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Image charge-induced ion cyclotron orbital frequency shift for orthorhombic and cylindrical FT-ICR ion traps
International Journal of Mass Spectrometry and Ion Processes, 125 (1993) 33--43 0168-1176/93/$06.00 © 1993 - Elsevier Science Publishers B.V., Amsterdam
33
Image charge-induced ion cyclotron orbital frequency shift for orthorhombic and cylindrical FT-ICR ion traps Xinzhen Xiang a, Peter B. Grosshans b, Alan G. Marshall a.c., aDepartment of Chemistry, The Ohio State University, 120 West 18th Avenue, Columbus, OH 43210, USA bExxon Research and Engineering Company, Room LF148, Route 22 East, Annandale, NJ 08801, USA CDepartment of Biochemistry, The Ohio State University, 484 West 12th Avenue, Columbus, OH 43210, USA (Received September 16 1992; accepted January 13 1993) Abstract A coherently orbiting ion packet induces a distribution of image charge on the ion trap electrodes; the resultant electric field from the image charge shifts the ICR orbital frequency of the ions in the ion packet. The effect of image charges upon the cyclotron motion of a trapped ion has previously been modeled for infinite parallel plate and spherical cavity geometries. In other prior work, an analytical method to solve for the charge induced on any of the electrodes of a Fourier transform ion cyclotron resonance (FT-ICR) ion trap has been presented. In this work, we report approximate analytical solutions for the frequency shift due to the induced image charges for both finite-size cylindrical and tetragonal traps as a function of ion axial and radial position. For a cubic trap, the mass-independent frequency shift at an ICR orbital radius equal to half the trap radius is ~ 10 -5 Hz for one ion. Thus, image charge-induced frequency shifts in the ppm range can be expected for typical ICR experiments involving 10~-I05 ions at low mass-to-charge ratio (60 ~< m/z <~ 600), whereas the shift can increase to ,~, ! u for 100 000 singly-charged 3000 u ions at 3.0 T. The image charge-induced shift for a cylindrical trap is higher than for a cubic trap of comparable size. Implications for mass calibration are discussed.
Key words: FT-ICR; Image charge-induced shift; Orthorhombic ion trap; Cylindrical ion trap.
Introduction One o f the most unique features of Fourier transform ion cyclotron resonance mass spectrometry ( F T - I C R M S ) [1,2] is its capability o f highly precise and accurate measurement of ionic mass. FT-ICR mass measurement errors at the sub-ppm level over a wide mass range have been reported by several research groups [3-6]. In the absence o f an electric field, the ion cyclotron orbital frequency, eoc, for an ion o f mass-to-charge ratio, m/q, moving in a magnetic field, B, is simply
qB ~o =
-
-
m
* Corresponding author.
(1)
However, in the presence of the electrostatic potential needed to trap the ions, the ion cyclotron orbital frequency shifts to a new value, a~+, and two additional natural motions arise: magnetron motion at frequency, a ~ ; and trapping oscillation at frequency, ogz. For a quadrupolar electrostatic potential [7-10] ~o~
~o~
m-+ = 2 - +
(2)
2
Equation (2) leads to the particularly simple mass calibration law [3];
m / z = cl -~ c2 V+
2
V+
(3)
in which Cl and c2 are parameters which are known
34
X. Xiang et al./Int. J. Mass Spectrom. Ion Processes 125 (1993) 33-43
analytically and which may be determined by numerically fitting Eq. (3) to experimental data for ions of known m/q, and v+ is the ion cyclotron frequency in Hertz. Equation (3) may be modified to account for Coulomb repulsions between ions of like charge and like or unlike mass [5,11,12]. Moreover, relativistic frequency shifts for ions of m/z <~ 20 have also been considered for ultrahighprecision experiments [13-16]. The additional shift in ion cyclotron orbital frequency due to the image charge induced on nearby electrodes was first considered by Wineland and Dehmelt [17], who calculated the image charge force by modeling the actual trap by a pair of grounded parallel plates. In this geometry, the instantaneous force on a lone ion is perpendicular to and directed toward the plate to which it is nearest. When this instantaneous force is averaged over one cyclotron orbit, there is a net radially outward-directed force which opposes the radially inward-directed magnetic force, qv x B, in which v is ion velocity. Some years later, a conductive sphere model was used to estimate the force on an ion due to its image charge induced on the electrodes of a Penning trap [18-20]. The radial electric force on an ion was incorporated into the Lorentz force equation to determine the magnitude of the image charged-induced ion cyclotron orbital frequency shift as a function of the ion cyclotron orbital radius and the number of other like m/q ions. In other prior work, Nikolaev and Gorshkov [21] calculated the induced image charge resulting from charge distributed uniformly along the central axis of a conductive cylinder extended infinitely along the direction of the applied magnetic field. In this paper, we calculate the image charge distribution induced by an ion on the electrodes of a finite-dimension orthorhombic or cylindrical trap. The analytical solutions for the image charge induced by a point charge on orthorhombic and cylindrical traps of finite dimensions are presented, and the magnitude of the shift in ion cyclotron orbital frequency for cubic and cylindrical traps of comparable size are numerically evaluated as a function of ion radial and axial position. The implications of the results for mass measurement
accuracy for singly- and multiply-charged ions are discussed.
Theory The well known Lorentz force equation F = q(E + v x B)
(4)
governs the motion of a charged particle in an electromagnetic field. In this equation, F is the instantaneous vector force on a particle of charge, q, moving with vector velocity, v, under the combined influence of an electric field, E, and a magnetic field B. In most calculations of ion trajectories in FT-ICR MS, B is assumed constant in magnitude and direction, and E is determined only by the potentials applied to the electrodes which define the closed inner surface of the ion trap. In principle, one could compute ion trajectories without the often invoked simplifying assumptions that the magnetic field is spatially homogeneous and temporally constant and that ion/ion interactions are negligible. However, even a treatment at this level of approximation is not complete: there exists another force on trapped ions, namely that due to the induced charge on the inner surfaces of the trap electrodes. This oversight results from the definition of the electric field: E = lim F~ q~0 q
(5)
in which Fe is the electric force felt by the point charge, q, and the limit is taken as the magnitude of the charge goes to zero. The surface charge induced by a point charge on a conductor is proportional to the magnitude of the point charge, and the force exerted by the induced surface charge density on the point charge is proportional to the product of the point charge and the induced surface charge density. Thus, the electric force on the point charge due to its image charge is proportional to q2, and this force is not incorporated in the electric field found by ordinary methods. A convenient way of calculating this force, for simple trap geometries, is by the image charge method [22]. In this method, the surface of the
X. Xiang et al.llnt. J. Mass Spectrom. Ion Processes 125 (1993) 33--43
x
Fig. 1. Rectilinear coordinate frame for an ion bounded by conductive orthorhombic trap plates. Top: the orthorhombic trap is defined by surfaces: x = 0, x = a; y = 0, y = b; and z = 0, z = c. F~ and Fe (see text) are parallel to the x and y axes, respectively. The position vector (x', y', z') of the point charge lies inside the trap. Bottom: the radial force, F, = FxCOS(~b) + Fy sin(~b), resulting from components, Fx and F~.
conductor divides space into two regions, the interior and the exterior. For the interior problem, the conductor is hollow and the charge is located inside the hollowed-out region. In the exterior problem, the charge is located outside of the conductor. In both cases, one is interested in solving Poisson's equation in the region of space in which the charge is located subject to the boundary conditions imposed by the conductor. The point is that a solution which satisfies Poisson's equation and the boundary conditions is the unique solution. Thus, one attempts to place a distribution of
8
Coulomb potential due to the point charge satisfies Poisson's equation, and that due to the imaginary charge distribution satisfies Laplace's equation. Thus, the sum of all the contributions to the electrostatic potential field satisfies Poisson's equation in the region of space that is of interest. As stated earlier, the method of images is useful for boundaries of simple geometry (e.g. a point charge inside a spherical conductive surface [22]). However, for tetragonal and cylindrical ion traps, no such simple solution is available, and an expression for the force on a point charge due to the induced surface charge density on the conductor must be obtained by other means. Here, we solve Poisson's equation for the geometry under consideration, and compute the induced surface charge density on the conductor from the derived potential inside the trap, which is in turn obtained by Green function methods [10]. Next, we employ a formula for the inward pressure exerted on a differential element of induced surface charge by the point charge; integration of this pressure over the entire interior surface of the conductor yields the net force on the conductor due to the point charge. By Newton's third law, that force must be equal and opposite to the net force exerted on the point charge by the conductor.
Induced image force in an orthorhombic trap Consider a point charge, q, located at position, r'(x',y',z'), inside an orthorhombic trap defined by surfaces x = 0 , x = a , y = 0 , y = b , z = 0 and z = c, as in Fig. l(top), the Green function is [10,22]:
sin(lnX~sin(lnX'~sin(l-~-) sin[lny~'sinClnz~b\ J G(r,r')= ~ ~. ~ \ a / \ a / n2e°abct=l'=l"=| ( ! ) 2 + ( b ) 2 + (n)2 "imaginary" charge in the region of space where there is no actual charge, such that the boundary conditions are satisfied and the simple Coulomb potential may be used to calculate the electrostatic field. In the region of space that is of interest, the
35
\ c /sin(~-~) (6)
the potential at any point r(x,y,z) inside the grounded orthorhombic trap is then
¢b(r,r') = qG(r,r')
(7)
and the induced charge density on the electrode at
x. Xianget aL/Int.J. MassSpectrom.Ion Processes125 (1993)33-43
36 x = a is consequently d*(r,r') I - s o ~ ~=~
a(x=a)=
• [lnx'~. [ l n y ~ . ['ny'~
(lnZ~sin(htz" ~ -\c/ \c/
/(-- l)' sln~--~--) sin ~-ff- ) sin ~--ff-) sin - ( ! ) 2 + (m)~ -b + (n)2
a2nbC l=l ~ l .=l
(8)
As illustrated in Fig. 1 (top), the force (~r2/2to) ds, acting on each element, ds = dydz, is in the x direction; the total force acting on the plate at x = a is found by integrating from y = 0 to b and from z = 0 to c:
Fx(x = a) =
32q2 rlb n 2eo a 4 b 2c2 d0
dy ~ dz ~ ~ ~ l(-l)lsln~-'a-)Sin~--ff")sm~-'~)sm~
• { l ' n x ' k . [m'ny'~. {t'ny"x. f m ' n z k . [n'rcz"~ oo I'(--l')" sln~----~ )sin~ ~ )sln~---~-jsln~--~ )sln~---~ ) (9) I=1 m = l n = l
From the relations
fo'sin ( _ ~ _ y )sink . f m nTy) " ~ dy = ~b t~,~,m,
(10a)
fOcsm . [[ mnz ~ . { mnz" ~ dz = ~c 6,,n' --~-) sm [ , T )
(lOb)
Eq. (9) becomes
,,
Fx(x=a) =
8q2 V ~ sin2(mnY'~sin2(nnz'~ ~ n2*0bc.ZZ--ln=,
k b /
\ c /l=l 12+[(b)2+(n)2]a2
l ' ( - l)r s i n ( ~ ) x
r=~ /,2 + [ ( b ) 2 + (n)21a 2
(11)
Let k~n = a x/(m/b) 2 + (n/c)2. Then, from the relation
1(-- l)l sm ~ - ' ~ - ) = Y. t~
1= 1
+
kmn2
x
hcx"
~ sinh ( - ~ )
sinh (km~x)
(12)
X. Xiang et al./lnt. J. Mass Spectrom. Ion Processes 125 (1993) 33-43
37
the final form of the force on the electrode x = a is
• -2f k ,~nx "~ . 2[rnrty "~ . 2{nnz "~ slnn t ) sin t-D--) sm t - U ) #
#
#
in which the negative sign denotes that the force is inward directed. Similarly, we obtain the net force on the electrode at x = 0:
2q2
sinh2Ik,..(Tr- rcx'~]sin2(mlrY'~sin2(~f- ) a ,].J \ b ,]
~
F.(x = 0)
..2"=i .L=,
(14)
sinh 2 (k~, re)
in which the force is again inward, which is the direction of increasing x. Thus, the net force along the x direction is the sum of these two components. By applying the relation sinh2(~x_ ' )
2 ' sinh2[km.(n ?')]= sinh (kr,.n) sinh Ikm.nX t a k.,)
-
(15)
we have the x component of the net force exerted by point charge on the ion trap: sinh(2karrX'
Fx =
eobc~'~'=l.=t
kojr)sin2(-~)sin2(~
-~) (16)
sinh (kin,n)
in which 0 < x' < a, 0 < y' < b and 0 < z' < c. By the same procedures, one may obtain the net force along the y direction: 2q2~, ~o sinh( 2k/"~y'
e,oacl=l ~
Fy=
b
, =I
k~,rt)sin2(hrX'~sin2(mrz'~ \ a / \ c I
(17)
sinh (kl, rO
in which kl, = b x/(l[a) 2 + (n[c) 2. The net force exerted on the ion in the x (or y) direction by its image charge is (by Newton's third law) simply -Fx(or -Fr). To compute the ion cyclotron orbital frequency shift due to the image charge force, we need the radial component of this force. From Fig. 1 (bottom), the radial outward force is F, = Fx cos(4b) + Fy sin(4b). Hence the analytical solution for the radial force due to the image charge of the ion is
i ]~sin t-~) [ .
{n
z'k
F, = 2q----~2
~'0 C m=l n : l
x
sinh(km, lZ)
cost4a]slnn . . . . . t[2km"lZX'ab k m ' l r ) s i n 2 ( - ~ ) +
sin(4b)sinh( 2kmJry'ba k m " ~ ) s i n 2 ( - ~ )
]
(18)
in which ~b = tan-'(y/x) Finally, the average radial force on the ion which should be employed in a calculation of image charged-induced cyclotron frequency shift is obtained from a numerically computed average over a
X. Xiang et al./Int. J. Mass Spectrom. Ion Processes 125 (1993) 33-43
38
cyclotron orbit:
F,(average) = ~1
charge• The induced charge density on the surface, p = a, may then be expressed as
fo2F, ddp
(19)
a(p
= a) = - e o
O~(r, r')
Op
Induced image force in a cylindrical trap The same method may be used to calculate the image force for a cylindrical trap. For the cylindrical trap defined by surfaces z = 0, z = c, p = a, shown in Fig. 2, we again start with the appropriate Green function [10,22]:
G(r,r') =
~
~ e""t¢-~'' sin
q
~
~ac
m=l
p=~
~ e'm(¢-¢3sin(n-~z) n=l
,.
• {.~z'~
x sln~----~)
m~, c
/
(21)
/nna\
sin[ - - ~ )
~ o C m=-oo n = l
X
I m ( ~ - ~ ) [ - /nrca\ [nnp>\ /nzra\ L I ' ~ - - - ~ ) K " ~ - - - - ~ )
/ mra'x -
/ nzrp > \-]
(20)
in which I.,(x) and K,.(x) are the linearly independent modified Bessel functions of order m and argument y, and-the symbol p < ( > ) means the smaller (larger) of p and p'; all other symbols have their usual meaning• Since we are concerned with the induced charge density on the cylindrical surface, p < = p', the coordinate of the point
The radial force on each differential element of area, ds = a d~bdz, is d E = (o'2/2e0) a dq~dz, directed radially inward• By symmetry, the net force is in the p' direction; thus, only the force component parallel to the p' direction contributes to the total force. Hence, the total radial force acting on the plate, which in turn is equal in magnitude and opposite in direction to the force acting on the charge, is given by the integral
=
_
_
a foc dz fo2~ dtk 2rm"2 cos(~b-~b') 4rCeo _
27zZeoaC2
;o Zf? d~b
Z e'm'*-¢')sin-
m\---~-j
~.
stn~----~-)
~ e...,,_~,)sin ( ~ _ ~ )
/ t T ~ a ' ~ m,=-oo n'=l
C
/. ,{n'r~p' "~ • /n'~z"~ ~ \
c
)
x s m ~ - - ~ j ~ ~ c o s_( ~_ b - ~ b ' ) ,
Fig. 2. Cylindrical c o o r d i n a t e frame for a n ion b o u n d e d by a c o n d u c t i v e cylinder. The cylinder is defined by surfaces, z = 0, z = c a n d p = a. p a n d q~ are c o n v e n t i o n a l cylindrical coordinates.
(22)
By performing the integration and simplification as before, the net radial image charge force in the
X. Xiang et al.llnt. J. Mass Spectrom. Ion Processes 125 (1993) 33--43
cylindrical trap may be expressed as Fr •
_ _neoac _
sin 2 m=l n=l
[ n~p' \
[ nrcp' \
[n~a\
(~_)
'mt "--C-)Sm+'t x
(23)
The negative sign indicates that the force on the trap is directed radially inward; thus, the force on the ion is radially outward.
39
The solution with the positive sign represents the ion cyclotron orbital frequency, o9+, whereas the negative sign solution represents the magnetron frequency, co_. Because the image charge force and the radial component of trapping force are small relative to the magnetic force, the square root in Eq. (25a) or (25b) may be expanded to first order in a Taylor series• The ICR orbital frequency shift due to the image force then becomes: Aog+ = o9+ (no image force) - o9+ (image force present) =
Ion cyclotron orbital frequency shift due to image charge
Now that the radial image force has been found for rectangular and cylindrical traps, we can return to the original problem of determining the shift in ion cyclotron orbital frequency due to that force• The equations of ion motion in the x - y plane are: mro92 = qrogB - FE (without image force)
(24a)
F, qBr
(26)
Combining Eq. (26) with Eqs. (18) and (23), we obtain the final frequency shift expression for the rectangular trap.
Ao9+ ~ 8 o ~ ~l .=1 sinh (kmnr0 × fo 2n de' [acos(¢')
mro92 = qrogB - F e - Ft (with image force) (24b)
in which the left-hand side of the Eq. (24b) represents the centripetal force; the first, second and third terms on the right-hand side correspond (see Fig. 3) to the magnetic force (Fn) arising from the static magnetic field along the z-direction, the force, FE, from the radial component of the electrostatic trapping field (F~ = (~qVTr)/a 2 in the quadrupolar potential approximation), and the force, Ft, due to image charge. Because the latter two terms, Fe and F~, are frequency independent, we may solve the quadratic Eq. (24a) or (24b) for fixed orbital radius to obtain the "natural" frequencies of ion motion:
× sinh(2kmnrCPaC°S(¢') ) • 2[mny"~ × sm ~,---~ ] + bsin(~b')
x
. . f 2k,..np" sin( ¢') ~ (_~)] s,nn[ ~ )sin 2
(27) in which we have substituted for x ' = a/2 + p' cos(¢') and y' = a/2 + p" sin(~b'). The corresponding expression for the frequency shift for the cylindrical trap takes the form
qrB +_-x/(qrB) 2 -- 4mrF~ o9=
2mr
(no image force)
(25a)
Am+-~q ~ ~sin2(~ f~) ~.8oacBr m=l n=l
[ nrrp' \ qrB +_ x/(qrB) 2 -- 4mr(FE + Fi) o9 =
x
2mr
(image force present)
(25h)
(28)
I-+1
40
X. Xiang et al./Int. J. Mass Spectrom. lon Processes 125 (1993) 33-43
IY
0
FE
c/6
N
-0.5x10 -4 .1.ox10 "4 ¢)
== o" ¢J
Cubic Trap .1.5x10 "4
II
-2.0x10-4
X
"
0.0
Fig. 3. Forces acting on a ion moving in a circular path in the x - y plane. The inward force, Ym, is the Lorentz magnetic force due to the homogeneous magnetic field along the z direction. FE is due to the radial component o f the trapping electric field, and F t is the force due to the induced image charge.
'
•
i
.
-
~
I
,
a/4 Ion cyclotron orbital radius
a/2
z=c/2
= ==
-0.001
z=c/3 z=c/6
o
-0.002
Cylindrical Trap
Results and discussion It.
-0.003
It is qualitatively clear that magnitude of the image charge induced on an electrode of an ICR trap, and thus the magnitude o f the (downward) shift in ICR orbital frequency resulting from that image charge, increases as the ion approaches the electrode. We therefore begin by examining the quantitative variation o f ICR orbital frequency with ion displacement from the center of the trap.
Effect of ion radius on image charge-induced ion cyclotron orbital frequency shift For convenience, numerical evaluation of the frequency shift of Eq. (26) was limited to a cubic trap, with xyz dimensions, a = b = c = I in. Figure 4 shows the image charge-induced ion cyclotron orbital frequency for a single ion as a function of ion cyclotron orbital radius, for a magnetron radius of zero (i.e. ICR orbit centered on the z axis). (Although it is convenient to derive the frequency shift in rad s - ' , Figures 4 and 5 display the frequency shift in typical experimental units of Hertz.) Because the frequency shift varies with ion cyclotron orbital phase angle, and because the variation in ICR orbital frequency during one orbit is much smaller than the ICR orbital frequency itself, we may predict the observed frequency shift as the
.
•
0.0
'
'
a}4 lon cyclotron
i
a/2
orbital radius
Fig. 4. Calculated image charge-induced ion cyclotron orbital frequency shift as a function o f ion cyclotron orbital radius for cubic (top) and cylindrical (bottom) ion traps, at each o f three axial positions (z = c/6, c/3, and c/2). z = c/2 is the trap midplane. The traps are similar in size (a = b = c = 1 in cubic, or p = 0.50in and c = I in cylindrical), and the magnetic field strength is 3 T. The reported frequency shift is calculated as a numerical average over 360 ° at 1° increments of ICR orbital phase.
equally weighted average over all ion cyclotron orbital phases (say, from 1° to 360 ° in 1° increments). For either the cubic or cylindrical trap, Figure 4 shows that the ion cyclotron orbital frequency shift due to image charge is negligible for an ion close to the central z axis o f the trap. As the ion cyclotron orbital radius increases, the frequency shift increases monotonically as the ion approaches a side plate of the trap: e.g., the frequency shift for a cubic trap increases by an order of magnitude as the ion cyclotron orbital radius increases from 0.1 to 0.8 of the trap radius. For cubic and cylindrical traps of the same aspect (i.e. length-to-width) ratio, the image charge-induced frequency shift for the cylindrical trap is greater than that for the cubic trap at large ICR orbital radius (say, r greater than about 0.8 of the trap radius), because the ion is on
X. Xiang et al./Int. J. Mass Spectrom. Ion Processes 125 (1993) 33-43
41
r = 0.25 rmax
m=lO0 u c
r = 0.50 rmax
_o
0
0
Q.
¢:
-0.5x10-4
.1.0x10 "8
J= t/J
r=0.75 r~x ~
-1.0xl0 -4 >, ¢} el)
oc
-1.5x10-4
o"
o u.
]
~
-2.0x10 "4
, c/2
0.0
u,. Q
~
-3.0x10
c
~
-4.0x10"8 0.o
r = 0.25 rmax
r = 0.50 rmax
0
0
.0.5x10 "4
/
,
,
,
m=lO,OOO~u ~ i
i
i
|
,
a/4 ion cyclotron orbital radius
/
i
,
a/2
m=100 u
Q.
\ r = 0.75 rmax
-1.0x10 -4 Cubic
-1.0x10"8 F_ .¢ (n
-2.0xl0 -8
Trap
[
.1.5x10 "4
u. ¢D
O" IL
z=le
e
,
position
Axial
A
-2.0x10"8
-2.0x10 "4
r
r
~
i
-4.0xlo "8 o.o
c/2
0.0 Axial
position
Fig. 5. Calculated image charge-induced ion cyclotron orbital frequency shift as a function of axial position, for each of three radial positions (r/r=~x = 0.25, 0.50, and 0.75; rm~x= a[2), at 3.0T, for the cylindrical (top) and cubic (bottom) traps of Figure 5.
the average closer to the walls of the cylinder than to the walls of a cubic trap of comparable size.
Effect of ion axial position on image charge-induced ion cyclotron orbital frequency shift Figure 5 shows the image charge-induced ion cyclotron orbital frequency shift as a function of the axial position of the ion, for various ICR orbital radii (again for zero magnetron radius). The frequency shift decreases as the ion approaches either trapping electrode, because less charge is induced on the "side" electrodes (i.e. the electric field lines from the ion terminate on the trapping electrode rather than on "side" electrodes. As for the radial-dependence discussed above, the image charge-induced ion cyclotron orbital frequency shift is less for a cubic than for a cylindrical trap of comparable size.
-3.0x10"8
n-
,
i
I
c/2 Axial Position
Fig. 6. Calculated image charge-induced relative ion cyclotron orbital frequency shift, lain+ I/m+ (see text) as a function of ion cyclotron orbital radius (top) or axial position (bottom) for a singly-charged ion of mass, 100, 1000, or 10000u, at 3.0T in a I in cubic trap. Although the absolute frequency shift is massindependent (see text), the relative frequency shift increases with increasing ion mass.
Mass-independence of the image charge-induced ion cyclotron orbital frequency shift Equations (27) and (28) clearly show that the magnitude of the image charge-induced ion cyclotron orbital frequency shift, law+ J, is independent of ion cyclotron orbital frequency, and thus independent of ionic mass, m, for singly-charged ions. Thus, the relative shift in frequency, ]Ato+ f/o+, is extremely small for low-mass ions (which have high ion cyclotron frequency). However, the relative shift increases with increasing ionic mass, illustrated in Figure 6 for singly-charged ions in a one-inch cubic trap. Qualitatively similar behavior will be found for traps of other shape. It is also worth noting that, at constant orbital radius, the magnitude of the absolute frequency shift varies inversely with the trap radial dimension, because
42
x. Xiang et al./Int. J. Mass Spectrom. Ion Processes125 (1993) 33--43
the ion packet is farther from the electrodes of a larger-radius trap.
Effect of number of ions on image charge-induced ion cyclotron orbital frequency shift
ions of charge, n, if both ion packets are spatially coherent. Thus, the ICR orbital frequency shift due to image charge depends not only on mass-tocharge ratio, m/q, but also on the charge, q itself.
Implications for mass calibration in FT-ICR M S All of the above calculations were based on a single orbiting ion. Because the induced charge is directly proportional to the number, N, of coherently orbiting ions, we can approximate the frequency shift for N ions as NAto+. Of course, the observed frequency shift due to image charge depends (see Figs. 4-6) on the radial and axial position of the ion orbit. Moreover, we have not considered the effect of finite magnetron radius (i.e. off-axis ions).
Experimental significance of image charge-induced ICR orbital frequency shifts Recent theory and experiment have shown that the F T - I C R M S detection limit for conventional broad-band detection is ~ 175 ions (for C6H~- at 3.0 T, based on signal-to-noise ratio of 3 : 1 for a I s undamped data acquisition period); the maximum ion number is about 32000 times larger ( ~ 5.6 million ions) [23]. However, we would not expect to observe an experimental image charge-induced shift in the ICR orbital frequency of (say) singlycharged N~- ions (v+ ~ 1.65 MHz at 3.0 T), even at relatively high ion density and large ICR orbital radius, because the predicted frequency shift ( ~ 10-4Hz per ion, from Figs. 4 and 5) for even 100000 ions barely reaches the I ppm level for NJunder those conditions. Figures 4-6 show that the image charge-induced relative frequency shift becomes much larger for singly-charged ions of higher mass (e.g. ~ 300 ppm, or ~ 1 u for singly-charged ions of 3 000 u). Furthermore, an ion carrying n charges will produce an n-fold higher image charge, and thus an n2-fold higher frequency shift. However, the ICR signal from such an ion will also be n times larger than for a singly-charged ion. Thus, for a given signal-tonoise level, N singly charged ions should produce approximately the same signal-to-noise level as N/n
For FT-ICR MS experiments involving a large (say, i> 100000) number of ions, of which some have relatively high m/q (say, ~ 500), image charge effects may be minimized by limiting the ICR orbital radius to (say) 20% of the maximum available trap radius. Alternatively, from the ICR orbital radius calculated for a given trap geometry from known values of the r.f. excitation voltage magnitude and duration [25-27] and measured number of ions [23], the present results (adapted, if necessary, to different trap dimensions) could be used to correct the observed ICR orbital frequencies for space charge effects. We propose to test that approach in future experiments. Correction for image charge could be significant in ultraprecise FT-ICR mass determinations for ions of m/z <<.20 or so [13,28,29], for which milliHertz precision is required.
Acknowledgments The authors wish to thank C.W. Ross, III for help with the numerical calculation of the frequency shift and Dr. L. Schweikhard for helpful discussion. This work was supported by grants (to A.G.M.) from the U.S.A. National Science Foundation (CHE-90-21058), U.S.A. Public Health Service (NIH GM-31683) and The Ohio State University.
References 1 A.G.Marshalland L. Schweikhard,Int. J. MassSpectrom. Ion Processes, 118/119 (1992) 37, 2 A.G. Marshall and P.B. Grosshans, Anal. Chem., 63 (1991) 215 3 E.B. Ledford Jr., D.L. Rempel and M.L. Gross, Anal. Chem., 56 (1984) 2744.
X. Xiang et al./Int. J. Mass Spectrom. Ion Processes 125 (1993) 33-43 4
5 6 7 8 9 10 11
12 13 14 15 16
J.T. Meek, W.G. Millen, T.J. Francl, G.W. Stockton, M.L. Thomson and R.S. Wayne, Proc. 35th ASMS Conference on Mass Spectrometry and Allied Topics, Denver, CO, 1987, p. 1122. M.J.C. Smith, 2nd ASMS Sanibel Conference, Sambel Island, FL, 1990. M. Wang and A.G. Marshall, Anal. Chem., 61 (1989) 1288. J. Byrne and P.S. Farago, Proc. Phys. Soc., 86 (1965) 801. R.D. Knight, Int. J. Mass Spectrom. Ion Phys., 51 (1983) 127. L.S. Brown and G. Gabrielse, Rev. Mod. Phys., 58 (1986) 233. P.B. Grosshans, P.J. Shields and A.G. Marshall, J. Chem. Phys., 94 (1991) 5341. S.-P. Chen and M.B. Comisarow, Rapid Commun. Mass Spectrom., 5 (1991) 450. S.-P. Chen and M.B. Comisarow, Rapid Commun. Mass Spectrom., 6 (1992) 1. E.T. Lippmaa, R. Pikver, E. Suurmaa, J. Past, J. Puskar, I. Koppel and A. Tammik, Phys. Rev. Lett., 54 (1985) 285. E.N. Nikolaev, Y.I. Neronov, M.V. Gorshkov and V.L. Talrose, Mass-Spektrom. Khim. Kinet., (1985) 228. V.L. Talrose and E.N. Nikolaev, Adv. Mass Spectrom., 10A (1985) 343. N.C. Hill, G.M. Alber, L. Schweikhard, T.L. Ricca and A.G. Marshall, Proc. 39th ASMS Conference on Mass
17 18 19 20 21 22 23 24 25 26 27 28 29
43 Spectrometry and Allied Topics, Nashville, TN, 1991, p. 1501. D. Wineland and H. Dehmelt, J. Appl. Phys., 46 (1975) 919. M.V. Vogel, H.J. Kluge and L. Schweikhard, Personal communication, 1991. L.S. Brown, K. Helmerson and J. Tan, Phys. Rev. A, 34 (1986) 2638. R.S. Van Dyck Jr., F.L. Moore, D.L. Farnham and P.B. Schwinberg, Phys. Rev. A, 40 (1989) 6308. E.N. Nikolaev and M.V. Gorshkov, Int. J. Mass Spectrom. Ion Processes, 64 (1985) 115. J.D. Jackson, Classical Electrodynamics, Wiley, New York, 1975. P.A. Limbach, P.B. Grosshans and A.G. Marshall, Anal. Chem., 65 (1993) 135. M.B. Comisarow, J. Chem. Phys., 69 (1978) 4097. P.B. Grosshans, P.J. Shields and A.G. Marshall, J. Am. Chem. Soc., 112 (1990) 1275. P.B. Grosshans and A.G. Marshall, Int. J. Mass Spectrom. Ion Processes, 100 (1990) 347. P.B. Grosshans and A.G. Marshall, Anal. Chem., 63 (1991) 2057. E.N. Nikolaev, Y.I. Neronov, M.V. Gorshkov and V.L. Talrose, JETP Lett., 39 (1984) 534. M.V. Gorshkov, G.M. Alber, L. Schweikhard and A.G. Marshall, Phys. Rev. A, (1993) in press.
33
Image charge-induced ion cyclotron orbital frequency shift for orthorhombic and cylindrical FT-ICR ion traps Xinzhen Xiang a, Peter B. Grosshans b, Alan G. Marshall a.c., aDepartment of Chemistry, The Ohio State University, 120 West 18th Avenue, Columbus, OH 43210, USA bExxon Research and Engineering Company, Room LF148, Route 22 East, Annandale, NJ 08801, USA CDepartment of Biochemistry, The Ohio State University, 484 West 12th Avenue, Columbus, OH 43210, USA (Received September 16 1992; accepted January 13 1993) Abstract A coherently orbiting ion packet induces a distribution of image charge on the ion trap electrodes; the resultant electric field from the image charge shifts the ICR orbital frequency of the ions in the ion packet. The effect of image charges upon the cyclotron motion of a trapped ion has previously been modeled for infinite parallel plate and spherical cavity geometries. In other prior work, an analytical method to solve for the charge induced on any of the electrodes of a Fourier transform ion cyclotron resonance (FT-ICR) ion trap has been presented. In this work, we report approximate analytical solutions for the frequency shift due to the induced image charges for both finite-size cylindrical and tetragonal traps as a function of ion axial and radial position. For a cubic trap, the mass-independent frequency shift at an ICR orbital radius equal to half the trap radius is ~ 10 -5 Hz for one ion. Thus, image charge-induced frequency shifts in the ppm range can be expected for typical ICR experiments involving 10~-I05 ions at low mass-to-charge ratio (60 ~< m/z <~ 600), whereas the shift can increase to ,~, ! u for 100 000 singly-charged 3000 u ions at 3.0 T. The image charge-induced shift for a cylindrical trap is higher than for a cubic trap of comparable size. Implications for mass calibration are discussed.
Key words: FT-ICR; Image charge-induced shift; Orthorhombic ion trap; Cylindrical ion trap.
Introduction One o f the most unique features of Fourier transform ion cyclotron resonance mass spectrometry ( F T - I C R M S ) [1,2] is its capability o f highly precise and accurate measurement of ionic mass. FT-ICR mass measurement errors at the sub-ppm level over a wide mass range have been reported by several research groups [3-6]. In the absence o f an electric field, the ion cyclotron orbital frequency, eoc, for an ion o f mass-to-charge ratio, m/q, moving in a magnetic field, B, is simply
qB ~o =
-
-
m
* Corresponding author.
(1)
However, in the presence of the electrostatic potential needed to trap the ions, the ion cyclotron orbital frequency shifts to a new value, a~+, and two additional natural motions arise: magnetron motion at frequency, a ~ ; and trapping oscillation at frequency, ogz. For a quadrupolar electrostatic potential [7-10] ~o~
~o~
m-+ = 2 - +
(2)
2
Equation (2) leads to the particularly simple mass calibration law [3];
m / z = cl -~ c2 V+
2
V+
(3)
in which Cl and c2 are parameters which are known
34
X. Xiang et al./Int. J. Mass Spectrom. Ion Processes 125 (1993) 33-43
analytically and which may be determined by numerically fitting Eq. (3) to experimental data for ions of known m/q, and v+ is the ion cyclotron frequency in Hertz. Equation (3) may be modified to account for Coulomb repulsions between ions of like charge and like or unlike mass [5,11,12]. Moreover, relativistic frequency shifts for ions of m/z <~ 20 have also been considered for ultrahighprecision experiments [13-16]. The additional shift in ion cyclotron orbital frequency due to the image charge induced on nearby electrodes was first considered by Wineland and Dehmelt [17], who calculated the image charge force by modeling the actual trap by a pair of grounded parallel plates. In this geometry, the instantaneous force on a lone ion is perpendicular to and directed toward the plate to which it is nearest. When this instantaneous force is averaged over one cyclotron orbit, there is a net radially outward-directed force which opposes the radially inward-directed magnetic force, qv x B, in which v is ion velocity. Some years later, a conductive sphere model was used to estimate the force on an ion due to its image charge induced on the electrodes of a Penning trap [18-20]. The radial electric force on an ion was incorporated into the Lorentz force equation to determine the magnitude of the image charged-induced ion cyclotron orbital frequency shift as a function of the ion cyclotron orbital radius and the number of other like m/q ions. In other prior work, Nikolaev and Gorshkov [21] calculated the induced image charge resulting from charge distributed uniformly along the central axis of a conductive cylinder extended infinitely along the direction of the applied magnetic field. In this paper, we calculate the image charge distribution induced by an ion on the electrodes of a finite-dimension orthorhombic or cylindrical trap. The analytical solutions for the image charge induced by a point charge on orthorhombic and cylindrical traps of finite dimensions are presented, and the magnitude of the shift in ion cyclotron orbital frequency for cubic and cylindrical traps of comparable size are numerically evaluated as a function of ion radial and axial position. The implications of the results for mass measurement
accuracy for singly- and multiply-charged ions are discussed.
Theory The well known Lorentz force equation F = q(E + v x B)
(4)
governs the motion of a charged particle in an electromagnetic field. In this equation, F is the instantaneous vector force on a particle of charge, q, moving with vector velocity, v, under the combined influence of an electric field, E, and a magnetic field B. In most calculations of ion trajectories in FT-ICR MS, B is assumed constant in magnitude and direction, and E is determined only by the potentials applied to the electrodes which define the closed inner surface of the ion trap. In principle, one could compute ion trajectories without the often invoked simplifying assumptions that the magnetic field is spatially homogeneous and temporally constant and that ion/ion interactions are negligible. However, even a treatment at this level of approximation is not complete: there exists another force on trapped ions, namely that due to the induced charge on the inner surfaces of the trap electrodes. This oversight results from the definition of the electric field: E = lim F~ q~0 q
(5)
in which Fe is the electric force felt by the point charge, q, and the limit is taken as the magnitude of the charge goes to zero. The surface charge induced by a point charge on a conductor is proportional to the magnitude of the point charge, and the force exerted by the induced surface charge density on the point charge is proportional to the product of the point charge and the induced surface charge density. Thus, the electric force on the point charge due to its image charge is proportional to q2, and this force is not incorporated in the electric field found by ordinary methods. A convenient way of calculating this force, for simple trap geometries, is by the image charge method [22]. In this method, the surface of the
X. Xiang et al.llnt. J. Mass Spectrom. Ion Processes 125 (1993) 33--43
x
Fig. 1. Rectilinear coordinate frame for an ion bounded by conductive orthorhombic trap plates. Top: the orthorhombic trap is defined by surfaces: x = 0, x = a; y = 0, y = b; and z = 0, z = c. F~ and Fe (see text) are parallel to the x and y axes, respectively. The position vector (x', y', z') of the point charge lies inside the trap. Bottom: the radial force, F, = FxCOS(~b) + Fy sin(~b), resulting from components, Fx and F~.
conductor divides space into two regions, the interior and the exterior. For the interior problem, the conductor is hollow and the charge is located inside the hollowed-out region. In the exterior problem, the charge is located outside of the conductor. In both cases, one is interested in solving Poisson's equation in the region of space in which the charge is located subject to the boundary conditions imposed by the conductor. The point is that a solution which satisfies Poisson's equation and the boundary conditions is the unique solution. Thus, one attempts to place a distribution of
8
Coulomb potential due to the point charge satisfies Poisson's equation, and that due to the imaginary charge distribution satisfies Laplace's equation. Thus, the sum of all the contributions to the electrostatic potential field satisfies Poisson's equation in the region of space that is of interest. As stated earlier, the method of images is useful for boundaries of simple geometry (e.g. a point charge inside a spherical conductive surface [22]). However, for tetragonal and cylindrical ion traps, no such simple solution is available, and an expression for the force on a point charge due to the induced surface charge density on the conductor must be obtained by other means. Here, we solve Poisson's equation for the geometry under consideration, and compute the induced surface charge density on the conductor from the derived potential inside the trap, which is in turn obtained by Green function methods [10]. Next, we employ a formula for the inward pressure exerted on a differential element of induced surface charge by the point charge; integration of this pressure over the entire interior surface of the conductor yields the net force on the conductor due to the point charge. By Newton's third law, that force must be equal and opposite to the net force exerted on the point charge by the conductor.
Induced image force in an orthorhombic trap Consider a point charge, q, located at position, r'(x',y',z'), inside an orthorhombic trap defined by surfaces x = 0 , x = a , y = 0 , y = b , z = 0 and z = c, as in Fig. l(top), the Green function is [10,22]:
sin(lnX~sin(lnX'~sin(l-~-) sin[lny~'sinClnz~b\ J G(r,r')= ~ ~. ~ \ a / \ a / n2e°abct=l'=l"=| ( ! ) 2 + ( b ) 2 + (n)2 "imaginary" charge in the region of space where there is no actual charge, such that the boundary conditions are satisfied and the simple Coulomb potential may be used to calculate the electrostatic field. In the region of space that is of interest, the
35
\ c /sin(~-~) (6)
the potential at any point r(x,y,z) inside the grounded orthorhombic trap is then
¢b(r,r') = qG(r,r')
(7)
and the induced charge density on the electrode at
x. Xianget aL/Int.J. MassSpectrom.Ion Processes125 (1993)33-43
36 x = a is consequently d*(r,r') I - s o ~ ~=~
a(x=a)=
• [lnx'~. [ l n y ~ . ['ny'~
(lnZ~sin(htz" ~ -\c/ \c/
/(-- l)' sln~--~--) sin ~-ff- ) sin ~--ff-) sin - ( ! ) 2 + (m)~ -b + (n)2
a2nbC l=l ~ l .=l
(8)
As illustrated in Fig. 1 (top), the force (~r2/2to) ds, acting on each element, ds = dydz, is in the x direction; the total force acting on the plate at x = a is found by integrating from y = 0 to b and from z = 0 to c:
Fx(x = a) =
32q2 rlb n 2eo a 4 b 2c2 d0
dy ~ dz ~ ~ ~ l(-l)lsln~-'a-)Sin~--ff")sm~-'~)sm~
• { l ' n x ' k . [m'ny'~. {t'ny"x. f m ' n z k . [n'rcz"~ oo I'(--l')" sln~----~ )sin~ ~ )sln~---~-jsln~--~ )sln~---~ ) (9) I=1 m = l n = l
From the relations
fo'sin ( _ ~ _ y )sink . f m nTy) " ~ dy = ~b t~,~,m,
(10a)
fOcsm . [[ mnz ~ . { mnz" ~ dz = ~c 6,,n' --~-) sm [ , T )
(lOb)
Eq. (9) becomes
,,
Fx(x=a) =
8q2 V ~ sin2(mnY'~sin2(nnz'~ ~ n2*0bc.ZZ--ln=,
k b /
\ c /l=l 12+[(b)2+(n)2]a2
l ' ( - l)r s i n ( ~ ) x
r=~ /,2 + [ ( b ) 2 + (n)21a 2
(11)
Let k~n = a x/(m/b) 2 + (n/c)2. Then, from the relation
1(-- l)l sm ~ - ' ~ - ) = Y. t~
1= 1
+
kmn2
x
hcx"
~ sinh ( - ~ )
sinh (km~x)
(12)
X. Xiang et al./lnt. J. Mass Spectrom. Ion Processes 125 (1993) 33-43
37
the final form of the force on the electrode x = a is
• -2f k ,~nx "~ . 2[rnrty "~ . 2{nnz "~ slnn t ) sin t-D--) sm t - U ) #
#
#
in which the negative sign denotes that the force is inward directed. Similarly, we obtain the net force on the electrode at x = 0:
2q2
sinh2Ik,..(Tr- rcx'~]sin2(mlrY'~sin2(~f- ) a ,].J \ b ,]
~
F.(x = 0)
..2"=i .L=,
(14)
sinh 2 (k~, re)
in which the force is again inward, which is the direction of increasing x. Thus, the net force along the x direction is the sum of these two components. By applying the relation sinh2(~x_ ' )
2 ' sinh2[km.(n ?')]= sinh (kr,.n) sinh Ikm.nX t a k.,)
-
(15)
we have the x component of the net force exerted by point charge on the ion trap: sinh(2karrX'
Fx =
eobc~'~'=l.=t
kojr)sin2(-~)sin2(~
-~) (16)
sinh (kin,n)
in which 0 < x' < a, 0 < y' < b and 0 < z' < c. By the same procedures, one may obtain the net force along the y direction: 2q2~, ~o sinh( 2k/"~y'
e,oacl=l ~
Fy=
b
, =I
k~,rt)sin2(hrX'~sin2(mrz'~ \ a / \ c I
(17)
sinh (kl, rO
in which kl, = b x/(l[a) 2 + (n[c) 2. The net force exerted on the ion in the x (or y) direction by its image charge is (by Newton's third law) simply -Fx(or -Fr). To compute the ion cyclotron orbital frequency shift due to the image charge force, we need the radial component of this force. From Fig. 1 (bottom), the radial outward force is F, = Fx cos(4b) + Fy sin(4b). Hence the analytical solution for the radial force due to the image charge of the ion is
i ]~sin t-~) [ .
{n
z'k
F, = 2q----~2
~'0 C m=l n : l
x
sinh(km, lZ)
cost4a]slnn . . . . . t[2km"lZX'ab k m ' l r ) s i n 2 ( - ~ ) +
sin(4b)sinh( 2kmJry'ba k m " ~ ) s i n 2 ( - ~ )
]
(18)
in which ~b = tan-'(y/x) Finally, the average radial force on the ion which should be employed in a calculation of image charged-induced cyclotron frequency shift is obtained from a numerically computed average over a
X. Xiang et al./Int. J. Mass Spectrom. Ion Processes 125 (1993) 33-43
38
cyclotron orbit:
F,(average) = ~1
charge• The induced charge density on the surface, p = a, may then be expressed as
fo2F, ddp
(19)
a(p
= a) = - e o
O~(r, r')
Op
Induced image force in a cylindrical trap The same method may be used to calculate the image force for a cylindrical trap. For the cylindrical trap defined by surfaces z = 0, z = c, p = a, shown in Fig. 2, we again start with the appropriate Green function [10,22]:
G(r,r') =
~
~ e""t¢-~'' sin
q
~
~ac
m=l
p=~
~ e'm(¢-¢3sin(n-~z) n=l
,.
• {.~z'~
x sln~----~)
m~, c
/
(21)
/nna\
sin[ - - ~ )
~ o C m=-oo n = l
X
I m ( ~ - ~ ) [ - /nrca\ [nnp>\ /nzra\ L I ' ~ - - - ~ ) K " ~ - - - - ~ )
/ mra'x -
/ nzrp > \-]
(20)
in which I.,(x) and K,.(x) are the linearly independent modified Bessel functions of order m and argument y, and-the symbol p < ( > ) means the smaller (larger) of p and p'; all other symbols have their usual meaning• Since we are concerned with the induced charge density on the cylindrical surface, p < = p', the coordinate of the point
The radial force on each differential element of area, ds = a d~bdz, is d E = (o'2/2e0) a dq~dz, directed radially inward• By symmetry, the net force is in the p' direction; thus, only the force component parallel to the p' direction contributes to the total force. Hence, the total radial force acting on the plate, which in turn is equal in magnitude and opposite in direction to the force acting on the charge, is given by the integral
=
_
_
a foc dz fo2~ dtk 2rm"2 cos(~b-~b') 4rCeo _
27zZeoaC2
;o Zf? d~b
Z e'm'*-¢')sin-
m\---~-j
~.
stn~----~-)
~ e...,,_~,)sin ( ~ _ ~ )
/ t T ~ a ' ~ m,=-oo n'=l
C
/. ,{n'r~p' "~ • /n'~z"~ ~ \
c
)
x s m ~ - - ~ j ~ ~ c o s_( ~_ b - ~ b ' ) ,
Fig. 2. Cylindrical c o o r d i n a t e frame for a n ion b o u n d e d by a c o n d u c t i v e cylinder. The cylinder is defined by surfaces, z = 0, z = c a n d p = a. p a n d q~ are c o n v e n t i o n a l cylindrical coordinates.
(22)
By performing the integration and simplification as before, the net radial image charge force in the
X. Xiang et al.llnt. J. Mass Spectrom. Ion Processes 125 (1993) 33--43
cylindrical trap may be expressed as Fr •
_ _neoac _
sin 2 m=l n=l
[ n~p' \
[ nrcp' \
[n~a\
(~_)
'mt "--C-)Sm+'t x
(23)
The negative sign indicates that the force on the trap is directed radially inward; thus, the force on the ion is radially outward.
39
The solution with the positive sign represents the ion cyclotron orbital frequency, o9+, whereas the negative sign solution represents the magnetron frequency, co_. Because the image charge force and the radial component of trapping force are small relative to the magnetic force, the square root in Eq. (25a) or (25b) may be expanded to first order in a Taylor series• The ICR orbital frequency shift due to the image force then becomes: Aog+ = o9+ (no image force) - o9+ (image force present) =
Ion cyclotron orbital frequency shift due to image charge
Now that the radial image force has been found for rectangular and cylindrical traps, we can return to the original problem of determining the shift in ion cyclotron orbital frequency due to that force• The equations of ion motion in the x - y plane are: mro92 = qrogB - FE (without image force)
(24a)
F, qBr
(26)
Combining Eq. (26) with Eqs. (18) and (23), we obtain the final frequency shift expression for the rectangular trap.
Ao9+ ~ 8 o ~ ~l .=1 sinh (kmnr0 × fo 2n de' [acos(¢')
mro92 = qrogB - F e - Ft (with image force) (24b)
in which the left-hand side of the Eq. (24b) represents the centripetal force; the first, second and third terms on the right-hand side correspond (see Fig. 3) to the magnetic force (Fn) arising from the static magnetic field along the z-direction, the force, FE, from the radial component of the electrostatic trapping field (F~ = (~qVTr)/a 2 in the quadrupolar potential approximation), and the force, Ft, due to image charge. Because the latter two terms, Fe and F~, are frequency independent, we may solve the quadratic Eq. (24a) or (24b) for fixed orbital radius to obtain the "natural" frequencies of ion motion:
× sinh(2kmnrCPaC°S(¢') ) • 2[mny"~ × sm ~,---~ ] + bsin(~b')
x
. . f 2k,..np" sin( ¢') ~ (_~)] s,nn[ ~ )sin 2
(27) in which we have substituted for x ' = a/2 + p' cos(¢') and y' = a/2 + p" sin(~b'). The corresponding expression for the frequency shift for the cylindrical trap takes the form
qrB +_-x/(qrB) 2 -- 4mrF~ o9=
2mr
(no image force)
(25a)
Am+-~q ~ ~sin2(~ f~) ~.8oacBr m=l n=l
[ nrrp' \ qrB +_ x/(qrB) 2 -- 4mr(FE + Fi) o9 =
x
2mr
(image force present)
(25h)
(28)
I-+1
40
X. Xiang et al./Int. J. Mass Spectrom. lon Processes 125 (1993) 33-43
IY
0
FE
c/6
N
-0.5x10 -4 .1.ox10 "4 ¢)
== o" ¢J
Cubic Trap .1.5x10 "4
II
-2.0x10-4
X
"
0.0
Fig. 3. Forces acting on a ion moving in a circular path in the x - y plane. The inward force, Ym, is the Lorentz magnetic force due to the homogeneous magnetic field along the z direction. FE is due to the radial component o f the trapping electric field, and F t is the force due to the induced image charge.
'
•
i
.
-
~
I
,
a/4 Ion cyclotron orbital radius
a/2
z=c/2
= ==
-0.001
z=c/3 z=c/6
o
-0.002
Cylindrical Trap
Results and discussion It.
-0.003
It is qualitatively clear that magnitude of the image charge induced on an electrode of an ICR trap, and thus the magnitude o f the (downward) shift in ICR orbital frequency resulting from that image charge, increases as the ion approaches the electrode. We therefore begin by examining the quantitative variation o f ICR orbital frequency with ion displacement from the center of the trap.
Effect of ion radius on image charge-induced ion cyclotron orbital frequency shift For convenience, numerical evaluation of the frequency shift of Eq. (26) was limited to a cubic trap, with xyz dimensions, a = b = c = I in. Figure 4 shows the image charge-induced ion cyclotron orbital frequency for a single ion as a function of ion cyclotron orbital radius, for a magnetron radius of zero (i.e. ICR orbit centered on the z axis). (Although it is convenient to derive the frequency shift in rad s - ' , Figures 4 and 5 display the frequency shift in typical experimental units of Hertz.) Because the frequency shift varies with ion cyclotron orbital phase angle, and because the variation in ICR orbital frequency during one orbit is much smaller than the ICR orbital frequency itself, we may predict the observed frequency shift as the
.
•
0.0
'
'
a}4 lon cyclotron
i
a/2
orbital radius
Fig. 4. Calculated image charge-induced ion cyclotron orbital frequency shift as a function o f ion cyclotron orbital radius for cubic (top) and cylindrical (bottom) ion traps, at each o f three axial positions (z = c/6, c/3, and c/2). z = c/2 is the trap midplane. The traps are similar in size (a = b = c = 1 in cubic, or p = 0.50in and c = I in cylindrical), and the magnetic field strength is 3 T. The reported frequency shift is calculated as a numerical average over 360 ° at 1° increments of ICR orbital phase.
equally weighted average over all ion cyclotron orbital phases (say, from 1° to 360 ° in 1° increments). For either the cubic or cylindrical trap, Figure 4 shows that the ion cyclotron orbital frequency shift due to image charge is negligible for an ion close to the central z axis o f the trap. As the ion cyclotron orbital radius increases, the frequency shift increases monotonically as the ion approaches a side plate of the trap: e.g., the frequency shift for a cubic trap increases by an order of magnitude as the ion cyclotron orbital radius increases from 0.1 to 0.8 of the trap radius. For cubic and cylindrical traps of the same aspect (i.e. length-to-width) ratio, the image charge-induced frequency shift for the cylindrical trap is greater than that for the cubic trap at large ICR orbital radius (say, r greater than about 0.8 of the trap radius), because the ion is on
X. Xiang et al./Int. J. Mass Spectrom. Ion Processes 125 (1993) 33-43
41
r = 0.25 rmax
m=lO0 u c
r = 0.50 rmax
_o
0
0
Q.
¢:
-0.5x10-4
.1.0x10 "8
J= t/J
r=0.75 r~x ~
-1.0xl0 -4 >, ¢} el)
oc
-1.5x10-4
o"
o u.
]
~
-2.0x10 "4
, c/2
0.0
u,. Q
~
-3.0x10
c
~
-4.0x10"8 0.o
r = 0.25 rmax
r = 0.50 rmax
0
0
.0.5x10 "4
/
,
,
,
m=lO,OOO~u ~ i
i
i
|
,
a/4 ion cyclotron orbital radius
/
i
,
a/2
m=100 u
Q.
\ r = 0.75 rmax
-1.0x10 -4 Cubic
-1.0x10"8 F_ .¢ (n
-2.0xl0 -8
Trap
[
.1.5x10 "4
u. ¢D
O" IL
z=le
e
,
position
Axial
A
-2.0x10"8
-2.0x10 "4
r
r
~
i
-4.0xlo "8 o.o
c/2
0.0 Axial
position
Fig. 5. Calculated image charge-induced ion cyclotron orbital frequency shift as a function of axial position, for each of three radial positions (r/r=~x = 0.25, 0.50, and 0.75; rm~x= a[2), at 3.0T, for the cylindrical (top) and cubic (bottom) traps of Figure 5.
the average closer to the walls of the cylinder than to the walls of a cubic trap of comparable size.
Effect of ion axial position on image charge-induced ion cyclotron orbital frequency shift Figure 5 shows the image charge-induced ion cyclotron orbital frequency shift as a function of the axial position of the ion, for various ICR orbital radii (again for zero magnetron radius). The frequency shift decreases as the ion approaches either trapping electrode, because less charge is induced on the "side" electrodes (i.e. the electric field lines from the ion terminate on the trapping electrode rather than on "side" electrodes. As for the radial-dependence discussed above, the image charge-induced ion cyclotron orbital frequency shift is less for a cubic than for a cylindrical trap of comparable size.
-3.0x10"8
n-
,
i
I
c/2 Axial Position
Fig. 6. Calculated image charge-induced relative ion cyclotron orbital frequency shift, lain+ I/m+ (see text) as a function of ion cyclotron orbital radius (top) or axial position (bottom) for a singly-charged ion of mass, 100, 1000, or 10000u, at 3.0T in a I in cubic trap. Although the absolute frequency shift is massindependent (see text), the relative frequency shift increases with increasing ion mass.
Mass-independence of the image charge-induced ion cyclotron orbital frequency shift Equations (27) and (28) clearly show that the magnitude of the image charge-induced ion cyclotron orbital frequency shift, law+ J, is independent of ion cyclotron orbital frequency, and thus independent of ionic mass, m, for singly-charged ions. Thus, the relative shift in frequency, ]Ato+ f/o+, is extremely small for low-mass ions (which have high ion cyclotron frequency). However, the relative shift increases with increasing ionic mass, illustrated in Figure 6 for singly-charged ions in a one-inch cubic trap. Qualitatively similar behavior will be found for traps of other shape. It is also worth noting that, at constant orbital radius, the magnitude of the absolute frequency shift varies inversely with the trap radial dimension, because
42
x. Xiang et al./Int. J. Mass Spectrom. Ion Processes125 (1993) 33--43
the ion packet is farther from the electrodes of a larger-radius trap.
Effect of number of ions on image charge-induced ion cyclotron orbital frequency shift
ions of charge, n, if both ion packets are spatially coherent. Thus, the ICR orbital frequency shift due to image charge depends not only on mass-tocharge ratio, m/q, but also on the charge, q itself.
Implications for mass calibration in FT-ICR M S All of the above calculations were based on a single orbiting ion. Because the induced charge is directly proportional to the number, N, of coherently orbiting ions, we can approximate the frequency shift for N ions as NAto+. Of course, the observed frequency shift due to image charge depends (see Figs. 4-6) on the radial and axial position of the ion orbit. Moreover, we have not considered the effect of finite magnetron radius (i.e. off-axis ions).
Experimental significance of image charge-induced ICR orbital frequency shifts Recent theory and experiment have shown that the F T - I C R M S detection limit for conventional broad-band detection is ~ 175 ions (for C6H~- at 3.0 T, based on signal-to-noise ratio of 3 : 1 for a I s undamped data acquisition period); the maximum ion number is about 32000 times larger ( ~ 5.6 million ions) [23]. However, we would not expect to observe an experimental image charge-induced shift in the ICR orbital frequency of (say) singlycharged N~- ions (v+ ~ 1.65 MHz at 3.0 T), even at relatively high ion density and large ICR orbital radius, because the predicted frequency shift ( ~ 10-4Hz per ion, from Figs. 4 and 5) for even 100000 ions barely reaches the I ppm level for NJunder those conditions. Figures 4-6 show that the image charge-induced relative frequency shift becomes much larger for singly-charged ions of higher mass (e.g. ~ 300 ppm, or ~ 1 u for singly-charged ions of 3 000 u). Furthermore, an ion carrying n charges will produce an n-fold higher image charge, and thus an n2-fold higher frequency shift. However, the ICR signal from such an ion will also be n times larger than for a singly-charged ion. Thus, for a given signal-tonoise level, N singly charged ions should produce approximately the same signal-to-noise level as N/n
For FT-ICR MS experiments involving a large (say, i> 100000) number of ions, of which some have relatively high m/q (say, ~ 500), image charge effects may be minimized by limiting the ICR orbital radius to (say) 20% of the maximum available trap radius. Alternatively, from the ICR orbital radius calculated for a given trap geometry from known values of the r.f. excitation voltage magnitude and duration [25-27] and measured number of ions [23], the present results (adapted, if necessary, to different trap dimensions) could be used to correct the observed ICR orbital frequencies for space charge effects. We propose to test that approach in future experiments. Correction for image charge could be significant in ultraprecise FT-ICR mass determinations for ions of m/z <<.20 or so [13,28,29], for which milliHertz precision is required.
Acknowledgments The authors wish to thank C.W. Ross, III for help with the numerical calculation of the frequency shift and Dr. L. Schweikhard for helpful discussion. This work was supported by grants (to A.G.M.) from the U.S.A. National Science Foundation (CHE-90-21058), U.S.A. Public Health Service (NIH GM-31683) and The Ohio State University.
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