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9. Trabajo preventivo de riesgo y delito
7.2.1 Variation of the Phase Function
Development of spectrometry always aims to pursue higher resolution.88 Dating back to the very first FT-ICR mass spectra in 1974,42-43 it was readily recognized that an absorption-mode spectral peak is inherently narrower than its corresponding magnitude-mode (which is phase independent),87 and in FT-NMR spectroscopy, the phasing problem was solved in the 1970’s.91-92 Although the conceptual understanding is the same for both, the complexity is very different. In NMR, the phase varies over the spectrum by typically 0.1–0.3 π, far less than one cycle (2 π). While in FT-ICR, because the excitation bandwidth is three orders of magnitude broader (from kHz to MHz), the variance of phase value is much larger (e.g., ~10,000 π for the m/z range from 200 to 2000 in a 12 T instrument). It is important to note that any peak can be perfectly phased at some Φ(ωi) between 0 and 2 π, and this means that a
small region around this peak, where phase variance is less than 2π (perhaps several Da), is also easily phased. Phase wrapping quickly complicates this (Figure 3.2), because Φ(ω i)= Φ(ω i)+2niπ for any integer ni at any ωi, and the
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phase variation for an entire spectrum is perhaps ~10,000 π. In order to solve the Φ(ω) for the entire m/z range, it is necessary to calculate all the individual values of ni before fitting the phases into a quadratic function Φ(ω). Note:
because every addition of a frequency/phase pair to the quadratic least squares fit also adds one value of ni, the solution to the phase function equation is not
unique being under-determined by exactly one degree of freedom. However, this lack of uniqueness is clearly characterized by the same phase wrapping problem in that the set of ni integer values can always vary by +/- an integer, but
their relative values remain constant.
7.2.2 Parameterization of the Phase Function
Because of linear frequency sweep excitation, the ions’ corresponding phase angle, Φ(ω), is a quadratic function of the excitation frequency (Eq.7.1) and that such an equation could be used for phase correction,95, 165 where A, B and C are the fitted constants.
2
( )
A
BC
(7.1)However, the previous published method is user-interactive;68 requiring manual peak selection and phase assignment, and it was assumed that the phase function varies from experiment to experiment if the instrument parameters are changed which means the function needs to be recalculated every time. These drawbacks could potentially hinder the application of phase correction in routine work. However, because the bulk of the manual algorithm could be easily automated, and computation for the phase function only has to be done when the pulse sequence parameters are changed, it was likely that the function could be parameterized by adding a few correction factors. If so, then the phase
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function could be applied directly to further scans without needing to be recalculated for each spectrum. Thus, a series of experiments were performed to test the variation of the phase function with particular experimental parameters such as trapping voltage, excitation amplitude, and space-charge. In the ICR trap, the reduced ion cyclotron frequency (including the electric field and the effect of space-charge) can be written in Eq.7.2a (with ωc and k
defined in Eq.7.2b and c):45, 60 1/ 2 [1 (1 4 / ) ]
2
c c k (7.2a) 0
c
qB
m
(7.2b) 2 2 0 iqV
q
G
k
ma
m
(7.2c) 2 2 0 2 4 3 0 0qB
V
V
m
m
a B
a B q
… (7.2d)in which B0 is magnetic field; α is a scaling factor which depends on the trap
geometry and ranges typically from 2 to 4; a is the characteristic dimension of the ICR trap usually defined as the distance between trapping plates; and V is the trapping potential applied to the ICR trapping plates. The last term in Eq.7.2c expresses the local space-charge component of the frequency shifts, where ρ represents the ion density, Gi the generalized ion cloud geometry
correction factor, and ε0 the permittivity of free space. As space-charge shifts
are largely independent of m/z,124, 133 the effect of electric field and space-charge are explored separately in the following section.
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7.2.3 Variation of the Phase Function with Trapping Voltage
Variation of the trapping voltage changes the electric field contribution to the ions’ frequencies (Eq.7.2a, and Eq.7.3c) and hence also alters the applied phase function. Thus, k in Eq.7.3a is substituted by the first term in Eq.7.2c, and Eq.7.2a can be expanded in a Taylor series to yield Eq.7.2d. In Eq.7.2d the third term can be ignored in routine calibration,118-119 because it is less than 1/106 of the second term in magnitude (the error caused by ignoring the 3rd term is sub-ppb). Additionally, because the phase variance over an entire spectrum is ~104 π, and because the phase function is quadratic, the order of magnitude error caused by neglecting the 3rd term of Eq.7.2d is squared (namely 10-12), so that neglecting of this term in calculation of the phase function is perfectly valid. Therefore, the “ω” in the initial phase function (Eq.7.1) is substituted by the first two items of Eq.7.2d, which gives Eq.7.3.
2 2 2 2 4 2 2 0 0 0 2 ( )
A
B
C A
V A V B V a B a B a B (7.3)The resulting equation is an optimized phase function with the correction factor for trapping potential, V (average of front and back plate, usually the same). The other parameters α, a, and B0 are the constants of the instrument which
can be easily acquired (see below).
7.2.4 Variation of the Phase Function with Total Ion Number
Frequency (and phase) of ions also vary with total ion number. Easterling and co-workers used the theoretical framework of the Eq.7.2 to demonstrate that the ion’s frequency shift is linear proportional to the number of ions in the ICR cell, which is mostly independent of m/z.124 In the hexapole (of FT-ICR),
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the number of ions increases linearly with the accumulation time. As the trapped ions are analyzed, the space-charge is regarded as a constant shift from the ideal cyclotron frequency for ions of all masses in the cell on average, a “mean-field” approximation. Thus, we can rewrite the phase function as follow: 2 2 0 2 0 0 0 0 0
,
i i newqB
V
q
G
st s
q
G
m
a B
B
B t
(7.4a) So, substituting Eq.7.4a into Eq.7.12 2 2
(
2
)
A
B
Ast
C
As t
Bst
( ) =
(7.4b)Here, s is defined as a space-charge factor with the unit of ion density per time;
t is the accumulation time in the hexapole; Eq.7.4b is the estimation of phase value with the “global” space-charge effect built in. Note: Eq.7.4 assumes that the beam current is constant, but Easterling et al. showed that space-charge frequency shifts are linear with total ion number,124 so the same excitation can be used for varying beam current based on total ion number by redefining ‘t’ as the total ion number and ‘s’ as a parameterization constant. Both methods introduce some error into the phase function because the total space-charge effect is usually varies which will be discussed below.
In the following experiment, the phase function for the mass range of 200-2000 was calculated first from a crude oil sample and applied to small molecules and large proteins separately. The instrument parameters were changed from normal to extreme conditions (Figure 7.1) in order to detect the variation of phase corrected absorption-mode spectra.
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Figure 7.1 Left: phase corrected absorption-mode spectra with the trapping potential from 0.1-10 volt; Right: ion accumulation time form 0.01-5 s (with the transient in Fig. J).