Hitachi F7000 Instruction Manual

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APPENDIX C  DETERMINATION COEFFICIENT 
OF CALIBRATION CURVE 
 
C.1    Calculation of Determination Coefficient 
 
The determination coefficient and other factors are calculated via the 
following formula. 
 
 
 
Difference DIFF  :  DIFF
n = Cn - Cstdn 
Relative difference RD  : 
RDDIFF
An=×100  
AA
Nn=∑ 
Student’s t-test  : 
tDIFF
DIFF
Nnn=

∑2
1
 
Relative coefficient  : 
()()
()
RCC CC
CCnnstdn
n= 
∑∑
∑
2
2
2
 
Determination coefficient  :  R
2 = (R)2 
 
An : Photometric or 
average value of 
standard 
Cn : Concentration on 
approximation curve 
versus A 
Cstdn :  Standard 
concentration (input 
value) 
N  :  Number of standards
↑ 
Data 
Conc → 
						

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C.2    Usage of Determination Coefficient 
 
The determination coefficient indicates the goodness of fit of the 
measured standards and the prepared calibration curve.    The closer 
this value is to “1,” the better the fit of the measured value and 
calibration curve.    If the value is far from “1,” then the standards must 
be measured or the calibration curve mode must be changed.   
Examples of determination coefficients upon changing the calibration 
curve mode are given below. 
 
 
 
 
With the above standard data, it can be seen that a quadratic curve 
provides a better result.    A calibration curve, which was formerly judged 
through skill or experience, can thus be judged more easily through 
numerical means. 
 
 x:  Standard measurement points
 
When calibration curve type is set 
to linear: 
Determination coefficient < 1 
When calibration curve type is set 
to quadratic: 
Determination coefficient ≓1 
↑ 
Data 
↑ 
Data 
Conc →
Conc → 
						

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APPENDIX D  INTEGRATION METHOD AND 
SMOOTHING 
 
D.1  Foreword 
 
The integration methods and smoothing used in the FL Solutions 
program are explained here. 
 
 
D.2  Integration Methods 
 
The FL Solution program comes with the following three integration 
methods. 
 
• Rectangular 
• Trapezoid 
• Romberg 
 
 
The Rectangular method is the simplest calculation method among the 
above three.    Since one sampling interval is equal to the width of each 
sectional area, the total value of all data points covering a peak is 
approximated as a target area. 
As shown in Fig. D-1, the target area is a total of the rectangular parts 
obtained from linear approximation of the curve drawn by following the 
data points.    If the number of peak data points is small, approximation 
will become rough. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Fig. D-1 
 
D.2.1 Rectangular 
Method  
						

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The Trapezoid method is an improved method of peak area calculation. 
Each section having a width equal to one sampling interval is indicated 
by a rectangle and a triangle on it.    And the target area is given by 
totaling the area values of all sections. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Fig. D-2 
 
 
When focusing attention on one section, the area I
r of its rectangular 
part can be expressed as follows. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Fig. D-3 
 
 
D.2.2 Trapezoid 
Method 
 
Ir= f1x
where, x :  Sampling interval 
 f
1 :  Left-side height of 
rectangle 
f1 
x
f2 
						

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To this rectangle, a triangular part is to be added. 
When replacing the sectional area with I
T, and the triangular area with It, 
I
T is given by the following formula. 
IIIff
xfxff
xTtr=+=−
+=+
⎛
⎝ ⎜⎞
⎠ ⎟21
112
22 
 
The above formula can be transformed to cover the entire peak as 
follows. 
If
fff
xTnn=++++⎛
⎝ ⎜⎞
⎠ ⎟ 1
2122K 
 
 
The Romberg method is the most exact integration used in this program. 
The aforementioned Trapezoid method has different step sizes 
(sampling intervals) for determining a peak area more exactly.     
This method capable of using a total of inherent errors can use both of 
two step sizes individually conceivable.    However, unlike a classic 
method of continuous approximation, the step size cannot be reduced 
freely in the same way as an increase in the X-axis direction for exact 
integration.    This is because a spectrum is handled as average data of 
the data points which define the spectrum. 
Yet, the Romberg method allows an increase in step size by a factor of 
2 or 4 when necessary.    This method is implemented as follows. 
 
(1)  At every data point, data is integrated by the Trapezoid method. 
(2)  At every two data points, data is integrated by the Trapezoid 
method. 
(3)  The results of the above (1) and (2) are combined with each other 
to obtain the following. 
 
IIIIRnnn=+2
3 
 
where, I
R  :  Integration by Romberg method 
 I
n  :  Trapezoid method at every data point 
 I
2n  :  Trapezoid integration at every two data point 
 
 
D.2.3 Romberg 
Method 
  
						

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D.3  SMOOTHING 
 
The FL Solutions program comes with the following three smoothing 
methods. 
 
• Savitsky-Golay smoothing 
• Mean smoothing 
• Median smoothing 
 
These are explained in detail below. 
 
 
For this method, refer to the following literature. 
 
Gorry,P.A. 
”General Least-Squares Smoothing and Differentiation by the 
Convolution (Savitsky -Golay) Method”   
Anal.Chem. 1990, 62, 570-573. 
 
D.3.1 Foreword 
 
D.3.2 Savitsky-Golay 
Smoothing  
						

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Average data is obtained from the numerical values at the specified data 
points and set for the center wavelength. 
 
Example) 
Number of data points :  7 
Number of times  :  1 
In case of 7 data points starting from 358.0 nm, the average value is 
set for the center wavelength of 358.6 nm.    At this time, data 
disappears at 3 points at the both sides on the spectrum. 
 
How to acquire data 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
When an even number (2n) of data points are specified, 
“2n” is replaced with “2n+1” for calculation.    This means that the same 
result is obtained when the number of data points is set at “8” and at “9.” 
 
 D.3.3 Mean 
Smoothing 
nm Data  Data 
358.0 22.41 ⎯ 
358.2 23.15 ⎯ 
358.4 23.82 ⎯ 
358.6 24.41  24.23 
358.8 24.91  24.71 
359.0 25.32  25.10 
359.2 25.60  25.39 
359.4 25.78  25.57 
359.6 25.85  25.65 
359.8 25.83  25.64 
360.0 25.70  25.54 
360.2 25.50  25.35 
360.4 25.23 
 
360.6 24.88 
 
360.8 24.47 
 

 
 
 

 
 
  
						

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Middle data is obtained from the numerical values at the specified data 
points and set for the center wavelength. 
 
Example) 
Number of data points :  7 
Number of times  :  1 
In case of 7 data points starting from 358.0 nm, the 4th smallest 
value is set for the center wavelength of 358.6 nm.    At this time, 
data disappears at 3 points at the both sides on the spectrum. 
 
How to acquire data 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
When an even number (2n) of data points are specified, 
“2n” is replaced with “2n+1” for calculation.    This means that the same 
result is obtained when the number of data points is set at “8” and at “9.” 
 
 D.3.4 Median 
Smoothing 
nm Data  Data 
358.0 22.41 ⎯ 
358.2 23.15 ⎯ 
358.4 23.82 ⎯ 
358.6 24.41  24.41 
358.8 24.91  24.91 
359.0 25.32  25.32 
359.2 25.60  25.60 
359.4 25.78  25.70 
359.6 25.85  25.70 
359.8 25.83  25.70 
360.0 25.70  25.70 
360.2 25.50  25.50 
360.4 25.23 
 
360.6 24.88 
 
360.8 24.47 
 

 
 
 

 
 
  
						

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APPENDIX E  DESCRIPTION OF 
FLUOROMETRY 
 
E.1  Description of Fluorescence 
 
 
 
 
 
 
Fig. E-1    Typical Organic Molecular Energy Level 
 
 
Figure E-1 illustrates the energy level transitions in an organic molecule 
in processes of light absorption and emission. 
When light strike an organic molecule in the ground state, it absorbs 
radiation of certain specific wavelengths to jump to an excited state.     
A part of the excitation (absorbed) energy is lost on vibration relaxation, 
i.e., radiationless transition to the lowest vibrational level takes place in 
the excited state.    And, eventually the molecule returns to the ground 
state while emitting fluorescent radiation. 
Excitation
Stable stage 
(ground state)Fluorescence
/phosphorescence
Unstable stage 
(excited state) 
Radiationless 
transition Radiationless 
transition 
Excited state 
Excited triplet state
Excitation ligh
t LightLight 
Ground state 
Absorption Fluorescence Phosphorescence 
						

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Also, if radiationless transition to the triplet state takes place, then 
phosphorescence is emitted during triplet-to-singlet transition (from the 
excited triplet state to the ground singlet state). 
Generally phosphorescence persists for 10
-4 sec or longer due to the 
selection rule imposed on the triplet-to-singlet transition.    In contrast, 
fluorescence persists for a period of 10
-8 to 10-9 sec in most cases. 
 
As mentioned above, a part of the radiation absorbed by a substance is 
lost as vibration energy, etc.; therefore, the fluorescence emitted from 
the substance has a longer wavelength than the excitation light (Stokes’ 
law). 
 
The ratio of the number of photons emitted during fluorescence to the 
number of photons absorbed is called the quantum efficiency of 
fluorescence.    The larger the quantum efficiency a substance has, the 
more fluorescence it emits.    Also, the intensity of fluorescence emitted 
from a substance is proportional to the quantity of light absorbed by it.   
When a dilute solution sample is measured, the fluorescence intensity is 
expressed by the following formula. 
 
F = KI
0cleεϕ 
 
where,  F  :  Intensity of fluorescence 
 K : Instrumental constant 
 I
0 :  Intensity of exciting radiation 
  c  :  Concentration of substance 
 l : Cell path length 
  ε  :  Absorptivity of substance 
 ϕ  :  Quantum efficiency of substance