Analysis devices, systems, methods, and programs

Abstract

To provide an analyzer, a system, a method and a program capable of calculating an index indicating an optimization degree of a model for measurement data.SOLUTION: An analysis device 400 calculates an index indicating an optimization degree of a model including information on electron density distribution for measurement data whose probability distribution function the measurement data follows is known, and includes: a measurement data acquisition unit 410 that acquires the measurement data; a calculation data acquisition unit 420 that acquires calculation data calculated from the model; and an index calculation unit 430 that calculates, for a residual defined by a predetermined formula including the measurement data and the calculation data, and an expected value of the residual defined based on the probability distribution function and the predetermined formula, an index GOF including a ratio between the residual and expected value of the residual.SELECTED DRAWING: Figure 3

Description

[Technical Field]

[0001] The present invention relates to an analytical apparatus, system, method, and program for calculating an index indicating the degree of optimization of a model. [Background technology]

[0002] Because X-ray reflectivity analysis deals with a large dynamic range of data, optimization calculations were performed using a logarithmically weighted least squares method (Non-Patent Literature 1). Similarly, CD (critical dimension) measurement using small-angle X-ray scattering analysis (CD-SAXS), which applies the theory of X-ray reflectivity analysis, also employed optimization calculations using a logarithmically weighted least squares method (Non-Patent Literature 2). In these analyses, the R factor, shown in the following formula (1), was used as an indicator of fitting reliability. Obs represents the measured intensity, and Calc represents the calculation intensity. This R factor is also used to evaluate the convergence of the fitting.

[0003]

number

[0004] On the other hand, in powder X-ray diffraction, an index called the S factor, shown in the following equations (2) to (4), was used to determine convergence (Non-Patent Literature 3, 4). S=1 means that the refinement is perfect, and generally, if S is less than 1.3, the analysis result is considered satisfactory. In this way, the S factor can be used to absolutely determine the convergence of the fitting. The S factor, which is an index for determining convergence, is defined as something separate from the minimization function.

[0005]

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[0006]

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[0007]

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[0008] Note that R is the R factor used in powder analysis, and R e is the estimated minimum R factor. Also, n represents the number of data points constituting the measurement profile, and p represents the number of fitting parameters. Also, w is a statistical weight parameter defined as the reciprocal of the error (= variance of statistical fluctuation) included in each data point.

Prior Art Documents

Non-Patent Documents

[0009]

Non-Patent Document 1

Non-Patent Document 2

Non-Patent Document 3

Non-Patent Document 4

Non-Patent Document 5

Summary of the Invention

Problems to be Solved by the Invention

[0010] As described above, when analyzing measurement data with a large dynamic range, such as X-ray reflectance and small-angle X-ray scattering, optimization calculations are generally performed using the least squares method with logarithmic weighting, without normalization by error as per the original definition of the least squares method. In other words, in such analyses, conventional methods have used the R factor calculated from the weighted sum of squared residuals as a criterion for determining convergence.

[0011] However, the value of this R factor changes depending on the quality of the measurement data. For example, data with good statistics will have a smaller R factor value. Also, the longer the measurement time, or the more the data used for fitting is limited to only the parts with relatively high intensity, the smaller the R factor value will be. Therefore, the convergence of the fitting cannot be absolutely determined based on the R factor value alone; only relative evaluation within the same data is possible. Furthermore, when using the R factor to determine whether the analysis has sufficiently converged in X-ray reflectance or small-angle X-ray scattering, it was necessary to rely on the experience of a skilled engineer.

[0012] Furthermore, the above S factor is used to determine the convergence of powder X-ray diffraction. The R factor is included in the S factor. e R wp These are parameters derived from the definition of and the definition of least squares, R e The derivation of the S-factor required the prerequisite of normalizing each data point by error (Non-Patent Literature 5). If error normalization is applied to measurement data with a large dynamic range, such as X-ray reflectance or small-angle X-ray scattering measurement data, the weight of signals in the high-intensity region becomes large, resulting in an analysis that ignores signals in the low-intensity region. In measurements such as X-ray reflectance and small-angle X-ray scattering, signals in the low-intensity region may also contain information necessary for analysis, and these cannot be ignored. Therefore, error normalization is not suitable for measurement data such as X-ray reflectance and small-angle X-ray scattering, and the S-factor has not been applied.

[0013] Therefore, there was a strong need for an absolute convergence evaluation index that could be applied to measurement data such as X-ray reflectance and small-angle X-ray scattering.

[0014] This invention has been made in view of these circumstances, and aims to provide an analysis device, system, method, and program for calculating an index that indicates the degree of optimization of a model with respect to measurement data. [Means for solving the problem]

[0015] (1) In order to achieve the above objective, the analysis apparatus of the present invention employs the following means. That is, an analysis apparatus according to one aspect of the present invention is an analysis apparatus that calculates an index indicating the degree of optimization of a model including electron density distribution information for measurement data whose probability distribution function is known, and is characterized by comprising: a measurement data acquisition unit that acquires the measurement data; a calculation data acquisition unit that acquires calculation data calculated from the model; and an index calculation unit that calculates an index GOF including the ratio of the residual to the expected value of the residual, for residuals defined by a predetermined mathematical formula including the measurement data and the calculation data, and the expected value of the residual defined based on the probability distribution function and the predetermined mathematical formula.

[0016] (2) Furthermore, an analysis apparatus according to one aspect of the present invention is characterized by comprising an index evaluation unit for evaluating the convergence of the GOF.

[0017] (3) Furthermore, in an analytical apparatus according to one aspect of the present invention, the predetermined mathematical formula is characterized in that it includes the sum of the squared residuals between the logarithm of the measurement data and the logarithm of the calculation data.

[0018] (4) Furthermore, in an analytical apparatus according to one aspect of the present invention, the predetermined mathematical formula is characterized in that it includes a weight parameter which is multiplied by a constant when the intensity of the measurement data is less than or equal to a predetermined value, and takes the logarithmic value of the measurement data when the intensity of the measurement data is greater than the predetermined value.

[0019] (5) In addition, in an analytical apparatus according to one aspect of the present invention, the probability distribution function is a Poisson distribution, and the index calculation unit approximates the expected value of the residual with a Gaussian distribution to obtain the result.

[0020] (6) Furthermore, an analysis apparatus according to one aspect of the present invention includes a model creation unit that creates the model and calculates the calculation data based on the created model, the calculation data acquisition unit acquires the calculation data calculated by the model creation unit, the model creation unit outputs the model when the index evaluation unit evaluates that the GOF has converged, the model creation unit updates the model parameters of the model, creates the model and calculates the calculation data when the index evaluation unit evaluates that the GOF has not converged, and the index calculation unit calculates the GOF based on the calculated calculation data.

[0021] (7) Furthermore, a system according to one aspect of the present invention is characterized by comprising an X-ray analyzer comprising an X-ray generating unit for generating X-rays, a sample stage for placing a sample, and a detector for detecting X-rays, and an analysis device according to any one of (1) to (6) above.

[0022] (8) Furthermore, a method according to one aspect of the present invention is a method for calculating an index indicating the degree of optimization of a model including electron density distribution information for measurement data whose probability distribution function is known, the method comprising: a measurement data acquisition step of acquiring the measurement data; a calculation data acquisition step of acquiring calculation data calculated from the model; and an index calculation step of calculating an index GOF that includes the ratio of the residual to the expected value of the residual, for residuals defined by a predetermined mathematical formula including the measurement data and the calculation data, and the expected value of the residuals defined based on the probability distribution function and the predetermined mathematical formula.

[0023] (9) Furthermore, a program according to one aspect of the present invention is a program for calculating an index indicating the degree of optimization of a model including information on the electron density distribution for measurement data whose probability distribution function is known, characterized in that it causes a computer to perform the following processes: acquiring the measurement data; acquiring calculation data calculated from the model; and calculating an index GOF including the ratio of the residual to the expected value of the residual, for residuals defined by a predetermined mathematical formula including the measurement data and the calculation data, and the expected value of the residual defined based on the probability distribution function and the predetermined mathematical formula. [Brief explanation of the drawing]

[0024] [Figure 1] This is a conceptual diagram showing an example of the configuration of an analysis system. [Figure 2] This is a cylinder representing an example of the configuration of a control device. [Figure 3] This is a block diagram showing an example of the configuration of the analysis device according to Embodiment 1. [Figure 4] This flowchart shows an example of the operation of the analysis device according to Embodiment 1. [Figure 5] This is a block diagram showing an example of the configuration of the analysis device according to Embodiment 2. [Figure 6] This flowchart shows an example of the operation of the analysis device according to Embodiment 2. [Figure 7] This is a block diagram showing an example of the configuration of the analysis device according to Embodiment 3. [Figure 8] This flowchart shows an example of the operation of the analysis device according to Embodiment 3. [Figure 9] (a) is a graph showing the pseudo-measured waveform and calculated waveform before model optimization. (b) is a graph showing the pseudo-measured waveform and calculated waveform after model optimization. [Figure 10] This graph shows the changes in the R factor and GOF values ​​corresponding to one diffraction line in the analysis process of the example. [Figure 11] This is a graph showing the measured waveform and the calculated waveform superimposed at a certain stage of the example. [Figure 12] This graph shows the values ​​of all R factors used in the analysis at a given stage. [Figure 13] This graph shows the values ​​of all R factors, RG factors, and GOF at a given stage used in the analysis. [Figure 14] This graph shows the values ​​of all R factors, RG factors, and GOF after optimization calculations used in the analysis. [Modes for carrying out the invention]

[0025] Next, embodiments of the present invention will be described with reference to the drawings. To facilitate understanding of the description, the same reference numerals are used for identical components in each drawing, and redundant descriptions are omitted.

[0026] [principle] X-ray reflectance and small-angle X-ray scattering have a large dynamic range in the measurement data; for example, the measured intensity is 1.0 to 1.0 × 10⁻⁶. 7 We handle data in the range of counts. In these analyses, even if the intensity is around 10 counts, if the period of the interference fringes is visible, it contains the information necessary for the analysis, so even parts with low intensity cannot be ignored during fitting.

[0027] To determine the shape of a sample using X-rays, it is necessary to solve an inverse problem using measurement data. Based on the shape and material of the sample to be measured, the sample is represented as a model that includes information on the electron density distribution. The calculated intensity is the result of calculating what kind of reflection or scattering occurs when X-rays are irradiated onto this model. The model is optimized using the least squares method so that the calculated intensity data (calculated data) and the measured intensity data (measured data) match well. At this time, an index is needed to check the degree of optimization, indicating how well the model has been optimized.

[0028] The present invention calculates an index GOF indicating the degree of optimization of a model including information on electron density distribution for measurement data whose probability distribution function is known. That the probability distribution function followed by the measurement data is known means that the variation width of the measurement data can be statistically estimated from the values of the measurement data. Specifically, it refers to the case where it is known from the nature of the measurement data that the measurement data follows a Poisson distribution, a Gaussian distribution, or the like.

[0029] GOF is represented, for example, by the following mathematical formula (5). Here, R (R-factor) is a residual defined by a predetermined mathematical formula including measurement data and calculation data. Also, R G (R G factor) is the expected value of the residual defined based on the probability distribution function followed by the measurement data and a predetermined mathematical formula. Note that GOF may be any formula as long as it defines a ratio of the residual including measurement data and calculation data to the expected value of the residual, and is not limited to the mathematical formula (5). The ratio of the residual including measurement data and calculation data to the expected value of the residual may be the ratio of R to R G or the ratio of R to R G . For example, a formula obtained by reversing the denominator and numerator of the mathematical formula (5) may be defined as GOF.

[0030]

Equation

[0031] Since the probability distribution function followed by the measurement data is known, the expected value of the residual defined based on the probability distribution function and a predetermined mathematical formula can be calculated. R G can be said to be the R-factor calculated from the residual of the statistical error between the measurement data and the measurement data. Also, R G can be said to be the minimum R-factor estimated from the measurement data. Specific examples of the predetermined mathematical formula and its expected value will be described in detail in the embodiments.

[0032] The GOF calculated as described above serves as an indicator of the degree of model optimization (the degree of model fitting). For example, it can be used to determine which of two models created for the same measurement data is more optimized. It can also be used to evaluate whether the model fitting has sufficiently converged, and if the model fitting has not sufficiently converged, the parameters can be updated, the model can be recreated, the calculation data can be recalculated, and the GOF can be determined again. Furthermore, if the model fitting is in an ideal state, the GOF can be defined to be 1 regardless of the quality of the measurement data. Therefore, the degree of model optimization can be absolutely determined even between different datasets. The GOF of the present invention is suitable as an indicator for analysis using the weighted least squares method. The detailed method for calculating the indicator of the present invention will be described in detail in the embodiments.

[0033] The following embodiments describe in detail a method for calculating the GOF using measurement data measured by an X-ray analyzer, a method for evaluating the convergence of the GOF, and a method for optimizing the model using the evaluation of the GOF convergence. Note that the present invention is not limited to measurement data of X-ray reflectance or small-angle X-ray scattering measured by an X-ray analyzer, but can be applied to measurement data measured by similar probes. Specifically, it can be applied to measurement data whose probability distribution function is known, such as measurement data from photons.

[0034] [Embodiment] [Overall System] Figure 1 is a conceptual diagram showing an example of the configuration of system 100. System 100 includes an X-ray analyzer 200, a control device 300, and an analysis device 400. By using this system 100, measurement data can be measured, and for measurement data whose probability distribution function is known, the Globe of Factor (GOF), an index indicating the degree of model optimization, can be calculated. Furthermore, the GOF can be evaluated and the model can be optimized.

[0035] In Figure 1, the control device 300 and the analysis device 400 are depicted as the same PC. However, as explained above, the method of the present invention can acquire measurement data and calculation data and perform index calculation independently of the X-ray analyzer 200 and the control device 300. Therefore, the analysis device 400 may be configured as a different device from the control device 300. The following describes the case where the control device 300 and the analysis device 400 are configured as different devices.

[0036] [X-ray analyzer] The X-ray analyzer 200 comprises an optical system that incidents X-rays onto a sample and detects the reflected X-rays generated from the sample. The X-ray analyzer 200 includes an X-ray generator 210 that generates X-rays from an X-ray focal point, i.e., an X-ray source; an incident optical unit 220; a goniometer 230; a sample stage 240 on which the sample is placed; and a detector 260 that detects the X-rays. The X-ray analyzer 200 may also include an exit optical unit 250. The X-ray generator 210, incident optical unit 220, goniometer 230, sample stage 240, exit optical unit 250, and detector 260 that make up the X-ray analyzer 200 can be general-purpose components, so their explanation is omitted. Note that the configuration shown in Figure 1 is just one example, and various other configurations can be adopted.

[0037] The X-ray analyzer 200 repeatedly moves its rotation axis and projects X-rays under predetermined conditions. This irradiates the sample with X-rays and acquires measurement data such as X-ray reflectance data and small-angle X-ray scattering data. The X-ray analyzer 200 transmits device information and the acquired measurement data to the control device 300.

[0038] [Control device] The control device 300 is connected to the X-ray analyzer 200 and controls the X-ray analyzer 200, as well as storing and displaying the acquired data. Within the system 100, if the analysis device 400 does not have the function to create a model and calculate computational data, the control device 300 or another device (e.g., a model creation device) has the function to create a model and calculate computational data.

[0039] Figure 2 is a block diagram showing an example of the configuration of the control device 300. The control device 300 is composed of a computer consisting of a CPU (Central Processing Unit), ROM (Read Only Memory), RAM (Random Access Memory), and memory connected to a bus. The control device 300 is connected to the X-ray analyzer 200 to receive information.

[0040] The control device 300 comprises a control unit 310, a device information storage unit 320, a measurement data storage unit 330, and a display unit 340. Each unit can send and receive information via the control bus L. The input device 510 and the display device 520 are connected to the CPU via appropriate interfaces.

[0041] The control unit 310 controls the operation of the X-ray analyzer 200. The device information storage unit 320 stores device information acquired from the X-ray analyzer 200. The device information may include information about the X-ray analyzer 200 such as the device name, type of radiation source, wavelength, and background.

[0042] The measurement data storage unit 330 stores the measurement data acquired from the X-ray analyzer 200. Along with the measurement data, it may also include information about the X-ray analyzer 200, such as the type of radiation source, wavelength, and background. The display unit 340 displays the measurement data on the display device 520. This allows the user to verify the measurement data. Furthermore, the user can give instructions and specifications to the control device 300, analysis device 400, etc., based on the measurement data.

[0043] [Analysis equipment] The analysis device 400 calculates a GOF index indicating the degree of optimization of a model that includes information on the electron density distribution, for measurement data whose probability distribution function is known. The control device 300 and the analysis device 400 are devices equipped with a CPU and memory, and may be PC terminals or servers on the cloud. Furthermore, not only the entire device, but also some devices or some functions within the device may be located on the cloud. The input device 510 is, for example, a keyboard and mouse, and provides input to the control device 300 and the analysis device 400. The display device 520 is, for example, a display, and displays the model, GOF, etc.

[0044] (Embodiment 1) Embodiment 1 describes the case where only the GOF is calculated. Figure 3 is a block diagram showing an example of the configuration of the analysis device 400 according to Embodiment 1. The analysis device 400 is composed of a computer with a CPU, ROM, RAM, and memory connected to a bus. The analysis device 400 may be connected to the X-ray analyzer 200 via the control device 300.

[0045] The analysis device 400 comprises a measurement data acquisition unit 410, a calculation data acquisition unit 420, and an index calculation unit 430. Each unit can send and receive information via the control bus L. If the analysis device 400 and the control device 300 have different configurations, the input device 510 and the display device 520 are also connected to the CPU of the analysis device 400 via an appropriate interface. In this case, the input device 510 and the display device 520 may be different from those connected to the control device 300.

[0046] Figure 4 is a flowchart illustrating an example of the operation of the analysis device 400 according to Embodiment 1. Figure 4 shows an example of operation when only the GOF is calculated. First, the analysis device 400 acquires measurement data using the measurement data acquisition unit 410 (step S1). Next, it acquires calculation data using the calculation data acquisition unit 420 (step S2). Next, it calculates the GOF using the index calculation unit 430 (step S3).

[0047] The measurement data acquisition unit 410 acquires measurement data. The measurement data acquisition unit 410 may acquire measurement data directly from the X-ray analyzer 200 or via the control device 300, or it may acquire measurement data that has been previously stored in a database or the like.

[0048] The calculation data acquisition unit 420 acquires calculation data calculated from the model. The model is a model that represents the characteristics of the sample on which measurement data was taken. Specifically, it includes information on the electron density distribution derived from the shape and material of the sample. It may also include information other than the electron density distribution, such as the refractive index of the sample. The calculation data is the result of calculating what kind of reflection and scattering occurs when X-rays are irradiated onto the model.

[0049] The index calculation unit 430 calculates an index called GOF, which includes the ratio of residuals to the expected value of residuals, based on residuals defined by a predetermined formula including measurement data and calculation data, and the expected value of residuals defined based on a probability distribution function and a predetermined formula. The predetermined formula, the formula defining the expected value of residuals, or the formula defining the index GOF are stored in advance. Alternatively, the system may be configured to allow the user to arbitrarily set these formulas by selecting or instructing a predetermined formula, etc.

[0050] Next, we will explain an example of how to calculate the GOF using measurement data measured by an X-ray analyzer. Let Obs be the X-ray count intensity of the X-ray reflectance and small-angle X-ray scattering measurement data measured by the X-ray analyzer, and Calc be the calculated X-ray intensity of the calculation data calculated from the model. We assume that the model has already been created. Furthermore, let R be a predetermined formula for fitting the model. This predetermined formula includes the measurement data and the calculation data, and defines the residuals for fitting the model. For example, R can be a formula obtained by dividing the sum of the squared residuals of the logarithms of Obs and Calc by the sum of the logarithmic intensities of Obs, as shown in formula (6) below. Thus, it is preferable that the predetermined formula includes the sum of the squared residuals of the logarithms of the measurement data and the logarithms of the calculation data. j Or Calc j This indicates the value of Obs or Calc at each data point j.

[0051]

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[0052] The probability distribution function f(x) of the X-ray count intensity Obs in the measured data is represented by a Poisson distribution with a standard deviation of √(Ideal), where Ideal is the theoretical mean of the X-ray count intensity. When the Poisson distribution is approximated by a Gaussian distribution, the probability distribution function f(x) is expressed by the following equation (7), where σ = √(Ideal). Thus, when the probability distribution function is a Poisson distribution, it is preferable to approximate the expected value of the residuals with a Gaussian distribution. This makes it easier to implement the software because the integral of the Gaussian distribution can be solved analytically. However, if the expected value of the residuals, including the measured data and the calculated data, can be calculated without approximating with another function, then approximation with another function is not necessary, and the expected value of the residuals can be calculated directly.

[0053]

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[0054] In equation (6) above, the given formula is defined to include the sum of squared residuals between the logarithm of Obs and the logarithm of Calc. Therefore, we consider the sum of squared residuals between the logarithm of Obs and the logarithm of Ideal, as shown in equation (8) below. In this case, the expected value of the sum of squared residuals between the logarithm of Obs and the logarithm of Ideal is expressed by equation (9) below.

[0055]

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[0056]

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[0057] In reality, the value of Ideal is unknown, so we substitute Obs for Ideal and perform the calculation. We calculate the expected value of the sum of squared residuals of the logarithm at each data point j, and divide this sum by the sum of logarithmic intensities. The ideal R for the above R is expressed by the following equation (10): G The value of can be calculated. That is, R G This is the expected value of the residuals, obtained by utilizing the properties of the probability distribution function, for residuals defined by a predetermined mathematical formula that includes the measured data and the calculated data. Therefore, R G σ can be defined in any way as long as it represents the expected value of the residual obtained using the properties of the probability distribution function. In this invention, the expected value of the residual obtained using the properties of the probability distribution function is defined as the expected value of the residual based on the probability distribution function and a predetermined formula. Note that σ in formula (10) is σ = √(Obs j )

[0058]

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[0059] The above R and R G In contrast, if we define GOF by, for example, the following formula (11), then GOF becomes an indicator of the degree of optimization of the model.

[0060]

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[0061] When actually calculating GOF, use R and R as described above. G There is no need to calculate each of them and then use them to calculate the GOF. For example, the above R and R G When defining GOF using formula (11), one may pre-define the following formula (12) as the formula for calculating GOF and use this formula to calculate GOF.

[0062]

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[0063] Furthermore, if a given formula is replaced with a different formula, the GOF will also change accordingly, but even if the given formula is the same, the GOF can be defined differently. G Alternatively, the GOF can be defined as, for example, the following formula (13). Thus, the GOF can be any index that includes the ratio of residuals to the expected value of residuals, where residuals are defined by a predetermined formula that includes measured data and calculated data, and the expected value of residuals is defined based on a probability distribution function and a predetermined formula.

[0064]

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[0065] The following describes another example of the given formula. As described above, when calculating residuals using the logarithm of measured or calculated data, the values ​​fluctuate greatly when the intensity of the measured or calculated data is small. For example, log0 may become -∞, or the values ​​of log2-log1 ≈ 0.3 and log3-log2 ≈ 0.17, which should be the same 1 count difference, may differ by almost double. In this way, as a method to reduce the effect of taking the logarithm for small values, the given formula can be changed to, for example, formula (14) below. W j Obs is the X-ray count intensity of the measurement data. j This is the weight defined by W. j It is defined, for example, as in equation (15), where e is the base of the natural logarithm.

[0066]

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[0067]

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[0068] At this time, the expected value R of the residual is defined based on the probability distribution function and the predetermined formula in equation (14).G These R and R can be defined as shown in the following formula (16). G For this, a GOF can be defined. Also, R and R G In contrast, when GOF is defined by formula (11), the formula for calculating GOF can be expressed, for example, as shown in formula (17) below.

[0069]

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[0070]

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[0071] The above approximation achieves more robust behavior by representing the residual in linear scale intensity when the intensity is less than or equal to e (≒2.7) counts, and taking the residual in logarithmic intensity when it exceeds that value. Thus, it is preferable that the given formula includes a weight parameter which is multiplied by a constant when the intensity of the measured data is less than or equal to a predetermined value, and takes the logarithmic value of the measured data when the intensity of the measured data is greater than the predetermined value. In formula (15), the predetermined value of the weight parameter is e, but the predetermined value may be different. Also, the formula showing the weight parameter may be any formula which is multiplied by a constant when the intensity of the measured data is less than or equal to a predetermined value, and takes the logarithmic value when the intensity of the measured data is greater than the predetermined value, and is not limited to formula (15). Although the count value is basically an integer value, there are cases where the count value is multiplied by a correction term, in which case the value multiplied by the correction term is considered to be the count value. Therefore, the count value can be a decimal value, and the predetermined value may also be a decimal value.

[0072] It is preferable to calculate the GOF directly from the formula defining the GOF, without calculating the residuals defined by a predetermined formula including the measured data and the calculated data, or the expected value of the residuals defined based on the probability distribution function and the predetermined formula. This allows for easy calculation of the GOF. In this way, the GOF can be calculated based on the measured data and the calculated data. By calculating the GOF in this manner, the user can confirm the degree of model optimization regardless of the quality of the measured data or the user's skill level.

[0073] (Embodiment 2) Embodiment 2 describes the case in which the convergence of GOF is evaluated. Since many of the procedures are the same as in Embodiment 1, only the differences will be described. Figure 5 is a block diagram showing an example of the configuration of the analysis device 400 according to Embodiment 2. As shown in Figure 5, it is preferable that the analysis device 400 includes a measurement data acquisition unit 410, a calculation data acquisition unit 420, and an index calculation unit 430, in addition to an index evaluation unit 440. Note that the analysis device 400 with the configuration in Figure 5 may also be called an index evaluation device.

[0074] Figure 6 is a flowchart illustrating an example of the operation of the analysis device 400 according to Embodiment 2. Figure 6 shows an example of the operation when evaluating the convergence of GOF after its calculation. First, the analysis device 400 acquires measurement data using the measurement data acquisition unit 410 (step T1). Next, it acquires calculation data using the calculation data acquisition unit 420 (step T2). Next, it calculates GOF using the index calculation unit 430 (step T3). The operation up to this point is the same as described above.

[0075] Next, the metric evaluation unit 440 evaluates the convergence of the GOF (step T4). In this way, the convergence of the GOF can be evaluated. The evaluation criteria for the convergence of the GOF may be set in advance, or the system may be configured so that the user can arbitrarily set the convergence evaluation criteria and predetermined thresholds by selecting them. The metric evaluation unit 440 may output the evaluation of the convergence of the GOF.

[0076] Next, we will explain an example of how to evaluate the convergence of the GoF (Gate of Functions). The evaluation of the convergence of the GoF differs depending on the definition of the GoF and the definition of convergence. For example, when the GoF is defined by equation (11), the value of the GoF is 1 when the fitting is perfect. Therefore, the convergence of the GoF can be defined by whether the value of the GoF is sufficiently close to 1. When evaluating convergence with such a definition, for example, predetermined thresholds a and b can be set, and convergence can be evaluated when the absolute value of the difference between the GoF and the predetermined threshold a is less than or equal to the predetermined threshold b. It is preferable that threshold a is the value of the GoF when the fitting is perfect.

[0077] Alternatively, convergence of the God of Factors (GOF) can be defined by whether the difference between GOF1 calculated based on an existing model and GOF2 calculated based on an updated model has become sufficiently small. When evaluating convergence using this definition, for example, a predetermined threshold c can be set, and convergence can be evaluated when the absolute value of the difference between GOF1 and GOF2 becomes less than or equal to c.

[0078] In this way, by evaluating the convergence of the GOF, users can determine whether or not they need to recreate the model.

[0079] (Embodiment 3) Embodiment 3 describes the case of optimizing the model. Since many of the procedures are the same as those in Embodiments 1 and 2, only the differences will be described. Figure 7 is a block diagram showing an example of the configuration of the analysis device 400 according to Embodiment 3. As shown in Figure 7, it is preferable that the analysis device 400 includes a measurement data acquisition unit 410, a calculation data acquisition unit 420, an index calculation unit 430, an index evaluation unit 440, and a model creation unit 415.

[0080] Figure 8 is a flowchart illustrating an example of the operation of the analysis device 400 according to Embodiment 3. Figure 8 shows an example of the operation when the index evaluation device optimizes a model. First, the analysis device 400 acquires measurement data using the measurement data acquisition unit 410 (step U1). Next, the model creation unit 415 sets the model parameters and analysis conditions (step U2). The model parameters or analysis conditions may be configured to be arbitrarily set by the user through selection, specification, input, etc.

[0081] Model parameters are parameters necessary to represent a sample as a model that includes information about the electron density distribution. Examples include parameters that indicate the shape of the sample and parameters that indicate the material of the sample. Specifically, model parameters include CD (Critical Dimension), depth or height, and the position of the center line. Analysis conditions include, for example, the range of measurement data used in the analysis, constraints on the model parameters, and the strength of the weights assigned to each diffraction line. Constraints on model parameters are conditions set to prevent the model parameters from becoming larger or smaller than a certain value, or to limit the range by which the parameter can change in each step of the iterative analysis so that the parameter value does not change abruptly.

[0082] Next, the model parameters are optimized (step U3). Model parameter optimization can be performed, for example, using the least squares method. Next, the model creation unit 415 creates the model (step U4). The model is created based on the optimized model parameters. Next, the calculation data is calculated (step U5). The calculation data is calculated based on the type of radiation source, wavelength, model shape, composition, etc. of the measured data.

[0083] Next, the index calculation unit 430 calculates the GOF (step U6). The calculation of the GOF is the same as in step S3. Next, the index evaluation unit 440 evaluates the convergence of the GOF (step U7). The evaluation of the convergence of the GOF is the same as in step T4.

[0084] Next, if the GOF convergence evaluation does not meet the conditions (step U8-NO), the process returns to step U2 to set (update) the model parameters and analysis conditions. Then, the model is created (recreated) using the updated model parameters, and the process up to step U7 is repeated.

[0085] On the other hand, if the GOF convergence evaluation satisfies the conditions (step U8-YES), the model is output as needed (step U9), and the process terminates. In this way, the model can be optimized. Note that if the GOF convergence is evaluated based on the GOF before and after the update, the GOF convergence may not be evaluated in the first loop. In such cases, the program may be configured to always perform a second loop.

[0086] The model creation unit 415 optimizes the model parameters. The model creation unit 415 also creates a model and calculates computational data based on the created model. The model is created based on the model parameters. The computational data is calculated based on the type of radiation source, wavelength, model shape, composition, etc., of the measured data. At this time, the computational data acquisition unit 420 acquires the computational data calculated by the model creation unit 415.

[0087] If the index evaluation unit 440 evaluates that the GOF has not converged, the model creation unit 415 updates the analysis conditions and model parameters, optimizes the model parameters, creates (recreates) the model, and recalculates the calculation data. At this time, the index calculation unit 430 recalculates the GOF based on the recalculated data. Furthermore, if the index evaluation unit 440 evaluates that the GOF has converged, the model creation unit 415 outputs the model as needed. This allows the analysis device 400 to optimize the model based on the evaluation of GOF convergence. In addition, the analysis device 400 can automatically optimize the model without requiring user confirmation of convergence. Note that the analysis device 400 configured in Figure 7 can also be called a model optimization device.

[0088] Next, we will explain an example of how to optimize a model using the evaluation of GOF convergence. First, set the analysis conditions and model parameters. Next, optimize the model parameters. Model parameter optimization can be done, for example, by the least squares method. Next, create a model based on the optimized model parameters. Next, calculate computational data from the model. Then, calculate the GOF using the measured data and computational data. The GOF is calculated according to the definition described above. Evaluate the convergence of the calculated GOF, and if it is evaluated as converged, output the model as needed and terminate.

[0089] On the other hand, the convergence of the calculated GOF is evaluated, and if it is determined that it has not converged, the analysis conditions and model parameters are updated. Next, the updated model parameters are optimized. Then, a model is created (updated) based on the optimized model parameters. Next, computational data is calculated from the updated model. Then, the GOF is calculated using the measured data and computational data. The model parameters may be updated randomly, or they may be updated based on the GOF value. Also, if there are multiple sets of measured data and computational data, such as multiple diffraction lines, all datasets can be combined to calculate a single GOF, or the GOF can be calculated for each corresponding set of measured and computational data. When calculating the GOF for each corresponding set of measured and computational data, the model parameters may be updated with weights that give more emphasis to diffraction lines or datasets whose GOF value is far from convergence, i.e., those with a far lower degree of optimization. For example, diffraction lines or datasets whose GOF value is far from convergence can be weighted by a constant multiplier of the residual.

[0090] In this way, by optimizing the model using the GOF convergence evaluation, it is possible to create an optimized model based on the objective metric GOF, regardless of the quality of the measurement data or the user's skill level.

[0091] [Examples] Using the index calculation device configured as described above, we investigated the analysis convergence process using simulation data. First, a calculated waveform was generated based on the analysis results (model) of an actual sample. Next, statistical noise (random numbers following a Poisson distribution) was added to the calculated waveform to generate a pseudo-measured waveform. Then, the generated pseudo-measured waveform was fitted using the least squares method with a simple cylinder as the initial model to reconstruct the model from which the waveform was generated.

[0092] Figure 9(a) is a graph showing the pseudo-measured waveform and calculated waveform before model optimization. Figure 9(b) is a graph showing the pseudo-measured waveform and calculated waveform after model optimization. The analysis was performed in the following three steps. 1. Perform trapezoidal fitting, where the diameter changes linearly at each depth. 2. Perform fittings with freely varying diameters at each depth. 3. For diffraction lines with a GOF value greater than 1, weighting is applied by multiplying the residuals by a constant, and fitting is performed to exceed the local minimum.

[0093] Figure 10 is a graph showing the changes in the R factor and GOF values ​​corresponding to one diffraction line during the analysis process of the embodiment. The GOF at the end of the analysis was 1.03, which is very close to 1, allowing the user to determine that the analysis had sufficiently converged, regardless of their skill level. In comparison, the R factor at the end of the analysis was 8.39%, and it is not possible to determine the end of the analysis based on this alone. Conventionally, skilled users would visually inspect the waveform at this state to determine the end of the analysis, so it was confirmed that the method of the present invention does not require user skill level for convergence determination.

[0094] Figure 11 is a graph overlaying the measured and calculated waveforms at a certain stage of the embodiment. The waveform plot in Figure 11 shows the overlay of the measured and calculated waveforms at a stage where the analysis has progressed to a certain extent. Note that the graph in Figure 11 displays only some of the waveforms used in the analysis. The actual simulation was performed using 162 diffraction lines. Looking at Figure 11, the waveforms appeared to match to a certain extent visually.

[0095] On the other hand, Figure 12 is a graph showing the values ​​of all the R factors used in the analysis at a certain stage. Figure 12 plots the R factor values ​​at the same stage as in Figure 11. As can be seen in Figure 12, the R factor values ​​fluctuate over a wide range because the intensity differs for each diffraction line. Therefore, it was not possible to determine from this alone which diffraction line showed a large discrepancy between the calculated and measured values.

[0096] Figure 13 shows all the R factors used in the analysis. G This is a graph of the factors and their values ​​at a certain stage of GOF. Figure 13 shows the R factor at the same stage as in Figure 11, plus R G The factors and GOF were also calculated and plotted for each diffraction line. Figure 13 shows that there are regions where the GOF is significantly greater than 1 (local minimum).

[0097] For diffraction lines with a GOF value greater than 1, the optimization calculation was performed again by adding weights that multiply the residuals by a constant. This is because larger residuals have a greater influence on the optimization calculation, resulting in an optimization result where the weighted diffraction lines are more strongly fitted. After the optimization calculation, the weights were removed and the GOF was calculated again. Figure 14 shows all the R factors used in the analysis, R G This is a graph of the factors and the values ​​of GOF after optimization calculation. As shown in Figure 14, the GOF was sufficiently close to 1 for all diffraction lines, indicating that convergence had occurred. This demonstrates that by using GOF, it was possible to converge to the correct solution beyond the local minimum.

[0098] Based on the above results, it has been confirmed that the index calculation device, system, method, and program of the present invention can calculate an absolute convergence evaluation index that can be applied to measurement data such as X-ray reflectance and small-angle X-ray scattering. [Explanation of Symbols]

[0099] 100 Systems 200 X-ray analyzer 210 X-ray generating unit 220 Incident Optical Unit 230 Goniometer 240 Sample stage 250 Output-side optical unit 260 detectors 300 Control device 310 Control Unit 320 Device information storage unit 330 Measurement data storage unit 340 Display section 400 Analysis equipment 410 Measurement data acquisition unit 415 Model Creation Department 420 Calculation Data Acquisition Unit 430 Indicator calculation section 440 Indicator Evaluation Department 510 Input device 520 Display device

Claims

1. An analysis device that calculates an index indicating the degree of optimization of a model including information on electron density distribution for measurement data whose probability distribution function is known, A measurement data acquisition unit that acquires the aforementioned measurement data, A calculation data acquisition unit that acquires calculation data calculated from the aforementioned model, The system includes an index calculation unit that calculates an index GOF which includes the ratio of the residual to the expected value of the residual, based on the residual defined by a predetermined formula including the measurement data and the calculation data, and the expected value of the residual defined based on the probability distribution function and the predetermined formula. An analytical apparatus characterized in that the formula for calculating the expected value of the residual includes the probability distribution function or a function that approximates the probability distribution function.

2. An analysis device that calculates an index indicating the degree of optimization of a model including information on the electron density distribution for measurement data whose probability distribution function is known, A measurement data acquisition unit that acquires the aforementioned measurement data, A calculation data acquisition unit that acquires calculation data calculated from the aforementioned model, The system includes an index calculation unit that calculates an index GOF which includes the ratio of the residual to the expected value of the residual, based on the residual defined by a predetermined formula including the measurement data and the calculation data, and the expected value of the residual defined based on the probability distribution function and the predetermined formula. The analytical apparatus is characterized in that the predetermined mathematical formula includes the sum of the squared residuals between the logarithm of the measurement data and the logarithm of the calculation data.

3. An analysis device that calculates an index indicating the degree of optimization of a model including information on the electron density distribution for measurement data whose probability distribution function is known, A measurement data acquisition unit that acquires the aforementioned measurement data, A calculation data acquisition unit that acquires calculation data calculated from the aforementioned model, The system includes an index calculation unit that calculates an index GOF which includes the ratio of the residual to the expected value of the residual, based on the residual defined by a predetermined formula including the measurement data and the calculation data, and the expected value of the residual defined based on the probability distribution function and the predetermined formula. The analysis device is characterized in that the predetermined formula includes a weight parameter which is multiplied by a constant when the intensity of the measurement data is less than or equal to a predetermined value, and takes the logarithmic value of the measurement data when the intensity of the measurement data is greater than the predetermined value.

4. The analysis apparatus according to any one of claims 1 to 3, further comprising an index evaluation unit for evaluating the convergence of the GOF.

5. The analysis apparatus according to any one of claims 1 to 3, characterized in that the probability distribution function is a Poisson distribution, and the index calculation unit calculates the expected value of the residual using a Gaussian distribution.

6. The system includes a model creation unit that creates the aforementioned model and calculates the aforementioned calculation data based on the created model, The calculation data acquisition unit acquires the calculation data calculated by the model creation unit, If the indicator evaluation unit evaluates that the GOF has converged, the model creation unit outputs the model. The analysis apparatus according to claim 4, characterized in that, if the index evaluation unit evaluates that the GOF has not converged, the model creation unit updates the model parameters of the model, creates the model and calculates the calculation data, and the index calculation unit calculates the GOF based on the calculated calculation data.

7. An X-ray analyzer comprising an X-ray generating unit for generating X-rays, a sample stage for placing a sample, and a detector for detecting X-rays, A system comprising an analysis device according to any one of claims 1 to 3.

8. A method for calculating an index indicating the degree of optimization of a model that includes information on the electron density distribution for measurement data, wherein the probability distribution function that the measurement data follows is known, A measurement data acquisition step for acquiring the aforementioned measurement data, A calculation data acquisition step involves obtaining calculation data calculated from the aforementioned model, The step includes calculating an index GOF that includes the ratio of the residual to the expected value of the residual, with respect to the residual defined by a predetermined formula including the measurement data and the calculation data, and the expected value of the residual defined based on the probability distribution function and the predetermined formula. A method characterized in that the formula for calculating the expected value of the residual includes the probability distribution function or a function that approximates the probability distribution function.

9. A program for calculating an index that indicates the degree of optimization of a model including information on the electron density distribution for measurement data whose probability distribution function is known, The process of acquiring the aforementioned measurement data, The process of obtaining the calculation data calculated from the aforementioned model, The computer is made to perform a process of calculating an index called GOF, which includes the ratio of the residual to the expected value of the residual, for residuals defined by a predetermined formula including the measurement data and the calculation data, and the expected value of the residual defined based on the probability distribution function and the predetermined formula. A program characterized in that the formula for calculating the expected value of the residual includes the probability distribution function or a function that approximates the probability distribution function.

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