J. Appl. Cryst. (2001). 34, 263±270 Daymond and Johnson  Stress-free lattice parameter 263 research papers Journal of Applied Crystallography ISSN 0021-8898 Received 30 October 2000 Accepted 5 February 2001 # 2001 International Union of Crystallography Printed in Great Britain ± all rights reserved The determination of a stress-free lattice parameter within a stressed material using elastic anisotropy M. R. Daymond* and M. W. Johnson ISIS Facility, Rutherford Appleton Laboratory, Chilton, Oxon OX11 0QX, England. Correspondence e-mail: mark.daymond@rl.ac.uk A method for the calculation of a stress-free lattice parameter from the analysis of diffraction data from stressed material is discussed, utilizing the elastic anisotropy of the material. The technique is demonstrated using data obtained during a uniaxial tension test on untextured austenitic (face-centred cubic) steel. The uncertainty in the calculated lattice parameter for various choices of number of diffraction peaks and different number of stress levels available for the calculation is considered. It is shown that when all the data are within the elastic regime, an accurate evaluation of the reference lattice parameter can be made. When some data are in the plastic regime, a more limited evaluation is possible. The use of plots of lattice parameter against À hkl [= (h 2 k 2 + h 2 l 2 +k 2 l 2 )/ (h 2 + k 2 + l 2 ) 2 ] as a method for monitoring plasticity as well as freedom from deviatoric stress is demonstrated. 1. Introduction One of the major issues that arises for strain measurements in polycrystalline materials carried out using both neutron and synchrotron X-ray techniques is the determination of a stress- free lattice spacing, or d -zero (d 0 ). The stress-free lattice spacing is usually determined by the measurement of a reference sample, which is assumed to be unstressed, to which the stressed sample is then compared. Typical examples of references include a powder of the material(s), a small `coupon' cut from a duplicate sample, or an area in the sample well away from the stress ®eld to be examined. The stress-free lattice spacing is also often calculated from known boundary conditions, e.g. plane stress, or from stress balance. In the case of a truly unstressed cubic lattice, the same lattice parameter (a 0 ) will be determined whichever diffrac- tion plane (d hkl 0 ) is used for the measurement. However, in a residual or applied stress ®eld, for the majority of engineering materials, differently oriented crystallites produce different strains as a result of the elastic anisotropy of the crystallite structure (Fig. 1). In this paper, we show that the relative values of the lattice parameter a determined from different diffraction peaks within the elastic regime provide suf®cient information to determine the lattice parameter that the material would have if no deviatoric component of stress were present. This will be demonstrated using data collected during a uniaxial load test on an untextured cubic system. In Fig. 2(a), we see that, in the region where the material is elastically stressed, the values of `a' which are calculated vary according to which set of hkl lattice planes has been used to determine the lattice parameter. It is this difference that is at the core of the method, and arises from the following obser- vation. If two lines are de®ned by the points {x 1 , y 1 }, {x 1 , y 2 }, {x 2 , y 3 } and {x 2 , y 4 } (as shown in Fig. 2b), then the `y' value at which they intersect (y 0 ) is totally independent of the values of x 1 and x 2 . The value of y 0 is in fact given by y 0 y 1 y 4  y 2 y 3 =y 1  y 4  y 2  y 3 : 1 Thus if we consider the `y' values to be the values of `a' determined by measurements on independent lattice planes and the values of `x' the different stress values at which they are made, we see that an ignorance of the actual stress values (x 1 , x 2 ) is no bar to determining the value of a 0 at which the stress is zero. The only requirement is that the pairs of values (y 1 , y 2 ) and (y 3 , y 4 ) are recorded at the same stress values. Thus, in principle, a 0 values may be obtained in a material where the composition is unknown, provided this composition does not vary, and no `stress-free' sample is available, simply by measuring the `a' value within the same gauge volume for different hkl values and at different, but unknown, stresses. Thus, for cubic materials, the measurement of lattice para- meters from a minimum of two different diffraction peaks at a minimum of two stress values allow a stress-free lattice para- meter (a 0 ) to be calculated. This is possible even though the measurements have been made at unknown stress values and the individual diffraction elastic constants for the two diffraction planes are unknown. A more general analysis is possible using multiple diffraction peaks and multiple stress levels, even where these are unknown, which provides an improved accuracy. The ideas behind the approach are described in this paper, and practical implications in terms of the number of measurements required to provide a given measurement accuracy, the effects of plasticity and other limitations are explored.