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There are two alternative parametrizations for the peaks. Both have
proven to yield comparable results. The latter reduces the correlation
between step and tails since the erf step approximates the asymptotic
value quite fast in comparison with the arctan. Furthermore the step
width of the erf can be fixed to 1.0 (in units of sigma) which reduces
the number of parameters by one. Finally the latter function is
analytically integrable whereas the integration of the former has to
be done numerical.
In both functions the volume but not the amplitude of the peaks is
fitted. The volume is the parameter of interest normally and in this
way it is not necessary to integrate the resulting function and to
estimate the errors of the obtained volume.
-
- tv fit function peak definition
continuous-exp-tail/arctan-step
- BG(x):
- background function see section
D.2 on page
-
- peak function of i-th peak
- P:
- position of i-th peak (parameter)
- V:
- volume of i-th peak (parameter)
- NORM:
- numeric INTEGRAL(GAUSM)
-
- modified gauss function of i-th peak
- dx:
- x - P
i
- S
i
:
-
σ
of gaussian part of i-th peak
(parameter)
- SL
i
:
- TL
i
⋅
S
iELi
- SR
i
:
- TR
i
⋅
S
iERi
- TL
i
:
- left tail of i-th peak (parameter)
- TR
i
:
- right tail of i-th peak (parameter)
- EL
i
:
- exponent of
σ
-weight of TL
i
[0..2]
- ER
i
:
- exponent of
σ
-weight of TR
i
[0..2]
-
- step function of i-th peak
STEPi(dx) = SHi ⋅({pi2} + arctan({SWi ⋅dxSi ⋅2}))
- SH
i
:
- step height of i-th peak (parameter)
- SW
i
:
- step width of i-th peak (parameter)
-
- tv
>
fit function peak definition additive-tail/erf-step
F(x) = BG(x) + ∑i=0peaknumber PEAKi(x)
- BG(x):
- background function see section
D.2 on page
-
- peak function of i-th peak
PEAKi(x) = {ViNORMi} ⋅(((1 + TAILi(x-Pi)) ⋅GAUSSi(x-Pi)) + STEPi(x-Pi))
- P
i
:
- position of i-th peak (parameter)
- V
i
:
- volume of i-th peak (parameter)
- NORM
i
:
- S
i ⋅((2 ⋅Pi) + TLi + TRi)
-
- gauss function of i-th peak
GAUSSi(dx)= exp({-dx22 ⋅Si2})
- S
i
:
-
σ
of gaussian part of i-th peak
(parameter)
-
- additional tail factor of i-th peak
TAILi(dx) = {{ - {TLi ⋅ ({|dx|Si})ELiFACFAC(ELi)}
-
- dx<0
- {TRi ⋅ ({|dx|Si})ERiFACFAC(ERi)}
-
- dx ≥0
}
- TL
i
:
- left tail of i-th peak (parameter)
- TR
i
:
- right tail of i-th peak (parameter)
- EL
i
:
- exponent of left tail [2..16]
- ER
i
:
- exponent of right tail [2..16]
-
- for simplification of integral
FACFAC(n) = {{ - (n-1)!!
-
- n ∈{3,5,7,...}
- (n-1)!! ⋅{</}SQRT>π2
-
- n ∈ {2,4,6,...}
}
-
- step function of i-th peak
STEPi(dx)= {12} ⋅SHi ⋅(1 - erf({dxSWi ⋅Si ⋅2}))
- SH
i
:
- step height of i-th peak (parameter)
- SW
i
:
- step width of i-th peak (parameter)
- erf:
-
erf(x) = {2π}∫0x exp(-t2)dt
Next: The background functions
Up: The fit- background- and
Previous: The fit- background- and
Andreas Fitzler
7/13/2000