ are the indices of the atoms in the direction of g, n = j − j′, L = nl0 and l0 is the distance between two atoms in the g direction (Warren, 1959). The mss is obtained as the square of the strain averaged over all atomic pairs with the distance of nl0: .
Several authors worked on the determination of the mean square strain, including Warren (1959), Krivoglaz & Ryaboshapka (1963) and Wilkens (1970). Warren (1959) assumed either random atomic displacements and/or stacking faults. Krivoglaz (1969) and Wilkens (1970) assumed dislocations as the main source of strain broadening. Krivoglaz & Ryaboshapka (1963) assumed a totally random distribution of dislocations in the entire crystal and obtained the following expression for small L values: , where D is the crystal size,  is the dislocation density, C is the contrast factor of dislocations (see more details below) and b is the Burgers vector of dislocations. The problem with this formula is that (i) it diverges as D tends to infinity and (ii) using this strain function, the strain Fourier transform does not decay as L tends to infinity. For small L values, however, this logarithmic expression enables to estimate the dislocation density (Ribárik, 2008). The logarithmic behavior of the mss for small L values was derived by a rather exact method by Groma (1998Groma ( , 2003, where it was shown that this is a general property of any dislocation configuration.

The Krivoglaz-Wilkens model of strain broadening caused by dislocations
There is a close correlation between strain broadening and the elastic stored energy of dislocations (Wilkens, S1969;Kocks & Scattergood, S1969;Groma, 1998). Based on that, Wilkens realized that the logarithmic singularity in eq. (55) is similar to the logarithmic singularity in the elastic stored energy of dislocations (Nabarro, S1952). In the case of the elastic stored energy of dislocations the logarithmic singularity is renormalized by introducing the effective outer cut-off radius of dislocations, Re (Nabarro, S1952). The physical meaning of Re is that the dislocations strain fields are screened in a dislocation ensemble and therefore do not reach out to longer distances than Re from any arbitrary position within the ensemble. Wilkens improved the Krivoglaz model by replacing D by R * in eq. (55). The restrictedly-random dislocation distribution was introduced, where the crystal is assumed to consist of separate regions with diameters of R * . Each R * diameter region consists of pairs of plus and minus straight parallel screw dislocations of the density  distributed randomly within these regions. There is no interaction between dislocations outside of these regions. Wilkens (1970) derived the mss in the entire L range: , where f is the strain function. In the following we call f the Wilkens function. It was developed numerically in the entire L range given in eqs. (A6-A8) in Appendix-A in Wilkens (1970): (57) where f( * )=f(*) and =  Fig. 8. In the Wilkens function the same logarithmic term is present as in the Krivoglaz model, but it does not diverge with the crystallite size, since it depends on the correlation-length parameter, R * , renormalizing the logarithmic singularity in eq. (55). Both the Krivoglaz and Wilkens models reveal a logarithmic singularity at small L values. This, however, has no significant effect on the shape of line profiles considerably because in the Fourier transform of the strain profiles, A D (L) the mss is multiplied by L 2 compensating this singularity to a great extent: L 2 logL→0, if L→0. We note that in most practical cases instead of R * the effective outer cut-off radius Re=R * exp(2) is used. The reason for this is that in the paper of Wilkens (1970) R * was used for easing calculus-technical processes. However, the physical quantity related to true distances is Re (Ungár et al. 1984;Wilkens, 1988;Hecker et al., 1997).
Inserting eq. (56) into (50)  The actual value of R * or Re depends on the actual value of the dislocation density, . In order to avoid this ambiguity Wilkens (1970) introduced the dimension free dislocation arrangement parameter: The value of M* or M is large when the dislocations are arranged uncorrelated randomly and the related strain fields are of long-range character. On the contrary, M* or M is small when opposite sign dislocations are in strong correlation close to each other and the related strain fields, due to screening, are of short range character. A schematic illustration is shown in in Fig. 9, where 12 dislocations are randomly distributed in Fig. 9a and in strong dipole configuration in Fig. 9b. In the case of random distribution the strain fields reach out much further than the average dislocation distance, ddisloc, whereas in the case of strong dipole configuration the strain fields are strongly screened and Re becomes shorter than ddisloc. The M value is in direct correlation with the average dislocation distance: The random or the strong correlated dislocation arrangements can be called weak or strong dipole character. In the first case M is larger than unity, M≫1, whereas in the second case it is close to unity or even smaller: M1.
(c) (d) Due to the reciprocity of length scales in crystal and reciprocal space, when the strain fields are of long or short range character the tail regions of diffraction peaks decay faster or slower and the related M or M* parameters are large or small, respectively. Fig. 10 shows strain profiles for the same dislocation density, , but for different M* values. The Fourier transform of the strain profile given by the Wilkens model is real, therefore its inverse Fourier transform, which is the strain profile itself, is symmetrical. Strain profiles are, however, not necessarily symmetrical. For example in the case of tensile deformed Cu single crystals (Ungár et al., 1984;Mughrabi et al., 1986;Gaál, 1973Gaál, , 1976Gaál, , 1984Jakobsen et al., 2006;Jakobsen et al., 2007) or in tensile deformed lath martensitic steels (Ungár et al., 2017;Harjo et al, 2017;Ódor et al., 2020) strain profiles are characteristically asymmetric. By characteristically asymmetry we mean that the peak asymmetry is of the opposite sense in patterns measured on planes normal (axial-case) or parallel (side-case) to the tensile direction. Characteristic peak asymmetry was explained by the composite model of heterogeneous microstructures developed by Mughrabi et al. (1986) for dislocation cell structures. A more fundamental theoretical description was given by Groma (1998) taking into account dislocationdislocation correlation and the formation of polarized dislocation dipoles.

Strain anisotropy: the concept of contrast factors
Strain anisotropy means that the broadening of the profiles show an anisotropic behavior as a function of the hkl indices: the width of the profiles is not a monotonous function of the length of the diffraction vector or its square, see for example the Williamson-Hall type plot (Williamson & Hall, 1953) of ball milled WC (Gillies & Lewis, S1968) in Fig. 11 or plastically deformed copper in Fig. 12a. Dislocations are extremely anisotropic lattice defects giving contrast in a diffraction experiment only under specific diffraction conditions. In TEM the visibility criterion of dislocations is gb0. The schematic drawing in Fig. 11 shows the same dislocation in two different diffraction conditions. One, (a), where g and b are parallel and the curved reflecting lattice planes are expected to cause large strain broadening. The other, (b), where g and b are normal to each other and the non-affected straight reflecting planes leave diffraction peaks narrow.
Line broadening can be visualized by the FWHM or the integral breadths,  (both in reciprocal space coordinates, K=d*=1/d) vs. K (Stokes & Wilson, 1944;Hall, 1949;Williamson & Hall, 1953): where D and  are related to the crystallite size and the microstrain. Line broadening is assumed Figure 11. The squared integral breadth,  vs. the squared d* of WC X-ray diffraction patterns. WC was ball-milled to different periods of times up to 16 h (Gillies & Lewis, 1968). K √ ̅ , where ̅ is the average contrast factor of Cu (Ungár & Borbély, 1996).
to be caused by microstrain whose spatial average is zero. Line broadening does not change lattice spacings. Therefore it is more appropriate to use √〈 , 2 〉 instead of  in eq. (14): In a texture free polycrystalline sample, an ideal powder specimen or in a single crystal where all possible slip systems are equally populated the contrast factor has to be isotropic for a particular hkl reflection. This means that the individual contrast factors, C have to be averaged over all possible permutations of hkl (Ungár & Tichy, 1999). Figure 13. The two schematic drawings (a) and (b) show the same edge dislocation (symbolized by the reversed T) with the lattice plane traces around them. The green arrow lines indicate in and outgoing radiation beams reflected on the blue highlighted planes. In (a) the reflecting planes are curved and gb0, thus strong strain broadening is occurring. In (b) the reflecting planes are unaffected and gb=0, in this case no strain broadening is occurring (this is true only schematically).
The longitudinal strain parallel to the diffraction vector g is [25]: where u is the displacement field of dislocations and  is the direction parallel to g. On the right hand side of equation (56) the only hkl dependent term is C. Tus the hkl dependence of C can be obtained from the expression: where =gradu is the distortion tensor. The pointed brackets indicate averaging over the crystal volume. Relation (64) shows that Cg 4 is a fourth order polynomial of the hkl indices: The average values of the contrast factors, C , is obtained by averaging P4(hkl) over all possible permutations of the hkl indices of any particular reflection. This means that we have to find the invariants of P4 with the requirement that they satisfy the symmetry of the particular crystal structure. For cubic crystals the only fourth order invariant except the trivial g 4 is: With this where A and B are constants. Using the notation: the average contrast factors of dislocations for cubic crystals will be: The constants A and B depend on the elastic constants of the crystal and the type of dislocations, e. g. edge and/or screw character (Ungár et al., 1999;Borbély et al., 2003;Martinez-Garcia et al., 2008). If B is not zero eq (69) can be given as: where ̅ h00 is the average contrast factor of the h00 reflections and q is a constant depending on the elastic constants of the crystal and the dislocation type. The q parameter is shown in Fig. 14 for edge and screw type dislocations in fcc and bcc crystals as a function of the elastic anisotropy, Ai, of crystal (Ungár et al., 1999). Note that for <h00> type Burger vectors B is zero, therefore in this case only eq. (69) is valid. In the case of a particular specimen at the corresponding Ai anisotropy parameter value (where Ai=2c44/(c11-c12) and c11, c12, and c44 are the elastic constants of the material) there is a window for the q parameter between the qedge and qscrew values belonging to edge or screw character dislocations. From the experimentally determined q parameter value the fraction of edge or screw character dislocations, fedge or fscrew, can be determined: For hexagonal crystals the average contrast factors are (Ungár & Tichy, 1999;Dragomir & Ungár, 2002): and a and c are the lattice constants of the hcp crystal.
For orthorhombic crystals the average contrast factors will be (Ungár & Tichy, 1999;Ribárik, 2008): ̅ = ̅ ℎ00 ( 0 2 + 1 1 2 + 2 2 2 + 3 3 2 + 4 4 2 + 5 5 2 ) , where (74) The constants Ch00 and Chk.0 are calculated on the basis of the crystallography of dislocations and from the elastic constants of the crystal (Ungár et al, 1999). The q, a1,a2, . . . and a5 parameters are global for all reflections in a particular diffraction pattern. They are related to slip modes and edge or screw dislocation character. Borbély et al. (2003) developed a freely available numerical procedure, ABIZC, for the calculation of individual or average contrast factors taking into account the elastic constants of the material, the lattice parameters and the relative orientations of the l, b, n and g vectors.
The CMWP procedure provides the weighted sum of squared residual (WSSR) and the goodness of fit (GoF) at the end of the fitting procedure listed in the *.sol file.