Abstract:
A method for determining atomic isotope masses in mass spectrometry, atomic isotope ratios being determined from molecular isotope ratios measured by means of an isotope mass spectrometer—ion correction, the determination of the atomic ratios being carried out by setting up and solving a system of equations which describes relationships between the atomic and the molecular ratios, and the system of equations having to have at least as many independent equations as there are atomic ratios. The entire system of equations is linearized by means of suitable numerical methods in a first step, in particular by means of a Taylor expansion or similar method, and in which the linearized system of equations is subsequently solved as a whole without transforming the individual equations.

Description:
STATEMENT OF RELATED APPLICATIONS 
   This application claims foreign priority benefits under 35 USC 119(a)–(d) or (f), or 365(b) on German patent application number 10344239.1, filed on 23 Sep. 2003. 
   BACKGROUND OF THE INVENTION 
   1. Technical Field 
   The invention relates to a method for determining atomic isotope masses in mass spectrometry, atomic isotope ratios being determined from molecular isotope ratios measured by means of an isotope mass spectrometer—ion correction, the determination of the atomic ratios being carried out by setting up and solving a system of equations which describes relationships between the atomic and the molecular ratios, and the system of equations having to have at least as many independent equations as there are atomic ratios. 
   2. Prior Art 
   In gas isotope mass spectrometry, gas molecules which are composed of a plurality of different atoms are measured in most cases. In this case, the ratios between various molecular isotope masses are measured. Atomic isotope ratios are mentioned when the mass ratios of two isotopes of a specific atom are to be designated. On the other hand, molecular isotopolog ratios are mentioned when the mass ratios of specific isotopologs at the molecular level are designated. 
   The determination of molecular isotopolog ratios is usually less desired than the determination of atomic isotope ratios. 
   In order to determine the atomic ratios from the molecular isotopolog ratios, it is known to set up various equations which describe theoretical and empirical relationships between the respective ratios. These equations are then solved individually and in a complicated manner for the atomic ratios sought and under certain circumstances each individual equation is solved numerically. If, during a further isotope mass measurement, other atoms or molecules play a part, new equations have to be set up and these in turn have to be solved individually. These steps have to be carried out independently for each type of gas which is measured or is to be measured. 
   BRIEF SUMMARY OF THE INVENTION 
   It is therefore an object of the present invention to specify a method in which at least one drawback of the prior art is avoided, the intention being in particular for a flexible possible determination, less susceptible to error, of the atomic isotope ratio to be possible. 
   This object is achieved by a method of the type mentioned at the beginning in which the entire system of equations is linearized by means of suitable numerical methods in a first step, in particular by means of a Taylor expansion or similar method, and in which the linearized system of equations is subsequently solved as a whole without transforming the individual equations. 
   The method according to the invention can advantageously be applied to all gases, the addition of further ratios being simply possible. 
   In a further refinement of the invention, in order to standardize the measurements, molecular ratios are determined from standard values of atomic ratios—inverse ion correction, the same linearized system of equations being used both for the inverse ion correction and for the ion correction. 
   The invention is also directed to an apparatus for determining atomic isotope masses in a mass spectrometer, in particular a computer, having an input interface, a computing unit and an output interface, it being possible for data about molecular isotope ratios which can be measured by means of an isotope mass spectrometer to be transmitted to the computer unit via the input interface, it being possible for atomic isotope ratios to be determined from the data about molecular isotope ratios in accordance with a method described in the present invention, and it being possible for the atomic isotope ratios to be output via the output interface. 
   Finally, the invention is directed to a computer program for determining atomic isotope masses in mass spectrometry, one of the methods described in the present application being implemented in a program by means of a suitable programming language. 
   Further features of the present invention emerge from the appended claims and the following description of a practical exemplary embodiment 

   DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS 
   In order to explain the method according to the invention, in the following text, by using the example of CO 2 , the method used hitherto in the prior art will be compared with the method according to the invention. In this case, under I, first of all general relationships of the method of the prior art will be explained, under II the new method according to the invention will be presented, and under III exemplary calculations will be carried out by using practical data. 
   I. General Relationships of the Conventional Method 
   In gas isotope mass spectrometry, measurements are often made on gas molecules which are composed of a number of different atoms. In each case, the ratios between different masses are measured on a molecular basis. However, the atomic isotope ratios are relevant to the end results and also the international standard and reference substances. 
   1. Ion Correction 
   In the measurement of CO 2 , for example, the ratios of the intensities of the mass 45 to the intensity of the mass 44 (R45) and of the intensities of the mass 46 to the intensity of the mass 44 (R46) are determined. 
   However, these molecular ratios are composed of the atomic ratios sought:
 
 R 13{circumflex over (=)}((amount of  13 C) divided by (amount of  12 C)),
 
 R 17{circumflex over (=)}((amount of  17 O) divided by (amount of  16 O)) and
 
 R 18{circumflex over (=)}((amount of  18 O) divided by (amount of  16 O)).
 
   In specific terms:
     the mass 44 consists only of  12 C 16 O 2 ,   the mass 45 of  12 C 16 O 17 O,  13 C 16 O 2 , and  12 C 16 O 17 O,   the mass 46 of  12 C 16 O 18 O,  13 C 17 O 16 O and  12 C 17 O 2 .   

   The masses 47 and 48 can likewise be composed of atomic ratios. The total frequency is, however, low, so that these masses are generally not measured. 
   It is easy to derive, and known from the literature, that:
 
 R 45 =R 13+2* R 17
 
 R 46=2* R 18+2* R 13* R 17 +R 17 2 
 
   In order to obtain the R13 and R18 sought (in rare cases also R17) from the known R45 and R46, information is obviously missing here. One remedy is provided by the following relationship being applied
 
 R 17 =K*R 18 λ 
 
   This equation cannot be derived exactly but is at best semi-empirical. There is generally disagreement about the factors K and λ. K can also be calculated from the R18 and R17 of the international standard in accordance with 
   
     
       
         
           K 
           = 
           
             
               R17 
               Standard 
             
             
               R18 
               Standard 
               λ 
             
           
         
       
     
   
   However, R17 of the international standard is likewise not very accurately determined. In any case, the parameters are uncertain; the generally accepted parameters will probably still be revised frequently in the future. 
   These equations are they were transformed manually to form the equation
 
0=−3 K   2 ( R 18) 2λ +2 K R 45  R 18 λ +2  R 18 −R 46
 
with only one unknown; this is solved numerically, for example in accordance with a Newton-Raphson algorithm; this gives R18 and then R13 in accordance with
 
 R 13 =R 45−2 K R 18 λ 
 
   For other parameters of the semi-empirical equation, the result is possibly completely different final formulae to be solved, for example for λ=0.5
 
 R 18=0.5*( R 46−2  R 17 R 13 −R 17 2 )
 
2. Inverse Ion Correction
 
   For the standardization, the first step in the data evaluation, it is necessary to convert the known or defined atomic ratios into the molecular ratios (inverse ion correction). This is carried out by insertion into the above system of equations, which means that different equations are used for ion correction and inverse ion correction. The disadvantage in this case, apart from the susceptibility of this procedure to error, is that possible changes, for example the introduction of further gases (see above), bring with them changes in both systems of equations. 
   3. Other Gases 
   For each gas, the above relationships must be and are derived separately and in a complicated manner. The disadvantage in the existing method is, in particular, the fact that for each type of gas the derivation of the final formula must be carried out independently, which is time-consuming and susceptible to error. 
   The method is also inflexible for one and the same gas, since any (parameter) changes in the initial equations must be made directly in the software source code in the case of a software implementation of the method. Furthermore, for the inverse ion correction, it is necessary to use a different system of equations from that of the ion correction, which is likewise time-consuming and susceptible to error. 
   II. New Method 
   1. General Approach 
   With the general gas
 
A 1   n1 A 2   n2 A 3   n3 A 4   n4 . . . 
 
where the A i  are the individual elements, with i as a natural number (i∈IN), n i  (i=0, 1 , 2, 3 . . . or i∈IN 0, ) the chemical stoichiometric numbers, and m i  (i∈IN) the mass of the lightest isotope of the element A i , it is true that:
 
           0   =         ∑     i   =   1       i   ⁢           ⁢   max       ⁢           ⁢       n   i     ⁢     R   ⁡     (       m   i     +   1     )           -     R   ⁡     (     1   +       ∑     i   =   1       i   ⁢           ⁢   max       ⁢       n   i     ⁢     m   i           )                     0   =         ∑     i   =   1       i   ⁢           ⁢   max       ⁢       n   i     ⁢     R   ⁡     (       m   i     +   2     )           +       ∑     i   =   1       i   ⁢           ⁢   max       ⁢       n   i     ⁢     R   ⁡     (       m   i     +   1     )       ⁢       ∑     j   =   1       i   ⁢           ⁢   max       ⁢     nj   ⁢           ⁢     R   ⁡     (       m   j     +   1     )               +       ∑     i   =   1       i   ⁢           ⁢   max       ⁢       Pos   ⁡     (       n   i     -   1     )       ⁢       R   ⁡     (       m   i     +   1     )       2         -     R   ⁡     (     2   +       ∑     i   =   1       i   ⁢           ⁢   max       ⁢       n   i     ⁢     m   i           )                     0   =         ∑     i   =   1       i   ⁢           ⁢   max       ⁢       n   i     ⁢     R   ⁡     (       m   i     +   3     )           +       ∑     i   =   1       i   ⁢           ⁢   max       ⁢       n   i     ⁢     R   ⁡     (       m   i     +   2     )       ⁢       ∑     j   =   1       i   ⁢           ⁢   max       ⁢     njR   ⁡     (       m   j     +   1     )             +       ∑     i   =   1       i   ⁢           ⁢   max       ⁢       n   i     ⁢     R   ⁡     (       m   i     +   1     )       ⁢     (       ∑     j   =   i       i   ⁢           ⁢   max       ⁢       njR   ⁡     (       m   j     +   1     )       ⁢       ∑     k   =   j       i   ⁢           ⁢   max       ⁢     njR   ⁡     (       m   j     +   1     )             )         +       ∑     i   =   1       i   ⁢           ⁢   max       ⁢       Pos   ⁡     (       n   i     -   1     )       ⁢       R   ⁡     (       m   i     +   1     )       2     ⁢       ∑     j   =   1       i   ⁢           ⁢   max       ⁢       n   j     ⁢     R   ⁡     (       m   j     +   1     )               +       ∑     i   =   1       i   ⁢           ⁢   max       ⁢       Pos   ⁡     (       n   i     -   2     )       ⁢       R   ⁡     (       m   i     +   1     )       3         -       R   ⁡     (     3   +       ∑     i   =   1       i   ⁢           ⁢   max       ⁢       n   i     ⁢     m   i           )       ⁢   etc   ⁢           ⁢   …             
In this case, R(x) are the ratios of the mass x (atomic or molecular) divided by the mass of the main isotope.
 
The function Pos(x) is defined as
   Pos ( x )=1 for  x&gt; 0 0 for x≦=0 
   The system of equations is simpler for real cases than it appears, since most terms are either 0 or are so small that they may be disregarded without any loss of accuracy (example: one constituent of carbon is always 14 C, but only in frequencies of 1e−10 of the main isotope, and can therefore be set equal to 0 without any loss of accuracy). 
   It is also entirely possible that, of the possible molecular masses or ratios, in spite of an adequate abundance (frequency), only a selection can be measured for metrological reasons. 
   In the general case, the system of equations is under-defined, possibly on account of the above restrictions, and is further supplemented with one or more equations of the form
 
0 =ƒ[R ( m   i   +n ),  R ( m   i   +m ),  R ( m   i   +k )]
 
where f can be any desired, general functional rule.
 
   It is critical that the number of atomic ratios used in the system of equations must be at least as large as the number of independent equations. 
   Each of the equations of the system of equations must, furthermore, be continuous and capable of continuous differentiation 
   This system of equations can also be represented in a simplified manner in vector notation as
 
 F ({right arrow over ( R   mol,    R   at, )})=0
 
   In accordance with the new method, this system of equations is now no longer transformed individually (that is to say newly each time for each type of gas, measured masses, etc,) into an equation to be solved numerically but is solved as a whole. This can be done by various methods. 
   Preference is given to a Newton or pseudo Newton method, for example Newton-Kantorowitsch. In this case, the effort on computation is quite low because of the fast iteration, and therefore the speed of the calculations is high. 
   Alternative methods are, for example, gradient methods or fixed point iterations. 
   2. Ion Correction 
   For the ion correction, the molecular ratios are known from the measurement.
 
F({right arrow over (R mol,  R at, )})
 
must therefore be sold for the atomic ratios. The first step consists in the linearisation of the general nonlinear system equations with the effect of a Taylor expansion around an initial vector {right arrow over (R at   0 )}
 
   
     
       
         
           
             F 
             ⁡ 
             
               ( 
               
                 
                   
                     R 
                     mol 
                   
                   → 
                 
                 , 
                 
                   
                     R 
                     
                       at 
                       , 
                     
                   
                   → 
                 
               
               ) 
             
           
           ≈ 
           
             
               
                 
                   ⅆ 
                   F 
                 
                 
                   ⅆ 
                   
                     
                       R 
                       
                         at 
                         , 
                       
                     
                     → 
                   
                 
               
               * 
               
                 ( 
                 
                   
                     
                       R 
                       
                         at 
                         , 
                       
                     
                     → 
                   
                   - 
                   
                     
                       R 
                       at 
                       0 
                     
                     → 
                   
                 
                 ) 
               
             
             + 
             
               F 
               ⁡ 
               
                 ( 
                 
                   
                     
                       R 
                       
                         mol 
                         ⁢ 
                         
                             
                         
                         , 
                       
                     
                     → 
                   
                   ⁢ 
                   
                     
                       R 
                       at 
                       0 
                     
                     → 
                   
                 
                 ) 
               
             
           
         
       
     
   
   The initial vector is ideally but, because of the fast convergence of this method, not necessarily, close to the actual ratios. This can be expediently be achieved by the known elementary ratios of the standard being used as a starting point. 
               ⅆ   F       ⅆ       R     at   ,       →         ,         
the derivative of the function with respect to the factor of the atomic ratios means that the system of equations is derived separately for each individual vector element. The result can be formulated most elegantly in matrix notation.
 
   The formation of the derivatives can be carried out exactly in a few cases (if the system of equations consists only of polynomials, for example), but in the general case numerically or by using approximation values. Because of the generality of the approach, an arbitrary numerical method is recommended. 
   One expedient method is for a slope vector to be calculated for each vector element in accordance with the following formula: 
   
     
       
         
           
             
               
                 
                   δ 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   F 
                 
                 → 
               
               
                 δ 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 
                   R 
                   
                     at 
                     , 
                     j 
                   
                 
               
             
             = 
             
               
                 
                   F 
                   ⁡ 
                   
                     ( 
                     
                       
                         
                           
                             R 
                             
                               mol 
                               , 
                             
                           
                           → 
                         
                         ⁢ 
                         
                           
                             R 
                             
                               at 
                               , 
                               
                                 i 
                                 = 
                                 j 
                               
                               , 
                             
                           
                           → 
                         
                       
                       , 
                       
                         [ 
                         
                           
                             R 
                             
                               at 
                               , 
                               j 
                             
                           
                           ( 
                           
                             1 
                             + 
                             δ 
                           
                           ] 
                         
                         ) 
                       
                     
                     ) 
                   
                 
                 - 
                 
                   F 
                   ⁡ 
                   
                     ( 
                     
                       
                         
                           
                             R 
                             
                               mol 
                               , 
                             
                           
                           → 
                         
                         ⁢ 
                         
                           
                             R 
                             
                               at 
                               , 
                               
                                 i 
                                 = 
                                 j 
                               
                               , 
                             
                           
                           → 
                         
                       
                       , 
                       
                         [ 
                         
                           
                             R 
                             
                               at 
                               , 
                               j 
                             
                           
                           ⁡ 
                           
                             ( 
                             
                               1 
                               - 
                               δ 
                             
                             ) 
                           
                         
                         ] 
                       
                     
                     ) 
                   
                 
               
               
                 2 
                 ⁢ 
                 δ 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 
                   R 
                   
                     at 
                     , 
                     j 
                   
                 
               
             
           
           , 
           
             j 
             = 
             1 
           
           , 
           2 
           , 
           
             3 
             ⁢ 
             
                 
             
             ⁢ 
             … 
           
         
       
     
   
   As can be seen immediately, the linearized form of the equation can also be written in matrix notation (what is known as the Jacobi matrix). 
   This linearized equation can be solved in accordance with the known Gauss algorithm; the solution is taken as an initial vector for the next iteration step. 
   If appropriate, it is possible to continue to use the derivative determined at first in the following iteration steps; the iteration proceeds faster if the derivative is determined anew each time. 
   Since this method then generally converges at least at the square of the convergence speed, for example 5 iteration steps are often adequate; however, a threshold value for the change between two successive iterations steps is worth recommending as a stop criterion for the convergence. 
   3. Inverse Ion Correction 
   One important advantage of the method according to the invention resides in the fact that, for the “inverse” ion correction (the calculation of the molecular ratios with given atomic ratios, necessary for the standardization), the same system of equations and the same algorithm as for the ion correction can be used. The system of equations merely has to be solved for the molecular ratios, not for the atomic ratios as in the case of the ion correction. 
   For the inverse ion correction, the number of equations is typically, but not necessarily, higher than the number of variables. In this case, one line which consists only of zeros is obtained in the matrix 
                 ⅆ   F       ⅆ       R     at   ,       →         .     ⁢                 
This line is then left out for the following calculations (see example).
 
III. Exemplary Calculation
 
   In the following text, by using practical numerical values, the conventional and the new method are compared in exemplary calculations. 
   1. Conventional Method (Using the Example of CO 2 ) 
   The system of equations for CO 2 , using the parameters K and λ from J. Sandrock, A. Studley, J. M. Hayes, Anal. Chem 1985 (57), 1444–1448, is:
 
0 =R 13+2 *R 17 −R 45
 
0=2 *R 18+2 *R 13 *R 17 +R 17 2   −R 46
 
0 =K*R 18 λ   −R 17 with  K= 0.0099235 and λ=0.516
 
Inverse Ion Correction
 
   The usual primary standard is VPDB (“Vienna PDB”: PDB for Pee Dee Belemite, a fossil found at the Pee Dee River; since this has recently no longer been available, the standard is now defined by the IAEA in Vienna (therefore “Vienna”)); for the CO 2  generated from this, the following atomic ratios can be applied:
 
R13 st =0.0112372
 
R17 st =0.000410850
 
R18 st =0.002088349
 
   Substitution in the equations
 
 R 45 std   =R 13 std +2 *R 17 std 
 
 R 46 std =2 *R 18 std +2 *R 13 std   *R 17 sdt   +R 17 std   2 
 
yields
 
R45 st =0.0120589
 
R46 st =0.0041861
 
Ion Correction
 
   The starting point, as in the new method explained below, will be molecular ratios which are 10 parts per thousand above those of the standard, that is to say
 
R45 sa =0.012179489
 
R46 sa =0.4227961409 e− 2
 
   For the ion correction, the system of equations must be transformed for this specific case; it follows that
 
0=−0.00029542755675( R 18) 1032 +20.0099235  R 45  R 18 0.516 +2  R 18− R 46
 
with only one unknown; this is solved numerically for R18, for example, in accordance with a Newton-Raphson algorithm. The individual steps in this algorithm will not be discussed here, since this is known in the prior art.
 
   The result for R18 is R18=0.0021092068 
   R13 also results from the transformed system of equations, specifically as
 
 R 13 =R 45−0.019847 *R 18 0.516 
 
that is to say R13=0.001135356448
 
2. New Method (Using the Example of CO 2 )
 
   In the case of CO 2 , according to the general approach:
     A 1 =C; A 2 =O   n 1 =1; n 2 =2   m 1 =12; m 2 =16
 
and thus, from the general system of equations explained above
 
0=1* R (12+1)+2* R (16+1)− R (1+12+2*16)
 
0=1* R (12+2)+2* R ( R (16+2)+1* R (12+1)*(2* R (17))+0* R (12+1) 2 +1 *R (17+1) 2   −R (2+12+2*16)
   

   From this, it then follows that
 
0 =R 13+2* R 17 −R 45
 
0=2* R 18+2* R 13* R 17 +R 17 2   −R 46
 
   The third equation needed to solve this system of equations with two unknowns is taken from the literature, specifically 
   
     
       
         
           0 
           = 
           
             
               
                 K 
                 * 
                 
                   R18 
                   λ 
                 
               
               - 
               
                 R17 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 with 
                 ⁢ 
                 
                   
                       
                   
                   ⁢ 
                   
                       
                   
                 
                 ⁢ 
                 K 
               
             
             = 
             
               
                 0.0099235 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 and 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 λ 
               
               = 
               0.516 
             
           
         
       
     
   
   These three equations and the atomic ratios for the standard are all that is needed. 
   Here, too, it will be assumed that the standard gas contains the same isotope ratios as the gas generated from VPDB
 
R13 st =0.0112372
 
R17 st =0.000410850
 
R18 st =0.002088349
 
Inverse Ion Correction
 
   By using this, first of all the molecular ratios for the standard are calculated. Since there are 3 independent equations but only 2 unknowns, the system of equations is even over-defined. 
   The atomic ratios of the standard are known, the molecular ratios have to be calculated by means of linearisation. 
   As mentioned, the system of equations is
 
0 =R 13+2* R 17 −R 45
 
0=2* R 18+2* R 13* R 17 +R 17 2   −R 46
 
0 =K*R 18 λ   −R 17 with  K= 0.0099235 and λ=0.516
 
or
 
0=0.0112372+2*0.000410850 −R 45
 
0=2*0.002088349+2*0.0112372*0.000410850+0.000410850 2   −R 46
 
0=0.516* R 18 0.0099235   −R 17
 
   These equations are differentiated numerically in accordance with the above scheme with respect to an initial vector of the molecular ratios. As the initial vector of the molecular ratio, it is recommended to derive this from that from the atomic ratios
 
R45 0 =R13 st =0.0112372
 
 R 46 0 =2* R 18 st =0.004176698
 
δ is assumed to be 1e−3 in this example.
 
   As an example, differentiate the first equation with respect to R45 
   
     
       
         
           
             
               
                 δ 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 F 
               
               → 
             
             
               δ 
               ⁢ 
               
                   
               
               ⁢ 
               R45 
             
           
           = 
           
             
               ( 
               
                 
                   
                     
                       
                         
                           
                             
                               
                                 .8329372 
                                 ⁢ 
                                 e 
                                 ⁢ 
                                 
                                   - 
                                 
                                 ⁢ 
                                 3 
                               
                               - 
                               
                                 .8104628 
                                 ⁢ 
                                 e 
                                 ⁢ 
                                 
                                   - 
                                 
                                 ⁢ 
                                 3 
                               
                             
                             
                               .224744 
                               ⁢ 
                               e 
                               ⁢ 
                               
                                 - 
                               
                               ⁢ 
                               7 
                             
                           
                         
                       
                       
                         
                           
                             
                               0 
                               - 
                               0 
                             
                             
                               .224744 
                               ⁢ 
                               e 
                               ⁢ 
                               
                                 - 
                               
                               ⁢ 
                               7 
                             
                           
                         
                       
                     
                   
                 
                 
                   
                     
                       
                         0 
                         - 
                         0 
                       
                       
                         .224744 
                         ⁢ 
                         e 
                         ⁢ 
                         
                           - 
                         
                         ⁢ 
                         7 
                       
                     
                   
                 
               
               ) 
             
             = 
             
               ( 
               
                 
                   
                     
                       
                         
                           1 
                         
                       
                       
                         
                           0 
                         
                       
                     
                   
                 
                 
                   
                     0 
                   
                 
               
               ) 
             
           
         
       
     
   
   Differentiation with respect to R46 is carried out in an analogous way, the linearized form of the system of equations is then (in matrix form) 
   
     
       
         
           
             
               
                 
                   
                     
                       ( 
                       
                         
                           
                             
                               - 
                               1 
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               - 
                               1 
                             
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                         
                       
                       ) 
                     
                     * 
                     
                       [ 
                       
                         
                           ( 
                           
                             
                               
                                 
                                   R45 
                                   st 
                                 
                               
                             
                             
                               
                                 
                                   R46 
                                   st 
                                 
                               
                             
                           
                           ) 
                         
                         - 
                         
                           ( 
                           
                             
                               
                                 0.0112372 
                               
                             
                             
                               
                                 0.004176698 
                               
                             
                           
                           ) 
                         
                       
                       ] 
                     
                   
                   + 
                 
               
             
             
               
                 
                   ( 
                   
                     
                       
                         0.0008217 
                       
                     
                     
                       
                         0.00219756484465 
                       
                     
                   
                   ) 
                 
               
             
           
           = 
           
             ( 
             
               
                 
                   0 
                 
               
               
                 
                   0 
                 
               
             
             ) 
           
         
       
     
   
   The third line of the matrix consists only of the zeros, and is therefore left out: 
   
     
       
         
           
             
               
                 
                   
                     
                       ( 
                       
                         
                           
                             
                               - 
                               1 
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               - 
                               1 
                             
                           
                         
                       
                       ) 
                     
                     * 
                     
                       [ 
                       
                         
                           ( 
                           
                             
                               
                                 
                                   R45 
                                   st 
                                 
                               
                             
                             
                               
                                 
                                   R46 
                                   st 
                                 
                               
                             
                           
                           ) 
                         
                         - 
                         
                           ( 
                           
                             
                               
                                 0.0112372 
                               
                             
                             
                               
                                 0.004176698 
                               
                             
                           
                           ) 
                         
                       
                       ] 
                     
                   
                   + 
                 
               
             
             
               
                 
                   ( 
                   
                     
                       
                         0.0008217 
                       
                     
                     
                       
                         0.00219756484465 
                       
                     
                   
                   ) 
                 
               
             
           
           = 
           
             ( 
             
               
                 
                   0 
                 
               
               
                 
                   0 
                 
               
             
             ) 
           
         
       
     
   
   If this is solved in accordance with the Gauss algorithm, it follows that
 
R45 st =0.0120589
 
 R 46 st =0.4186100405 e− 2
 
   Since the original system of equations is already strictly linear with respect to R45 and R46, the final result already follows in the first iteration step. 
   Ion Correction 
   According to a measurement listed by way of example, the measured molecular ratios are to be in each case 10 parts per thousand higher than the standard values, that is to say
 
R45 sa =0.012179489
 
 R 46 sa =0.4227961409 e− 2
 
   Using the initial values (here, as recommended above, the values known from the standard are used)
 
R13 sa,0 =0.0112372
 
R17 sa,0 =0.000410850
 
R18 sa,0 =0.002088349
 
   The form of the system of equations linearized in accordance with the above rule is 
   
     
       
         
           
             
               
                 
                   
                     ( 
                     
                       
                         
                           1 
                         
                         
                           2 
                         
                         
                           0 
                         
                       
                       
                         
                           0.0008217 
                         
                         
                           
                             
                               .2329610 
                               ⁢ 
                               e 
                             
                             - 
                             1 
                           
                         
                         
                           2 
                         
                       
                       
                         
                           0 
                         
                         
                           
                             - 
                             1 
                           
                         
                         
                           .1015149221 
                         
                       
                     
                     ) 
                   
                   * 
                 
               
             
             
               
                 
                   
                     [ 
                     
                       
                         ( 
                         
                           
                             
                               
                                 
                                   
                                     0.0112372 
                                   
                                 
                                 
                                   
                                     0.000410850 
                                   
                                 
                               
                             
                           
                           
                             
                               0.002088349 
                             
                           
                         
                         ) 
                       
                       - 
                       
                         ( 
                         
                           
                             
                               
                                 
                                   
                                     R13 
                                   
                                 
                                 
                                   
                                     R17 
                                   
                                 
                               
                             
                           
                           
                             
                               R18 
                             
                           
                         
                         ) 
                       
                     
                     ] 
                   
                   + 
                   
                     ( 
                     
                       
                         
                           
                             
                               
                                 
                                   
                                     .120589 
                                     ⁢ 
                                     e 
                                   
                                   - 
                                   3 
                                 
                               
                             
                             
                               
                                 
                                   
                                     .4186100404 
                                     ⁢ 
                                     e 
                                   
                                   - 
                                   4 
                                 
                               
                             
                           
                         
                       
                       
                         
                           
                             
                               .271 
                               ⁢ 
                               e 
                             
                             - 
                             10 
                           
                         
                       
                     
                     ) 
                   
                 
               
             
           
           = 
           
             ( 
             
               
                 
                   
                     
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                     
                   
                 
               
               
                 
                   0 
                 
               
             
             ) 
           
         
       
     
   
   The factor 0.271e−10 is different from zero only because of rounding errors. 
   Using this, the result of the first iteration is
 
 R 13=0.1135355425 e− 1,
 
 R 17=0.4129673746 e− 3,
 
 R 18=0.2109207035 e− 2
 
   Of the further iterations, only the final results are reproduced here: 
   2nd Iteration
 
 R 13=0.1135356448 e− 1,
 
 R 17=0.4129622625 e− 3,
 
 R 18=0.2109206842 e− 2
 
3rd Iteration
 
 R 13=0.1135356448 e− 1,
 
 R 17=0.4129622624 e− 3,
 
 R 18=0.2109206842 e− 2
 
   Even after 3 iterations, in this example the final value has been reached within the context of the computing accuracy used here for R13 and R18 (no longer any changes between the last two iteration steps).