Space Charge Induced Nonlinear Effects in QuadrupoleIon Traps
Supplementary Information
Dan Guo,1§Yuzhuo Wang,1§Xingchuang Xiong,2Hua Zhang,3Xiaohua Zhang,3 Tao Yuan,4Xiang Fang2 and Wei Xu1*
1Department of Biomedical Engineering, Beijing Institute of Technology, Beijing 100081, China
2National Institute of Metrology, Beijing 100013, China
3Beijing Purkinje General Instrument Co.,Ltd, Beijing, China.
4KunShanInnowave Communication Technology Co.,Ltd., Jiangshu, China
*Corresponding Author:
Wei Xu
School of Life Science
Beijing Institute of Technology
Haidian, Beijing, 100081, China
Email:
Phone: +86-010-68918123
§These authors have equal contribution to this work.
- The calculation of electric field and electric potential
According to the Gaussian theorem, the electric field (E) generated by a number of point charges (q) have the following relationship,
(S1)
whereS is an integral surface, is the permittivityofvacuum, is the sum of charge inside the integral surface, S.
Whenthe shape of an ion cloud in a 3D ion trap is sphere, the direction of the electric field is along the direction of r in the spherical coordinate system. Therefore,a spherical surface was taken as the integral surface and equation S1 can be written as,
(S2)
where u is the distance to the center of the ion trap, q is the charge of anion, N is the total number of ions, a is the standard deviation.
After calculating the integration, the expression of the electric field, Ewithin the3D ion trap can be derived as,
(S3)
whereErf is the error function.
In a linear ion trap, the length of the ion cloud along z-axis is much biggerthan the radius of the ion cloud, so the direction of the electric field is approximately along the r-direction in the cylindrical coordinate system near the center of the ion trap (z = 0). Takinga cylindrical surface as the integral surface, equation S1 can be written as,
(S4)
in which is the ion line density.
After calculating the integration in Equation S4, the expression of the electric field, Ein a linear ion trap can be derived as,
(S5)
The electric potential is the integration of the electric field. We can assume the electric potential at infinity is zero.
In a 3D ion trap:
(S6)
(S7)
In a linear ion trap:
(S8)
(S9)
2.Polynomial fitting:
Since the function of the electric field is complex, polynomial fitting was used to modelthe electric field function in this study. Under the condition of qz=0.46,N=106,m=195 Da, T=300K, z0=3.535 mm andq=1× e, the polynomial function in the 3D ion trap is shown in Equation S10 and Equation S11 shows the polynomial function in the linear ion trap under the condition of qx=0.36,N=3×106,m=195 Da, T=300K, z0=4 mm andq=1× e. Fig. S1 (a) and (b) show image of the polynomial fitting within the 3D ion trap and linear ion trap, respectively.
In the 3D ion trap:
(S10)
In the linear ion trap:
(S11)
(a)
(b)
Figure S1.Polynomial fitting of the electric field strength.(a)the polynomial fitting of the electric field strength within a 3D ion trap, qz=0.46,N=106,m=195 Da, T=300K, z0=3.54 mm,q=1× e; (b) the polynomial fitting of the electric field strength within a linear ion trap, qx=0.26,N=3×106,m=195 Da, T=300K, x0=4 mm,q=1× e.
3. The correspondence between space charge induced electric field and high-order field components
Theoretically, the electrical filed induced by space charge can be fitted by an infinite order polynomial, and the ion motion equation can be written as,
(S12)
where=z in a 3D ion trap and =x in a linear ion trap, is the ion secular frequency. Since the function of electric field induced by space charge is odd in both 3D and linear ion traps, the space charge effect can be equivalent to a summation of even-order fields.
Considering the effect of even-order fields, the ion motion equation can be written as,
(S13)
where, , , … , are the amplitudes of high-order fields (for example, represents quadrupole, represents octopole).
Comparing the coefficients of equation S12 and equation S13, the coefficients of the equivalent high-order field component induced by space charge effect can be derived as,
(S14)
Asan estimation, the electric field induced by space charge was expanded using an 11th order polynomial fitting method. When trapped 1×106ions in the 3D ion trap, T=300K, m/z = 195 Da,q=1×eandqz= 0.36, the results are,, , , , .
When trapped 2×106ions in thelinear ion trap, T=300K, m/z = 195 Da and q=1× e, qx= 0.36, the results are, , , , , .
The relative magnitudes of these even-order fields induced by space charge are relatively large compare with those created by physical geometry distortions of ion trap electrodes. However, the effects induced by positive and negative even-order fields would cancel each other.1 Therefore, the relative frequency shifts originated from space charge are relatively comparable with those introduced by physical geometry distortions, especially at large ion motion amplitudes.
4.The harmonic balance method:
The ion motion equation without damping and excitation is:
(S15)
The solution of equation S15 can be written as the summation of a series of harmonic functions with base frequency ,
With the first-order approximation, the solution of Equation S15 can be written as,1,2
(S16)
Substituting equation S16 back into equation S15, the coefficient of term and the constant term are zero, so,
(S17)
(S18)
After solving this equation set Equation (S17) and (S18), ion motion angular frequency can bederived:
(S19)
When considering buffer gas damping and AC excitation, the ion motion equation becomes:
(S20)
With the first-order approximation, the solution of Equation S18 can be written as,
(S21)
Substituting equation S21 back into equation S20, the coefficient of term and the constant term are zero, the coefficient of term equals. As is always zero in this case, so the constant term was left out.
(S22)
(S23)
Ion motion amplitudes (a0, a1, a2) can be calculated by solving this equation set (S22, S23).
5. The error estimation of polynomialfitting and harmonic balance method
The ion motion equation (Equation15 in the main text) can be solved directly by numerical methods usingthe “Mathematica” program, where the electric field induced by space charge is expressed by Equation 10. After solving the equation and applying the Fourier Transform to the ion trajectory, the ion motion frequency will be obtained. The differences between the results obtained by two methods are shown in table S1 and S2.
Table S1.The calculation accuracy of ion frequency shift (KHz) under different ion abundance in 3D ion traps.qz =0.46, T=300K, m/z=195 Da, q=1× e, r0=5 mm, z0= 3.536 mm, z/ z0=1.
Numerical Value / theoretical Value / Relative ErrorN=500k / -1.5 / -1.47 / 2%
N=1000k / -2.62 / -2.59 / 1.15%
N=1500k / -3.58 / -3.54 / 1.12%
Table S2.The calculation accuracy of ion frequency shift (KHz) under different ion abundance in 2D ion traps.qx =0.36,T=300K, m/z=195 Da, q=1× e, x0=4 mm, x/x0=1.
Numerical Value / theoretical Value / Relative ErrorN=1000k / -0.57 / -0.54 / 5.26%
N=2000k / -1.12 / -1.08 / 3.57%
N=3000k / -1.67 / -1.62 / 2.99%
6. The calculation of
(S24)
where m is the mass of anion, is the AC voltage, q is the charge of anion, is the radius of the ion trap, in a 3D ion trap, in a linear ion trap.3, 4
7.Detailed explanation of Figure 2 in the main text
Aone-dimensional harmonic oscillator motion equation can be written as,
(S25)
whereis the spring constant, . The solution is:
(S26)
where . The frequency of the harmonic oscillator is proportional to the square root of the spring constant.
In our case, the ion motion equation can be written as,
(S27)
where the parameters have the same definitionsas those in the main text. The frequency of an ion is proportional to the term , and the frequency shift would directly relate to the electric field induced by space charge (E(u) in Equation 10) divided by distance (u). The absolution value of E(u)/u has a maximum at zero and decreases as ion motion amplitude (a1) increases, so the frequency shift decreases as ion motion amplitude increases (as shown in Figure 2).
8.Estimation of mass resolution
At a particular excitation frequency, ions whose frequenciesare close to the excitation frequency can get higher amplitudes and be ejected. We can define mass resolution as follows,
(S28)
where , and have the same definitionsas those in the main text.For simplicity, the first three terms of (Equation 19) were taken into account, which can be written as,
. (S29)
where and are the coefficients of Taylor expansion of .
In a 3D ion trap, , . So Equation S29 can be written as,
(S30)
In a linear ion trap, , . So Equation S29 can be written as,
(S31)
Under typical ion trap conditions,and in a 3D ion trap and in a linear ion trap, mass resolution increases with increasing aand . When temperature increases, a gets bigger and the resolution increases. When q increases, the change of a is small comparing to the change of , so the resolution increases too.
9. Ion secular frequency shifts in the linear ion trap
Figure S2. Ion secular frequency shifts with different ion abundance and qx=0.36 (a); at different qz values and N=2×106 (b). T=300K, r0= 4 mm, m/z= 195 Da, q=1× e.
Reference:
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