Fit functions

     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  

$F(x)\; =\; BG(x)\; +\; \sum$_i=0^peaknumber PEAK_i(x))

BG(x):

background function see section D.2 on page

peak function of i-th peak

$PEAKi(x)=\; \{V$_iNORM_i} ⋅(GAUSM_i(x-P_i) + STEP_i(x-P_i))

P$$_i :

position of i-th peak (parameter)

V$$_i :

volume of i-th peak (parameter)

NORM$$_i :

numeric INTEGRAL(GAUSM$$_i )

modified gauss function of i-th peak

$GAUSMi(dx)\; =\; \{\{$ exp($ \{SL$$$_iS_i²} $\cdot$(dx + $\{SL$_i2}))      dx<SL_i $exp($ \{-dx22$ \cdot $S$$_i²})      SL_i<dx<SR_i $exp($ \{-SR$$_iS_i²} $\cdot$(dx - $\{SR$_i2}))      SR_i<dx }

dx:

x - P$$_i

S$$_i :

$\sigma$ of gaussian part of i-th peak (parameter)

SL$$_i :

TL$$_i $\cdot$ S$$_i^EL_i

SR$$_i :

TR$$_i $\cdot$ S$$_i^ER_i

TL$$_i :

left tail of i-th peak (parameter)

TR$$_i :

right tail of i-th peak (parameter)

EL$$_i :

exponent of $\sigma$-weight of TL$$_i [0..2]

ER$$_i :

exponent of $\sigma$-weight of TR$$_i [0..2]

step function of i-th peak

$STEP$_i(dx) = SH_i ⋅({p_i2} + arctan({SW_i ⋅dxS_i ⋅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  

$F(x)\; =\; BG(x)\; +\; \sum$_i=0^peaknumber PEAK_i(x)

BG(x):

background function see section D.2 on page

peak function of i-th peak

$PEAK$_i(x) = {V_iNORM_i} ⋅(((1 + TAIL_i(x-P_i)) ⋅GAUSS_i(x-P_i)) + STEP_i(x-P_i))

P$$_i :

position of i-th peak (parameter)

V$$_i :

volume of i-th peak (parameter)

NORM$$_i :

S $$_i ⋅((2 ⋅P_i) + TL_i + TR_i)

gauss function of i-th peak

$GAUSS$_i(dx)= exp({-dx²2 ⋅S_i²})

S$$_i :

$\sigma$ of gaussian part of i-th peak (parameter)

additional tail factor of i-th peak

$TAIL$_i(dx) = {{ $\{TL$_i $\cdot$ ($\{|dx|S$_i})^EL_iFACFAC(EL_i)}      dx<0 $\{TR$_i $\cdot$ ($\{|dx|S$_i})^ER_iFACFAC(ER_i)}      dx $\ge$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

step function of i-th peak

$STEP$_i(dx)= {12} ⋅SH_i ⋅(1 - erf({dxSW_i ⋅S_i ⋅2}))

SH$$_i :

step height of i-th peak (parameter)

SW$$_i :

step width of i-th peak (parameter)

erf:

$erf(x)\; =\; \{2\pi \}\int$₀^x exp(-t²)dt