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Dependence of hydrocarbon mass spectra on the structure of fragments
International Journal of Mass Spectrometry and Ion Processes, 54 (1983) 255-261 Elsevier Science Publishers B.V., Amsterdam - Printed in The Netherlands
DEPENDENCE STRUCTURE
OF HYDROCARBON OF FRAGMENTS
MASS
SPECTRA
255
ON THE
GR. ALEXANDRU Institute
of Physics
(Received
and Nuclear
Engineering,
Bucharest,
P. 0. Box M. G. 6 (Romania)
11 May 1983)
ABSTRACT For the mass-spectrometric fragmentation processes of hydrocarbons, a simple characteristic equation depending separately on the structure of the fragments and on the structure of the dissociating molecule has been determined.
INTRODUCTION
The characteristic equation for mass-spectrometric fragmentation processes [l] depends on the structure of the fragments and on the structure of the dissociating molecule in a very sophisticated way on the scattering amplitudes of the fragments or the respective molecule. The computations are considerably simplified if one takes into account the impulse approximation [2], assuming that the energy of excitation of the molecule is greater than the bonding energy of the atoms in the molecule and the principle of similitude [3], that considers that the fragmentation matrix elements of similar molecules are identical. In this paper, in the impulse approximation and considering similar molecules with very high mass, the characteristic equation is determined from the mass spectrometric fragmentation processes of hydrocarbons, appearing as a product of two functions, one of which depends on the fragment structure only and the other is determined from the scattering amplitude of a carbon atom on the dissociating molecule. APPROXIMATION
OF IMPULSES
In the impulse approximation and when considering only elastic processes (M + m + M + m), the scattering amplitude f of hydrocarbon m on hydro0168-l
176/83/$03.00
6 1983 Elsevier Science Publishers B.V.
256
carbon M has the form [2] f=
0)
N&ln(PML
where m and M also denote the respective rest masses, and N,, is the number of carbon atoms from hydrocarbon m (only the C-C bonds will be considered). Thus c,(p)
= /d3xP, (x’) exp( - ip’x’)
where WY is the energy of the state ly), T is the temperature, Fa is the spatial coordinate of the carbon atom (Y, 2 = X:,exp( - WY/T) of hydrocarbon m, p’ is the transmission of the impulse and f, represents the elastic scattering amplitude of a carbohydron mch (from hydrocarbon m) on hydrocarbon M. The denomination of carbohydron, from hydrocarbon M containing Nmc carbon atoms and Nmh hydrogen atoms, is given to a carbon atom surrounded by n,,,,, hydrogen atoms representing the average number of hydrogen atoms on a carbon atom ( nmh = N,JN,,.,=, mch = m/N,,). If we also consider the non-elastic processes, eqn. (I) becomes [2] f-i
Nmc+ Nlnc( &lx - 1) lGnb)12 +
[
+m(P)
the volume of hydrocarbon
]c,(p)l’+
=jd3Xd3X’Pz(X’,
9)
fc
Ii
Ill
where V, represents $-C,(p) In
l/2
(4)
m
exp[ -ip’(x’-Z)]
(5)
and
( )[ fN~,_l)
P,(;.;‘)=~~exp -T Y
N
mc
- 2)S(z’B- x”)lY) atp=
I
I
(6)
wheref, is the (elastic and non-elastic) scattering amplitude of a carbohydron m ch on hydrocarbon M. As is seen from eqns. (3) and (6), the functions P,(T) and P,(Z, x”) may be determined from the computations, if the spatial distribution of the carbon atoms, the temperature T, the energies W, and the states ly } of hydrocarbon m, are known. The amplitudes f, and f, may be also written in the impulse approximation
257
as functions of NM,, V,, cM(p), C,(p) and of the scattering amplitude of two carbohydrons f, from the hydrocarbons m and M. If the hydrocarbons m and A4 are ionized having the charges 4, and qM, and qM/Nhl,, the carbohydrons mch and A&, have the charges q,/N,, respectively . The amplitude f,is determined from the scattering of two carbon atoms or of two methane molecules by performing the respective mass and charge corrections. SIMILAR
MOLECULES
The mass-spectrometric fragmentation represented by the relationship L(m,
process
[M+
+
(M -
m) + m +] is
Pmo, r’> = CIF12
(7)
where m is the fragment rest mass, pm0 is the time component of the fragment four-momentum p,, C is the normalization constant, and F = (m+, M - m; out IM+) is the respective matrix element. For simplification, the probability of the positive charge remaining on the fragment m was introduced in F. The name “similar” is applied to those molecules whose matrix elements F for A4 > M, do not depend on M [F - 0( A4 - m)], where A4, is the mass. of the lightest molecule from the respective family. It is obvious that P depends on M, in a sophisticated way. The normal hydrocarbons are known to be similar [3]. In the main, any molecule may have a similar one with very high mass. Thus, we can write the matrix element I; [l]
XP
Reg(S’)exp[b(-5’)lsin r= (M+m)”
+-g(t)
(t’ - M2)W
cosd-t)
I
dt’)
dtJ
- c>
where, in the coordinate system in which j?i,,, = 0, the transmission square impulse [ has the form t=
-M2+2Mp,,-m*
w(--)
=
RefoW 1 - 4nfott>
(8) of the (9)
(10)
and
f. M2 + 2Mp,,
=f re p resents the elastic scattering
+ m2/A4{~
amplitude of the process Mf + m - + M+ + m- [2]. The function g(t) consists of a real constant g, an integral having the form of that in eqn. (8) and a term that considers the non-elastic processes from the intermediate states [4]. If one considers the similar molecule with mass M + cc the mentioned integrals are annihilated and in the approximation. in which the non-elastic processes from the intermediary states are neglected, eqn. (8) takes the form
Fbh
Pan,,
T) = g 03s cph
where r&m, p,,,
0
P,,~
T) is determined
(12) from eqn. (10). Considering
the equality
PI fohb Pnl,, r) = ev[Wm,p,,,
T)]
sin a(m,
P,,,
T)
(13)
where Wm, P,,, T) is the (real) phase of the scattering process M+ + m- * AI++ m-, resulting in ~(rn, pmo, T) = S(m, pmo, T) and
IfAm, PIno, T)J2 = sin2S(m, Introducing IF(m,
pmo, T)
04)
eqn. ( 14) into eqn. ( 12)
&-,,, T)12=g2[l
-Kilf(m, K,,,T)12]
(15)
where f(m, K , T) is determined in the impulse approximation from eqn. (1) andK,=Jm. If in the intermediate state, one considers the recombination processes (formation of isomers) and multiple dissociations (into three and several fragments), the function S(m, K,, T) becomes complex (non-elastic scattering) and in the impulse approximation, f( m, K,, r) is determined from eqn. (4). Therefore,
in the impulse approximation,
In this approximation, P=
=-
- dll
1 Nmc
+ 2mctl
-2m2
eqn. (7) takes the form
from eqn. (9)
Pkicho - K2h
+ 2rn/m
(17)
259
is obtained, where M,, is the rest mass of the carbohydron from hydroMC,.,= me,,, carbon A4 (Mch is the same for all similar hydrocarbons), =p,,,/N,, because m is a fragment of M. PMcho EXPERIMENTAL
Using the notation
and introducing g2 in the normalization L(m,
JL,
T) = C[l
- &.Jf,(m,
K,,
constant
C eqn. (16) becomes
T)12]
(19)
From eqn. (I 8), IV,, = 1 (fragment composed of a carbohydron), S, = Ki and from eqn. ( 19), since L is known experimentally (only for N,, = l.), f, may be determined. Then, computing S,.,, from eqns. (18), (5) and (6) the mass spectrum L( m, K,, T) may be determined integrally. Taking into account that S, depends only on the fragment m and, does not depend on the molecule M, eqn. (19) may be checked experimentally without performing the computations from eqns. (5) and (6). For this, we denote with a, and u2 the abundance of fragments m, and m 2, respectively, from the mass spectrum of molecule A4 and with b, and b; the same magnitudes from the mass spectrum of molecule M’ (A4 and M’ are not similar). From eqn. (19), we obtain S -=ml S m2
C,-a, s;, _ C,-b, C, - a2 ’ $A, C, - b,
(20)
that can be easily checked experimentally. Table 1 shows the abundances of the fragments from the mass spectra of n-octane and 4-methylheptane obtained at T = 25°C and with the ionization
TABLE
1
The abundances of the fragments and of Cmethylheptane
(only the C-C
bonds) from the mass spectra of n-octane
Mass number of ions
C-C-C-C-C-C-C-C c-c-c-c
7
-c-c-c
15
29
43
57
71
85
99
2.7
58
155
63
39
32
0.3
2.5
53
149
45
91
8
1.5
260 TABLE
2
The ratios of the fragment structure functions (only the C-C bonds) from the mass spectra of n-octane and of 4-methylheptane Hydrocarbon
SdS29
%/S43
s29/s43
G/S43
%,/S43
%,/S43
%/S43
n-octane 4-methylheptane
1.19 1.17
1.78 1.73
1.50 I .48
1.47 1.52
1.59 1.29
1.63 1.70
1.79 1.73
energy Ei = 800 eV. Both mass spectra are normalized at the same value C = C, = C, = 350 (e.g., the sum of the fragment abundances, without the molecular ion, is equal to 350). The experimental data were obtained with a parallel beam mass spectrometer [5]. In Table 2 there are shown the ratios S,,/Smz determined according to eqn. (20) and Table 1. As is seen in Table 2, the experimental data confirms eqn. (19) predicting for identical fragments the same values S,,, despite the nature of the dissociating hydrocarbon. It is obvious that the respective mass spectra should be obtained at the same temperature T and with an ionization energy high enough to ensure the validity of the impulse approximation. The functions Si5, S,, and S,, (their ratios for excluding the unknown function f,) are, within an accuracy less than 2%, equal for n-octane and 4-methylheptane, respectively. The ratios S,,/S,, (linear fragments with four carbon atoms) for n-octane and 4-methylheptane are also equal, though in the latter case the respective fragment results from a second dissociation of an excited fragment with five carbon atoms. The relatively large difference of 3.5% is probably due to these multiple dissociations. The ratios S,,/S,, are different for n-octane (1.59) and 4-methylheptane (1.29) because the respective fragments are different at the moment of breaking, i.e., the atoms are linear in the first case and have a perpendicular bond to the other bonds in the second case. Therefore, in the case of 4-methylheptane, the excited fragment with five atoms of carbon, may be dissociated later or may pass in the respective normal hydrocarbon. it results that, when the functions From the ratios S,,/S,, and S&&, 1~1 are relatively low, the functions S,,, depend a little on the nature of the fragment. In the general case, one should take into account that the mass spectrometer simultaneously records several types of fragments with the same mass.
261 CONCLUSIONS
The fact that the mass-spectrometric fragmentation process of hydrocarbons depends separately on the fragment structure, in the impulse approximation, allows one to determine the mass spectrum of a hydrocarbon from the mass spectra of other hydrocarbons, containing the respective fragments. The simple relation (151, which is obtained without the impulse approximation is also very interesting for an understanding of the mass-spectrometric fragmentation process. REFERENCES 1 Gr. Alexandru, Int. J. Mass Spectrom. Ion Phys., 25 (1977) 1. 2 M.L. Goldberger and K.M. Watson, Collision Theory, Chap. 11, Wiley-Interscience, York, 1964. 3 Gr. Alexandru, Int. J. Mass Spectrom. Ion Phys., 32 (1980) 369. 4 Gr. Alexandru, Int. J. Mass Spectrom. Ion Phys., 6 (1971) 125. 5 Gr. Alexandru, Rev. Sci. Instrum., 39 (1968) 10, 1571.
New
DEPENDENCE STRUCTURE
OF HYDROCARBON OF FRAGMENTS
MASS
SPECTRA
255
ON THE
GR. ALEXANDRU Institute
of Physics
(Received
and Nuclear
Engineering,
Bucharest,
P. 0. Box M. G. 6 (Romania)
11 May 1983)
ABSTRACT For the mass-spectrometric fragmentation processes of hydrocarbons, a simple characteristic equation depending separately on the structure of the fragments and on the structure of the dissociating molecule has been determined.
INTRODUCTION
The characteristic equation for mass-spectrometric fragmentation processes [l] depends on the structure of the fragments and on the structure of the dissociating molecule in a very sophisticated way on the scattering amplitudes of the fragments or the respective molecule. The computations are considerably simplified if one takes into account the impulse approximation [2], assuming that the energy of excitation of the molecule is greater than the bonding energy of the atoms in the molecule and the principle of similitude [3], that considers that the fragmentation matrix elements of similar molecules are identical. In this paper, in the impulse approximation and considering similar molecules with very high mass, the characteristic equation is determined from the mass spectrometric fragmentation processes of hydrocarbons, appearing as a product of two functions, one of which depends on the fragment structure only and the other is determined from the scattering amplitude of a carbon atom on the dissociating molecule. APPROXIMATION
OF IMPULSES
In the impulse approximation and when considering only elastic processes (M + m + M + m), the scattering amplitude f of hydrocarbon m on hydro0168-l
176/83/$03.00
6 1983 Elsevier Science Publishers B.V.
256
carbon M has the form [2] f=
0)
N&ln(PML
where m and M also denote the respective rest masses, and N,, is the number of carbon atoms from hydrocarbon m (only the C-C bonds will be considered). Thus c,(p)
= /d3xP, (x’) exp( - ip’x’)
where WY is the energy of the state ly), T is the temperature, Fa is the spatial coordinate of the carbon atom (Y, 2 = X:,exp( - WY/T) of hydrocarbon m, p’ is the transmission of the impulse and f, represents the elastic scattering amplitude of a carbohydron mch (from hydrocarbon m) on hydrocarbon M. The denomination of carbohydron, from hydrocarbon M containing Nmc carbon atoms and Nmh hydrogen atoms, is given to a carbon atom surrounded by n,,,,, hydrogen atoms representing the average number of hydrogen atoms on a carbon atom ( nmh = N,JN,,.,=, mch = m/N,,). If we also consider the non-elastic processes, eqn. (I) becomes [2] f-i
Nmc+ Nlnc( &lx - 1) lGnb)12 +
[
+m(P)
the volume of hydrocarbon
]c,(p)l’+
=jd3Xd3X’Pz(X’,
9)
fc
Ii
Ill
where V, represents $-C,(p) In
l/2
(4)
m
exp[ -ip’(x’-Z)]
(5)
and
( )[ fN~,_l)
P,(;.;‘)=~~exp -T Y
N
mc
- 2)S(z’B- x”)lY) atp=
I
I
(6)
wheref, is the (elastic and non-elastic) scattering amplitude of a carbohydron m ch on hydrocarbon M. As is seen from eqns. (3) and (6), the functions P,(T) and P,(Z, x”) may be determined from the computations, if the spatial distribution of the carbon atoms, the temperature T, the energies W, and the states ly } of hydrocarbon m, are known. The amplitudes f, and f, may be also written in the impulse approximation
257
as functions of NM,, V,, cM(p), C,(p) and of the scattering amplitude of two carbohydrons f, from the hydrocarbons m and M. If the hydrocarbons m and A4 are ionized having the charges 4, and qM, and qM/Nhl,, the carbohydrons mch and A&, have the charges q,/N,, respectively . The amplitude f,is determined from the scattering of two carbon atoms or of two methane molecules by performing the respective mass and charge corrections. SIMILAR
MOLECULES
The mass-spectrometric fragmentation represented by the relationship L(m,
process
[M+
+
(M -
m) + m +] is
Pmo, r’> = CIF12
(7)
where m is the fragment rest mass, pm0 is the time component of the fragment four-momentum p,, C is the normalization constant, and F = (m+, M - m; out IM+) is the respective matrix element. For simplification, the probability of the positive charge remaining on the fragment m was introduced in F. The name “similar” is applied to those molecules whose matrix elements F for A4 > M, do not depend on M [F - 0( A4 - m)], where A4, is the mass. of the lightest molecule from the respective family. It is obvious that P depends on M, in a sophisticated way. The normal hydrocarbons are known to be similar [3]. In the main, any molecule may have a similar one with very high mass. Thus, we can write the matrix element I; [l]
XP
Reg(S’)exp[b(-5’)lsin r= (M+m)”
+-g(t)
(t’ - M2)W
cosd-t)
I
dt’)
dtJ
- c>
where, in the coordinate system in which j?i,,, = 0, the transmission square impulse [ has the form t=
-M2+2Mp,,-m*
w(--)
=
RefoW 1 - 4nfott>
(8) of the (9)
(10)
and
f. M2 + 2Mp,,
=f re p resents the elastic scattering
+ m2/A4{~
amplitude of the process Mf + m - + M+ + m- [2]. The function g(t) consists of a real constant g, an integral having the form of that in eqn. (8) and a term that considers the non-elastic processes from the intermediate states [4]. If one considers the similar molecule with mass M + cc the mentioned integrals are annihilated and in the approximation. in which the non-elastic processes from the intermediary states are neglected, eqn. (8) takes the form
Fbh
Pan,,
T) = g 03s cph
where r&m, p,,,
0
P,,~
T) is determined
(12) from eqn. (10). Considering
the equality
PI fohb Pnl,, r) = ev[Wm,p,,,
T)]
sin a(m,
P,,,
T)
(13)
where Wm, P,,, T) is the (real) phase of the scattering process M+ + m- * AI++ m-, resulting in ~(rn, pmo, T) = S(m, pmo, T) and
IfAm, PIno, T)J2 = sin2S(m, Introducing IF(m,
pmo, T)
04)
eqn. ( 14) into eqn. ( 12)
&-,,, T)12=g2[l
-Kilf(m, K,,,T)12]
(15)
where f(m, K , T) is determined in the impulse approximation from eqn. (1) andK,=Jm. If in the intermediate state, one considers the recombination processes (formation of isomers) and multiple dissociations (into three and several fragments), the function S(m, K,, T) becomes complex (non-elastic scattering) and in the impulse approximation, f( m, K,, r) is determined from eqn. (4). Therefore,
in the impulse approximation,
In this approximation, P=
=-
- dll
1 Nmc
+ 2mctl
-2m2
eqn. (7) takes the form
from eqn. (9)
Pkicho - K2h
+ 2rn/m
(17)
259
is obtained, where M,, is the rest mass of the carbohydron from hydroMC,.,= me,,, carbon A4 (Mch is the same for all similar hydrocarbons), =p,,,/N,, because m is a fragment of M. PMcho EXPERIMENTAL
Using the notation
and introducing g2 in the normalization L(m,
JL,
T) = C[l
- &.Jf,(m,
K,,
constant
C eqn. (16) becomes
T)12]
(19)
From eqn. (I 8), IV,, = 1 (fragment composed of a carbohydron), S, = Ki and from eqn. ( 19), since L is known experimentally (only for N,, = l.), f, may be determined. Then, computing S,.,, from eqns. (18), (5) and (6) the mass spectrum L( m, K,, T) may be determined integrally. Taking into account that S, depends only on the fragment m and, does not depend on the molecule M, eqn. (19) may be checked experimentally without performing the computations from eqns. (5) and (6). For this, we denote with a, and u2 the abundance of fragments m, and m 2, respectively, from the mass spectrum of molecule A4 and with b, and b; the same magnitudes from the mass spectrum of molecule M’ (A4 and M’ are not similar). From eqn. (19), we obtain S -=ml S m2
C,-a, s;, _ C,-b, C, - a2 ’ $A, C, - b,
(20)
that can be easily checked experimentally. Table 1 shows the abundances of the fragments from the mass spectra of n-octane and 4-methylheptane obtained at T = 25°C and with the ionization
TABLE
1
The abundances of the fragments and of Cmethylheptane
(only the C-C
bonds) from the mass spectra of n-octane
Mass number of ions
C-C-C-C-C-C-C-C c-c-c-c
7
-c-c-c
15
29
43
57
71
85
99
2.7
58
155
63
39
32
0.3
2.5
53
149
45
91
8
1.5
260 TABLE
2
The ratios of the fragment structure functions (only the C-C bonds) from the mass spectra of n-octane and of 4-methylheptane Hydrocarbon
SdS29
%/S43
s29/s43
G/S43
%,/S43
%,/S43
%/S43
n-octane 4-methylheptane
1.19 1.17
1.78 1.73
1.50 I .48
1.47 1.52
1.59 1.29
1.63 1.70
1.79 1.73
energy Ei = 800 eV. Both mass spectra are normalized at the same value C = C, = C, = 350 (e.g., the sum of the fragment abundances, without the molecular ion, is equal to 350). The experimental data were obtained with a parallel beam mass spectrometer [5]. In Table 2 there are shown the ratios S,,/Smz determined according to eqn. (20) and Table 1. As is seen in Table 2, the experimental data confirms eqn. (19) predicting for identical fragments the same values S,,, despite the nature of the dissociating hydrocarbon. It is obvious that the respective mass spectra should be obtained at the same temperature T and with an ionization energy high enough to ensure the validity of the impulse approximation. The functions Si5, S,, and S,, (their ratios for excluding the unknown function f,) are, within an accuracy less than 2%, equal for n-octane and 4-methylheptane, respectively. The ratios S,,/S,, (linear fragments with four carbon atoms) for n-octane and 4-methylheptane are also equal, though in the latter case the respective fragment results from a second dissociation of an excited fragment with five carbon atoms. The relatively large difference of 3.5% is probably due to these multiple dissociations. The ratios S,,/S,, are different for n-octane (1.59) and 4-methylheptane (1.29) because the respective fragments are different at the moment of breaking, i.e., the atoms are linear in the first case and have a perpendicular bond to the other bonds in the second case. Therefore, in the case of 4-methylheptane, the excited fragment with five atoms of carbon, may be dissociated later or may pass in the respective normal hydrocarbon. it results that, when the functions From the ratios S,,/S,, and S&&, 1~1 are relatively low, the functions S,,, depend a little on the nature of the fragment. In the general case, one should take into account that the mass spectrometer simultaneously records several types of fragments with the same mass.
261 CONCLUSIONS
The fact that the mass-spectrometric fragmentation process of hydrocarbons depends separately on the fragment structure, in the impulse approximation, allows one to determine the mass spectrum of a hydrocarbon from the mass spectra of other hydrocarbons, containing the respective fragments. The simple relation (151, which is obtained without the impulse approximation is also very interesting for an understanding of the mass-spectrometric fragmentation process. REFERENCES 1 Gr. Alexandru, Int. J. Mass Spectrom. Ion Phys., 25 (1977) 1. 2 M.L. Goldberger and K.M. Watson, Collision Theory, Chap. 11, Wiley-Interscience, York, 1964. 3 Gr. Alexandru, Int. J. Mass Spectrom. Ion Phys., 32 (1980) 369. 4 Gr. Alexandru, Int. J. Mass Spectrom. Ion Phys., 6 (1971) 125. 5 Gr. Alexandru, Rev. Sci. Instrum., 39 (1968) 10, 1571.
New