JP2007248108A - Peak position judging method of spectrum and analyzer - Google Patents

Abstract

<P>PROBLEM TO BE SOLVED: To realize a peak position judging method of a spectrum for judging a spectrum peak having a gentle shape as a peak while keeping peak judging precision, and analyzer. <P>SOLUTION: Even when the absolute value of the minimum value of second derivative value data 53 is judged to be contained in an error range not exceeding a threshold value T<SB>2</SB>by a peak position judging means 57, it is further judged whether the magnitude of the difference between the maximum value adjacent to a first derivative value and the minimum value exceeds a threshold value T<SB>1</SB>and the minimum value of the secondary differential value data 53 is judged as the spectrum peak when the magnitude of the difference exceeds the threshold value T<SB>1</SB>. Accordingly, even the spectrum peak having the gentle peak shape along with a steep peak is detected while keeping judging precision. <P>COPYRIGHT: (C)2007,JPO&INPIT

Description

  The present invention relates to a spectrum peak position determination method and an analyzer for determining a spectrum peak at which the spectrum has a maximum from a spectrum obtained by detecting a detected line such as Auger electrons in the analyzer.

  In recent years, analyzers such as Auger electron analyzers have been used as effective methods for analyzing elemental compositions present on the surface of a sample. An Auger electron analyzer irradiates a sample surface with an electron beam, detects Auger electrons generated from the sample, and generates an energy spectrum (hereinafter referred to as a spectrum) that is a count value using the kinetic energy of the Auger electrons as a parameter. get. Based on this spectrum, the element composition existing on the sample surface is specified and quantified.

  At this time, the spectrum contains a large amount of secondary electrons as well as Auger electrons from the sample surface. Secondary electrons form a continuous spectrum with continuous energy in the spectrum, while Auger electrons have energy inherent to the elements present on the sample surface and form a spectral peak at this spectral position in the spectrum. To do.

  The Auger electron analyzer identifies and quantifies the elemental composition present on the sample surface by detecting this spectral peak from the spectrum. Here, the determination of the peak position should be performed accurately because the spectrum includes a lot of continuous spectra of secondary electrons and the count value, which is the number of detected electrons, includes a statistical error. Is not easy.

As a method for accurately obtaining the peak position, a second derivative value of the spectrum is calculated, and the magnitude of the second derivative minimum value at the spectrum position where the second derivative value is minimized is set based on the statistical error. When the threshold value is exceeded, there is a method in which this spectral position is set as a peak position (for example, see Patent Document 1).
Japanese Patent Laid-Open No. 10-318837, (pages 2 to 4, FIG. 1)

  However, according to the background art, there is a case where the peak position of the Auger electron count value is not determined to be a peak in the spectrum. That is, when the peak shape of the count value of Auger electrons formed in the spectrum is gentle, it is determined that the secondary differential minimum value is small and the error range is within the threshold, and a determination omission that is not determined as the spectrum peak occurs.

  On the other hand, reducing the threshold value for peak determination leads to erroneous determination of peak noise caused by statistical errors as the peak position of the Auger electron count value, resulting in a decrease in peak determination accuracy. It is not preferable.

  From these facts, it is important how to realize a spectrum peak position determination method and an analysis apparatus for determining a peak with respect to a gently shaped spectrum peak while maintaining peak determination accuracy.

  The present invention has been made to solve the above-described problems caused by the background art, and a peak position determination method and analysis for determining a peak with respect to a gently shaped spectrum peak while maintaining peak determination accuracy. An object is to provide an apparatus.

  In order to solve the above-described problems and achieve the object, a spectrum peak position determination method according to the first aspect of the present invention provides a spectrum peak determination method for determining a spectrum peak at which the spectrum has a maximum from the obtained spectrum. A method for determining a peak position, wherein the spectrum is differentiated, and a first derivative error obtained by the differentiation is calculated from a statistical error of data forming the spectrum, and an extreme value of the first derivative value is calculated. An index indicating the magnitude of the difference is calculated, and it is determined that the spectrum peak exists when the value of the index exceeds a threshold set based on the first-order differential error.

  In the first aspect of the present invention, the spectrum is differentiated, and the first derivative error of the first derivative obtained by this differentiation is calculated from the statistical error of the data forming the spectrum, and the extreme value of the first derivative is calculated. An index indicating the magnitude of the difference is calculated, and it is determined that a spectrum peak exists when the value of this index exceeds a threshold set based on the first-order differential error.

  A spectrum peak position determination method according to a second aspect of the present invention is the peak position determination method according to the first aspect, wherein the calculation is a second-order differential value obtained by second-order differentiation of the spectrum. The second-order differential error is calculated from the statistical error.

According to the second aspect of the present invention, the secondary differential value and the secondary differential error with a small background included in the spectrum are obtained, and these are used for determining the position of the spectral peak.
The spectrum peak position determination method according to claim 3 is the peak position determination method according to claim 2, wherein the calculation obtains a second derivative minimum value of the second derivative value, In the determination, the position on the spectrum having the second-order differential minimum value is set as the position of the spectrum peak.

According to the third aspect of the present invention, the position of the spectrum where the minimum value of the secondary differential value exists is set as the position of the spectrum peak, and the position of the spectrum peak is determined with high accuracy.
A spectrum peak position determination method according to a fourth aspect of the present invention is the spectrum peak position determination method according to any one of the first to third aspects, wherein the index is adjacent to the primary differential value. It is the size of the difference between the local maximum value and the local minimum value.

In the invention described in claim 4, the index is a large value considering both the maximum value and the minimum value.
A spectrum peak position determining method according to a fifth aspect of the present invention is the peak position determining method according to any one of the first to fourth aspects, wherein the differentiation is performed using a Savitzky-Golay method. It is characterized by being.

In the invention according to claim 5, the differentiation is performed by simple four arithmetic operations.
A spectrum peak position determination method according to a sixth aspect of the present invention is the peak position determination method according to any one of the first to fifth aspects, wherein the statistical error is a magnitude of a square root of the data. It is characterized by being made.

In the invention described in claim 6, the standard deviation is used as the statistical error.
The spectrum peak position determination method according to the invention described in claim 7 is the spectrum peak position determination method according to any one of claims 1 to 6, wherein the calculation includes calculating the first-order differential error, It is calculated from the data using the law of error propagation.

In the invention described in claim 7, the first-order differential error is a value theoretically obtained from the statistical error.
The spectrum peak position determination method according to the invention described in claim 8 is the spectrum peak position determination method according to any one of claims 2 to 7, wherein the calculation includes calculating the second-order differential error. And calculating from the data using the law of error propagation.

In the invention according to the eighth aspect, the second-order differential error is a value theoretically obtained from the statistical error.
The spectrum peak position determination method according to the invention described in claim 9 is the spectrum peak position determination method according to any one of claims 1 to 8, wherein the spectrum is a count value of the number of electrons. It is a spectrum using as a data.

According to the ninth aspect of the present invention, the spectral peak of Auger electrons is determined.
The analyzer according to the invention of claim 10 is an analyzer for determining a spectrum peak at which the spectrum shows a maximum from the obtained spectrum, the differentiating means for differentiating the spectrum, and the differentiation. A calculation means for calculating a primary differential error of the acquired primary differential value from a statistical error of data forming the spectrum; and an index calculation means for calculating an index indicating a magnitude of a difference between extreme values of the primary differential value; And a peak position determining means for determining that the spectrum peak is present when a value of the index exceeds a threshold set based on the first derivative error.

  In the invention described in claim 10, the spectrum is differentiated by the differentiating means, and the first derivative error of the first derivative value obtained by the differentiation is calculated by the calculating means from the statistical error of the data forming the spectrum, and the index An index indicating the magnitude of the difference between the extreme values of the first derivative value is calculated by the calculating means, and when the index value exceeds a threshold set based on the first derivative error by the peak position determining means, a spectrum peak is detected. It is determined that it exists.

  The analyzer according to claim 11 is the analyzer according to claim 10, wherein the differentiating means second-order differentiates the spectrum, and the calculating means is acquired by the second-order differentiation. The step of calculating a second-order differential error of the second-order differential value from the statistical error is also executed.

  The analyzer according to claim 12 is the analyzer according to claim 11, wherein the calculating means obtains a secondary differential minimum value of the secondary differential value, and the peak position determining means is The position on the spectrum having the second derivative minimum value is the position of the spectrum peak.

  The analyzer according to the invention described in claim 13 is the analyzer according to any one of claims 10 to 12, wherein the index calculation means includes the adjacent maximum value and minimum value of the primary differential value. The difference is defined as the index.

  The analyzer according to claim 14 is the analyzer according to any one of claims 10 to 13, wherein the spectrum is a spectrum having a count value of the number of electrons as data. The analyzer according to any one of claims 10 to 13.

  According to the present invention, the magnitude of the difference between the extreme values (maximum value and minimum value) of the first derivative of the spectrum is calculated as an index, and when this magnitude exceeds the threshold value based on the first derivative error, Since it is determined that a spectrum peak exists, it is possible to perform a determination to distinguish a peak having a gentle peak shape from a peak-shaped noise caused by a statistical error. Here, by combining with the information of the second derivative value, accurate spectral peak position and peak size quantitative information is acquired, and the peak shape is gentle while maintaining peak judgment accuracy for steep peaks. The spectral peak can also be detected.

  DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS A best mode for carrying out a spectrum peak position determining method and an analyzing apparatus according to the present invention will be described below with reference to the accompanying drawings. Note that the present invention is not limited thereby.

First, the overall configuration of an Auger electron analyzer 90 which is an analyzer according to the present embodiment will be described. FIG. 1 is a cross-sectional configuration diagram showing the overall configuration of the Auger electron analyzer 90. The Auger electron analyzer 90 includes a lens barrel unit 10, a sample chamber 20, an electron spectrometer 30, an electron beam control unit 40, a control unit 41, a display unit 42, and an input unit 43. Here, the inside of the lens barrel part 10, the sample chamber 20 and the electron spectrometer 30 is brought into an ultrahigh vacuum state of the order of 10 ⁻⁶ Pa to 10 ⁻⁸ Pa by a pump (not shown). The lens barrel unit 10 includes an electron gun 1, a focusing lens 2, an objective lens 3, and the like, and the sample chamber 20 includes a sample 21, a sample stage 22, and the like.

  The electron gun 1 accelerates electrons emitted from the heated filament with an acceleration voltage applied between the anode and the electron gun and irradiates the sample 21. The electron beam 4 indicates a travel route along which electrons travel by this irradiation.

  The focusing lens 2 focuses the electron beam 4 emitted from the electron gun 1 and travels in the direction of the sample 21. The objective lens 3 is a coil that focuses the electron beam 4 emitted from the electron gun 1 with a spread on the surface of the sample 21, and has a focal position on the traveling path of the electron beam 4, for example, on the surface of the sample 21.

The electron beam control unit 40 controls the electron gun 1, the focusing lens 2, and the objective lens 3 so that the electron beam 4 has a predetermined current value and focuses on a target position on the sample 21.
The sample stage 22 is a stage on which the sample 21 is placed at the irradiation position of the electron beam 4. This stage is movable in the XY directions substantially orthogonal to the traveling direction of the electron beam 4, and the sample 21 is disposed at a position desired by the operator.

  The electron spectrometer 30 detects electrons generated from the surface of the sample 21 by irradiation of the electron beam 4. Further, the electron spectrometer 30 performs electrical scanning for selecting incident electrons for each kinetic energy (hereinafter abbreviated as energy), and outputs the count value information of the incident electrons to the control unit 41 for each energy.

  The display unit 42 includes an LCD or the like, and displays various control information or count value information input from the electron spectrometer 30. The input unit 43 includes a keyboard or a mouse and inputs various control information.

  The control unit 41 includes hardware such as a CPU and a memory. The control unit 41 controls the electron beam control unit 40, forms a spectrum based on the count value information from the electron spectrometer 30, and determines a spectrum peak from the spectrum. I do. FIG. 2 is a functional block diagram illustrating a functional configuration of the control unit 41. The control unit 41 includes spectrum information 50 in the memory 59, primary differential value information 51, primary differential error information 52, secondary differential value information 53, and secondary differential error information 54, as well as differential means 55, error calculation. Each means of means 56, index calculation means 58 and peak position determination means 57 is included. The information and the means will be described in detail in the description of the operation of the control unit 41.

  Next, the operation of the control unit 41 of the Auger electron analyzer 90, particularly the operation of determining the peak position from the spectrum will be described with reference to FIG. FIG. 3 is a flowchart showing the operation of the control unit 41. First, the control unit 41 acquires the spectrum information 50 from the electron spectrometer 30 (step S301). FIG. 5 is an explanatory diagram showing an example of the spectrum information 50. The horizontal axis represents the energy of electrons and is in the range of about 1500 to 2500 eV. The vertical axis indicates the count value (data) of electrons, which is a value of about tens of thousands to hundreds of thousands. This spectrum is formed with the count value of electrons as data.

  Here, the count value of electrons includes secondary electrons and Auger electrons generated from the surface of the sample 21. The secondary electrons form a continuous spectrum in which energy is continuous in the spectrum information 50 shown in FIG. 5, and form a background of Auger electrons.

  In addition, Auger electrons have specific energy inherent to the elements present on the surface of the sample 21. Therefore, a portion where Auger electrons are present forms a peak protruding from the spectrum of secondary electrons forming the background. The shape of the peak is affected by the type of element that generates Auger electrons, the surface state of the sample 21, and the like, and there are various shapes from a steep one to a gentle one.

  Returning to FIG. 3, the threshold coefficient input from the input unit 43 is input to the error calculation unit 56 by the operator (step S <b> 302). This threshold coefficient is used when the threshold value is calculated by the error calculation means 56 described later. The threshold coefficient a of the primary differential value and the threshold coefficient b of the secondary differential value are calculated by the operator via the input unit 43. Input to means 56. Note that the value of the threshold coefficient is approximately 0.5 to 3 and is determined experimentally.

Thereafter, the differentiation means 55 performs differentiation using the spectrum information 50 (step S303), and obtains primary differential value information 51 and secondary differential value information 53. The differentiating means 55 performs a differentiation operation using, for example, an algorithm of the Savitzky-Golay method. In this differential operation, if the spectrum value at spectrum position i (i = 1, 2,...) Of spectrum information 50 is P _i , the weighting factor is f _j , and the operation result is S _i ,

The calculation is performed using the relational expression. Note that j is a variable consisting of an integer value indicating the calculation range. According to equation (1), the differential operation can be performed by weighted linear addition (convolution operation) of the spectrum values P _i . In addition, as the weighting factor f _j , values for the number of terms in the calculation range are given as a predetermined table. The secondary differential value information 53 is obtained from the spectrum information 50 by the same calculation using a different weighting factor g _i instead of the weighting factor f _j .

  FIG. 6 is an explanatory diagram showing spectra of the primary differential value information 51 and the secondary differential value information 53 obtained by differentiating the spectrum information 50. FIG. 6A shows only the spectral portion near the peak where Auger electrons exist from the spectral information 50 shown in FIG. FIG. 6B is a diagram showing a primary differential value obtained by differentiating the spectral portion shown in FIG. The first derivative value has a value of zero at the peak position of the spectrum portion shown in FIG. 6A, and has a maximum and a minimum at an inflection point existing in the vicinity of the peak position. FIG. 6C is a diagram showing a secondary differential value obtained by second-order differentiation of the spectrum shown in FIG. This secondary differential value becomes zero at the peak position of the primary differential value shown in FIG. 6B, and becomes maximum and minimum at an inflection point existing in the vicinity of this peak position. As a result, the secondary differential value has a minimum value at the peak position of the spectrum portion shown in FIG.

  Here, the diagrams shown in FIGS. 6A to 6C schematically show spectra, primary differential values, and secondary differential values, and actually include statistical noise and background of secondary electrons. It is a waste. In particular, since the background is not sufficiently removed with the first-order differential value, the peak position of the spectral portion shown in FIG. 6A may not match the zero point of the first-order differential value.

  On the other hand, in the secondary differential value, the background is sufficiently removed empirically, and the peak position of the spectrum portion and the minimum position of the secondary differential value coincide with each other. As will be described later, the peak position of the spectral portion is thereby determined by the minimum position of the secondary differential value.

Returning to FIG. 3, the error calculation means 56 calculates the errors of the primary and secondary differential values using the spectrum information 50 and sets a threshold value (step S304). First, the count value of the spectrum information 50 has a statistical error according to a normal distribution. If the count value is n, it is considered that a true value is generally present in a range of ± n ^1/2 corresponding to the standard deviation.

Here, the primary differential error information 52 and the secondary differential error information 54, which are errors of the primary and secondary differential values, are calculated based on the error of the count value using the error propagation law. Since the calculation for deriving the primary and secondary differential values is a linear addition obtained by multiplying the weighting factor f _j , the error α _i of the primary differential value at the spectral position i (i = 1, 2,...) When the statistical error of the count value given by the [delta] _i,

It can obtain | require with the relational expression. Similarly, the error σ _i of the secondary differential value is represented by g _{i as the} weighting coefficient used for obtaining the secondary differential value and δ _{i as} the statistical error of the count value.

It can obtain | require with the relational expression. Note that j is a variable consisting of an integer value indicating the calculation range.
Then, the threshold value T ₁ of the primary differential value and the threshold value T ₂ of the secondary differential value are set using the primary threshold coefficient a and the secondary threshold coefficient b input in step S302. The threshold value T ₁ of the primary differential value is, for example, a value obtained from an equation of T ₁ = 2aα, and the threshold value T ₂ of the secondary differential value is, for example, a value obtained from an equation of T ₂ = −bσ. In FIGS. 6B and 6C, threshold values T ₁ and T ₂ are shown by broken lines. Further, the error α _i of the primary differential value and the error σ _i of the secondary differential value are similarly obtained from the statistical error δ _i , and the threshold coefficients a and b are also of the same level, so Become.

  Thereafter, the index calculation means 58 obtains the minimum value of the secondary differential value using the secondary differential value information 53 (step S305), and then uses the primary differential value information 51 to determine the maximum value of the primary differential value and A minimum value is obtained (step S306). Note that the minimum value of the secondary differential value, and the maximum value and minimum value of the primary differential value are the minimum position 61 of the secondary differential value and the maximum of the primary differential value shown in FIGS. This corresponds to the count values at the position 62 and the minimum position 63.

  Thereafter, the index calculating means 58 calculates the difference between the maximum value and the minimum value of the primary differential value obtained in step S306, which is an index (step S307), and determines the magnitude from peak to peak.

Or then, the peak position determining means 57 compares the absolute value of both the minimum value and the threshold T ₂ of the second derivative value, the absolute value of the minimum value of the secondary differential value exceeds the absolute value of the threshold T ₂ It is determined whether or not (step S308). If the absolute value of the minimum value of the secondary differential value exceeds the absolute value of the threshold T ₂ (Yes in step S308), the minimum position of the secondary differential value having this minimum value is set as the spectrum peak position of the spectrum. (Step S310), the process proceeds to Step S312 described later.

The peak position determining means 57, one absolute value of the minimum value of the secondary differential value, (negative step S308). If not exceed the absolute value of the threshold T _2, present in the spectral position sandwiching the minimum value order the size of the difference between the differentiated value further determines whether exceeding the threshold value T ₁ (step S309). The magnitude of the difference of the primary differential value, if exceeding the threshold T ₁ (step S309: Yes), the process proceeds to step S310, the spectral peak position of the spectrum minimum positions of the second derivative having the minimum value Further, the determination in step S308 is repeated. Incidentally, in FIG. 6 (c), the absolute value of the minimum value of the secondary differential value, if not exceed the absolute value of the threshold T ₂ is indicated by a solid line, the absolute value of the minimum value of the secondary differential value but if it exceeds the absolute value of the threshold T ₂ is indicated by a broken line.

  Here, it is known that the difference between the maximum value and the minimum value of the primary differential value is less likely to decrease even when the spectrum has a gentle peak shape compared to the absolute value of the minimum value of the secondary differential value. Yes. FIG. 7 is a diagram showing a primary differential value and a secondary differential value when the spectrum peak has a Gaussian function type. In FIG. 7, the peak width at half maximum (FWHM) is Δ, and as an example of an index indicating the magnitude of the primary differential value, the magnitude between the maximum value and the minimum extreme value is PP, and the index indicating the magnitude of the secondary differential value. As an example, the size between the maximum value and the minimum value is shown as PP2.

Here, the size PP1 is the half width and
PP1∝1 / Δ
The size PP2 is the half width and
PP2∝1 / Δ ²
It is theoretically required to have the following relationship. Therefore, the larger the half-value width Δ of the spectral peak and the gentler the peak shape, the smaller the index value PP2 of the second derivative value becomes, compared to the index value PP1 of the first derivative value. . On the other hand, as described above, the statistical errors of the primary differential value and the secondary differential value and the threshold values T ₁ and T ₂ have the same magnitude, and therefore the absolute value of the minimum value of the secondary differential value is small. Even if it is a value and is buried in the statistical error, the difference of the primary differential value exceeds the threshold value T ₁ and may be determined as a spectrum peak.

Returning to FIG. 3, if the magnitude of the difference of the primary differential value does not exceed the threshold value T ₁ (No in step S 309), the peak position determination unit 57 sets the peak value within the range included in the statistical error. Therefore, the minimum position of the secondary differential value is not set as the spectrum peak position of the spectrum (step S311).

After that, the peak position determination means 57 compares the absolute value of the minimum value of the secondary differential value and the absolute value of the threshold value T ₂ at all spectrum positions, and further compares the magnitude of the difference of the primary differential value and the threshold value T ₁ . (Step S312), and if the comparison has not been completed at all the spectral positions (No at step S312), the process proceeds to step S308 and the comparison is continued. If the comparison is completed at all spectrum positions (Yes at Step S312), the spectrum peak position of the obtained spectrum is displayed on the display unit 42 (Step S313), and this process is terminated.

As described above, when the peak position determination unit 57 determines that the absolute value of the minimum value of the secondary differential value information 53 does not exceed the threshold value T ₂ and is included in the error range, the primary differential value is further increased. It is determined whether or not the magnitude of the difference between the adjacent local maximum value and local minimum value exceeds the threshold value T _1, and when it exceeds, this local minimum value of the secondary differential value information 53 is determined to be a spectrum peak. A spectral peak having a gentle peak shape along with the peak can also be detected while maintaining the determination accuracy.

In the present embodiment, the difference between the maximum value and the minimum value of adjacent primary differential values is used as an index. However, the absolute value of the maximum value or the minimum value can also be used.
In the present embodiment, an example using an Auger electron analyzer has been shown. However, not only an Auger electron analyzer but also error information is generated based on a probability event, such as EPMA, ESCA, and fluorescent X-ray analyzer. It can also be applied to the determination of spectral peaks.

It is a cross-sectional block diagram which shows the whole structure of an Auger electron analyzer. It is a functional block diagram which shows the functional structure of the control part concerning embodiment. It is a flowchart which shows operation | movement of the control part concerning embodiment (the 1). It is a flowchart which shows operation | movement of the control part concerning embodiment (the 2). It is explanatory drawing which shows an example of a spectrum. It is explanatory drawing which shows an example of the maximum part of a spectrum, the primary differential value of this part, and a secondary differential value. It is explanatory drawing which shows the relationship between the half value width of a spectrum peak, and the magnitude | size of a primary and secondary differential value.

Explanation of symbols

DESCRIPTION OF SYMBOLS 1 Electron gun 2 Focusing lens 3 Objective lens 4 Electron beam 10 Barrel part 20 Sample chamber 21 Sample 22 Sample stage 30 Electron spectrometer 40 Electron beam control part 41 Control part 42 Display part 43 Input part 50 Spectral information 51 Primary differential value information 52 primary differential error information 53 secondary differential value information 54 secondary differential error information 55 differential means 56 error calculation means 57 peak position determination means 58 index calculation means 59 memory 61 minimum position of the secondary differential value 62 maximum position of the primary differential value 63 Minimum position of first derivative 90 Auger electron analyzer

Claims (14)

A spectrum peak position determination method for determining a spectrum peak where the spectrum shows a maximum from the obtained spectrum,
Differentiating the spectrum;
Calculating a first derivative error of a first derivative value obtained by the differentiation from a statistical error of data forming the spectrum;
Calculating an index indicating the magnitude of the difference between the extreme values of the first derivative,
A spectrum peak position determination method, wherein the spectrum peak is determined to exist when the index value exceeds a threshold set based on the first derivative error.   The peak position of the spectrum according to claim 1, wherein the calculation includes calculating a second-order differential error of a second-order differential value obtained by second-order differentiation of the spectrum from the statistical error. Judgment method.   The calculation is characterized in that a secondary differential minimum value of the secondary differential value is obtained, and in the determination, a position on the spectrum having the secondary differential minimum value is set as a position of the spectrum peak. Item 3. A method for determining a peak position of a spectrum according to Item 2.   The spectrum peak position determination method according to any one of claims 1 to 3, wherein the index is a magnitude of a difference between adjacent maximum and minimum values of the primary differential value.   5. The spectrum peak position determination method according to claim 1, wherein the differentiation is performed using a Savitzky-Golay method.   6. The spectrum peak position determination method according to claim 1, wherein the statistical error is a size of a square root of the data.   The spectrum peak position determination method according to any one of claims 1 to 6, wherein in the calculation, the first-order differential error is calculated from the data using a law of error propagation.   The spectrum peak position determination method according to any one of claims 2 to 7, wherein in the calculation, the second-order differential error is calculated from the data using a law of error propagation.   The spectrum peak position determination method according to claim 1, wherein the spectrum is a spectrum using a count value of the number of electrons as data. From the obtained spectrum, an analyzer for determining a spectrum peak showing the maximum of the spectrum,
Differentiating means for differentiating the spectrum;
Calculating means for calculating a first derivative error of a first derivative value obtained by the differentiation from a statistical error of data forming the spectrum;
Index calculating means for calculating an index indicating the magnitude of the difference between the extreme values of the primary differential values;
Peak position determination means for determining that the spectrum peak exists when the value of the index exceeds a threshold set based on the first derivative error;
An analysis apparatus comprising:   The differentiating means performs second order differentiation of the spectrum, and the calculating means also executes a step of calculating a second order differential error of a second order differential value obtained by the second order differentiation from the statistical error. The analyzer according to claim 10, wherein the analyzer is characterized in that:   The calculating means obtains a secondary differential minimum value of the secondary differential value, and the peak position determining means sets the position on the spectrum having the secondary differential minimum value as the position of the spectrum peak. The analyzer according to claim 11, wherein the analyzer is characterized.   The analyzer according to any one of claims 10 to 12, wherein the index calculating means uses the difference between the adjacent maximum value and minimum value of the primary differential value as the index.   The analyzer according to any one of claims 10 to 13, wherein the spectrum is a spectrum using a count value of the number of electrons as data.

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