Method of calculating the structure of an inhomogeneous sample
View Patent ↗A method is provided of calculating the structure of an inhomogeneous sample in which an electron beam is used to cause excitation of x-rays from the sample under known conditions of beam energy and geometry with respect to the sample. Notably the beam current is unknown. Measured x-ray intensity data for the sample corresponding to one or more sets of beam conditions and beam currents are firstly obtained, together with comparative x-ray intensity data for samples having known structures. A beam current factor for each beam condition is estimated and effective x-ray intensity data for each of the sets of conditions are then calculated using the measured and comparative x-ray intensity data and the beam current factor. The structure of the sample is then calculated for each of the sets of conditions using the effective x-ray intensity data. Predicting x-ray intensity data are produced corresponding to the calculated structure and compared with the effective x-ray intensity data. These steps are repeated using revised beam current factors until the predicted and effective x-ray intensity data achieve a predetermined similarity condition.
1. A method, implemented in a processor, of calculating the structure of an inhomogeneous sample in which an electron beam is used to cause excitation of x-rays from the sample under known conditions of beam energy and geometry with respect to the sample, and wherein the beam current is unknown, the method comprising:
i) obtaining measured x-ray intensity data for the sample, the data corresponding to one or more sets of beam conditions and beam currents, and inputting the measured x-ray intensity data into the processor;
ii) obtaining comparative x-ray intensity data for samples having known structures and inputting the comparative x-ray intensity data into the processor;
iii) estimating, in the processor, a beam current factor for each beam condition;
iv) calculating, in the processor, effective x-ray intensity data for each of the sets of conditions using the measured and comparative x-ray intensity data and the beam current factor;
v) calculating, in the processor, the structure of the sample for each of the sets of conditions using the effective x-ray intensity data;
vi) predicting, in the processor, x-ray intensity data corresponding to the calculated structure;
vii) comparing, in the processor, the predicted and effective x-ray intensity data; and
viii) repeating steps (iii) to (vii) in the processor using beam current factors revised in accordance with step (vii) until the predicted and effective x-ray intensity data achieve a predetermined similarity criterion.
2. A method according to claim 1 , wherein the structure comprises one or each of the physical dimensions and composition of the sample.
3. A method according to claim 1 , wherein the sample is a multi-layered sample.
4. A method according to claim 1 , wherein the calculation of the structure according to step (v) uses structure data defining physical parameters and composition parameters of the structure.
5. A method according to claim 4 , wherein part of the structure data comprises unknown physical parameters or unknown composition parameters or unknown physical and composition parameters.
6. A method according to claim 1 , wherein the comparative x-ray intensity data comprises data normalised with respect to a particular element.
7. A method according to claim 1 , wherein the comparative x-ray intensity data are obtained by a method selected from the group of: calculation according to a model, physical measurement of standard samples of pure elemental composition, physical measurement of standard samples of compounds of known composition wherein corresponding pure elemental contributions are calculated.
8. A method according to claim 1 , wherein the processor calculates an initial estimate of the beam current in step (iii) by:
a) using structure data defining an assumed structure of the sample to calculate assumed x-ray intensity data for the sample, the structure data including assumed values for any unknown parameters of physical structure or composition; and,
b) dividing the sum of the values for the assumed x-ray intensity data by the sum of the values of the measured x-ray intensity data for the particular set of conditions and beam current in question.
9. A method according to claim 1 , wherein the x-ray intensity data are expressed as k-ratios and wherein an initial estimate of the beam current in step (iii) is calculated as the inverse of the sum of the measured x-ray intensity data.
10. A method according to claim 1 , wherein the predicted x-ray intensity data are expressed as k-ratios.
11. A method according to claim 1 , wherein the processor performs the calculation of the structure in step (v) by a structure solver model.
12. A method according to claim 1 , wherein the comparison step (vii) includes calculating, in the processor, the sum of the squared differences between the values of the predicted and effective x-ray intensity data.
13. A method according to claim 1 , wherein the similarity criterion is met when the difference between the values of the predicted and effective x-ray intensity data does not reduce upon further repetitions according to step (viii).
14. A method according to claim 1 , further comprising investigating, in the processor, the calculated structure by representing the predicted k-ratios as a vector of M dimension and the unknown parameters of the structure, including the beam current factor, as a vector of N dimension and using numerical differentiation of a theoretical model for x-ray emission so as to form a Jacobian matrix of partial derivatives relating the k-ratios and the unknown structure parameters; and calculating a condition number relating to the degree of solubility of the matrix.
15. A method according to claim 14 , further comprising comparing, in the processor, the condition number with a predetermined threshold so as to indicate to a user whether the method is soluble for a given set of experimental conditions.
16. A computer program comprising;
a non-transitory computer readable storage medium; and
program code, stored on the medium, adapted to perform the steps of the method of claim 1 when the program is executed upon a computer.
17. A system for calculating the structure of an inhomogeneous sample in which an electron beam is used to cause excitation of x-rays from the sample under known conditions of beam energy and geometry with respect to the sample, and wherein the beam current is unknown, the system comprising:
a source of data taken from the sample; and
a processor in communication with the source, the processor being configured for:
i) obtaining measured x-ray intensity data for the sample, the data corresponding to one or more sets of beam conditions and beam currents;
ii) obtaining comparative x-ray intensity data for samples having known structures;
iii) estimating a beam current factor for each beam condition;
iv) calculating effective x-ray intensity data for each of the sets of conditions using the measured and comparative x-ray intensity data and the beam current factor;
v) calculating the structure of the sample for each of the sets of conditions using the effective x-ray intensity data;
vi) predicting x-ray intensity data corresponding to the calculated structure;
vii) comparing the predicted and effective x-ray intensity data; and
viii) repeating steps (iii) to (vii) using beam current factors revised in accordance with step vii) until the predicted and effective x-ray intensity data achieve a predetermined similarity criterion.
18. A system according to claim 17 , wherein the processor is configured such that the structure comprises one or each of the physical dimensions and composition of the sample.
19. A system according to claim 17 , wherein the processor is configured to accommodate a multi-layered sample.
20. A system according to claim 17 , wherein the processor is configured such that the calculation of the structure according to step (v) uses structure data defining physical parameters and composition parameters of the structure.
21. A system according to claim 20 , wherein the processor is configured such that part of the structure data comprises unknown physical parameters or unknown composition parameters or unknown physical and composition parameters.
22. A system according to claim 17 , wherein the processor is configured such that the comparative x-ray intensity data comprises data normalised with respect to a particular element.
23. A system according to claim 17 , wherein the processor is configured such that the comparative x-ray intensity data are obtained by a method selected from the group of: calculation according to a model, physical measurement of standard samples of pure elemental composition, physical measurement of standard samples of compounds of known composition wherein corresponding pure elemental contributions are calculated.
24. A system according to claim 17 , wherein the processor is configured to calculate an initial estimate of the beam current in step (iii) by:
c) using structure data defining an assumed structure of the sample to calculate assumed x-ray intensity data for the sample, the structure data including assumed values for any unknown parameters of physical structure or composition; and
d) dividing the sum of the values for the assumed x-ray intensity data by the sum of the values of the measured x-ray intensity data for the particular set of conditions and beam current in question.
25. A system according to claim 17 , wherein the processor is configured such that the x-ray intensity data are expressed as k-ratios and wherein an initial estimate of the beam current in step (iii) is calculated as the inverse of the sum of the measured x-ray intensity data.
26. A system according to claim 17 , wherein the processor is configured such that the predicted x-ray intensity data are expressed as k-ratios.
27. A system according to claim 17 , wherein the processor is configured to perform the calculation of the structure in step (v) by a structure solver model.
28. A system according to claim 17 , wherein the processor is configured such that the comparison step (vii) includes calculating the sum of the squared differences between the values of the predicted and effective x-ray intensity data.
29. A system according to claim 17 , wherein the processor is configured such that the similarity criterion is met when the difference between the values of the predicted and effective x-ray intensity data does not reduce upon further repetitions according to step (viii).
30. A system according to claim 17 , wherein the processor is configured for investigating the calculated structure by representing the predicted k-ratios as a vector of M dimension and the unknown parameters of the structure, including the beam current factor, as a vector of N dimension and using numerical differentiation of a theoretical model for x-ray emission so as to form a Jacobian matrix of partial derivatives relating the k-ratios and the unknown structure parameters; and calculating a condition number relating to the degree of solubility of the matrix.
31. A system according to claim 30 , wherein the processor is configured for comparing the condition number with a predetermined threshold so as to indicate to a user whether the method is soluble for a given set of experimental conditions.