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Some lineshape transformations for Fourier transform resolution enhancement of photoelectron spectra
Journal of Electron Spectroscopy and ReIated Phenomena 12 (1977) 213-219
@ Elsevler Scientific Pubhshmg Company, Amsterdam - Prmted m The Netherlands
Short commumcation Some lineshape transformations photoelectron spectra
for
Fourier
transform
resolution
enhancement
of
J LLOYD Nattonal Chemical Research Laboratory, Councdfor SclentiJicand Industrral Research, P 0 Pretorq 0001 (Republic of South Africa)
(Fwst received 7 February
Box 395,
1977, m final form 1 May 1977)
Several reports ’- 3 on ways of lmprovmg the resolution of photoelectron and, more recently, Beatham and Orchard2 have spectra have appeared Werthelml constructed accurate instrument response functions for monochromatlsed and achromatlc sources respectively and proposed various methods to allow direct apphcatlon wlthout catastrophic amplification of background noise The study by Plreaux3 provrdes an alternatlve to these m the form of numerical derlvatlves which use only the characterlstlcs of the data to obtam narrower line widths In mvestlgatmg such methods for our own use we have found that denvatlve methods can be of value m peak location but that the response function approach IS more reliable, particularly when noise levels are high We have, however, also found that lmeshape transformations m Fourier space allow construction of simple yet efficient resolution enhavcement filters which produce results of quality at least comparable to that of other methods Since resolution enhancement techmques are of some Importance we believe this hltherto untried approach may be of interest and present below a descrlptlon of its characterlstlcs along with some examples of its apphcation An observed kinetic energy spectrum, f(E), can be considered to arlse from a combmatlon of both uniform and non-umform dlstortlon processes acting on some mtrmslc spectrum, g(E) For the achromatic X-ray sources considered here the umform dlstortlon results from a convolution of g(E) with the Lorentzlan multlplet of the lomzmg radiation and a Gaussian slit function” 2 Combmmg these processes mto a smgle lmeshape or instrument response function, I(E), gives the Fourier space equlvalent as a product of the component transforms4 P(w)
= L(w)
Z(E), and hence L(o),
G(o)
1s usually known and m the ideal case the above equation
11) can
214 be solved directly to obtam back mto energy space The noise, whose transform IS character of L(w) resuIts m a second lmeshape function, transform equation becomes
where
E;‘(o)
= L’(o)F(o)/L(w)
F’(o)
1s hopefully
G(o), the enhanced data bemg obtained by mverslon presence of non-umform contrlbutlons such as random essentially flat, prevents this because the exponential drastic over amphficatlon of the noise To compensate, L’(W), must be Introduced and the practical Fourier
(2)
a more
useful approxlmatlon
to the mtrmslc
transform
G(o) If the power spectral densrty of the noise 1s known then an optlmlsed filter can be constructed’ This 1s rarely practical and the smusoldal characterlstlcs of the enhanced data severely reduce Its mterpretlve value A slmllar crltlclsm’ apphes to The treated data band-hmltmg methods*, m which L’(o) IS a rectangular function where w, 1s the cut-off, and one again 1s now convoluted with sm (~nxo,)/nxo,, mtroduces an oscdlatory character mto the data A conslderatlon of this problem for apphcatlons m NMR spectroscopy led Ernst’ to suggest the use of mtermedlate lme shape transformations as an alternatlve The basic Idea 1s that If L(w) can be wrItten as an exponential fun&on L(o)
= exp (-
Iw~/~},
fhen one can obtain resolution tions of the type exp (-
(3) enhancement
iawl”} -+ exp (-
with reduced osclllatlon
I#W’}
by transforma(4)
o! and B are lmewldth parameters, for example, for a Lorentzlan function I = 1 and o! = zr: The order of superscrlpts 1s k < I and for slgmficant resolution enhancement o! > p Two specific transformations are consldered m thls,study The first represents the rnstrument response function as a single Lorentzlan of width, rl, and thus F’(o)
= exp
1
-
7&-,202 4 In 2
+ ~r,lol
1
F(w)
(5)
This appears at first sight a rather crude approxlmatron to the true sltuatlon’ but, for reasons which are discussed later, It does m fact form a very useful working approxlmatlon The maJor advantage lies m the final Gausslan shape bemg nonoscillatory m character thus no “wiggles” are mtroduced mto the data The second transformation 1s based on an accurate Instrument response function’ and includes an ai,2 separation, d, for the lomzmg radlatlon A cubic lmeshape of width, TC, IS used for the final lrneshape and eqn (2) becomes
n3rflo13 + 7c2r,02 4 ln 2 + ~rh-4 8
>
W-4/(3
+ S exp (-
Q7~dm)) (6)
215 d, Ts and lYl should now, of course, be determined from known values for the X-ray radlatlon and sht settings used It should be noted that mverslon of the cubic transform results m a function which does go below zero but the oscillatory character IS much smaller than with a sme function Before consldermg the apphcatlon of the above equations it IS important that the general philosophy behind all resolution enhancement methods be clearly understood In any real set of data points the mformatlon content will be fixed by the relevant noise levels present For example, the characterlstlc tall of an mtrmslc Lorentzlan lme will be completely submerged by noise after instrumental broadenmg has occurred Regardless of how accurately the instrument response function IS known this cannot be recovered by numerlcal analysis Resolution enhancement must therefore be viewed solely as a means to mcrease the accesslblhty of mformatlon already present m data and not as some scheme which creates mformatlon A factor often neglected m this analysis IS that the processes described by eqn (1) are completely linear The response function parameter values may therefore be considered as upper hmlts and it IS perfectly rigorous to use smaller values If the quality of the treated data IS improved For example, Fig 1 shows the effect of usmg a smaller a 1,2 separation than the published value2 to obtam an enhanced spectrum of better quality yet comparable mformatron content Most of the important mformatlon contamed m a data transform ~111reside m the lower range of w Band-hmltmg methods exploit this m a rather severe way
(b)
(a)
L t-2rv-I
L_
Figure 1 Resolution enhancementof gold 4flevels usmg eqn (5) (a) rl = 0 41, lTg = 0 776, lTC = 0 62, d = -0 33, (b) I?, = 0.41, rg = 0 776, rC = 0 62, d = -0 2 The data spectrumwas obtained usmg Mg Ku radlatlon at the widest sht settmgs glvmg a Gaussian sht function of N 1 0-eV width
216 /
(a)
(bl/
200
/
/ /
IO 0
/
t
_&Is!--,
/
/ i’ c-
\
/’ --_
-o-o<,
\
-.
\
-0
\ ‘\‘O
*\
‘y
TYPICAL CUT OFF VALUE FOR BAND LlMlTlNG
3
‘pp \
I 15
\
“\, lo,
\
\
\ \
\
“\ \
1
, 20
*\ ,
W
Figure 2 Comparison between an accurate response function and eqns (4) and (5) usmg various parameters (a) The true response function r~ = 0 41, rg = 0 5, (b) II’1= 0 41, Ts = 0 5, re = 0 4, W rl = 0 41, rs = 0 5, rC = 0 5, (d) I?1 = 0 41, rg = 0 7, IYc = 0 7, (e) The Lorentzlan-Gausnan transformation rl = 0 9, rg = 0 6
are zero In Fig 2 it 1s shown how lmeshape transformations differ from this by comparmg L- l(w) and L’(cD)/L(o) for the exponential part of a representattve Instrument response function Since values greater than 1 correspond to enhancement while those below 1 correspond to smoothmg it can clearly be seen how the present method permits maximum retention of mformatlon without excessive build up of noise yet closely approximates the true response function over the mltlal range of w If the response function parameters are used then the value of lYCrequired to give a transition from enhancement to smoothmg at some fixed point, oC, IS given by by mtroducmg
a cut-off
beyond
which
all components
(7) In many cases TC wllI turn out larger than lYg reflectmg the level of mformatlon present When this occurs It may well prove more effective to exceed the hmlts of the true response function by usmg a larger value for rg As shown m Fig 2 It IS a relatively simple matter to do this while still mamtammg a good approxlmatlon to L(w) over the mltlal range of o) Providing an art&la1 response function never exceeds the true value, then the only significant dlstortlon mtroduced 1s that mherent m the final hneshape It can be seen from Fig 2 that the simple Lorentzlan-Gausslan transformation falls
217
(a)
(d)
Flgure 3 Resolution enhancement of gold 4flevels as m Fig 1 (a) Data after removal of scattermg, (b) Lorentzlan-Gausslan transformation I?1 = 1 1, re = 0 6, (c) I?1 = 0 41, rg = 0 58, d = -0 33data cut off at w = 1 18, (d) I71 = 0 41, rg = 0 58, d = -0 20, JTc = 0 53
m this respect The drstortlon induced, however, 1s not particularly severe and mamfests Itself as a dip m the tails of single peaks (see Fig 3) To provide some examples of the above methods two cases will be dIscussed The first uses well defined gold 4f levels and 1s mtended to be a standard example, while the second IS derived from our own research Interests and shows the vaIue of the techniques The Fourier transforms were calculated usrng standard algorithms’ and m all cases a correctlon for melastlc scattering has been apphed The only presmoothmg used was a g-point quadratlc polynomla13, Its mam function bemg to facilitate the scattering correction In Fig 3 the results for typical parameter sets on gold 4f levels are shown It can be seen that the lmeand a “band hmltmg” case 1s included for comparison widths have been reduced by - 0 5 eV m all cases but that the present methods produce less distortion m the tails The essentially flat response of the LorentzlanGaussian transformation beyond the tails 1s clearly evident
218
Figure 4 Resolution enhancement of chlorme (2~) levels m a mlvture of chlorate and perchlorate speaes resultmg from Mg Ku lrradlatlon of sodturn perchlorate (a) Data after scattermg removal, (b) Lorentzlan-Gausslan transformation I?1 = 1 1, rg = 0 6, (c) rl = 0 41, rg = 0 5, d = -0 33cutoffatw=10,(d)lT~=041,r,=07,rc=07,d=-033
The second example shows chlorme 2p levels m a sample of sodium perchlorate after prolonged exposure to Mg Ku radiation A chlorate intermediate IS present and In attemptmg to derive the kmetlcs mvolved’ It was necessary to determme peak widths and separations for the two component doublets Some of the parameter values used are shown m Fig 4 and once again the dn-ect apphcatlon of eqn (4) provides results of comparable quahty with the multiple lmeshape transformatron producing the best results A factor not applied m the present study makes the most simple transformation even more useful The lmeshape functions involved are completely even and hence a simple cosme series can be used for the data transform1 This IS not only computatlonally simpler but also removes most requirements for data pre-treatment smce the transform IS less sensltlve to dlscontmultles The slmphclty of the parameters allows apphcatlon to cases where the response fun&Ion IS not accurately known and, as shown here, one can obtam results of quahty comparable to those
219 obtained by more exact methods It therefore provides a very powerful tool for routme appllcatlons particularly when high noise levels cannot be avoided The methods presented here represent an attempt to overcome an obvious hmltatlon in the band-llmltmg approach It 1s not claimed that the degree of resolution enhancement obtainable will be slgmficantly larger since this 1s determined by the noise levels The amount of dlstortlon Introduced should be lower, however It 1s m thrs hght that the value of these methods should be assessed and where they offer the greatest advantages ACKNOWLEDGEMENTS
The spectra used here were kindly supplied by Dr R G Copperthwalte who suggested this study Dr IS G R Pachler provided many useful dlscusslons on the problems mvolved and the Council for Sclentlfic and Industrial Research 1s to be thanked for the award of a post-doctoral fellowshlp
REFERENCES
6 7
8
G K Werthelm, J Electron Spectrosc Reiat Phenom , 6 (1975) 239 N Beatham and A F Orchard, I Electron Spectrosc Relat Phenom , 9 (1976) 129 J -J Pneaux, Appl Spectrosc , 30 (1976) 219 R Bracewell, The Fourrer Transform and rts Applrcatrons, McGraw-HJ11, New York, 1965, ch 3 R R Ernst, m J Waugh (Ed ), Advances m Magnetrc Resonartce, Vol 2, Academic Press, New York, 1966 I? IS used throughout to represent the full width of a function of half-maxlmum he&t SubscrIpts are used to deslgnate the type, e g g for GaussJan and c for cubic The data transforms were calculated using the FFTP Fourier transform subroutme produced by The International Mathematical and Statlstlcal Library Inc Houston, Texas R G Copperthwalte and J Lloyd, J Chem Sot Dalton Trarrs, 11 (1977) 1117
@ Elsevler Scientific Pubhshmg Company, Amsterdam - Prmted m The Netherlands
Short commumcation Some lineshape transformations photoelectron spectra
for
Fourier
transform
resolution
enhancement
of
J LLOYD Nattonal Chemical Research Laboratory, Councdfor SclentiJicand Industrral Research, P 0 Pretorq 0001 (Republic of South Africa)
(Fwst received 7 February
Box 395,
1977, m final form 1 May 1977)
Several reports ’- 3 on ways of lmprovmg the resolution of photoelectron and, more recently, Beatham and Orchard2 have spectra have appeared Werthelml constructed accurate instrument response functions for monochromatlsed and achromatlc sources respectively and proposed various methods to allow direct apphcatlon wlthout catastrophic amplification of background noise The study by Plreaux3 provrdes an alternatlve to these m the form of numerical derlvatlves which use only the characterlstlcs of the data to obtam narrower line widths In mvestlgatmg such methods for our own use we have found that denvatlve methods can be of value m peak location but that the response function approach IS more reliable, particularly when noise levels are high We have, however, also found that lmeshape transformations m Fourier space allow construction of simple yet efficient resolution enhavcement filters which produce results of quality at least comparable to that of other methods Since resolution enhancement techmques are of some Importance we believe this hltherto untried approach may be of interest and present below a descrlptlon of its characterlstlcs along with some examples of its apphcation An observed kinetic energy spectrum, f(E), can be considered to arlse from a combmatlon of both uniform and non-umform dlstortlon processes acting on some mtrmslc spectrum, g(E) For the achromatic X-ray sources considered here the umform dlstortlon results from a convolution of g(E) with the Lorentzlan multlplet of the lomzmg radiation and a Gaussian slit function” 2 Combmmg these processes mto a smgle lmeshape or instrument response function, I(E), gives the Fourier space equlvalent as a product of the component transforms4 P(w)
= L(w)
Z(E), and hence L(o),
G(o)
1s usually known and m the ideal case the above equation
11) can
214 be solved directly to obtam back mto energy space The noise, whose transform IS character of L(w) resuIts m a second lmeshape function, transform equation becomes
where
E;‘(o)
= L’(o)F(o)/L(w)
F’(o)
1s hopefully
G(o), the enhanced data bemg obtained by mverslon presence of non-umform contrlbutlons such as random essentially flat, prevents this because the exponential drastic over amphficatlon of the noise To compensate, L’(W), must be Introduced and the practical Fourier
(2)
a more
useful approxlmatlon
to the mtrmslc
transform
G(o) If the power spectral densrty of the noise 1s known then an optlmlsed filter can be constructed’ This 1s rarely practical and the smusoldal characterlstlcs of the enhanced data severely reduce Its mterpretlve value A slmllar crltlclsm’ apphes to The treated data band-hmltmg methods*, m which L’(o) IS a rectangular function where w, 1s the cut-off, and one again 1s now convoluted with sm (~nxo,)/nxo,, mtroduces an oscdlatory character mto the data A conslderatlon of this problem for apphcatlons m NMR spectroscopy led Ernst’ to suggest the use of mtermedlate lme shape transformations as an alternatlve The basic Idea 1s that If L(w) can be wrItten as an exponential fun&on L(o)
= exp (-
Iw~/~},
fhen one can obtain resolution tions of the type exp (-
(3) enhancement
iawl”} -+ exp (-
with reduced osclllatlon
I#W’}
by transforma(4)
o! and B are lmewldth parameters, for example, for a Lorentzlan function I = 1 and o! = zr: The order of superscrlpts 1s k < I and for slgmficant resolution enhancement o! > p Two specific transformations are consldered m thls,study The first represents the rnstrument response function as a single Lorentzlan of width, rl, and thus F’(o)
= exp
1
-
7&-,202 4 In 2
+ ~r,lol
1
F(w)
(5)
This appears at first sight a rather crude approxlmatron to the true sltuatlon’ but, for reasons which are discussed later, It does m fact form a very useful working approxlmatlon The maJor advantage lies m the final Gausslan shape bemg nonoscillatory m character thus no “wiggles” are mtroduced mto the data The second transformation 1s based on an accurate Instrument response function’ and includes an ai,2 separation, d, for the lomzmg radlatlon A cubic lmeshape of width, TC, IS used for the final lrneshape and eqn (2) becomes
n3rflo13 + 7c2r,02 4 ln 2 + ~rh-4 8
>
W-4/(3
+ S exp (-
Q7~dm)) (6)
215 d, Ts and lYl should now, of course, be determined from known values for the X-ray radlatlon and sht settings used It should be noted that mverslon of the cubic transform results m a function which does go below zero but the oscillatory character IS much smaller than with a sme function Before consldermg the apphcatlon of the above equations it IS important that the general philosophy behind all resolution enhancement methods be clearly understood In any real set of data points the mformatlon content will be fixed by the relevant noise levels present For example, the characterlstlc tall of an mtrmslc Lorentzlan lme will be completely submerged by noise after instrumental broadenmg has occurred Regardless of how accurately the instrument response function IS known this cannot be recovered by numerlcal analysis Resolution enhancement must therefore be viewed solely as a means to mcrease the accesslblhty of mformatlon already present m data and not as some scheme which creates mformatlon A factor often neglected m this analysis IS that the processes described by eqn (1) are completely linear The response function parameter values may therefore be considered as upper hmlts and it IS perfectly rigorous to use smaller values If the quality of the treated data IS improved For example, Fig 1 shows the effect of usmg a smaller a 1,2 separation than the published value2 to obtam an enhanced spectrum of better quality yet comparable mformatron content Most of the important mformatlon contamed m a data transform ~111reside m the lower range of w Band-hmltmg methods exploit this m a rather severe way
(b)
(a)
L t-2rv-I
L_
Figure 1 Resolution enhancementof gold 4flevels usmg eqn (5) (a) rl = 0 41, lTg = 0 776, lTC = 0 62, d = -0 33, (b) I?, = 0.41, rg = 0 776, rC = 0 62, d = -0 2 The data spectrumwas obtained usmg Mg Ku radlatlon at the widest sht settmgs glvmg a Gaussian sht function of N 1 0-eV width
216 /
(a)
(bl/
200
/
/ /
IO 0
/
t
_&Is!--,
/
/ i’ c-
\
/’ --_
-o-o<,
\
-.
\
-0
\ ‘\‘O
*\
‘y
TYPICAL CUT OFF VALUE FOR BAND LlMlTlNG
3
‘pp \
I 15
\
“\, lo,
\
\
\ \
\
“\ \
1
, 20
*\ ,
W
Figure 2 Comparison between an accurate response function and eqns (4) and (5) usmg various parameters (a) The true response function r~ = 0 41, rg = 0 5, (b) II’1= 0 41, Ts = 0 5, re = 0 4, W rl = 0 41, rs = 0 5, rC = 0 5, (d) I?1 = 0 41, rg = 0 7, IYc = 0 7, (e) The Lorentzlan-Gausnan transformation rl = 0 9, rg = 0 6
are zero In Fig 2 it 1s shown how lmeshape transformations differ from this by comparmg L- l(w) and L’(cD)/L(o) for the exponential part of a representattve Instrument response function Since values greater than 1 correspond to enhancement while those below 1 correspond to smoothmg it can clearly be seen how the present method permits maximum retention of mformatlon without excessive build up of noise yet closely approximates the true response function over the mltlal range of w If the response function parameters are used then the value of lYCrequired to give a transition from enhancement to smoothmg at some fixed point, oC, IS given by by mtroducmg
a cut-off
beyond
which
all components
(7) In many cases TC wllI turn out larger than lYg reflectmg the level of mformatlon present When this occurs It may well prove more effective to exceed the hmlts of the true response function by usmg a larger value for rg As shown m Fig 2 It IS a relatively simple matter to do this while still mamtammg a good approxlmatlon to L(w) over the mltlal range of o) Providing an art&la1 response function never exceeds the true value, then the only significant dlstortlon mtroduced 1s that mherent m the final hneshape It can be seen from Fig 2 that the simple Lorentzlan-Gausslan transformation falls
217
(a)
(d)
Flgure 3 Resolution enhancement of gold 4flevels as m Fig 1 (a) Data after removal of scattermg, (b) Lorentzlan-Gausslan transformation I?1 = 1 1, re = 0 6, (c) I?1 = 0 41, rg = 0 58, d = -0 33data cut off at w = 1 18, (d) I71 = 0 41, rg = 0 58, d = -0 20, JTc = 0 53
m this respect The drstortlon induced, however, 1s not particularly severe and mamfests Itself as a dip m the tails of single peaks (see Fig 3) To provide some examples of the above methods two cases will be dIscussed The first uses well defined gold 4f levels and 1s mtended to be a standard example, while the second IS derived from our own research Interests and shows the vaIue of the techniques The Fourier transforms were calculated usrng standard algorithms’ and m all cases a correctlon for melastlc scattering has been apphed The only presmoothmg used was a g-point quadratlc polynomla13, Its mam function bemg to facilitate the scattering correction In Fig 3 the results for typical parameter sets on gold 4f levels are shown It can be seen that the lmeand a “band hmltmg” case 1s included for comparison widths have been reduced by - 0 5 eV m all cases but that the present methods produce less distortion m the tails The essentially flat response of the LorentzlanGaussian transformation beyond the tails 1s clearly evident
218
Figure 4 Resolution enhancement of chlorme (2~) levels m a mlvture of chlorate and perchlorate speaes resultmg from Mg Ku lrradlatlon of sodturn perchlorate (a) Data after scattermg removal, (b) Lorentzlan-Gausslan transformation I?1 = 1 1, rg = 0 6, (c) rl = 0 41, rg = 0 5, d = -0 33cutoffatw=10,(d)lT~=041,r,=07,rc=07,d=-033
The second example shows chlorme 2p levels m a sample of sodium perchlorate after prolonged exposure to Mg Ku radiation A chlorate intermediate IS present and In attemptmg to derive the kmetlcs mvolved’ It was necessary to determme peak widths and separations for the two component doublets Some of the parameter values used are shown m Fig 4 and once again the dn-ect apphcatlon of eqn (4) provides results of comparable quahty with the multiple lmeshape transformatron producing the best results A factor not applied m the present study makes the most simple transformation even more useful The lmeshape functions involved are completely even and hence a simple cosme series can be used for the data transform1 This IS not only computatlonally simpler but also removes most requirements for data pre-treatment smce the transform IS less sensltlve to dlscontmultles The slmphclty of the parameters allows apphcatlon to cases where the response fun&Ion IS not accurately known and, as shown here, one can obtam results of quahty comparable to those
219 obtained by more exact methods It therefore provides a very powerful tool for routme appllcatlons particularly when high noise levels cannot be avoided The methods presented here represent an attempt to overcome an obvious hmltatlon in the band-llmltmg approach It 1s not claimed that the degree of resolution enhancement obtainable will be slgmficantly larger since this 1s determined by the noise levels The amount of dlstortlon Introduced should be lower, however It 1s m thrs hght that the value of these methods should be assessed and where they offer the greatest advantages ACKNOWLEDGEMENTS
The spectra used here were kindly supplied by Dr R G Copperthwalte who suggested this study Dr IS G R Pachler provided many useful dlscusslons on the problems mvolved and the Council for Sclentlfic and Industrial Research 1s to be thanked for the award of a post-doctoral fellowshlp
REFERENCES
6 7
8
G K Werthelm, J Electron Spectrosc Reiat Phenom , 6 (1975) 239 N Beatham and A F Orchard, I Electron Spectrosc Relat Phenom , 9 (1976) 129 J -J Pneaux, Appl Spectrosc , 30 (1976) 219 R Bracewell, The Fourrer Transform and rts Applrcatrons, McGraw-HJ11, New York, 1965, ch 3 R R Ernst, m J Waugh (Ed ), Advances m Magnetrc Resonartce, Vol 2, Academic Press, New York, 1966 I? IS used throughout to represent the full width of a function of half-maxlmum he&t SubscrIpts are used to deslgnate the type, e g g for GaussJan and c for cubic The data transforms were calculated using the FFTP Fourier transform subroutme produced by The International Mathematical and Statlstlcal Library Inc Houston, Texas R G Copperthwalte and J Lloyd, J Chem Sot Dalton Trarrs, 11 (1977) 1117