the analyte *i*. Combining Eq. (8) with Eq. (24) leads to the general form of concentration

1 *ii ij j j WR W*

Eq. (25) is usually applied if influence coefficients are determined using *Ri* calculated from theory. The total matrix correction term expressed by Eq. (24) can also be combined with Eq. (5). Then, we obtain the general equation that can be used during calibration if the matrix

> 1 *i ii ij j i j W KI W B*

intercept *Bi* are determined from multiple-regression analysis on a large suite of standards. One of the simplest algorithms was proposed by Lachance and Traill in 1966 (Lachance and Traill, 1966). The algorithm can be easily derived from the Sherman equation if the following assumptions are made: the sample is infinitely thick, the monochromatic excitation is applied and enhancement effects are negligible. Then, the Sherman equation

1 () () 4 sin (, ) *i i i i i*

(, ) (, ) (, ) *i ii i j j i*

() ( ) (, ) sin sin *i ii*

() ( ) (, ) sin sin *j j i*

element *i*. Assuming that the sum of the element weight fractions in the specimen equals

0

*j*

1 2

1 2

>

 *W W* (28)

*i*) depends on both matrix elements *j* and analyte

*i*

(27)

correction term is calculated from theory for each standard specimen:

Eq. (26) is also used in empirical algorithms if influence coefficients

1

The total mass-attenuation coefficient can be expressed as follows:

The total mass attenuation coefficient *χ*(

*d I QqW I*

*i i*

*j i*

,

  *ij* is influence coefficient describing the matrix effect of the interfering element *j* on

*W* (24)

(25)

(26)

*ij*, slope *Ki* and

(29)

*M*

fractions of matrix elements *Wj*:

Where

based algorithm:

simplifies to:

Where:

using iterative approach (Węgrzynek et al., 1993). In multielement analysis, the measurements are usually performed for a few elements and then relationship between *χ*(*,i*) and wavelength is established. If minor and/or major elements are present in a sample, discontinuities in the relationship between *χ*(*,i*) and wavelength resulting from absorption edges are observed. Then, at least two measurements for each wavelength region are performed or appropriate iteration procedure can be applied (Markowicz and Haselberger, 1992). Quantification based on E-T method, including accuracy and calibration can be found in Ref. (Markowicz et al., 1992; Markowicz and Van Grieken, 2002; Węgrzynek et al., 2003b).

Fig. 4. Measurements in emission-transmission method.

#### **8. Influence coefficient algorithms**

Many influence coefficient algorithms have been developed. They were reviewed and discussed in one chapter of "Handbook of X-Ray Spectrometry," edited by Van Grieken and Markowicz (de Vries and Vrebos, 2002) and numerous papers (Rousseau, 2001, 2002, 2004, 2006), (Willis and Lachance, 2000, 2004). The algorithms can be divided in different ways. The influence coefficients can be calculated from theory (using Sherman equation) or from the measurements, therefore the algorithms are generally divided into two groups: theoretical and empirical influence coefficients algorithms (Fig. 5). The algorithms can use a single or more than one coefficient per matrix element. Moreover, the influence coefficients can be constant or can vary with composition of the sample.

Fig. 5. General division of influence coefficient algorithms.

using iterative approach (Węgrzynek et al., 1993). In multielement analysis, the measurements

wavelength is established. If minor and/or major elements are present in a sample,

*,*

edges are observed. Then, at least two measurements for each wavelength region are performed or appropriate iteration procedure can be applied (Markowicz and Haselberger, 1992). Quantification based on E-T method, including accuracy and calibration can be found in Ref. (Markowicz et al., 1992; Markowicz and Van Grieken, 2002; Węgrzynek et al., 2003b).

Many influence coefficient algorithms have been developed. They were reviewed and discussed in one chapter of "Handbook of X-Ray Spectrometry," edited by Van Grieken and Markowicz (de Vries and Vrebos, 2002) and numerous papers (Rousseau, 2001, 2002, 2004, 2006), (Willis and Lachance, 2000, 2004). The algorithms can be divided in different ways. The influence coefficients can be calculated from theory (using Sherman equation) or from the measurements, therefore the algorithms are generally divided into two groups: theoretical and empirical influence coefficients algorithms (Fig. 5). The algorithms can use a single or more than one coefficient per matrix element. Moreover, the influence coefficients

*,i*) and

*i*) and wavelength resulting from absorption

are usually performed for a few elements and then relationship between *χ*(

discontinuities in the relationship between *χ*(

Fig. 4. Measurements in emission-transmission method.

can be constant or can vary with composition of the sample.

Fig. 5. General division of influence coefficient algorithms.

**8. Influence coefficient algorithms**

In general, the total matrix correction term *Mi* is expressed as a linear combination of weight fractions of matrix elements *Wj*:

$$\mathcal{M}\_i = \mathbf{1} + \sum\_j \alpha\_{ij} \mathcal{W}\_j \tag{24}$$

Where *ij* is influence coefficient describing the matrix effect of the interfering element *j* on the analyte *i*. Combining Eq. (8) with Eq. (24) leads to the general form of concentration based algorithm:

$$\mathcal{W}\_i = \mathcal{R}\_i \left[ \mathbf{1} + \sum\_j \alpha\_{ij} \mathcal{W}\_j \right] \tag{25}$$

Eq. (25) is usually applied if influence coefficients are determined using *Ri* calculated from theory. The total matrix correction term expressed by Eq. (24) can also be combined with Eq. (5). Then, we obtain the general equation that can be used during calibration if the matrix correction term is calculated from theory for each standard specimen:

$$\mathcal{W}\_{i} = K\_{i} I\_{i} \left[ \mathbf{1} + \sum\_{j} \alpha\_{ij} \mathcal{W}\_{j} \right] + B\_{i} \tag{26}$$

Eq. (26) is also used in empirical algorithms if influence coefficients *ij*, slope *Ki* and intercept *Bi* are determined from multiple-regression analysis on a large suite of standards. One of the simplest algorithms was proposed by Lachance and Traill in 1966 (Lachance and Traill, 1966). The algorithm can be easily derived from the Sherman equation if the following assumptions are made: the sample is infinitely thick, the monochromatic excitation is applied and enhancement effects are negligible. Then, the Sherman equation

$$I\_i = \frac{d\Omega}{4\pi\sin\phi} Q\_i q\_i \mathcal{V}\_i \tau\_i(\mathcal{X}) I\_0(\mathcal{X}) \frac{1}{\mathcal{X}(\mathcal{X}, \mathcal{X}\_i)}\tag{27}$$

The total mass-attenuation coefficient can be expressed as follows:

$$\mathcal{X}(\mathcal{X}, \mathcal{X}\_{i}) = \mathcal{W}\_{i}\mathcal{X}\_{i}(\mathcal{X}, \mathcal{X}\_{i}) + \sum\_{j} \mathcal{W}\_{j}\mathcal{X}\_{j}(\mathcal{X}\_{i}\mathcal{X}\_{i}) \tag{28}$$

Where:

simplifies to:

$$\begin{aligned} \chi\_i(\lambda, \lambda\_i) &= \frac{\mu\_i(\lambda)}{\sin \phi\_1} + \frac{\mu\_i(\lambda\_i)}{\sin \phi\_2} \\\\ \chi\_j(\lambda, \lambda\_i) &= \frac{\mu\_j(\lambda)}{\sin \phi\_1} + \frac{\mu\_j(\lambda\_i)}{\sin \phi\_2} \end{aligned} \tag{29}$$

The total mass attenuation coefficient *χ*(,*i*) depends on both matrix elements *j* and analyte element *i*. Assuming that the sum of the element weight fractions in the specimen equals

Quantification in X-Ray Fluorescence Spectrometry 151

1 *WR W ii i*

*ij*

1 *i i*

*W R W*

For example, to determine influence coefficient describing influence of Cr on Ni in FeNiCr, the binary system NiCr is taken into account, and *RNi* for this system has to be determined. The relative intensity can be determined based on actual measurements. In this case, the pure element specimens and suitable binary specimens must be available. Therefore, *Ri* is usually calculated from theory using fundamental parameter method. In the first step, the composition of hypothetical binary specimen is assumed. In next step, *Ri* is calculated from theory for actual measurement conditions (voltage of X-ray tube, incident and take-off

influence coefficients can be determined for any quantitative composition of binary systems. Table 1 shows influence coefficients calculated for various compositions of binary systems. The example is given for determination of Ni in ternary system FeNiCr. Thus, in this case the secondary fluorescence does not exist. As can be observed, if monochromatic excitation is applied, the coefficients are constants, i.e. they do not vary with composition. A different

> Monochromatic excitation, Mo K

0.2 0.8 0.0672 2.47 0.0883 1.58 0.5 0.5 0.2238 2.47 0.2759 1.63 0.8 0.2 0.5355 2.47 0.5972 1.70 Table 1. Influence coefficient calculated from *Ri* for determination of Ni in ternary system

A similar situation is observed in determination of Fe in FeNiCr if polychromatic excitation

Cr on Fe is positive because only absorption effect is observed in this case. The coefficient describing influence of Ni on Fe is negative because the enhancement effect dominates.

Summarizing, the influence coefficients can be treated as constants only if monochromatic excitation is applied and there are no enhancement effects. Fig. 6 presents results for determination of Cr in stainless steel before and after matrix correction using constant

*NiFe*

0.2 0.8 0.0561 3.21 0.0740 2.13 0.5 0.5 0.1920 3.21 0.2396 2.17 0.8 0.2 0.4873 3.21 0.5510 2.26

*j*

*NiCr RNi*

*FeNi* are variable. The coefficient describing influence of

*ij*:

angles, medium: air, helium or vacuum). Finally,

situation is observed for polychromatic excitation, where both

*RNi*

FeNiCr using monochromatic and polychromatic excitation.

*FeCr* and

Moreover, it strongly depends on specimen composition.

Eq. (35) can be rewritten to obtain

composition of the specimen.

*WNi WFe WCr*

is applied (Table 2). Both

*j j* (35)

(36)

*NiFe* and

Polychromatic excitation, Mo target X-ray tube operated at 45kV

> *NiFe*

*NiCr* vary with

*NiCr*

*ij* is calculated from Eq. (36). The

one (then *Wi* = 1 - *Wj*), the weight fraction of analyte element *i* can be eliminated from Eq. (28) and then *χ*(,*i*) is given by:

$$\mathcal{X}(\mathcal{X}, \mathcal{Y}\_i) = \mathcal{X}\_i(\mathcal{X}, \mathcal{Y}\_i) \left[ \mathbf{1} + \sum\_j \alpha\_{ij} \mathcal{W}\_j \right] \tag{30}$$

Where:

$$\alpha\_{ij} = \frac{\mathbb{X}\_{j}(\mathbb{X}, \mathbb{X}\_{i})}{\mathbb{X}\_{i}(\mathbb{X}, \mathbb{X}\_{i})} - 1 \tag{31}$$

Finally, if the absolute intensity *Ii* is replaced by the relative radiation intensity *Ri* the Lachance-Traill equation can be obtained from the simplified Sherman formula (Eq. (27)):

$$\mathcal{W}\_i = R\_i \left[ 1 + \sum\_j \alpha\_{ij} \mathcal{W}\_j \right] \tag{32}$$

In Lachance-Traill algorithm, the influence coefficient *ij* corrects for the absorption effects of the matrix element *j* on the analyte *i* in the case of monochromatic excitation of wavelength . The coefficient can be positive or negative. If the analyte is determined in presence of a heavier matrix element, then *χi*(,*i*)< *χj*(,*i*) and *ij* is positive. If the analyte is determined in presence of a lighter matrix element, then *χi*(,*i*)> *χj*(,*i*) and *ij* is negative. The influence coefficients can be calculated in different ways:

The influence coefficients can be calculated directly from the Eq. (31) only if monochromatic excitation is applied (or the effective wavelength is used although it is composition dependent) and there are no enhancement effects. The determination of Ni in ternary system FeNiCr is shown here as an example. The analyte is excited by molybdenum radiation (Mo K) and following measurement geometry is assumed: *1* = 60° and *2* = 40°. In this case, the influence coefficient *NiFe* describing influence of Fe on Ni is calculated from the following equation:

$$\alpha\_{\rm NiFe} = \frac{\mu\_{\rm Fe}(\lambda\_{\rm Mo})}{\sin \phi\_1} + \frac{\mu\_{\rm Fe}(\lambda\_{\rm Ni})}{\sin \phi\_2} - 1 = \frac{37.4}{\frac{0.87}{\sin \phi\_1} + \frac{363.3}{\sin \phi\_2}}{\frac{44.6}{0.87} + \frac{59.8}{0.64}} - 1 = 3.21\tag{33}$$

The influence coefficient *NiCr* describing influence of Cr on Ni is calculated in the same way. Finally, the following equation is obtained for determination of Ni in FeNiCr using aforementioned measurement conditions:

$$\mathcal{W}\_{\rm Ni} = \mathcal{R}\_{\rm Ni} \left[ \mathbf{1} + \mathbf{3.21} \times \mathcal{W}\_{\rm Fe} + \mathbf{2.47} \times \mathcal{W}\_{\rm Cr} \right] \tag{34}$$

The influence coefficients can also be calculated from relative radiation intensity of the analyte. In this case, the influence coefficient *ij* is calculated for binary systems. Then Eq. (32) simplifies to:

$$\mathcal{W}\_i = \mathcal{R}\_i \left[ \mathbf{1} + \alpha\_{ij} \mathcal{W}\_j \right] \tag{35}$$

Eq. (35) can be rewritten to obtain *ij*:

150 X-Ray Spectroscopy

one (then *Wi* = 1 - *Wj*), the weight fraction of analyte element *i* can be eliminated from Eq.

( , ) ( , )1 *i ii ij j*

(, ) 1 (, ) *j i*

*i i*

Finally, if the absolute intensity *Ii* is replaced by the relative radiation intensity *Ri* the Lachance-Traill equation can be obtained from the simplified Sherman formula (Eq. (27)):

> 1 *ii ij j j WR W*

of the matrix element *j* on the analyte *i* in the case of monochromatic excitation of

,*i*)< *χj*(,*i*) and

It can be determined from multiple-regression analysis on a large suite of standards.

The influence coefficients can be calculated directly from the Eq. (31) only if monochromatic excitation is applied (or the effective wavelength is used although it is composition dependent) and there are no enhancement effects. The determination of Ni in ternary system FeNiCr is shown here as an example. The analyte is excited by molybdenum radiation (Mo K) and

( ) () 37.4 363.3

sin sin 0.87 0.64

Finally, the following equation is obtained for determination of Ni in FeNiCr using

The influence coefficients can also be calculated from relative radiation intensity of the

1 2

*Fe Mo Fe Ni*

*Ni Mo Ni Ni*

1 2

*NiFe* describing influence of Fe on Ni is calculated from the following equation:

sin sin 0.87 0.64 1 1 3.21 ( ) ( ) 44.6 59.8

*1* = 60° and

*NiCr* describing influence of Cr on Ni is calculated in the same way.

*WR W W Ni Ni* 1 3.21 2.47 *Fe Cr* (34)

*ij* is calculated for binary systems. Then Eq.

,*i*)> *χj*(,*i*) and

. The coefficient can be positive or negative. If the analyte is determined in

*ij*

*j*

 *W*

(30)

(31)

(32)

*ij* corrects for the absorption effects

*ij* is positive. If the analyte is

*2* = 40°. In this case, the influence

*ij* is negative.

(33)

(28) and then *χ*(

Where:

wavelength

coefficient

The influence coefficient

(32) simplifies to:

,

*i*) is given by:

In Lachance-Traill algorithm, the influence coefficient

determined in presence of a lighter matrix element, then *χi*(

The influence coefficients can be calculated in different ways:

It can be calculated from the relative radiation intensity *Ri*.

presence of a heavier matrix element, then *χi*(

It can be calculated directly from the Eq. (31).

following measurement geometry is assumed:

*NiFe*

aforementioned measurement conditions:

analyte. In this case, the influence coefficient

$$
\alpha\_{ij} = \frac{\mathcal{W}\_i / \mathcal{R}\_i - 1}{\mathcal{W}\_j} \tag{36}
$$

For example, to determine influence coefficient describing influence of Cr on Ni in FeNiCr, the binary system NiCr is taken into account, and *RNi* for this system has to be determined. The relative intensity can be determined based on actual measurements. In this case, the pure element specimens and suitable binary specimens must be available. Therefore, *Ri* is usually calculated from theory using fundamental parameter method. In the first step, the composition of hypothetical binary specimen is assumed. In next step, *Ri* is calculated from theory for actual measurement conditions (voltage of X-ray tube, incident and take-off angles, medium: air, helium or vacuum). Finally, *ij* is calculated from Eq. (36). The influence coefficients can be determined for any quantitative composition of binary systems. Table 1 shows influence coefficients calculated for various compositions of binary systems. The example is given for determination of Ni in ternary system FeNiCr. Thus, in this case the secondary fluorescence does not exist. As can be observed, if monochromatic excitation is applied, the coefficients are constants, i.e. they do not vary with composition. A different situation is observed for polychromatic excitation, where both *NiFe* and *NiCr* vary with composition of the specimen.

Table 1. Influence coefficient calculated from *Ri* for determination of Ni in ternary system FeNiCr using monochromatic and polychromatic excitation.

A similar situation is observed in determination of Fe in FeNiCr if polychromatic excitation is applied (Table 2). Both *FeCr* and *FeNi* are variable. The coefficient describing influence of Cr on Fe is positive because only absorption effect is observed in this case. The coefficient describing influence of Ni on Fe is negative because the enhancement effect dominates. Moreover, it strongly depends on specimen composition.

Summarizing, the influence coefficients can be treated as constants only if monochromatic excitation is applied and there are no enhancement effects. Fig. 6 presents results for determination of Cr in stainless steel before and after matrix correction using constant

Quantification in X-Ray Fluorescence Spectrometry 153

matrix elements. Therefore, applying binary coefficients to multielement samples leads to an incomplete matrix correction. It can be explained using simple example: the enhancement of Fe by Ni in two different matrices - Al and Cr matrix. In Cr matrix, Ni radiation is strongly absorbed by Cr. Therefore, Ni excites Fe to a lesser extent in Cr matrix than in Al matrix. In

applied. Therefore, a better solution is the use of multielement coefficient algorithms, e.g.

In general, the constant coefficient algorithms (e.g. Lachance-Traill algorithm) give satisfactory results but concentration range should be limited (0-10%). The algorithms using binary variable coefficients (e.g. Claisse-Quintin algorithm) give excellent results over medium concentration range of 0-40%. Multielement coefficient algorithms can be used in a

A number of influence coefficient algorithms have been proposed for correction of matrix effects. They can be divided in several ways: by the way of calculating influence coefficients (theoretical or empirical), by the variables used in matrix correction term (concentration or intensity based) or by the analytical context of determination of coefficients (constant or variable, binary or multielement). The general scheme of division and sub-divisions of

In the next part of the chapter, various influence coefficient algorithms will be reviewed. The application of these algorithms for calibration and analysis of unknown specimen discussed here is valid for any of the algorithms. Some differences are observed in the use of

In the case of theoretical influence coefficient algorithms, the coefficients are determined from theory e.g. using *Ri* calculated from Sherman equation for given measurement conditions. In the next stage, the matrix correction term (Eq. (24)) is calculated for all standard specimens and for a given analyte. Then, the calibration graph is plotted: the measured radiation intensity of the analyte multiplied by the corresponding matrix correction versus weight fraction of analyte. Then, slope *Ki* and intercept *Bi* are determined

In the case of empirical algorithms, the influence coefficients, slope *Ki* and intercept *Bi* are determined from multiple-regression analysis on a large suite of standards. If there are *n* matrix elements, *n*+2 coefficients have to be determined (*n* influence coefficients, slope *Ki* and intercept *Bi*). For the calculation of *n*+2 coefficients, *n*+2 standards are required. In practice, a much larger number of standards is used: 2*n* or even 3*n*. It should be noted, that the matrix element can be included in matrix correction term only if its influence on the analyte is significant. Otherwise, the matrix element should be omitted in matrix correction term to obtain correct values of other influence coefficients. Moreover, the concentration of all elements in the reference specimens should cover composition of unknown sample. Only

Analysis of unknown sample is similar for both theoretical and empirical algorithms. In this step, a set of equations has to be solved for the unknowns: *Wi*, *Wj*, … *Wn*. If there are *n* linear equations with *n* unknowns, the set of equations can be solved algebraically. However, an iterative procedure is usually applied, especially in the case of a set of nonlinear equations. General scheme of using theoretical influence algorithm is presented in

*FeNi* is used if binary coefficient algorithm is

both cases, the same influence coefficient

full concentration range between 0 and 100%.

Lachance-Claisse, Broll-Tertain or Rousseau algorithm.

influence coefficient algorithms is presented in Fig. 5.

by least-squares fit using set of standard specimens.

then, accurate results can be obtained.

Fig. 7.

theoretical and empirical algorithms during the calibration stage.

coefficients (e.g. Lachance-Traill algorithm) and variable linear coefficients (e.g. Claisse-Quintin algorithm). Before matrix correction, the mean error of Cr determination equals 1.11 % Cr. Considerable improvement is obtained after matrix correction with constant coefficient - the mean error is equal to 0.35 % Cr. Even better results are obtained for linear coefficients - the mean error equals 0.14 % Cr.

Table 2. Influence coefficient calculated from *Ri* for determination of Fe in ternary system FeNiCr using polychromatic excitation: Mo target X-ray tube operated at 45kV.

Fig. 6. Determination of Cr in stainless steel before and after matrix correction using (a) Lachance-Traill algorithm and (b) Claisse-Quintin algorithm (see subchapters 8.1 and 8.2).

The coefficients calculated from *Ri* for binaries are the so-called binary coefficients, i.e. they describe influence of matrix element on analyte but they do not take into account other

coefficients (e.g. Lachance-Traill algorithm) and variable linear coefficients (e.g. Claisse-Quintin algorithm). Before matrix correction, the mean error of Cr determination equals 1.11 % Cr. Considerable improvement is obtained after matrix correction with constant coefficient - the mean error is equal to 0.35 % Cr. Even better results are obtained for linear

0.2 0.8 0.3017 -0.42 0.5 0.5 0.5985 -0.33 0.8 0.2 0.8431 -0.26 Table 2. Influence coefficient calculated from *Ri* for determination of Fe in ternary system

*FeCr* *FeNi*

coefficients - the mean error equals 0.14 % Cr.

*WFe WCr WNi RFe*

0.2 0.8 0.0717 2.24 0.5 0.5 0.2329 2.29 0.8 0.2 0.5405 2.40

FeNiCr using polychromatic excitation: Mo target X-ray tube operated at 45kV.

Fig. 6. Determination of Cr in stainless steel before and after matrix correction using (a) Lachance-Traill algorithm and (b) Claisse-Quintin algorithm (see subchapters 8.1 and 8.2). The coefficients calculated from *Ri* for binaries are the so-called binary coefficients, i.e. they describe influence of matrix element on analyte but they do not take into account other matrix elements. Therefore, applying binary coefficients to multielement samples leads to an incomplete matrix correction. It can be explained using simple example: the enhancement of Fe by Ni in two different matrices - Al and Cr matrix. In Cr matrix, Ni radiation is strongly absorbed by Cr. Therefore, Ni excites Fe to a lesser extent in Cr matrix than in Al matrix. In both cases, the same influence coefficient *FeNi* is used if binary coefficient algorithm is applied. Therefore, a better solution is the use of multielement coefficient algorithms, e.g. Lachance-Claisse, Broll-Tertain or Rousseau algorithm.

In general, the constant coefficient algorithms (e.g. Lachance-Traill algorithm) give satisfactory results but concentration range should be limited (0-10%). The algorithms using binary variable coefficients (e.g. Claisse-Quintin algorithm) give excellent results over medium concentration range of 0-40%. Multielement coefficient algorithms can be used in a full concentration range between 0 and 100%.

A number of influence coefficient algorithms have been proposed for correction of matrix effects. They can be divided in several ways: by the way of calculating influence coefficients (theoretical or empirical), by the variables used in matrix correction term (concentration or intensity based) or by the analytical context of determination of coefficients (constant or variable, binary or multielement). The general scheme of division and sub-divisions of influence coefficient algorithms is presented in Fig. 5.

In the next part of the chapter, various influence coefficient algorithms will be reviewed. The application of these algorithms for calibration and analysis of unknown specimen discussed here is valid for any of the algorithms. Some differences are observed in the use of theoretical and empirical algorithms during the calibration stage.

In the case of theoretical influence coefficient algorithms, the coefficients are determined from theory e.g. using *Ri* calculated from Sherman equation for given measurement conditions. In the next stage, the matrix correction term (Eq. (24)) is calculated for all standard specimens and for a given analyte. Then, the calibration graph is plotted: the measured radiation intensity of the analyte multiplied by the corresponding matrix correction versus weight fraction of analyte. Then, slope *Ki* and intercept *Bi* are determined by least-squares fit using set of standard specimens.

In the case of empirical algorithms, the influence coefficients, slope *Ki* and intercept *Bi* are determined from multiple-regression analysis on a large suite of standards. If there are *n* matrix elements, *n*+2 coefficients have to be determined (*n* influence coefficients, slope *Ki* and intercept *Bi*). For the calculation of *n*+2 coefficients, *n*+2 standards are required. In practice, a much larger number of standards is used: 2*n* or even 3*n*. It should be noted, that the matrix element can be included in matrix correction term only if its influence on the analyte is significant. Otherwise, the matrix element should be omitted in matrix correction term to obtain correct values of other influence coefficients. Moreover, the concentration of all elements in the reference specimens should cover composition of unknown sample. Only then, accurate results can be obtained.

Analysis of unknown sample is similar for both theoretical and empirical algorithms. In this step, a set of equations has to be solved for the unknowns: *Wi*, *Wj*, … *Wn*. If there are *n* linear equations with *n* unknowns, the set of equations can be solved algebraically. However, an iterative procedure is usually applied, especially in the case of a set of nonlinear equations. General scheme of using theoretical influence algorithm is presented in Fig. 7.

Quantification in X-Ray Fluorescence Spectrometry 155

The algorithm was discussed in detail in the previous section. The influence coefficients can be determined from the relative radiation intensity or from multiple-regression analysis on a large suite of standards. They can also be calculated directly from the Eq. (31) if monochromatic excitation is applied and enhancement effects are not observed. If polychromatic beam is applied, then the effective wavelength can be used, however it is also

> 1 *i ii ij j j j e*

Where *Ei* is a constant determined during calibration. The algorithm proposed by de Jongh (de Jongh, 1973) uses theoretical coefficients calculated from fundamental parameters. This algorithm looks similar to the Lachance-Traill equation but the eliminated element *e* is the same for all equations (in Lachance-Traill algorithm, the analyte is eliminated from each

around an average composition. They are multielement coefficients rather than binary coefficients, i.e. the influence coefficient describes influence of matrix element *j* on analyte *i* in the presence of all matrix elements. The advantage of the de Jongh algorithm is that one element can be arbitrarily eliminated from the correction procedure, so there is no need to measure it. For example, in ferrous alloys, iron is often the major element and it is usually determined by difference, and therefore, can be eliminated from the correction procedure.

> *ii ij j k j k i*

In this algorithm, two different coefficients are used for correction of absorption effect (coefficient *aij*) and enhancement effect (coefficient *bij*) and only one coefficient is used for each matrix element. The coefficient *aij* is used if absorption is the dominant effect (then *bik* = 0). The coefficients *bik* are used if enhancement is the dominant effect (then *aij* = 0). The coefficients are determined experimentally on the basis of measured reference samples. The serious disadvantage of the algorithm is the fact that it is not clear which matrix element should be assigned an *aij* and which one a *bik*. Therefore, the Rasberry-Heinrich algorithm is not considered to be generally applicable, but it gives satisfactory results for FeNiCr alloys.

> *i ii ij j i j W I k aI B*

Where *ki* and *aij* are the correction coefficients, *Bi* is a background term. The coefficients are determined by a least-squares fit on the basis of measured reference samples. The correction

*b W R aW W*

1

It is for this system that the algorithm was originally developed.

**The Lucas-Tooth and Price algorithm (1964)**

*ii* = 0). The influence coefficients are calculated by a Taylor series expansion

1 *ik*

*W*

(39)

(40)

(38)

*W ER W*

composition dependent.

equation, i.e.

**The de Jongh algorithm (1973)**

**The Rasberry-Heinrich algorithm (1974)**

Fig. 7. General scheme of using theoretical influence algorithm.

#### **8.1 Constant coefficient algorithms**

The previous section shows that influence coefficients can be treated as constants only if: monochromatic excitation is applied and there are no enhancement effects. However, the monochromatic excitation is rarely used in practice and enhancement effects are usually observed in multielement specimens. Therefore, to obtain accurate results with constant coefficients, the concentration ranges of analyte and matrix elements should be limited. The constant coefficients are calculated for a given composition range or a series of reference samples that cover composition of unknown sample. If composition range is wide, then the sample can be diluted by pelletizing with a binder or by fusion.

#### **The Lachance-Traill algorithm (1966)**

$$\mathcal{W}\_i = R\_i \left[ 1 + \sum\_j \alpha\_{ij} \mathcal{W}\_j \right] \tag{37}$$

The algorithm was discussed in detail in the previous section. The influence coefficients can be determined from the relative radiation intensity or from multiple-regression analysis on a large suite of standards. They can also be calculated directly from the Eq. (31) if monochromatic excitation is applied and enhancement effects are not observed. If polychromatic beam is applied, then the effective wavelength can be used, however it is also composition dependent.

#### **The de Jongh algorithm (1973)**

154 X-Ray Spectroscopy

Fig. 7. General scheme of using theoretical influence algorithm.

sample can be diluted by pelletizing with a binder or by fusion.

The previous section shows that influence coefficients can be treated as constants only if: monochromatic excitation is applied and there are no enhancement effects. However, the monochromatic excitation is rarely used in practice and enhancement effects are usually observed in multielement specimens. Therefore, to obtain accurate results with constant coefficients, the concentration ranges of analyte and matrix elements should be limited. The constant coefficients are calculated for a given composition range or a series of reference samples that cover composition of unknown sample. If composition range is wide, then the

> 1 *ii ij j j WR W*

(37)

**8.1 Constant coefficient algorithms**

**The Lachance-Traill algorithm (1966)**

$$\mathcal{W}\_i = E\_i R\_i \left[ \mathbf{1} + \sum\_{j} \alpha\_{ij} \mathbf{W}\_j \right] \tag{38}$$

Where *Ei* is a constant determined during calibration. The algorithm proposed by de Jongh (de Jongh, 1973) uses theoretical coefficients calculated from fundamental parameters. This algorithm looks similar to the Lachance-Traill equation but the eliminated element *e* is the same for all equations (in Lachance-Traill algorithm, the analyte is eliminated from each equation, i.e. *ii* = 0). The influence coefficients are calculated by a Taylor series expansion around an average composition. They are multielement coefficients rather than binary coefficients, i.e. the influence coefficient describes influence of matrix element *j* on analyte *i* in the presence of all matrix elements. The advantage of the de Jongh algorithm is that one element can be arbitrarily eliminated from the correction procedure, so there is no need to measure it. For example, in ferrous alloys, iron is often the major element and it is usually determined by difference, and therefore, can be eliminated from the correction procedure.

#### **The Rasberry-Heinrich algorithm (1974)**

$$\mathcal{W}\_i = R\_i \left[ \mathbf{1} + \sum\_j a\_{ij} \mathcal{W}\_j + \sum\_k \frac{b\_{ik}}{\mathbf{1} + \mathcal{W}\_i} \mathcal{W}\_k \right] \tag{39}$$

In this algorithm, two different coefficients are used for correction of absorption effect (coefficient *aij*) and enhancement effect (coefficient *bij*) and only one coefficient is used for each matrix element. The coefficient *aij* is used if absorption is the dominant effect (then *bik* = 0). The coefficients *bik* are used if enhancement is the dominant effect (then *aij* = 0). The coefficients are determined experimentally on the basis of measured reference samples. The serious disadvantage of the algorithm is the fact that it is not clear which matrix element should be assigned an *aij* and which one a *bik*. Therefore, the Rasberry-Heinrich algorithm is not considered to be generally applicable, but it gives satisfactory results for FeNiCr alloys. It is for this system that the algorithm was originally developed.

#### **The Lucas-Tooth and Price algorithm (1964)**

$$\mathcal{W}\_{i} = I\_{i} \left[ k\_{i} + \sum\_{j} a\_{ij} I\_{j} \right] + B\_{i} \tag{40}$$

Where *ki* and *aij* are the correction coefficients, *Bi* is a background term. The coefficients are determined by a least-squares fit on the basis of measured reference samples. The correction

Quantification in X-Ray Fluorescence Spectrometry 157

system (*Wi* = 0.30, *Wj* = 0.35, *Wk* = 0.35). The applicability of COLA algorithm is similar to the Claisse-Quintin algorithm. The algorithm corrects for both absorption and enhancement

Binary influence coefficients are based on an approximation: the total matrix effect on the analyte *i* equals the sum of the effects of each matrix element j and each effect is calculated independently of each other. The binary coefficient algorithms give accurate results if composition range is limited, i.e. 0-10% for the constant coefficient algorithms and 0-40% for binary variable coefficient algorithms. The coefficients are calculated for a given composition range rather than for a given sample composition. The multielement coefficients are exact and are calculated for each individual reference and unknown specimens. Therefore, the multielement coefficient algorithms are applicable in full composition range from 0 to 100%. The most important algorithms were proposed by Broll

> 1 *i ii ij ij j j i*

*R*

*ij j*

*W*

*ij j j*

*W*

(43)

(44)

(45)

*W W R W*

1

1

1

effects, respectively. The multielement influence coefficients can be easily calculated for reference specimens of known composition. In the case of unknown specimens, the iteration is used. In Rousseau algorithm, a first estimate of the composition of the unknown specimen is calculated using the Claisse-Quintin algorithm, then the influence coefficients are calculated for this composition. An exact estimate of composition is finally obtained by applying the iterative process to Eq. (44). Willis and Lachance (2004) showed that the aforementioned multielement coefficient algorithms give the same high-quality results and

The influence coefficients algorithms are widely applied for quantitative analysis of infinitely thick samples. Influence coefficients algorithms can be deduced from the Sherman

*W R W*

*i i j j ij j j*

*i i*

*W R*

1

**8.4 Influence coefficient algorithms for intermediate-thickness samples**

*j*

*ij ij*

*W*

*ij* are influence coefficients correcting for absorption and enhancement

effects over a broad range of concentration.

**The Broll-Tertain algorithm (1983)**

**The Rousseau algorithm (1984)**

Where

*ij* and

**The Lachance-Claisse algorithm (1995)**

no algorithm can outperform any of the others.

**8.3 Variable multielement coefficient algorithms**

and Tertian (1983), Rousseau (1984), and Lachance and Claisse (1995).

is done using intensities rather than concentrations. The algorithm assumes that the matrix effect of an element *j* on the analyte *i* is proportional to its intensity. Because this approximation is unreliable (the measured intensities are modified by matrix effects and are not directly proportional to concentrations), the algorithm has a limited range of applicability. However, determination of a single element is possible using Lucas-Tooth and Price algorithm.

#### **8.2 Variable binary coefficient algorithms**

#### **The Claisse-Quintin algorithm (1967)**

$$\begin{aligned} \mathcal{W}\_i &= \mathcal{R}\_i \left[ \mathbf{1} + \sum\_j \alpha\_{\text{lin},ij} \mathcal{W}\_j + \sum\_j \sum\_k \alpha\_{ijk} \mathcal{W}\_j \mathcal{W}\_k \right] \\\\ \alpha\_{\text{lin},ij} &= \alpha\_{ij} + \alpha\_{\text{jj}} \mathcal{W}\_M \end{aligned} \tag{41}$$

In 1967, Claisse and Quintin proposed an improved version of Lachance-Traill algorithm (Claisse and Quintin, 1967). The algorithm uses linear coefficients *lin,ij* instead of constant coefficients and can be applied for polychromatic excitation in a medium range of concentration. The cross-product coefficient *ijk* included in matrix correction term corrects for the simultaneous presence of both *j* and *k* and compensates for the fact that the total matrix effect correction cannot be represented as a sum of binary matrix effect corrections. *WM* is matrix concentration (*WM* = 1 – *Wi*) proposed by Tertian instead of *Wj* in the original equation in order to obtain high quality results (Tertian, 1976). Equations for calculating accurate and valid theoretical binary influence coefficients in the Claisse–Quintin algorithm were first proposed by Rousseau and Claisse (1974) and later improved by Rousseau (1984 b). The influence coefficients are calculated from *Ri* using fundamental parameters at two binaries (*W*i = 0.2 and 0.8). The algorithm introduces a theoretical mean relative error of 0.04% on the calculated concentrations in the case of cement samples prepared as pressed powder pellets (medium concentration range of 0 - 40%). For a large concentration range (0– 100%), e.g. in the case of alloy analysis, the Claisse–Quintin algorithm introduces 0.3% relative error (Rousseau, 2001).

#### **The COLA algorithm (1981)**

$$\begin{aligned} \mathcal{W}\_i &= R\_i \left[ 1 + \sum\_j \alpha\_{\text{hyp}, ij} \mathcal{W}\_j + \sum\_j \sum\_k \alpha\_{ijk} \mathcal{W}\_j \mathcal{W}\_k \right] \\\\ \alpha\_{\text{hyp}, ij} &= \alpha\_{ij, 1} + \frac{\alpha\_{ij, 2} \mathcal{W}\_M}{1 + \alpha\_{ij, 3} \left( 1 - \mathcal{W}\_M \right)} \end{aligned} \tag{42}$$

In 1981, Lachance proposed an algorithm called COLA (comprehensive Lachance) with a new approximation to the binary influence coefficients (Lachance, 1981). This time, hyperbolic influence coefficients were applied. The influence coefficients are calculated from *Ri* using fundamental parameters at three binaries (*W*i = 0.001, 0.5 and 0.999). Similarly to the Claisse-Quintin algorithm, the cross-product coefficients are calculated from a ternary

is done using intensities rather than concentrations. The algorithm assumes that the matrix effect of an element *j* on the analyte *i* is proportional to its intensity. Because this approximation is unreliable (the measured intensities are modified by matrix effects and are not directly proportional to concentrations), the algorithm has a limited range of applicability. However, determination of a single element is possible using Lucas-Tooth and

> , 1 *i i lin ij j ijk j k j j k*

In 1967, Claisse and Quintin proposed an improved version of Lachance-Traill algorithm

coefficients and can be applied for polychromatic excitation in a medium range of

for the simultaneous presence of both *j* and *k* and compensates for the fact that the total matrix effect correction cannot be represented as a sum of binary matrix effect corrections. *WM* is matrix concentration (*WM* = 1 – *Wi*) proposed by Tertian instead of *Wj* in the original equation in order to obtain high quality results (Tertian, 1976). Equations for calculating accurate and valid theoretical binary influence coefficients in the Claisse–Quintin algorithm were first proposed by Rousseau and Claisse (1974) and later improved by Rousseau (1984 b). The influence coefficients are calculated from *Ri* using fundamental parameters at two binaries (*W*i = 0.2 and 0.8). The algorithm introduces a theoretical mean relative error of 0.04% on the calculated concentrations in the case of cement samples prepared as pressed powder pellets (medium concentration range of 0 - 40%). For a large concentration range (0– 100%), e.g. in the case of alloy analysis, the Claisse–Quintin algorithm introduces 0.3%

> , 1 *ii hyp ij j ijk j k j j k*

,2

*ij M*

*ij M W*

,3 1 1

In 1981, Lachance proposed an algorithm called COLA (comprehensive Lachance) with a new approximation to the binary influence coefficients (Lachance, 1981). This time, hyperbolic influence coefficients were applied. The influence coefficients are calculated from *Ri* using fundamental parameters at three binaries (*W*i = 0.001, 0.5 and 0.999). Similarly to the Claisse-Quintin algorithm, the cross-product coefficients are calculated from a ternary

 *W WW*

*W*

, ,1

*hyp ij ij*

 *W WW*

*ijk* included in matrix correction term corrects

(41)

(42)

*lin,ij* instead of constant

*lin i*, *j ij ijjWM*

(Claisse and Quintin, 1967). The algorithm uses linear coefficients

Price algorithm.

**8.2 Variable binary coefficient algorithms The Claisse-Quintin algorithm (1967)**

concentration. The cross-product coefficient

relative error (Rousseau, 2001). **The COLA algorithm (1981)**

*W R*

*W R*

system (*Wi* = 0.30, *Wj* = 0.35, *Wk* = 0.35). The applicability of COLA algorithm is similar to the Claisse-Quintin algorithm. The algorithm corrects for both absorption and enhancement effects over a broad range of concentration.

#### **8.3 Variable multielement coefficient algorithms**

Binary influence coefficients are based on an approximation: the total matrix effect on the analyte *i* equals the sum of the effects of each matrix element j and each effect is calculated independently of each other. The binary coefficient algorithms give accurate results if composition range is limited, i.e. 0-10% for the constant coefficient algorithms and 0-40% for binary variable coefficient algorithms. The coefficients are calculated for a given composition range rather than for a given sample composition. The multielement coefficients are exact and are calculated for each individual reference and unknown specimens. Therefore, the multielement coefficient algorithms are applicable in full composition range from 0 to 100%. The most important algorithms were proposed by Broll and Tertian (1983), Rousseau (1984), and Lachance and Claisse (1995).

#### **The Broll-Tertain algorithm (1983)**

$$\mathcal{W}\_{i} = R\_{i} \left[ \mathbf{1} + \sum\_{j} \left( \alpha\_{ij} - \varepsilon\_{ij} \frac{\mathcal{W}\_{i}}{R\_{i}} \right) \mathcal{W}\_{j} \right] \tag{43}$$

**The Rousseau algorithm (1984)**

$$\mathcal{W}\_i = R\_i \frac{1 + \sum\_j \alpha\_{ij} \mathcal{W}\_j}{1 + \sum\_j \varepsilon\_{ij} \mathcal{W}\_j} \tag{44}$$

**The Lachance-Claisse algorithm (1995)**

$$\mathcal{W}\_i = R\_i \left[ 1 + \sum\_j \frac{\alpha\_{ij} - \varepsilon\_{ij}}{1 + \sum\_j \varepsilon\_{ij} \mathcal{W}\_j} \mathcal{W}\_j \right] \tag{45}$$

Where *ij* and *ij* are influence coefficients correcting for absorption and enhancement effects, respectively. The multielement influence coefficients can be easily calculated for reference specimens of known composition. In the case of unknown specimens, the iteration is used. In Rousseau algorithm, a first estimate of the composition of the unknown specimen is calculated using the Claisse-Quintin algorithm, then the influence coefficients are calculated for this composition. An exact estimate of composition is finally obtained by applying the iterative process to Eq. (44). Willis and Lachance (2004) showed that the aforementioned multielement coefficient algorithms give the same high-quality results and no algorithm can outperform any of the others.

#### **8.4 Influence coefficient algorithms for intermediate-thickness samples**

The influence coefficients algorithms are widely applied for quantitative analysis of infinitely thick samples. Influence coefficients algorithms can be deduced from the Sherman

Quantification in X-Ray Fluorescence Spectrometry 159

In the next paper (Sitko, 2008b), two algorithms of constant and linear coefficients for simultaneous determination of composition and thickness of thin films were proposed. This time, the coefficients are not calculated directly from theory but from the relative radiation intensity (calculated from theory) of hypothetical pure element films and binary films. The potential of the algorithms was demonstrated with hypothetical ternary and binary systems:

Many methods, both empirical and theoretical, have been proposed for quantitative XRF analysis. Method selection usually depends on sample type (thin or bulk, alloys or rocks, etc.), method of sample preparation (without treatment, fusion, etc.), expected results (quantitative or semi-quantitative analysis, determination of a single element or multielement analysis) and availability of standard samples. If the compensation methods are applied, then complicated sample preparation is required, only one or few elements can be quantitatively determined (standard addition, internal standard, Compton scatter) and additional matrix correction may be required (dilution). In influence coefficient algorithm, the matrix correction and calibration is clear, therefore, interpretation of data is very easy. The fundamental parameter methods are like "a black box". Nevertheless, the serious advantage is their versatility: the analysis of bulk, thin samples and multilayers is possible. Fundamental parameter methods are usually considered to be less accurate than the influence coefficient algorithms. This results from the fact that the fundamental parameter methods are usually used with only a few standards. The accuracy of fundamental parameter methods is very similar to that of influence coefficient algorithms when the same

Araujo, M.F.; Van Espen, P.; Van Grieken, R. (1990). Determination of sample thickness via

Bao, S.X. (1997). A power function relation between mass attenuation coefficient and Rh Kα

Broll, N.; Tertian, R. (1983). Quantitative X-ray fluorescence analysis by use of fundamental

De Boer, D.K.G. (1990). Calculation of X-ray fluorescence intensities from bulk and

De Boer, D.K.G.; Borstrok, J.J.M.; Leenaers, A.J.G.; Van Sprang, H.A.; Brouwer, P.N. (1993).

Claisse, F.; Quintin, M. (1967). Generalization of the Lachance–Traill method for the

de Jongh, W.K. (1973). X-ray fluorescence analysis applying theoretical matrix correction.

influence coefficients, *X-ray Spec*trometry, Vol. 12, pp. 30–37.

multilayer samples, *X-Ray Spectrometry*, Vol. 19, pp. 145–154.

multilayer samples, *X-Ray Spectrometry*, Vol. 22, pp. 33–38.

Stainless steel, *X-ray Spectrometry*, Vol. 2, pp. 151–158.

scattered radiation in X-ray fluorescence spectrometry with filtered continuum

Compton peak intensity and its application to XRF analysis, *X-Ray Spectrometry*,

How accurate is the fundamental parameter approach? XRF analysis of bulk and

correction of the matrix effect in X-ray fluorescence analysis, *Canadian Journal of*

FeCrNi, FeCr, FeNi, CrNi and experimental data of FeNi and Cu films.

standards (many standards similar to the unknown) are used in both cases.

excitation, *X-Ray Spectrometry*, Vol. 19, pp. 29–33.

**9. Conclusions**

**10. References**

Vol. 26, pp. 23-27.

*Spectroscopy*, Vol. 12, pp. 129–134.

equation because exponential term equals zero for thick samples. In consequence, the matrix correction term can be expressed as a linear combination of matrix elements concentrations. The analysis of intermediate-thickness samples is much more complicated because the exponential term in Eq. (9) cannot be neglected. However, Sitko (2005) proposed a certain approximation to eliminate the exponential term:

$$1 - \exp\left(-\mathbf{x}\right) = \frac{1}{k\_1 + k\_0/\mathbf{x}}\tag{46}$$

Where *k1* and *k0* are constants, *x* = *χ*(*,i*)*t*. The proposed approximation allows not only for eliminating the exponential term but also transforming Sherman equation to a form in which the matrix correction term is expressed as a linear combination of all matrix elements concentrations. Thus, the proposed approximation was a base for the first empirical influence coefficient algorithm dedicated for the analysis of intermediate-thickness samples (Sitko, 2005):

$$\mathcal{W}\_i \mathcal{W}\_i m = I\_i \left[ a\_i + c\_i m + \sum\_j \alpha\_{ij} \mathcal{W}\_j m \right] + B\_i \tag{47}$$

Where *Wim* and *Wjm* are the masses per unit area of the analyte *i* and the matrix element *j*, respectively; *ij* is the influence coefficient; *ai* and *ci* are the coefficients dependent on the range of thickness of calibration samples. The coefficients *ai*, *ci*, *ij* and *B*i are calculated by the least-squares fit on the basis of reference samples of various thickness and composition. The influence coefficients in this algorithm are treated as constants. Therefore, to obtain accurate results, sample thickness and composition range should be limited. To overcome this inconvenience, the theoretical influence coefficient algorithm for intermediate-thickness samples was proposed (Sitko, 2006 b):

$$\begin{aligned} \mathcal{W}\_i m &= K\_i I\_i \left[ a\_i + c\_i m + \sum\_j \alpha\_{ij} \mathcal{W}\_j m \right] + B\_i \\\\ \alpha\_{ij} &= \frac{\alpha\_{ij} c\_i - S\_{ij} \left( a\_i f m + c\_i \right)}{1 + \sum\_j S\_{ij} \mathcal{W}\_j} \end{aligned} \tag{48}$$

Where *Ki*, *Bi* are the calibration constants calculated by the least-squares fit on the basis of experimental results for reference samples; *ai* and *ci* are the coefficients dependent on the thickness and total mass-attenuation coefficient of the sample; *ij* is the absorption influence coefficient; *Sij* is the enhancement term. The coefficients *ai*, *ci*, *ij* and *Sij* are multielement coefficients and are calculated directly from theory. The proposed theoretical influence algorithm can be applied in analysis of samples of any thickness and is more general in its form than algorithms for thick samples. It is worth emphasizing that the algorithm takes the well-known form for thick samples, e.g. the Lachance-Claisse algorithm (if sample thickness → ∞ then *ai* → 0 and *ci* → 1). The theoretical influence algorithm is more flexible than the previously proposed empirical algorithm and can be applied for a wide range of thickness and composition.

In the next paper (Sitko, 2008b), two algorithms of constant and linear coefficients for simultaneous determination of composition and thickness of thin films were proposed. This time, the coefficients are not calculated directly from theory but from the relative radiation intensity (calculated from theory) of hypothetical pure element films and binary films. The potential of the algorithms was demonstrated with hypothetical ternary and binary systems: FeCrNi, FeCr, FeNi, CrNi and experimental data of FeNi and Cu films.

#### **9. Conclusions**

158 X-Ray Spectroscopy

equation because exponential term equals zero for thick samples. In consequence, the matrix correction term can be expressed as a linear combination of matrix elements concentrations. The analysis of intermediate-thickness samples is much more complicated because the exponential term in Eq. (9) cannot be neglected. However, Sitko (2005) proposed a certain

eliminating the exponential term but also transforming Sherman equation to a form in which the matrix correction term is expressed as a linear combination of all matrix elements concentrations. Thus, the proposed approximation was a base for the first empirical influence coefficient algorithm dedicated for the analysis of intermediate-thickness samples

> *i ii i ij j i j Wm I a cm Wm B*

Where *Wim* and *Wjm* are the masses per unit area of the analyte *i* and the matrix element *j*,

the least-squares fit on the basis of reference samples of various thickness and composition. The influence coefficients in this algorithm are treated as constants. Therefore, to obtain accurate results, sample thickness and composition range should be limited. To overcome this inconvenience, the theoretical influence coefficient algorithm for intermediate-thickness

> *i ii i i ij j i j*

> > *ij i ij i i*

*c S am c S W*

*j*

Where *Ki*, *Bi* are the calibration constants calculated by the least-squares fit on the basis of experimental results for reference samples; *ai* and *ci* are the coefficients dependent on the

coefficients and are calculated directly from theory. The proposed theoretical influence algorithm can be applied in analysis of samples of any thickness and is more general in its form than algorithms for thick samples. It is worth emphasizing that the algorithm takes the well-known form for thick samples, e.g. the Lachance-Claisse algorithm (if sample thickness → ∞ then *ai* → 0 and *ci* → 1). The theoretical influence algorithm is more flexible than the previously proposed empirical algorithm and can be applied for a wide range of thickness

*ij j*

*Wm KI a cm Wm B*

1

*ij*

thickness and total mass-attenuation coefficient of the sample;

coefficient; *Sij* is the enhancement term. The coefficients *ai*, *ci*,

*ij* is the influence coefficient; *ai* and *ci* are the coefficients dependent on the

*,i*)

range of thickness of calibration samples. The coefficients *ai*, *ci*,

1 1 exp *x k kx*

1 0

*t*. The proposed approximation allows not only for

(47)

*ij* and *B*i are calculated by

*ij* is the absorption influence

*ij* and *Sij* are multielement

(46)

(48)

approximation to eliminate the exponential term:

Where *k1* and *k0* are constants, *x* = *χ*(

(Sitko, 2005):

respectively;

and composition.

samples was proposed (Sitko, 2006 b):

Many methods, both empirical and theoretical, have been proposed for quantitative XRF analysis. Method selection usually depends on sample type (thin or bulk, alloys or rocks, etc.), method of sample preparation (without treatment, fusion, etc.), expected results (quantitative or semi-quantitative analysis, determination of a single element or multielement analysis) and availability of standard samples. If the compensation methods are applied, then complicated sample preparation is required, only one or few elements can be quantitatively determined (standard addition, internal standard, Compton scatter) and additional matrix correction may be required (dilution). In influence coefficient algorithm, the matrix correction and calibration is clear, therefore, interpretation of data is very easy. The fundamental parameter methods are like "a black box". Nevertheless, the serious advantage is their versatility: the analysis of bulk, thin samples and multilayers is possible. Fundamental parameter methods are usually considered to be less accurate than the influence coefficient algorithms. This results from the fact that the fundamental parameter methods are usually used with only a few standards. The accuracy of fundamental parameter methods is very similar to that of influence coefficient algorithms when the same standards (many standards similar to the unknown) are used in both cases.

#### **10. References**

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energy-dispersive X-ray fluorescence spectrometry, *X-Ray Spectrometry*, Vol. 21, pp.

element analysis by X-ray fluorescence: An extension of the Compton scattering

**9**

*Japan*

Kengo Moribayashi

*Japan Atomic Energy Institute*

**The Interaction of High Brightness**

**X-Rays with Clusters or Bio-Molecules**

The three dimensional (3D) structures of bio-molecules such as proteins are important for the development of new drugs. At present, the structures are analyzed as follows. (i) The diffraction patterns of bio-molecules are produced by irradiation of synchrotron radiation xrays onto crystallized bio-molecules. (ii) The structures are reproduced using computer simulations. However, there are a lot of bio-molecules which are very difficult to be crystallized. For such bio-molecules, Neutze et al. proposed that the diffraction patterns are produced from the irradiation of x-rays onto single bio-molecules. Then, the intensity of xrays is required to be very bright. They also suggested that free electron x-ray laser (XFEL) light pulses, which have been developed by US, EU, and Japan, can have enough brightness x-rays (Neutze et al., 2000). When we use single molecules for the analysis of 3D structures, the study of the damage and the destruction of bio-molecules due to the irradiation of XFEL light pulses is indispensable (Hau-Riege et al., 2004, 2007, Jurek et al., 2004, Kai & Moribayashi, 2009, Kai, 2010, Moribayashi & Kai, 2009, Moribayashi, 2008, 2009, 2010, Neutze et al., 2000, Ziaja et al., 2006). We define the damage and the destruction as the ionization and the movement of atoms in a target, respectively (Hau-Riege et al., 2004, Moribayashi, 2008). This comes from the fact that places of the atoms are and are not changed due to the movement and the ionization, respectively. The change of the places means that the reconstruction of the 3D structure cannot be executed. The damage and the destruction mainly occur through the following occurrences: (i) the atoms in the target are ionized through the x-ray absorption or Compton scattering. (ii) From these ionization processes, free electrons, quasi-free electrons and ions are produced and move, where we define 'a free electron' and 'a quasi-free electron' as an electron, which is ionized from an atom, outside and inside the target, respectively (Hau-Riege et al., 2004, Moribayashi, 2008). (iii) Quasi-free electrons promote the ionization of the other atoms through electron impact

ionization processes. (iv) Other ionization processes, such as Auger, also occur.

Before experiments of diffraction patterns start, simulations of the damage and the destruction play an important role. The simulations have been executed using various methods, such as molecular dynamics (MD) (Neutze et al., 2000, Jurek et al., 2004), rate equations (Hau-Riege et al., 2004, Hau-Riege et al., 2007, Kai & Moribayashi, 2009, Kai, 2010, Moribayashi, 2008, Moribayashi & Kai, 2009), and kinetic Boltzmann equations (Ziaja et al., 2006). All of these methods have both advantages and disadvantages. In the MD, accurate simulation can be executed for bio-molecules of small size, having up to 10,000 atoms.

**1. Introduction**

## **The Interaction of High Brightness X-Rays with Clusters or Bio-Molecules**

Kengo Moribayashi *Japan Atomic Energy Institute Japan*

#### **1. Introduction**

162 X-Ray Spectroscopy

Sitko, R. (2008b). Determination of thickness and composition of thin films by X-ray

Sitko, R. (2009). Quantitative X-ray fluorescence analysis of samples of less than 'infinite

Szalóki, I.; Somogyi, A.; Braun, M.; Tóth, A. (1999). Investigation of geochemical

Tertian, R. (1976). An accurate coefficient method for X-ray fluorescence analysis, *Advances*

Van Dyck, P.M.; Török, S.B.; Van Grieken, R.E. (1986). Enhancement effect in X-ray

Vrebos, B.A.R. (2006). Compensation methods, in: B. Beckhoff, B. Kanngieβer, N. Langhoff,

Węgrzynek, D.; Hołyńska, B.; Pilarski, T. (1993). The fundamental parameter method for

Węgrzynek, D.; Markowicz, A.; Chinea-Cano, E.; Bamford, S. (2003b). Evaluation of the

Willis, J.E. (1989). Enhancement between layers in multiple-layer thin-film samples, *X-Ray*

Willis, J.P.; Lachance, G.R. (2000). Resolving apparent differences in mathematical

Willis, J.P.; Lachance, G.R. (2004). Comparison between some common influence coefficient

*Spectrometry*, Vol. 37, pp. 265–272.

*Spectrometry*, Vol. 28, pp. 399–405.

*Chemistry*, Vol. 58, pp. 1761–1766.

*in X-Ray Analysis*, Vol. 19, pp. 85–111.

1st ed., Springer, Berlin, 2006, pp. 358–369.

1172

119–128.

Vol. 32, pp. 317–335.

*Spectrometry*, Vol. 18, pp. 143–149.

*The Rigaku Journal*, Vol. 17, pp. 23-33.

algorithms, *X-Ray Spectrometry*, Vol. 33, pp. 181–188.

fluorescence spectrometry using theoretical influence coefficient algorithms, *X-Ray*

thickness': Difficulties and possibilities, *Spectrochimica Acta Part* B, Vol. 64, pp. 1161-

composition of lake sediments using ED-XRF and ICP-AES techniques, *X-Ray*

fluorescence analysis of environmental samples of medium thickness, *Analytical*

R. Wedell and H. Wolff (Eds.), Handbook of Practical X-Ray fluorescence Analysis,

energy-dispersive X-ray fluorescence analysis of intermediate thickness samples with the use of monochromatic excitation, *X-Ray Spectrometry*, Vol. 22, pp. 80–85. Węgrzynek, D.; Markowicz, A.; Chinea-Cano, E. (2003a). Application of the backscatter

fundamental parameter method for in situ element determination using a portable energy-dispersive X-ray fluorescence spectrometer, *X-Ray Spectrometry*, Vol. 32, pp.

uncertainty of element determination using the energy-dispersive X-ray fluorescence technique and the emission–transmission method, *X-Ray Spectrometry*,

expressions relating intensity to concentration in X-ray fluorescence spectrometry,

The three dimensional (3D) structures of bio-molecules such as proteins are important for the development of new drugs. At present, the structures are analyzed as follows. (i) The diffraction patterns of bio-molecules are produced by irradiation of synchrotron radiation xrays onto crystallized bio-molecules. (ii) The structures are reproduced using computer simulations. However, there are a lot of bio-molecules which are very difficult to be crystallized. For such bio-molecules, Neutze et al. proposed that the diffraction patterns are produced from the irradiation of x-rays onto single bio-molecules. Then, the intensity of xrays is required to be very bright. They also suggested that free electron x-ray laser (XFEL) light pulses, which have been developed by US, EU, and Japan, can have enough brightness x-rays (Neutze et al., 2000). When we use single molecules for the analysis of 3D structures, the study of the damage and the destruction of bio-molecules due to the irradiation of XFEL light pulses is indispensable (Hau-Riege et al., 2004, 2007, Jurek et al., 2004, Kai & Moribayashi, 2009, Kai, 2010, Moribayashi & Kai, 2009, Moribayashi, 2008, 2009, 2010, Neutze et al., 2000, Ziaja et al., 2006). We define the damage and the destruction as the ionization and the movement of atoms in a target, respectively (Hau-Riege et al., 2004, Moribayashi, 2008). This comes from the fact that places of the atoms are and are not changed due to the movement and the ionization, respectively. The change of the places means that the reconstruction of the 3D structure cannot be executed. The damage and the destruction mainly occur through the following occurrences: (i) the atoms in the target are ionized through the x-ray absorption or Compton scattering. (ii) From these ionization processes, free electrons, quasi-free electrons and ions are produced and move, where we define 'a free electron' and 'a quasi-free electron' as an electron, which is ionized from an atom, outside and inside the target, respectively (Hau-Riege et al., 2004, Moribayashi, 2008). (iii) Quasi-free electrons promote the ionization of the other atoms through electron impact ionization processes. (iv) Other ionization processes, such as Auger, also occur.

Before experiments of diffraction patterns start, simulations of the damage and the destruction play an important role. The simulations have been executed using various methods, such as molecular dynamics (MD) (Neutze et al., 2000, Jurek et al., 2004), rate equations (Hau-Riege et al., 2004, Hau-Riege et al., 2007, Kai & Moribayashi, 2009, Kai, 2010, Moribayashi, 2008, Moribayashi & Kai, 2009), and kinetic Boltzmann equations (Ziaja et al., 2006). All of these methods have both advantages and disadvantages. In the MD, accurate simulation can be executed for bio-molecules of small size, having up to 10,000 atoms.

The Interaction of High Brightness X-Rays with Clusters or Bio-Molecules 165

On the other hand, in XFEL light pulses, we may need to consider this second term on the

damage. Namely, the damage and the destruction change the diffraction patterns due to (i)

rays scattered by electrons bounded in the atoms with those by quasi-free and free electrons, which changes *Io* [see the second term of the right side of Eq.(2)], and (iii) the movement of atoms, which changes *ri*. For the movement of the atoms, the distances over which the atoms move become one of the factors for the decision of the highest resolving power of the 3D structures obtained from the experiments. Therefore, we need to control these distances to be smaller than the desired resolving power during the irradiation of XFEL light pulses on the target. However, the movement of atoms may be able to be controlled using a short pulse of XFEL light pulses (Neutze et al., 2000) or a tamper target (Hau-Riege et al., 2007), where a tamper target has been defined as a bio-molecule surrounded by multi-layers of water. Hau-Riege et al. (Hau-Riege et al., 2007) showed from their simulation that the movement of atoms can be controlled using a tamper target and a pulse of 50 fs as diffraction patterns change little. On the other hand, it is almost impossible to control the movement of electrons. This means that the effect of the movement of atoms on the analysis of 3D structures is much smaller than that of the movement of electrons. In our simulations, we calculate the change of the electronic states of the atoms and the movement of free and

 We have developed two models, that is, the rate equation and the MCN models. Using the rate equations, we can roughly estimate the damage for the large size of spherical targets. The rate equations are suitable to research the most suitable experimental conditions. On the other hand, in the MCN model, we can treat atoms and electrons individually in various shapes with smaller number of atoms. The MCN model is suitable to reproduce

process occurs, respectively. We calculate the change of both of ionized and excited states of the atoms and the production of free and quasi-free electrons using rates or cross sections of these ionization processes as a function of times. We use the same rates or cross sections as those given in several papers (Bell et al., 1983, Henke et al., 1993, Kai & Moribayashi, 2009,

*2*

 *xa | f |r|i | ,*

where | *i* > and | *f* > are the wave functions for the initial and final states, respectively

*h*'  *h*'

*h*  *sin*2

(Cowan, 1968). On the other hand, the cross sections of Compton scattering (

*atom*(*r*) and the places of atoms because of the larger

*ion*(*r*)] (Hau-Riege et al., 2007).], (ii) the interference of x-

*'* are the x-ray energies before and after the

*xa*) are roughly calculated by

(3)

), (4)

*CS*) are

*'* ), the electron impact ionization (C + *e*- C+ + 2 *e*- ), and

C+ + *e*- ), the Compton

*atom*(*r)* to the electron

ionization processes, which reduce *Io*. [Then, we should change

quasi-free electrons. However, we ignore the movement of atoms.

Atomic processes treated here are the x-ray absorption (e.g., C + *h*

experimental results of diffraction patterns or photo-electron spectroscopy.

 and *h*

determined by the Klein-Nishina formula (Klein & Nishina, 1929), that is,

right side of Eq.(2), the change of

density according to ionized states [

**2.1 Atomic processes**

the Auger (C+\* C2+ + *e*- ), where *h*

C+ + *e*- + *h*

Moribayashi, 2008). The x-ray absorption cross sections (

*d CS d* 1 2 *rc* 2 (*h*')2 (*h*) 2 ( *h*

scattering (C + *h*

However, MD is unsuitable for larger sizes because it takes too much time to calculate the damage and the movement of electrons and ions. The rate equations and the kinetic Boltzmann equations can treat bio-molecules larger size using spherically symmetry models, that is, one dimensional models. We have developed the Monte Carlo and the Newton equation (MCN) model (Moribayashi, 2010), which is almost the same as the MD except for the treatment of the movement of ions, as well as the rate equation (Moribayashi, et al., 1998, 2004, 2005, Moribayashi, 2007a, 2008, 2009, Moribayashi & Kai, 2009).

The 3D structures are investigated from diffraction patterns, which come from the irradiation of XFEL light pulses onto the bio-molecules. The shape of the diffraction patterns changes according to the x-ray flux which irradiates the bio-molecules. Generally speaking, larger numbers of x-rays produce better diffraction patterns. However, as the number of xrays increases, bio-molecules are damaged, that is, the atoms in bio-molecules such as C, N, O are more often ionized and more highly charged ions are produced. The highly charged ions cause a Coulomb explosion, as a result, the bio-molecules are destroyed (Neutze et al., 2000). The damage and destruction appear as noise for the analysis of the three-dimensional structures. Namely, it is very important to know the x-ray flux irradiating the bio-molecules for the study of the 3D structures of bio-molecules. We have proposed the measurement of the x-ray flux using the x-ray emission from the hollow atoms ((Moribayashi et al., 2004, Moribayashi, 2008) and the energy loss of the photo-electrons (Moribayashi, 2009).

In this chapter, we discuss (i) the damage of bio-molecules or clusters (see Sec.3) and (ii)the measurement methods of x-ray fluxes (see Sec.4) by the irradiation of XFEL light pulses onto bio-molecules or clusters through the simulations using rate equations and the MCN models (see Sec.2). In our previous paper (Moribayashi and Kai, 2009, Moribayashi, 2008, 2009, 2010), we have only treated carbons in targets. On the other hand, in this chapter, we treat mixtures which have carbon, nitrogen and oxygen atoms. Then, the densities of C, N, and O are 1.8 1022/cm3, 6 1021/cm3, and 6 1021/cm3, respectively. We decided these densities according to the similar value to the existent ratio among these elements in bio-molecules.

#### **2. Simulation methods**

The analysis of 3D structures of bio-molecules is executed based on diffraction patterns, which come from x-rays scattered by electrons bounded in atoms. The intensity of the diffraction patterns (*Io*) is given by

$$|I\_o(\vec{k}) \propto I\_i \, | \, F(\vec{k}) \, |^2,\tag{1}$$

where *Ii* is the intensity of XFEL light pulses and *F*(*k*) defined by

$$F(\vec{k}) = \int \rho(\vec{r}) e^{i\vec{k}\cdot\vec{r}} \, d\vec{r} = \sum\_{i} e^{i\vec{k}\cdot\vec{r\_{i}}} \int \rho\_{atom}(\vec{r}) e^{i\vec{k}\cdot\vec{r}} \, d\vec{r} + \int \rho\_{fe}(\vec{r}) e^{i\vec{k}\cdot\vec{r}} \, d\vec{r} \tag{2}$$

is the structure factor as a function of wave number vectors (*k*) with *k* = *Ki* - *Kf*. Here *r*, (*r*), *atom*(*r*), and *fe*(*r*) are the place of an atom in the target, the electron density in the biomolecule, the electron density in the atom, and the density of free and quasi-free electrons, respectively and *Ki* and *Kf* are the wave number vectors of the incident and scattered x-rays, respectively. The change of *atom*(*r*) and the places of atoms during the irradiation of x-rays are also conventionally ignored because of the small amount of the damage and destruction. On the other hand, in XFEL light pulses, we may need to consider this second term on the right side of Eq.(2), the change of *atom*(*r*) and the places of atoms because of the larger damage. Namely, the damage and the destruction change the diffraction patterns due to (i) ionization processes, which reduce *Io*. [Then, we should change *atom*(*r)* to the electron density according to ionized states [*ion*(*r*)] (Hau-Riege et al., 2007).], (ii) the interference of xrays scattered by electrons bounded in the atoms with those by quasi-free and free electrons, which changes *Io* [see the second term of the right side of Eq.(2)], and (iii) the movement of atoms, which changes *ri*. For the movement of the atoms, the distances over which the atoms move become one of the factors for the decision of the highest resolving power of the 3D structures obtained from the experiments. Therefore, we need to control these distances to be smaller than the desired resolving power during the irradiation of XFEL light pulses on the target. However, the movement of atoms may be able to be controlled using a short pulse of XFEL light pulses (Neutze et al., 2000) or a tamper target (Hau-Riege et al., 2007), where a tamper target has been defined as a bio-molecule surrounded by multi-layers of water. Hau-Riege et al. (Hau-Riege et al., 2007) showed from their simulation that the movement of atoms can be controlled using a tamper target and a pulse of 50 fs as diffraction patterns change little. On the other hand, it is almost impossible to control the movement of electrons. This means that the effect of the movement of atoms on the analysis of 3D structures is much smaller than that of the movement of electrons. In our simulations, we calculate the change of the electronic states of the atoms and the movement of free and quasi-free electrons. However, we ignore the movement of atoms.

 We have developed two models, that is, the rate equation and the MCN models. Using the rate equations, we can roughly estimate the damage for the large size of spherical targets. The rate equations are suitable to research the most suitable experimental conditions. On the other hand, in the MCN model, we can treat atoms and electrons individually in various shapes with smaller number of atoms. The MCN model is suitable to reproduce experimental results of diffraction patterns or photo-electron spectroscopy.

#### **2.1 Atomic processes**

164 X-Ray Spectroscopy

However, MD is unsuitable for larger sizes because it takes too much time to calculate the damage and the movement of electrons and ions. The rate equations and the kinetic Boltzmann equations can treat bio-molecules larger size using spherically symmetry models, that is, one dimensional models. We have developed the Monte Carlo and the Newton equation (MCN) model (Moribayashi, 2010), which is almost the same as the MD except for the treatment of the movement of ions, as well as the rate equation (Moribayashi,

The 3D structures are investigated from diffraction patterns, which come from the irradiation of XFEL light pulses onto the bio-molecules. The shape of the diffraction patterns changes according to the x-ray flux which irradiates the bio-molecules. Generally speaking, larger numbers of x-rays produce better diffraction patterns. However, as the number of xrays increases, bio-molecules are damaged, that is, the atoms in bio-molecules such as C, N, O are more often ionized and more highly charged ions are produced. The highly charged ions cause a Coulomb explosion, as a result, the bio-molecules are destroyed (Neutze et al., 2000). The damage and destruction appear as noise for the analysis of the three-dimensional structures. Namely, it is very important to know the x-ray flux irradiating the bio-molecules for the study of the 3D structures of bio-molecules. We have proposed the measurement of the x-ray flux using the x-ray emission from the hollow atoms ((Moribayashi et al., 2004,

et al., 1998, 2004, 2005, Moribayashi, 2007a, 2008, 2009, Moribayashi & Kai, 2009).

Moribayashi, 2008) and the energy loss of the photo-electrons (Moribayashi, 2009).

**2. Simulation methods**

*atom*(*r*), and

respectively. The change of

diffraction patterns (*Io*) is given by

In this chapter, we discuss (i) the damage of bio-molecules or clusters (see Sec.3) and (ii)the measurement methods of x-ray fluxes (see Sec.4) by the irradiation of XFEL light pulses onto bio-molecules or clusters through the simulations using rate equations and the MCN models (see Sec.2). In our previous paper (Moribayashi and Kai, 2009, Moribayashi, 2008, 2009, 2010), we have only treated carbons in targets. On the other hand, in this chapter, we treat mixtures which have carbon, nitrogen and oxygen atoms. Then, the densities of C, N, and O are 1.8 1022/cm3, 6 1021/cm3, and 6 1021/cm3, respectively. We decided these densities according to the similar value to the existent ratio among these elements in bio-molecules.

The analysis of 3D structures of bio-molecules is executed based on diffraction patterns, which come from x-rays scattered by electrons bounded in atoms. The intensity of the

> *k* ) |2

> > *rki*

*atom*(*r*) and the places of atoms during the irradiation of x-rays

*fe*(*r*) are the place of an atom in the target, the electron density in the bio-

, (1)

(2)

(*r*),

*fe*

*k* ) *Ii* | *F*(

)()( )( )( *rderrdererderkF rki*

molecule, the electron density in the atom, and the density of free and quasi-free electrons, respectively and *Ki* and *Kf* are the wave number vectors of the incident and scattered x-rays,

are also conventionally ignored because of the small amount of the damage and destruction.

*rki i*

*rki*

is the structure factor as a function of wave number vectors (*k*) with *k* = *Ki* - *Kf*. Here *r*,

*atom*

*Io*(

*i*

where *Ii* is the intensity of XFEL light pulses and *F*(*k*) defined by

Atomic processes treated here are the x-ray absorption (e.g., C + *h* C+ + *e*- ), the Compton scattering (C + *h* C+ + *e*- + *h'* ), the electron impact ionization (C + *e*- C+ + 2 *e*- ), and the Auger (C+\* C2+ + *e*- ), where *h* and *h'* are the x-ray energies before and after the process occurs, respectively. We calculate the change of both of ionized and excited states of the atoms and the production of free and quasi-free electrons using rates or cross sections of these ionization processes as a function of times. We use the same rates or cross sections as those given in several papers (Bell et al., 1983, Henke et al., 1993, Kai & Moribayashi, 2009, Moribayashi, 2008). The x-ray absorption cross sections (*xa*) are roughly calculated by

$$
\sigma\_{\text{xa}} \propto \left|  \right|^2,\tag{3}
$$

where | *i* > and | *f* > are the wave functions for the initial and final states, respectively (Cowan, 1968). On the other hand, the cross sections of Compton scattering (*CS*) are determined by the Klein-Nishina formula (Klein & Nishina, 1929), that is,

$$\frac{d\sigma\_{\rm CS}}{d\Omega} = \frac{1}{2} r\_c^2 \frac{(h\nu')^2}{\left(h\nu\right)^2} (\frac{h\nu}{h\nu'} + \frac{h\nu'}{h\nu} - \sin^2\theta),\tag{4}$$

The Interaction of High Brightness X-Rays with Clusters or Bio-Molecules 167

and

*SLJ SLJ*

(9)

(10)

*m,k* is the transition rate via

*ek k rav 0 P N A dt ,* (11)

*g A*

*SLJ*

*av*

*0 0*

*k1 k1 k k*

where *N*k is the population of singly inner-shell ionized atoms or hollow atoms.

*dN N N ( k 1,2,...,n), dt*

where *N*0, *N*1, *N*2,, *N*n are the populations of the ground state of the atom and the ions,

the transition processes such as photo-ionization, electron impact ionization from the *m*'th to

fluorescent x-ray photons per volume (*Pek*) from singly inner-shell ionized states of the

The Monte Carlo and Newton equation (MCN) model employed here is almost the same method as that treated in the MD (Jurek et al., 2004) except for the movement of atoms or ions as mentioned in Sec.1. In the MCN model, we can treat the change of the electric states of atoms and electrons individually in various shapes such as bio-molecules. The MCN model is applied to reproduce the experimental results of the diffraction pattern and photo

For the reconstruction of the 3D structures of bio-molecules, a lot of pulses are required. It should be noted that the production and the movement of electrons depend on the initial values of the random numbers (seeds) and that we can demonstrate the calculations of the damage and the electron distributions for different pulses using different initial seeds for the random number generated. We will show the results averaged by a few hundred

*k* is the decay rate via transition processes in the *k*'th state. The number of

*0*

*dN N , dt*

*k*

singly inner-shell ionized atoms and hollow atoms of the ions,

**2.3 Monte Carlo and Newton equation (MCN) model**

S*L*J and 1*s*22*s*22*pn*-1 S'*L'*J', respectively.

atoms or the hollow atoms is given by

**2.2 Rate equations**

the *k*'th state and

electron spectroscopy.

pulses.

the following rate equations:

*S,L.J*

*A , g*

*S,L,J*

respectively, where *g*SLJ expresses the statistical weight and *Ar*SLJS'L'J' and *En* SLJS'L'J'(2*p*1*s*) are the radiative transition probability and the energy difference between the states of 1*s* 2*s*22*p*n

We calculate the damage, the x-ray emission from the hollow atoms, photo-electron spectroscopy for the various parameters of XFEL light pulses using rate equations. With these atomic rates, the population dynamics of the various atomic may be investigated by

where *rc*, , and, are the radius of a classical electron, the scattering angle, and the solid angle, respectively and *h*' is given by

$$h\,\nu'=\frac{h\,\nu}{1+\frac{h\,\nu}{m\_\circ c^2}(1-\cos\theta)},\tag{5}$$

where *me* and *c* are the mass of an electron and the light speed, respectively. Then, the rates of the x-ray absorption (*Rxa*) and Compton scattering (*RCS*) are given by

$$R\_{\chi a} = \frac{I\sigma\_{\chi a}}{h\nu} \qquad \text{and} \qquad R\_{\text{CS}} = \frac{I\sigma\_{\text{CS}}}{h\nu} , \tag{6}$$

respectively (Moribayashi et al., 1998, 1999, 2004, 2005, Moribayashi, 2008), where *I* is the intensity of the x-rays. On the other hand, Auger rates are roughly given by

$$A\_d \propto \quad | < f \mid \frac{1}{r\_{12}} \mid i > |^2 \quad \text{ } \tag{7}$$

where *r12* is the length between an electron transferred from an excited state to the ground state and that ionized from an ion (Cowan, 1968). We have used the Auger rates given in our previous paper (Moribayashi, 2008). For the cross sections of the electron impact ionization processes (*e*), we employ the data given by Bell et al. (Bell et al., 1983).

Though we treat isolated atoms, that is, an isolated system, atoms in bio-molecules form a condensed matter system. Therefore, we have compared some atomic data of isolated atoms with those of molecules or solids. Photo-absorption measurements, which correspond to photo-ionization cross sections, in the foil targets of C show good agreement with those in the isolated C atom (Henke et al., 1993) at x-ray energies larger than 350 eV where inner-shell ionization dominates. Further, it was reported that photo-ionization cross sections of several molecules such as CO2 and C3H6 almost equal to the sum of the cross sections for the constituent atoms in this x-ray energy region (Henke et al., 1993, Hatano, 1999). Coville and Thomas calculated the lifetimes due to Auger processes (~ 1/*A*a) of singly inner-shell ionized atoms of 14 molecules containing C, N, O and compared them with those of the isolated atoms (Coville & Thomas, 1991). They showed that the lifetimes of singly inner-shell ionized atoms of the molecules are 0.6 to 0.85 times shorter than that of the isolated atom. Their results agreed with the measured lifetimes to within 25 % except for CO2. However, for CO2, a newer experimental result showed good agreement with their lifetime (Neeb et al., 1991).

We encountered numerical difficulties in treating a large number of coupled rate equations associated with multiple energy levels in the singly inner-shell ionized atoms and hollow atoms in obtaining x-ray spectra due to the decay from these excited states. We have employed the approximation as follows (Moribayashi et al., 2004, 2005, Moribayashi, 2007a, 2008). Namely, the atomic data are averaged over the quantum numbers of spin angular momentum (*S*), orbital angular momentum (*L*), and total angular momentum *(J*). The averaged transition energy (*E*av) and the averaged atomic data of *A*a and *Ar* are given by

$$E\_{\rm avn}(2p \to 1 \text{s}) = \frac{\sum\_{S,L,l} \text{g}\_{\text{S},L} \sum\_{S'L',l'} A\_{rS\text{L}\mid S'L',l'} E\_{\rm{vS\text{L}\mid S'L'}\mid l'}}{\sum\_{S,L,l} \text{g}\_{\text{S}\mid L} \sum\_{S'L',l'} A\_{rS\text{L}\mid S'L'} \text{j} \tag{8}$$

and

$$A\_{\rm av} = \frac{\sum\_{S,L,f} g\_{SLf} A\_{SLf}}{\sum\_{S,L,f} g\_{SLf}},$$

respectively, where *g*SLJ expresses the statistical weight and *Ar*SLJS'L'J' and *En* SLJS'L'J'(2*p*1*s*) are the radiative transition probability and the energy difference between the states of 1*s* 2*s*22*p*n S*L*J and 1*s*22*s*22*pn*-1 S'*L'*J', respectively.

#### **2.2 Rate equations**

166 X-Ray Spectroscopy

*mec* 2 (1 cos

where *me* and *c* are the mass of an electron and the light speed, respectively. Then, the rates

 *xa CS xa CS I I R andR , h h*

respectively (Moribayashi et al., 1998, 1999, 2004, 2005, Moribayashi, 2008), where *I* is the

 *2 a 12 1 A | f | |i | , r*

where *r12* is the length between an electron transferred from an excited state to the ground state and that ionized from an ion (Cowan, 1968). We have used the Auger rates given in our previous paper (Moribayashi, 2008). For the cross sections of the electron impact

Though we treat isolated atoms, that is, an isolated system, atoms in bio-molecules form a condensed matter system. Therefore, we have compared some atomic data of isolated atoms with those of molecules or solids. Photo-absorption measurements, which correspond to photo-ionization cross sections, in the foil targets of C show good agreement with those in the isolated C atom (Henke et al., 1993) at x-ray energies larger than 350 eV where inner-shell ionization dominates. Further, it was reported that photo-ionization cross sections of several molecules such as CO2 and C3H6 almost equal to the sum of the cross sections for the constituent atoms in this x-ray energy region (Henke et al., 1993, Hatano, 1999). Coville and Thomas calculated the lifetimes due to Auger processes (~ 1/*A*a) of singly inner-shell ionized atoms of 14 molecules containing C, N, O and compared them with those of the isolated atoms (Coville & Thomas, 1991). They showed that the lifetimes of singly inner-shell ionized atoms of the molecules are 0.6 to 0.85 times shorter than that of the isolated atom. Their results agreed with the measured lifetimes to within 25 % except for CO2. However, for CO2, a newer

experimental result showed good agreement with their lifetime (Neeb et al., 1991).

*avn*

We encountered numerical difficulties in treating a large number of coupled rate equations associated with multiple energy levels in the singly inner-shell ionized atoms and hollow atoms in obtaining x-ray spectra due to the decay from these excited states. We have employed the approximation as follows (Moribayashi et al., 2004, 2005, Moribayashi, 2007a, 2008). Namely, the atomic data are averaged over the quantum numbers of spin angular momentum (*S*), orbital angular momentum (*L*), and total angular momentum *(J*). The averaged transition energy (*E*av) and the averaged atomic data of *A*a and *Ar* are given by

*S,L.J S'L'J'*

*S,L,J S'L'J'*

*SLJ rSLJS'L'J' nSLJS'L'J'*

*g AE*

*SLJ rSLJ ,S'L'J7*

*E (2p 1s) , g A* (8)

*e*), we employ the data given by Bell et al. (Bell et al., 1983).

are the radius of a classical electron, the scattering angle, and the solid

, (5)

(6)

(7)

)

>

where *rc*,

, and,

ionization processes (

angle, respectively and *h*

' is given by

*h*

of the x-ray absorption (*Rxa*) and Compton scattering (*RCS*) are given by

intensity of the x-rays. On the other hand, Auger rates are roughly given by

' *h*

1 *h*

> We calculate the damage, the x-ray emission from the hollow atoms, photo-electron spectroscopy for the various parameters of XFEL light pulses using rate equations. With these atomic rates, the population dynamics of the various atomic may be investigated by the following rate equations:

$$\begin{aligned} \frac{dN\_0}{dt} &= -\beta\_0 N\_{0'}\\ \frac{dN\_k}{dt} &= \alpha\_{k-1} N\_{k-1} - \beta\_k N\_k \ (k = 1, 2, \dots, n), \end{aligned} \tag{10}$$

where *N*0, *N*1, *N*2,, *N*n are the populations of the ground state of the atom and the ions, singly inner-shell ionized atoms and hollow atoms of the ions, *m,k* is the transition rate via the transition processes such as photo-ionization, electron impact ionization from the *m*'th to the *k*'th state and *k* is the decay rate via transition processes in the *k*'th state. The number of fluorescent x-ray photons per volume (*Pek*) from singly inner-shell ionized states of the atoms or the hollow atoms is given by

$$P\_{ek} = \int\_0^{\infty} \mathcal{N}\_k A\_{nw} dt \,\,,\tag{11}$$

where *N*k is the population of singly inner-shell ionized atoms or hollow atoms.

#### **2.3 Monte Carlo and Newton equation (MCN) model**

The Monte Carlo and Newton equation (MCN) model employed here is almost the same method as that treated in the MD (Jurek et al., 2004) except for the movement of atoms or ions as mentioned in Sec.1. In the MCN model, we can treat the change of the electric states of atoms and electrons individually in various shapes such as bio-molecules. The MCN model is applied to reproduce the experimental results of the diffraction pattern and photo electron spectroscopy.

For the reconstruction of the 3D structures of bio-molecules, a lot of pulses are required. It should be noted that the production and the movement of electrons depend on the initial values of the random numbers (seeds) and that we can demonstrate the calculations of the damage and the electron distributions for different pulses using different initial seeds for the random number generated. We will show the results averaged by a few hundred pulses.

The Interaction of High Brightness X-Rays with Clusters or Bio-Molecules 169

where *P*(*b*) is the probability that the corresponding processes occur as a function of *b* and *bmax* is the maximum *b* where the process occurs. When we assume *P*(*b*) to be a step function with value 0 outside of *bmax*, = *bmax*2. Only when *b* becomes smaller than (/)1/2, we

The Coulomb forces due to ions and electrons act on free and quasi-free electrons. The

the velocity of the *i*'th electron, the charge of the *l*'th ion, and the distances between the *i*'th electron and the *j*'th free and quasi-free electron (the *l*'th ion), respectively. In order to avoid the divergence near *rij(l*) = 0, we use an approximation where *rij(l*) is approximately replaced

In the case of a spherical target with a radius of 100 nm, the number of atoms is larger than 107. In our calculation using the MCN developed here, it takes about 12 hours to calculate the damage and the movement of free and quasi-free electrons for the number of atoms of only 8000 and the x-ray flux of 3 1020 photons/pulse/mm2. Therefore, it takes too much time to execute the 3D calculation for the damage of bio-molecules when we treat a target

When we study the irradiation of XFEL light pulses with the clusters or bio-molecules, the uniform space charge, *Qe* (*r*) is produced from electrons escaped from the target, that is.

> ( ) 4/3 , ( ) ( ) 4/3 , ( )

with *Dpe* = *Ne*/*Vt*, where *r*0, *e, Nee*, and *Vt* are the radius of the target, the charge of an electron, the number of the electrons which escape from the target, and the volume of the target, respectively. This comes from the Gauss law for the sphere. Then, in our first approximation (which we call SSM1), we use Eq.(15) for the space charge, where the uniform charge distribution in the spherical targets is assumed. In the case of ellipsoids, we define an escaped electron as an electron, which has a value of *r* larger than that of the atom furthest from the center of the target (*ralm*), that is, *r*0 = *ralm*. Then, the force acting on an

3

0 0 0

*e pe*

( ) 1 ( ) ,( ) 4 3

*Q re F r rD e r r r*

( ) () . ( ) 4

Here, the force is directed toward the center. It should be noted that Eq. (16) follows the Gauss law in the case of the uniform charge distribution in spherical targets.  Then, spherically symmetric models become useful.

*e pe e pe*

*r ij* 4 0*rij* 3

*ql*

*e r il*

*ij(l*) are the dielectric constant in vacuum, the mass of an electron,

0

4 0*ril* 3

*l*

*dt e*2

*ji*

judge that the particle impact process occurs (Moribayashi, 2011)

 *F me d v ei*

, *ql*, and *r*

**2.3.4 Spherically symmetric models**

where *0*, *me*, *vei*

by (*rij(l*)

2 + *as*

electron becomes

movement of these electrons is solved by the Newton's equations, that is,

2)1/2 (Jurek et al. 2004, Moribayashi, 2010, 2011).

,)( max 0 *bdbbP b*

(13)

, (14)

(15)

(16)

#### **2.3.1 Initial electron energies**

The initial energies and velocities of electrons produced from these ionization processes should be mentioned because they contribute significantly not only to the movement of free and quasi-free electrons but also to the treatment of electron impact ionization processes. (i) The x-ray absorption processes: The initial electron energy corresponds to the value that subtracted a bound energy (*EB*) of atoms or ions from the x-ray energy. Since the x-ray energy treated here is much larger than *EB* of H, C, N, and O, which are main elements of bio-molecules, the initial electron energy is almost the same as the x-ray energy. (ii) The Compton scattering: The value of is determined randomly by treating the right side of Eq.(4) multiplied by *d* as a weighting factor and the initial electron energy is *h* – *h*' – *EB*. (iii) Auger: We employed the initial electron energy calculated by Cowan's code (Cowan, 1968). (iv) Electron impact ionization processes: We calculate the initial electron energy from the binary encounter dipole (BED) theory (Kim et al., 2000) or use the data given by Nakazaki et al. (Nakazaki et al*.*, 1991). After the initial electron energy is determined, the initial direction of the electron velocity is given randomly except for that due to Compton scattering. In Compton scattering, the initial direction is determined from the electron energy, and the momentum conservation law.

#### **2.3.2 Monte Carlo**

The x-ray absorption, Auger, and Compton scattering processes are treated using the Monte Carlo method as follows (Moribayashi, 2007b, 2009, 2010): (i) just when an XFEL light pulse begins to irradiate a target, we start the calculation and set the time of *t* = 0. We also set the neutral and the ground states for ionized and excited states of all atoms in the target, respectively. (ii) We calculate the transition rates [*Rifp* (*m*)] of all the possible ionization processes according to the ionized and excited states of all the atoms and random numbers [*NR* (*m*)]. One random number is given to each atom at the time interval between *t* and *t* + *t*, where *Rifp* (*m*) and *NR* (*m*) are the transition rate from the *i*'th state to the *f*'th one of the *m*'th atom due to the *p*'th ionization process and the random number given to the *m*'th atom, respectively. (iii) Only when

$$\sum\_{\rho} \sum\_{f} R\_{\circ \circ}(m) \Delta t > N\_{\aleph}(m),\tag{12}$$

one process for the *m*'th atom occurs. When this equation is satisfied, the state where the ionization occurs is chosen randomly among all the possible transitions using the respective *Rijp* (*m*) as weighting factors. (iv) The value of *t* increases by *t* and procedures (ii) and (iii) are executed. (v) We reiterate procedures (ii) - (iv) until the XFEL light pulse passes through the target.

#### **2.3.3 Electron movements**

As for the electron impact ionization process, it is judged that the process occurs only when a quasi-free electron crosses the area of a cross section according to an ionized state of an atom. The center of the cross section is located at the place of the atomic nucleus and the cross section is perpendicular to the direction of the electron velocity (Jurek et al., 2004, Moribayashi, 2009, 2010, 2011). Specifically, we use the relationship of cross sections with impact parameters (*b*) where *b* is defined as the perpendicular distance between the path of an incident ion and the center of the atom. The electron impact cross section () is given by

$$
\sigma = \pi \int\_0^{b\_{\text{max}}} P(b)bdb,\tag{13}
$$

where *P*(*b*) is the probability that the corresponding processes occur as a function of *b* and *bmax* is the maximum *b* where the process occurs. When we assume *P*(*b*) to be a step function with value 0 outside of *bmax*, = *bmax*2. Only when *b* becomes smaller than (/)1/2, we judge that the particle impact process occurs (Moribayashi, 2011)

The Coulomb forces due to ions and electrons act on free and quasi-free electrons. The movement of these electrons is solved by the Newton's equations, that is,

$$\vec{F} = m\_e \frac{d\vec{v}\_{ei}}{dt} = -\sum\_{j \neq i} \frac{e^2 \vec{r}\_{\vec{y}}}{4\pi\varepsilon\_0 r\_{\vec{y}}^3} + \sum\_l \frac{q\_l e \vec{r}\_{il}}{4\pi\varepsilon\_0 r\_{il}^3},\tag{14}$$

where *0*, *me*, *vei* , *ql*, and *r ij(l*) are the dielectric constant in vacuum, the mass of an electron, the velocity of the *i*'th electron, the charge of the *l*'th ion, and the distances between the *i*'th electron and the *j*'th free and quasi-free electron (the *l*'th ion), respectively. In order to avoid the divergence near *rij(l*) = 0, we use an approximation where *rij(l*) is approximately replaced by (*rij(l*) 2 + *as* 2)1/2 (Jurek et al. 2004, Moribayashi, 2010, 2011).

#### **2.3.4 Spherically symmetric models**

168 X-Ray Spectroscopy

The initial energies and velocities of electrons produced from these ionization processes should be mentioned because they contribute significantly not only to the movement of free and quasi-free electrons but also to the treatment of electron impact ionization processes. (i) The x-ray absorption processes: The initial electron energy corresponds to the value that subtracted a bound energy (*EB*) of atoms or ions from the x-ray energy. Since the x-ray energy treated here is much larger than *EB* of H, C, N, and O, which are main elements of bio-molecules, the initial electron energy is almost the same as the x-ray energy. (ii) The

(iii) Auger: We employed the initial electron energy calculated by Cowan's code (Cowan, 1968). (iv) Electron impact ionization processes: We calculate the initial electron energy from the binary encounter dipole (BED) theory (Kim et al., 2000) or use the data given by Nakazaki et al. (Nakazaki et al*.*, 1991). After the initial electron energy is determined, the initial direction of the electron velocity is given randomly except for that due to Compton scattering. In Compton scattering, the initial direction is determined from the electron

The x-ray absorption, Auger, and Compton scattering processes are treated using the Monte Carlo method as follows (Moribayashi, 2007b, 2009, 2010): (i) just when an XFEL light pulse begins to irradiate a target, we start the calculation and set the time of *t* = 0. We also set the neutral and the ground states for ionized and excited states of all atoms in the target, respectively. (ii) We calculate the transition rates [*Rifp* (*m*)] of all the possible ionization processes according to the ionized and excited states of all the atoms and random numbers [*NR* (*m*)]. One random number is given to each atom at the time interval between *t* and *t* +

*t*, where *Rifp* (*m*) and *NR* (*m*) are the transition rate from the *i*'th state to the *f*'th one of the *m*'th atom due to the *p*'th ionization process and the random number given to the *m*'th atom,

one process for the *m*'th atom occurs. When this equation is satisfied, the state where the ionization occurs is chosen randomly among all the possible transitions using the respective

are executed. (v) We reiterate procedures (ii) - (iv) until the XFEL light pulse passes through

As for the electron impact ionization process, it is judged that the process occurs only when a quasi-free electron crosses the area of a cross section according to an ionized state of an atom. The center of the cross section is located at the place of the atomic nucleus and the cross section is perpendicular to the direction of the electron velocity (Jurek et al., 2004, Moribayashi, 2009, 2010, 2011). Specifically, we use the relationship of cross sections with impact parameters (*b*) where *b* is defined as the perpendicular distance between the path of

*f p*

an incident ion and the center of the atom. The electron impact cross section (

*Rijp* (*m*) as weighting factors. (iv) The value of *t* increases by

as a weighting factor and the initial electron energy is *h*

is determined randomly by treating the right side of

*Rifp* (*m*)*t NR* (*m*), (12)

*t* and procedures (ii) and (iii)

) is given by

 – *h*' – *EB*.

**2.3.1 Initial electron energies**

Compton scattering: The value of

and the momentum conservation law.

Eq.(4) multiplied by *d*

energy,

the target.

**2.3.2 Monte Carlo**

respectively. (iii) Only when

**2.3.3 Electron movements**

In the case of a spherical target with a radius of 100 nm, the number of atoms is larger than 107. In our calculation using the MCN developed here, it takes about 12 hours to calculate the damage and the movement of free and quasi-free electrons for the number of atoms of only 8000 and the x-ray flux of 3 1020 photons/pulse/mm2. Therefore, it takes too much time to execute the 3D calculation for the damage of bio-molecules when we treat a target with a radius around 100 nm. Then, spherically symmetric models become useful.

When we study the irradiation of XFEL light pulses with the clusters or bio-molecules, the uniform space charge, *Qe* (*r*) is produced from electrons escaped from the target, that is.

$$\begin{aligned} Q\_{\epsilon}(r) &= 4 \,/\, 3\pi r^3 D\_{pc} e\_{\prime} \qquad & \quad (r < r\_0) \\ Q\_{\epsilon}(r) &= 4 \,/\, 3\pi r\_0^3 D\_{pc} e\_{\prime} \qquad & \quad (r \ge r\_0) \end{aligned} \tag{15}$$

with *Dpe* = *Ne*/*Vt*, where *r*0, *e, Nee*, and *Vt* are the radius of the target, the charge of an electron, the number of the electrons which escape from the target, and the volume of the target, respectively. This comes from the Gauss law for the sphere. Then, in our first approximation (which we call SSM1), we use Eq.(15) for the space charge, where the uniform charge distribution in the spherical targets is assumed. In the case of ellipsoids, we define an escaped electron as an electron, which has a value of *r* larger than that of the atom furthest from the center of the target (*ralm*), that is, *r*0 = *ralm*. Then, the force acting on an electron becomes

$$F(r) = \frac{Q\_e(r)e}{4\pi\varepsilon\_0 r^2} = \frac{1}{3\varepsilon\_0} r D\_{pe} e^2,\qquad \text{ } \{r < r\_0\} \tag{16}$$

$$F(r) = \frac{Q\_e(r\_0)e}{4\pi\varepsilon\_0 r^2}.\tag{17}$$

Here, the force is directed toward the center. It should be noted that Eq. (16) follows the Gauss law in the case of the uniform charge distribution in spherical targets. Namely,

The Interaction of High Brightness X-Rays with Clusters or Bio-Molecules 171

(a) (b) (c)

(d) (e) (f)

(g) (h) (i)

= 100 fs,

= 0.1 nm, (d - f)

= 10 fs,

intensity of x-rays passes through the bio-molecule.

Fig. 1. Population of the charge of C (a, d,g), N (b, e, h), and O (c, f, i) atoms as a function of time for the x-ray flux of 1022/pulse/mm2, the radius of bio-molecules of 10 nm, and (a - c)

= 0.1 nm, (g - i)

Figures 2 (a - f) show the population of C with different charge states as a function of time for x-ray fluxes of 1019 - 1021/pulse/mm2 for the wavelength of an XFEL light pulse of (a – c) 0.1 nm and (d – f) 0.06 nm, respectively. X-ray fluxes correspond to the time scale of the photo absorption ionization processes because the rates of photo absorption ionization processes (*R*ap) are in proportion to the x-ray flux. Smaller x-ray fluxes produce smaller damage. However, the resolution powers or the intensities for the diffraction pattern correspond to x-ray fluxes. Intensities for the diffraction patterns (*I*O) are given by Eq.(1) using the structure factor defined in Eq.(2). Therefore, we need to study the intensities of the

numbers written in the figures express charge (Lower figures). Upper figures: the x-ray intensity of an XFEL light pulse as a function of time. The time of 0 is set when the peak

= 10 fs,

= 0.06 nm, respectively. The

electric fields (*F*(*r*)/*e*) are produced from the charge, which exists inside the place of interest, and the charge outside it can be ignored because of cancellation. We treat Eq. (16) instead of Eq. (14) for the movement of electrons in the SSM1. In Eq. (16), there is no divergence near *r* = 0, which often appears for the point charge, because *F* (0) = 0. This approximation is useable only when the number of quasi-free electrons is too small to effect on the charge distribution. Since the SSM1 is useful for the saving of calculation time, we examine the limits of application of the SSM1. When we consider that quasi-free electrons effect on the charge distribution, the charge distribution for *r* becomes nonuniform. Then, in our second approximation (SSM2), we estimate the charge *Qes* (*r*) by counting the total charge inside the place where the electron of interest exists and we use *Qes* (*r*) instead of *Qe* (*r*) in Eq. (16).

#### **3. X-ray damage**

Here, we study (i) the relationship between the damage and the parameters of XFEL light pulses such as pulse widths, wavelenths, and x-ray fluxes using rate equations in order to research the most suitable experimental conditions and (ii) the free and qausi-free electron movement in the target using MCN model in order to aim at the reproduction of the experimental data.

#### **3.1 The most suitable XFEL parameters**

We have been studying the role of atomic processes such as photo-ionization, Auger, radiative transition, and electron impact ionization processes for the damage of biomolecules irradiated by an XFEL light pulse. By considering these roles, we have constructed the models mentioned in Sec.2..

We have calculated the changes of the electronic states of atoms or ions in a bio-molecule as a function of time. However, the number of the electronic states is too large. Then, for clearer figures, we show the changes of charge determined from the electronic states.

Figures 1 (a – i) show the populations of the charge of C, N, and O as a function of time for the pulse width () and wavelength (*)* of (a – c) an XFEL light pulse of 100 fs and 0.1 nm, (d – f) 10 fs and 0.1 nm, and (g – i) 10 fs and 0.06 nm, respectively. The x-ray flux of the XFEL light pulses and the radius of bio-molecules are 1022/pulse/mm2 and 10 nm, respectively. A gauss type time function is employed for the fluxes of the XFEL light pulses (see upper sides of Fig.1) and the time of 0 is set when the peak intensity of x-rays is located in the bio-molecule. We have found that (i) the damage becomes larger as the atomic number increases, (ii) a shorter value of produces smaller damage (see Figs.1 (a – c ) and (d - f)), and (iii) a shorter value of also produces smaller damage (see Figs.1 (d f) and (g – i)). The reason why a shorter pulse produces smaller damage is due to the fact that time scale of Auger processes is about 10 fs. Namely, photo-absorption ionization and Auger processes occur only once in the case of = 10 fs and several times in the case of 100 fs, respectively. Auger-electrons also give more significant contribution to the damage in the case of = 100 fs. On the other hand, the dependence of wavelength on the damage comes from the inner-shell photo-ionization cross sections. The photo absorption ionization cross sections for = 0.1 nm is about 10 times larger than that for = 0.06 nm (Henke et al, 1993).

electric fields (*F*(*r*)/*e*) are produced from the charge, which exists inside the place of interest, and the charge outside it can be ignored because of cancellation. We treat Eq. (16) instead of Eq. (14) for the movement of electrons in the SSM1. In Eq. (16), there is no divergence near *r* = 0, which often appears for the point charge, because *F* (0) = 0. This approximation is useable only when the number of quasi-free electrons is too small to effect on the charge distribution. Since the SSM1 is useful for the saving of calculation time, we examine the limits of application of the SSM1. When we consider that quasi-free electrons effect on the charge distribution, the charge distribution for *r* becomes nonuniform. Then, in our second approximation (SSM2), we estimate the charge *Qes* (*r*) by counting the total charge inside the place where the electron of interest exists and we use

Here, we study (i) the relationship between the damage and the parameters of XFEL light pulses such as pulse widths, wavelenths, and x-ray fluxes using rate equations in order to research the most suitable experimental conditions and (ii) the free and qausi-free electron movement in the target using MCN model in order to aim at the reproduction of the

We have been studying the role of atomic processes such as photo-ionization, Auger, radiative transition, and electron impact ionization processes for the damage of biomolecules irradiated by an XFEL light pulse. By considering these roles, we have

We have calculated the changes of the electronic states of atoms or ions in a bio-molecule as a function of time. However, the number of the electronic states is too large. Then, for

Figures 1 (a – i) show the populations of the charge of C, N, and O as a function of time

nm, (d – f) 10 fs and 0.1 nm, and (g – i) 10 fs and 0.06 nm, respectively. The x-ray flux of the XFEL light pulses and the radius of bio-molecules are 1022/pulse/mm2 and 10 nm, respectively. A gauss type time function is employed for the fluxes of the XFEL light pulses (see upper sides of Fig.1) and the time of 0 is set when the peak intensity of x-rays is located in the bio-molecule. We have found that (i) the damage becomes larger as the

f) and (g – i)). The reason why a shorter pulse produces smaller damage is due to the fact that time scale of Auger processes is about 10 fs. Namely, photo-absorption ionization and

100 fs, respectively. Auger-electrons also give more significant contribution to the damage

comes from the inner-shell photo-ionization cross sections. The photo absorption

= 100 fs. On the other hand, the dependence of wavelength on the damage

= 0.1 nm is about 10 times larger than that for

*)* of (a – c) an XFEL light pulse of 100 fs and 0.1

produces smaller damage (see Figs.1 (a –

= 10 fs and several times in the case of

= 0.06 nm

also produces smaller damage (see Figs.1 (d -

clearer figures, we show the changes of charge determined from the electronic states.

*Qes* (*r*) instead of *Qe* (*r*) in Eq. (16).

**3.1 The most suitable XFEL parameters**

constructed the models mentioned in Sec.2..

atomic number increases, (ii) a shorter value of

Auger processes occur only once in the case of

c ) and (d - f)), and (iii) a shorter value of

) and wavelength (

**3. X-ray damage**

experimental data.

for the pulse width (

in the case of

(Henke et al, 1993).

ionization cross sections for

Fig. 1. Population of the charge of C (a, d,g), N (b, e, h), and O (c, f, i) atoms as a function of time for the x-ray flux of 1022/pulse/mm2, the radius of bio-molecules of 10 nm, and (a - c) = 100 fs, = 0.1 nm, (d - f) = 10 fs, = 0.1 nm, (g - i) = 10 fs, = 0.06 nm, respectively. The numbers written in the figures express charge (Lower figures). Upper figures: the x-ray intensity of an XFEL light pulse as a function of time. The time of 0 is set when the peak intensity of x-rays passes through the bio-molecule.

Figures 2 (a - f) show the population of C with different charge states as a function of time for x-ray fluxes of 1019 - 1021/pulse/mm2 for the wavelength of an XFEL light pulse of (a – c) 0.1 nm and (d – f) 0.06 nm, respectively. X-ray fluxes correspond to the time scale of the photo absorption ionization processes because the rates of photo absorption ionization processes (*R*ap) are in proportion to the x-ray flux. Smaller x-ray fluxes produce smaller damage. However, the resolution powers or the intensities for the diffraction pattern correspond to x-ray fluxes. Intensities for the diffraction patterns (*I*O) are given by Eq.(1) using the structure factor defined in Eq.(2). Therefore, we need to study the intensities of the

The Interaction of High Brightness X-Rays with Clusters or Bio-Molecules 173

(a) (b)

(c) (d)

the number of electrons [*Ne* (*r*)] at the interval between *r* and *r* +

*r*. (ii) The electron distribution *Fed* (*r*) is given by

nm for

pulse of XFEL are 3 1020 photons/pulse/mm2, 2000, 1 Å, and 10 fs, respectively.

Fig. 3. Electron distribution defined by Eq. (17) vs. *r* at (a) *t* = 1 fs (b) *t* = 3 fs, (c) *t* = 6 fs, and (d) *t* = 9 fs for a spherical target. The calculation methods are the MCN (), the SSM1 (), and the SSM2 (). The x-ray fluxes, the number of atoms in a target, a wavelength, and a

Figure 3 shows the free and quasi-free electron distribution as a function of *r* for a spherical target at *t* = 1, 3, 6, and 9 fs calculated by the MCN, SSM1, and the SSM2, where *r* is the length from the center of the target. We use the constant x-ray flux for the XFEL light pulses as a function of time and set the time of *t* = 0 just when an XFEL light pulse begins to irradiate the target. The electron distribution treated here is defined as follows: (i) we count

> *Fed* (*r*) *Ne* (*r*) 4

The results calculated by the SSM1 and the SSM2 show good agreement with those of the MCN at *t* =1 fs [see Figs.3 (a)]. Then, a lot of quasi-free electrons can escape from the target. Then, the distribution becomes smaller as *r* increases. We predict from Fig.3 (a) as follows: (i) since quasi-free electrons are accelerated toward *r* = 0, the electrons become concentrated near *r* = 0. (ii) As the charge becomes smaller near *r* = 0, the acceleration becomes weaker as time progresses. (iii) This reduces the invasion of quasi-free electrons into *r* = 0. As a result,

(*r r*/2)2 . (17)

*r*, where we take to be 0.1

diffraction patterns including the damage in order to propose the best parameter for the experiments in future.

Fig. 2. The same as figure 1(a) for the x-ray flux of (a, d)1021/pulse/mm2, (b, e) 1020/pulse/mm2, (c, f) 1019/pulse/mm2 and the radius is 10 nm. The wavelengths are are (a - c) 0.1 nm and (d - f) 0.06 nm.

#### **3.2 Electron movement**

We treat model clusters with spheres at a solid density (3 1022/cm3). We decide places inside and outside the target from the number and the density of atoms. Then, the places of the atoms are assigned randomly on the condition that they are located inside the target and that lengths among the atoms are larger than 2.7 Å, which is almost the same as the length between carbons in proteins. Then, we attempt to apply our MCN models to one bio-molecule, that is, a lysozyme which has elements of H, C, N, O and, S. We use the place coordinate data of a lysozyme in the protein data bank (PDB) (http://www.pdb.org/pdb/home/home.do), in which we employ 2LZM as PDB ID.

For the parameters of XFEL light pulses, it is estimated that x-ray fluxes around 1020 photons/pulse/mm2 and wavelength around 1 Å are required (Neutze et al*.,* 2000). In this paper, we treat x-ray fluxes of 1020 to 5 1020 photons/ pulse/mm2, a wavelength of 1 Å, a pulse of 10 fs, and the number of atoms of 1000 - 8000.

diffraction patterns including the damage in order to propose the best parameter for the

(a) (b) (c)

(d) (e) (f)

Fig. 2. The same as figure 1(a) for the x-ray flux of (a, d)1021/pulse/mm2, (b, e)

1020/pulse/mm2, (c, f) 1019/pulse/mm2 and the radius is 10 nm. The wavelengths are are (a

We treat model clusters with spheres at a solid density (3 1022/cm3). We decide places inside and outside the target from the number and the density of atoms. Then, the places of the atoms are assigned randomly on the condition that they are located inside the target and that lengths among the atoms are larger than 2.7 Å, which is almost the same as the length between carbons in proteins. Then, we attempt to apply our MCN models to one bio-molecule, that is, a lysozyme which has elements of H, C, N, O and, S. We use the place coordinate data of a lysozyme in the protein data bank (PDB) (http://www.pdb.org/pdb/home/home.do), in which we employ 2LZM as PDB ID. For the parameters of XFEL light pulses, it is estimated that x-ray fluxes around 1020 photons/pulse/mm2 and wavelength around 1 Å are required (Neutze et al*.,* 2000). In this paper, we treat x-ray fluxes of 1020 to 5 1020 photons/ pulse/mm2, a wavelength of 1 Å, a

experiments in future.

pulse of 10 fs, and the number of atoms of 1000 - 8000.

**3.2 Electron movement**

Fig. 3. Electron distribution defined by Eq. (17) vs. *r* at (a) *t* = 1 fs (b) *t* = 3 fs, (c) *t* = 6 fs, and (d) *t* = 9 fs for a spherical target. The calculation methods are the MCN (), the SSM1 (), and the SSM2 (). The x-ray fluxes, the number of atoms in a target, a wavelength, and a pulse of XFEL are 3 1020 photons/pulse/mm2, 2000, 1 Å, and 10 fs, respectively.

Figure 3 shows the free and quasi-free electron distribution as a function of *r* for a spherical target at *t* = 1, 3, 6, and 9 fs calculated by the MCN, SSM1, and the SSM2, where *r* is the length from the center of the target. We use the constant x-ray flux for the XFEL light pulses as a function of time and set the time of *t* = 0 just when an XFEL light pulse begins to irradiate the target. The electron distribution treated here is defined as follows: (i) we count the number of electrons [*Ne* (*r*)] at the interval between *r* and *r* + *r*, where we take to be 0.1 nm for *r*. (ii) The electron distribution *Fed* (*r*) is given by

$$F\_{ed}(r) = \frac{N\_e(r)}{4\pi(r + \Delta r/2)^2}.\tag{17}$$