- Email: [email protected]
A TOF mass spectrometer with higher resolution and sensitivity via elimination of chromatic TOF aberrations of higher orders
International Journal of Mass Spectrometry 376 (2015) 23–26
Contents lists available at ScienceDirect
International Journal of Mass Spectrometry journal homepage: www.elsevier.com/locate/ijms
Short communication
A TOF mass spectrometer with higher resolution and sensitivity via elimination of chromatic TOF aberrations of higher orders Seitkerim B. Bimurzaev ∗ Almaty University of Power Engineering and Telecommunication, Almaty, Kazakhstan
a r t i c l e
i n f o
Article history: Received 24 June 2014 Received in revised form 2 November 2014 Accepted 3 November 2014 Available online 22 November 2014 Keywords: Time-of-flight mass spectrometer Time-of-flight chromatic aberration Time-of-flight focusing Spatial and time-of-flight focusing Immersion objective
a b s t r a c t The expressions describing the time-of-flight (TOF) and conditions of TOF focusing of ion packets by energies in the ion source with two accelerating gaps have been obtained. The first gap is the ionization region with a uniform electrostatic field, whereas the second gap is the immersion objective with a nonuniform electrostatic field, where the role of the “cathode” (emitting surface) is played by the exit window of the ionization region. Numerical calculations were used to obtain the ratios between geometric and electrical parameters of three-electrode immersion objectives with two-dimensional and rotational symmetry, determining conditions of four orders of TOF focusing by energy simultaneously with spatial focusing of ion packets. The schemes of time-of-flight mass spectrometers of high resolution and high sensitivity with direct and orthogonal ion injection are presented. © 2014 Elsevier B.V. All rights reserved.
1. Introduction It is well known that the TOF mass spectrometer of simple structure [1] consisting of an ion source, a field-free drift space and an ion detector has a relatively low resolution and sensitivity. The main factor limiting its resolution is the initial energy spread of ions in the packet generated by the ion source. A low sensitivity is caused by the fact that a uniform electric field in the accelerating gap of the ion source formed by flat fine-structure grids cannot provide spatial focusing of ion packets. The presence of fine-structure grids also reduces sensitivity. To improve the resolution of TOF mass spectrometer, the authors [1] used the ion source providing TOF focusing of ions by energy in the plane coinciding with the plane of the detector. This method enabled them to eliminate some terms in the expansion of the total time-of-flight of ions in powers of the initial energy spread. However, in this case the first-order TOF chromatic aberration typical of any emission system remains unchanged [2]. It is this aberration that determines the width (in the direction of movement) of the ion packet in the detector plane [3]:
z ≈ f
ε , ˚0
∗ Tel.: +7 7014651014. E-mail address: [email protected] http://dx.doi.org/10.1016/j.ijms.2014.11.007 1387-3806/© 2014 Elsevier B.V. All rights reserved.
(1)
where f is the focal distance from the ion source, qε is the initial energy spread of ions, q˚0 is the drift energy of ions, and q is the ion charge. To reduce the influence of the width on the resolution in the TOF mass reflectron [3] a temporary primary focus is created near the source. Then, in the image plane of the detector, the ion reflector creates an image of the ion packet of a width close to its own width in the plane of the temporary primary focus. This paper considers a possibility of creation of a simple-circuit TOF mass spectrometer (without an ion reflector) with high resolution and sensitivity. To solve this problem, the ion source must have two accelerating gaps – the ionization region with a uniform electrostatic field and a system of electrodes forming a non-uniform electrostatic field. This field is directly adjacent to the exit window of the ionization region, forming the immersion objective [4], in which the role of the “cathode” (emitting surface) is played by the exit window of the ionization region. Only such mutual arrangement of the accelerating gaps provides elimination of the first-order TOF chromatic aberration of the ionization region. In addition, the non-uniform field of the immersion objective enables us to get high-quality TOF focusing of ion packets in the plane of the detector simultaneously with spatial focusing. TOF chromatic aberrations play an important role in TOF focusing of charged particle beams. In electron-optical systems with a straight optical axis TOF geometrical aberrations are effectively reduced by simple diaphragming, i.e. using rather narrow paraxial beams. TOF chromatic aberrations remain unchanged and impose principal limitations on the quality of TOF focusing.
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S.B. Bimurzaev / International Journal of Mass Spectrometry 376 (2015) 23–26
2. Time-of-flight
(3)
Dtε = −
Let us consider an ion source consisting of an ionization region with a uniform electric field and an accelerating gap in the form of an immersion objective with a non-uniform electric field. In order to study TOF chromatic aberrations of the ion source it is sufficient to consider the motion of particles along its main optical axis z and to determine the dependence of the time of flight of particles on their initial energies. Let us first determine the time of flight of ions in the ionization region. Any ion of mass m and initial energy qU0 moving in the ionization region will increase its energy to the value [1]: mz˙ 2 = q(U0 + sE), 2
(2)
where E = U1 /s1 is the strength of the electric field in the ionization region of width s1 created by the pushing-out impulse U1 , s is the path length of the ion in the field of the pushing-out impulse. When ions leave the ionization region, they have different energies due to two factors: 1) different initial energies qU0 ; 2) different path lengths s (0 ≤ s ≤ s1 ). In order to simplify the calculations let us rewrite the Eq. (2) as mz˙ 2 = qε (0 ≤ ε ≤ U0 + U1 ), 2
(4)
Dtε = (5)
where the sign ± shows the direction of the initial speed of the ion. For two ions formed in one point and moving in different directions at the same absolute values of initial speeds, the difference in the time of flight caused by the “turn-around time” is expressed as [1] t1 =
2 E
2mU0 , q
(5)
which also follows from (4). In this case the maximum possible time of flight of any ion in the ionization region can be written as
t1 max =
2m 1 √ ( ε + 3 U0 ). q E
(6)
T=
0
(z
5 k/2 ε
1
0
k=1
˚0
(1)
2˚0 , ˚u
1 (2) (1) Dtε = − (z − zT ), 2
−
˚ u 3˚u
(14)
,
4 ˚0 3 ˚u
(2)
zT = zu +
J1 =
1 2
z zu
z zu
1 ˚0 J2 = − 2 ˚u 3 ˚0 J3 = 4 ˚ u
(15)
(16)
1 − (1 + J1 )
˚0 ˚u
+ J2 + J3
2
˚ u
,
(17)
˚0 (zu − z)˚ dz, ˚ ˚
˚ − ˚
˚0 ˚
u
˚
z zu
z
˚0 ˚
˚0 ˚
zu
(18)
+
˚u dz, ˚0
˚ − ˚
u
˚
˚ −˚ u ˚
(19)
˚ + u ˚0
2
−
dz,
˚u ˚0
(20)
2 dz.
(21)
The total time of flight t = t1max + t2 of the ion in the ion source, where the plane of the exit window of the ionization region coincides with the plane z = zu of ion emission from the immersion objective, can be determined as (22)
where T and t are determined by Eqs. (8)–(9) with the only difference that the first-order TOF chromatic aberration coefficient for the ion source is expressed as
1 E
1+3
U0 ε
1 − ˚u
,
(23)
(8) 3. Conditions for elimination of TOF chromatic aberrations As it is seen from (23), if the condition (k)
Dtε
(9) E = ˚u
is the total TOF chromatic aberration of the immersion objective, (k) where Dtε (k = 1, 2, . . ., 5) is the aberration coefficient of the k-th order: Dtε = −
˚u
2˚0 (1 + J1 ), ˚u
(1)
zT = zu +
(1)
is the time of flight of the central ion (with ε = 0) and t =
˚u
5
Dtε = 2˚0 (0) − zT )
(13)
3 ˚ 2 8 ˚0 u
(7)
Here 1
(12)
t = T + t,
According to [5,6], the time of flight of the ion in the immersion objective from its emission point z = zu to an arbitrary plane z = const can be written as t2 = T + t.
,
(0)
1 J0 = 2
(4)
˚u
zT = zu + J0 ,
where qε is the total energy spread of ions in the packet formed by the ionization region. Then the time-of-flight of an ion in the ionization region can be written as [1] t1 =
˚u
Here and further 0 = 2q˚0 /m is speed of the central ion in the drift space with potential ˚0 , index “u” denotes the values in the emission point z = zu of ions, primes denote differentiation with (0) (1) (2) respect to z. The values of zT , zT and zT determining (according to (8), (11) and (13)) positions of the efficient plane of ion emission and reference planes of the TOF focusing of ions by energy are functions of the axial distribution of the accelerating potential ˚ = ˚(z) [7]:
where
2m 1 √ ( ε ± U0 ), q E
˚ 2 ˚ 0 u
3 (2) (z − zT ), 8
Dtε = −
(3)
4 3
(10) (11)
1+3
U0 ε
(24)
is fulfilled, the first-order TOF chromatic aberration coefficient is (1) equal to zero (Dtε = 0). (1)
(2)
If the condition z = zT or z = zT is fulfilled, it follows from the Eqs. (11) and (13) that the TOF chromatic aberration coeffi(2) (4) cient of the second-order (Dtε = 0) or the fourth-order (Dtε = 0) is, respectively, equal to zero.
S.B. Bimurzaev / International Journal of Mass Spectrometry 376 (2015) 23–26
25
Both coefficients are equal to zero simultaneously if the condition (1)
(2)
z = zT = zT
(25)
is fulfilled. In case of the flat surface of the exit window in the ionization region (˚u = 0), it follows from the Eqs. (12) and (14), that the third-order TOF chromatic aberration coefficient is equal to zero (3) (Dtε = 0) and the fifth-order TOF chromatic aberration coefficient is expressed as 8 15
(5)
Dtε =
˚ 3 ˚ 0 u ˚u
˚u
.
(26)
Therefore, if conditions (24)–(25) are satisfied, in the TOF mass spectrometer with the immersion objective, where the role of the “cathode” is played by the flat exit window in the ionization region, it is possible to get ion TOF focusing by energy up to the fourth order. 4. Spatial-time-of-flight focusing If the focal plane z = zF of the immersion objective coincides with reference planes of the TOF focusing, i.e. if condition (1)
(2)
z = zT = zT = zF
(27)
is fulfilled in the plane of the detector coinciding with the focal plane z = zF of the immersion objective, the TOF focusing of ions by energy to the fourth order inclusively is achieved simultaneously with spatial focusing. The location of plane z = zF is determined from the equation p , p
zF = z −
(28)
where p = p(z) is a partial solution of the paraxial equation
˚p +
1 ˚ p + Qp = 0 2
(29)
satisfying the following initial conditions pu = −
pu = 1,
2Qu . ˚u
(30)
Here Q =
1 ˚ + Qϕ 4
(31)
where Qϕ = Qϕ (z) is a quadrupole component of the electrostatic field. It is known that for systems with rotational symmetry Qϕ = 0, and for systems with two-dimensional symmetry Qϕ = (1/4)˚ . 5. Time-of-flight dispersion and mass resolution (1)
The plane z = zT is said to be the main reference plane of the TOF focusing of the immersion objective. Let us rewrite the Eq. (8) taking into account (25) as follows T = T0 −
1
0
(1)
(z − zT ),
(32)
where T0 =
1
0
(1)
(0)
(zT − zT )
(33)
is the time of flight of the central ion from the point of its emission (1) z = zu to the main reference plane of TOF focusing z = zT . This time is called the time interval of focusing. The dependence of the time interval of focusing on the ion mass determines the value of TOF dispersion by masses of the immersion objective Dtm = m
∂T0 1 = T0 . 2 ∂m
(34)
Fig. 1. Scheme of TOF mass spectrometer with a three-electrode immersion objective with rotational symmetry for direct injection of ions 1, 2, 3 are electrodes of the immersion objective, 4 are ion trajectories, d is the diameter of the cylinder.
The time interval of focusing also determines the effective drift distance L of the immersion objective L = 0 T0 .
(35)
If conditions (24) and (25) are fulfilled simultaneously, the limiting resolution R by masses of the TOF mass spectrometer with the immersion objective is determined by the equation R=
˚0 Dtm = R0 ε t
5/2 ,
(36)
where R0 =
L (5)
(37)
2Dtε
is a dimensionless coefficient. 6. Calculations of time-of-flight mass spectrometers Three-electrode immersion objectives with two types of symmetry: rotational and two-dimensional have been studied. In the immersion objective with rotational symmetry the accelerating non-uniform electrostatic field is created by two coaxial cylinders of equal diameter, whereas in the immersion objective with twodimensional symmetry it is created by two pairs of flat plates. In both cases, the role of the “cathode” is played by the exit window of the ionization region. The ratio between geometric and electrical parameters for the three-electrode immersion objective satisfying the condition (27) was obtained by numerical calculations. The values of electrical quantities are given in units of ˚0 (potential of drift space), and the values of the geometric quantities are given in units of d (the diameter of the cylinder in the rotational symmetry system or the distance between the plates in the two-dimensional system). The immersion objective with rotational symmetry (Fig. 1) has the following parameters: V1 = 0, V2 = 0.111˚0 , V3 = ˚0 , where V1 , V2 and V3 are potentials on the first, second and third electrodes, respectively, R0 = 1.24, zF = 8.06d, l = 0.743d is the width of the second electrode, d is the diameter of the cylinder. Here and further the plane z = 0 coincides with the plane of the exit window of the ionization region. The immersion objective with two-dimensional symmetry (Fig. 2) has the following parameters: V1 = 0, V2 = 0.121˚0 , V3 = ˚0 , R0 = 3.47, zF = 7.74d, l = 0.873d. Here d is the distance between the electrode plates. According to the method of formation of ion packets, the TOF mass spectrometer can be divided into two types: 1) TOF mass spectrometer with a conventional (direct) ion injection; 2) TOF mass spectrometer with orthogonal ion injection. In the first type, ions formed in the ionization region, under the action of the pushingout pulse generate pulsed ion packets in the form of flat discs, and enter the accelerating gap of the ion source. In the second type, the stationary ion beam in the form of a “cord” is transformed into pulsed packets in the form of short segments in the direction perpendicular to the movement of the stationary ion flow under
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S.B. Bimurzaev / International Journal of Mass Spectrometry 376 (2015) 23–26
are not worse than the same parameters of TOF mass reflectrons. Such high resolution is achieved due to possibility of eliminating of the first-order TOF chromatic aberration and high-quality TOF focusing of ion packets by energy in such a device. In addition, the non-uniform electrostatic field of the immersion objective provides spatial focusing of ion packets, which increases sensitivity of the device. Fig. 2. Scheme of TOF mass spectrometer with a three-electrode immersion objective with two-dimensional symmetry for orthogonal injection of ions. 1, 2, 3 are electrodes of the immersion objective, 4 are ion trajectories, d is the distance between electrode plates.
the action of periodically accelerating electric field, the so-called orthogonal accelerator. Figs. 1 and 2 show schematics of TOF mass spectrometer with direct and orthogonal ion injection, which use an immersion objective of rotational or two-dimensional symmetry, respectively. 7. Conclusions In conclusion it should be noted that the results of this research lay a physical basis for creation of a simple TOF mass spectrometer (without an ion reflector) whose resolution and sensitivity
Acknowledgments This work was supported in part by the Ministry of Education and Science of the Republic of Kazakhstan, grant no. 5 IPS GF3. References [1] W.C. Wiley, I.H. McLaren, Rev. Sci. Instrum. 26 (12) (1955) 1150. [2] E.K. Zavoisky, S.D. Fanchenko, Rep. USSR Acad. Sci. 108 (2) (1956) 218. [3] B.A. Mamyrin, V.I. Karataev, D.V. Shmikk, V.A. Zagulin, Sov. Phys. J. Exp. Theor. Phys. 64 (1) (1973) 82. [4] V.M. Kel’man, S.Ya. Yavor, Electronnaya Optika, AN SSSR, Moscow-Leningrad, 1968, 488 pp. [5] Z.Z. Ibraeva, A.A. Sapargaliev, E.M. Yakushev, Sov. Phys. Zhurnal Tekhnicheskoi Fiziki 55 (1985) 2170. [6] S.B. Bimurzaev, R.S. Bimurzaeva, Nucl. Instr. Methods A 645 (2011) 219. [7] S.B. Bimurzaev, R.S. Bimurzaeva, B.T. Sarkeev, Sov. Phys. Radiotekhn. Elektron. 36 (11) (1991) 2186.
Contents lists available at ScienceDirect
International Journal of Mass Spectrometry journal homepage: www.elsevier.com/locate/ijms
Short communication
A TOF mass spectrometer with higher resolution and sensitivity via elimination of chromatic TOF aberrations of higher orders Seitkerim B. Bimurzaev ∗ Almaty University of Power Engineering and Telecommunication, Almaty, Kazakhstan
a r t i c l e
i n f o
Article history: Received 24 June 2014 Received in revised form 2 November 2014 Accepted 3 November 2014 Available online 22 November 2014 Keywords: Time-of-flight mass spectrometer Time-of-flight chromatic aberration Time-of-flight focusing Spatial and time-of-flight focusing Immersion objective
a b s t r a c t The expressions describing the time-of-flight (TOF) and conditions of TOF focusing of ion packets by energies in the ion source with two accelerating gaps have been obtained. The first gap is the ionization region with a uniform electrostatic field, whereas the second gap is the immersion objective with a nonuniform electrostatic field, where the role of the “cathode” (emitting surface) is played by the exit window of the ionization region. Numerical calculations were used to obtain the ratios between geometric and electrical parameters of three-electrode immersion objectives with two-dimensional and rotational symmetry, determining conditions of four orders of TOF focusing by energy simultaneously with spatial focusing of ion packets. The schemes of time-of-flight mass spectrometers of high resolution and high sensitivity with direct and orthogonal ion injection are presented. © 2014 Elsevier B.V. All rights reserved.
1. Introduction It is well known that the TOF mass spectrometer of simple structure [1] consisting of an ion source, a field-free drift space and an ion detector has a relatively low resolution and sensitivity. The main factor limiting its resolution is the initial energy spread of ions in the packet generated by the ion source. A low sensitivity is caused by the fact that a uniform electric field in the accelerating gap of the ion source formed by flat fine-structure grids cannot provide spatial focusing of ion packets. The presence of fine-structure grids also reduces sensitivity. To improve the resolution of TOF mass spectrometer, the authors [1] used the ion source providing TOF focusing of ions by energy in the plane coinciding with the plane of the detector. This method enabled them to eliminate some terms in the expansion of the total time-of-flight of ions in powers of the initial energy spread. However, in this case the first-order TOF chromatic aberration typical of any emission system remains unchanged [2]. It is this aberration that determines the width (in the direction of movement) of the ion packet in the detector plane [3]:
z ≈ f
ε , ˚0
∗ Tel.: +7 7014651014. E-mail address: [email protected] http://dx.doi.org/10.1016/j.ijms.2014.11.007 1387-3806/© 2014 Elsevier B.V. All rights reserved.
(1)
where f is the focal distance from the ion source, qε is the initial energy spread of ions, q˚0 is the drift energy of ions, and q is the ion charge. To reduce the influence of the width on the resolution in the TOF mass reflectron [3] a temporary primary focus is created near the source. Then, in the image plane of the detector, the ion reflector creates an image of the ion packet of a width close to its own width in the plane of the temporary primary focus. This paper considers a possibility of creation of a simple-circuit TOF mass spectrometer (without an ion reflector) with high resolution and sensitivity. To solve this problem, the ion source must have two accelerating gaps – the ionization region with a uniform electrostatic field and a system of electrodes forming a non-uniform electrostatic field. This field is directly adjacent to the exit window of the ionization region, forming the immersion objective [4], in which the role of the “cathode” (emitting surface) is played by the exit window of the ionization region. Only such mutual arrangement of the accelerating gaps provides elimination of the first-order TOF chromatic aberration of the ionization region. In addition, the non-uniform field of the immersion objective enables us to get high-quality TOF focusing of ion packets in the plane of the detector simultaneously with spatial focusing. TOF chromatic aberrations play an important role in TOF focusing of charged particle beams. In electron-optical systems with a straight optical axis TOF geometrical aberrations are effectively reduced by simple diaphragming, i.e. using rather narrow paraxial beams. TOF chromatic aberrations remain unchanged and impose principal limitations on the quality of TOF focusing.
24
S.B. Bimurzaev / International Journal of Mass Spectrometry 376 (2015) 23–26
2. Time-of-flight
(3)
Dtε = −
Let us consider an ion source consisting of an ionization region with a uniform electric field and an accelerating gap in the form of an immersion objective with a non-uniform electric field. In order to study TOF chromatic aberrations of the ion source it is sufficient to consider the motion of particles along its main optical axis z and to determine the dependence of the time of flight of particles on their initial energies. Let us first determine the time of flight of ions in the ionization region. Any ion of mass m and initial energy qU0 moving in the ionization region will increase its energy to the value [1]: mz˙ 2 = q(U0 + sE), 2
(2)
where E = U1 /s1 is the strength of the electric field in the ionization region of width s1 created by the pushing-out impulse U1 , s is the path length of the ion in the field of the pushing-out impulse. When ions leave the ionization region, they have different energies due to two factors: 1) different initial energies qU0 ; 2) different path lengths s (0 ≤ s ≤ s1 ). In order to simplify the calculations let us rewrite the Eq. (2) as mz˙ 2 = qε (0 ≤ ε ≤ U0 + U1 ), 2
(4)
Dtε = (5)
where the sign ± shows the direction of the initial speed of the ion. For two ions formed in one point and moving in different directions at the same absolute values of initial speeds, the difference in the time of flight caused by the “turn-around time” is expressed as [1] t1 =
2 E
2mU0 , q
(5)
which also follows from (4). In this case the maximum possible time of flight of any ion in the ionization region can be written as
t1 max =
2m 1 √ ( ε + 3 U0 ). q E
(6)
T=
0
(z
5 k/2 ε
1
0
k=1
˚0
(1)
2˚0 , ˚u
1 (2) (1) Dtε = − (z − zT ), 2
−
˚ u 3˚u
(14)
,
4 ˚0 3 ˚u
(2)
zT = zu +
J1 =
1 2
z zu
z zu
1 ˚0 J2 = − 2 ˚u 3 ˚0 J3 = 4 ˚ u
(15)
(16)
1 − (1 + J1 )
˚0 ˚u
+ J2 + J3
2
˚ u
,
(17)
˚0 (zu − z)˚ dz, ˚ ˚
˚ − ˚
˚0 ˚
u
˚
z zu
z
˚0 ˚
˚0 ˚
zu
(18)
+
˚u dz, ˚0
˚ − ˚
u
˚
˚ −˚ u ˚
(19)
˚ + u ˚0
2
−
dz,
˚u ˚0
(20)
2 dz.
(21)
The total time of flight t = t1max + t2 of the ion in the ion source, where the plane of the exit window of the ionization region coincides with the plane z = zu of ion emission from the immersion objective, can be determined as (22)
where T and t are determined by Eqs. (8)–(9) with the only difference that the first-order TOF chromatic aberration coefficient for the ion source is expressed as
1 E
1+3
U0 ε
1 − ˚u
,
(23)
(8) 3. Conditions for elimination of TOF chromatic aberrations As it is seen from (23), if the condition (k)
Dtε
(9) E = ˚u
is the total TOF chromatic aberration of the immersion objective, (k) where Dtε (k = 1, 2, . . ., 5) is the aberration coefficient of the k-th order: Dtε = −
˚u
2˚0 (1 + J1 ), ˚u
(1)
zT = zu +
(1)
is the time of flight of the central ion (with ε = 0) and t =
˚u
5
Dtε = 2˚0 (0) − zT )
(13)
3 ˚ 2 8 ˚0 u
(7)
Here 1
(12)
t = T + t,
According to [5,6], the time of flight of the ion in the immersion objective from its emission point z = zu to an arbitrary plane z = const can be written as t2 = T + t.
,
(0)
1 J0 = 2
(4)
˚u
zT = zu + J0 ,
where qε is the total energy spread of ions in the packet formed by the ionization region. Then the time-of-flight of an ion in the ionization region can be written as [1] t1 =
˚u
Here and further 0 = 2q˚0 /m is speed of the central ion in the drift space with potential ˚0 , index “u” denotes the values in the emission point z = zu of ions, primes denote differentiation with (0) (1) (2) respect to z. The values of zT , zT and zT determining (according to (8), (11) and (13)) positions of the efficient plane of ion emission and reference planes of the TOF focusing of ions by energy are functions of the axial distribution of the accelerating potential ˚ = ˚(z) [7]:
where
2m 1 √ ( ε ± U0 ), q E
˚ 2 ˚ 0 u
3 (2) (z − zT ), 8
Dtε = −
(3)
4 3
(10) (11)
1+3
U0 ε
(24)
is fulfilled, the first-order TOF chromatic aberration coefficient is (1) equal to zero (Dtε = 0). (1)
(2)
If the condition z = zT or z = zT is fulfilled, it follows from the Eqs. (11) and (13) that the TOF chromatic aberration coeffi(2) (4) cient of the second-order (Dtε = 0) or the fourth-order (Dtε = 0) is, respectively, equal to zero.
S.B. Bimurzaev / International Journal of Mass Spectrometry 376 (2015) 23–26
25
Both coefficients are equal to zero simultaneously if the condition (1)
(2)
z = zT = zT
(25)
is fulfilled. In case of the flat surface of the exit window in the ionization region (˚u = 0), it follows from the Eqs. (12) and (14), that the third-order TOF chromatic aberration coefficient is equal to zero (3) (Dtε = 0) and the fifth-order TOF chromatic aberration coefficient is expressed as 8 15
(5)
Dtε =
˚ 3 ˚ 0 u ˚u
˚u
.
(26)
Therefore, if conditions (24)–(25) are satisfied, in the TOF mass spectrometer with the immersion objective, where the role of the “cathode” is played by the flat exit window in the ionization region, it is possible to get ion TOF focusing by energy up to the fourth order. 4. Spatial-time-of-flight focusing If the focal plane z = zF of the immersion objective coincides with reference planes of the TOF focusing, i.e. if condition (1)
(2)
z = zT = zT = zF
(27)
is fulfilled in the plane of the detector coinciding with the focal plane z = zF of the immersion objective, the TOF focusing of ions by energy to the fourth order inclusively is achieved simultaneously with spatial focusing. The location of plane z = zF is determined from the equation p , p
zF = z −
(28)
where p = p(z) is a partial solution of the paraxial equation
˚p +
1 ˚ p + Qp = 0 2
(29)
satisfying the following initial conditions pu = −
pu = 1,
2Qu . ˚u
(30)
Here Q =
1 ˚ + Qϕ 4
(31)
where Qϕ = Qϕ (z) is a quadrupole component of the electrostatic field. It is known that for systems with rotational symmetry Qϕ = 0, and for systems with two-dimensional symmetry Qϕ = (1/4)˚ . 5. Time-of-flight dispersion and mass resolution (1)
The plane z = zT is said to be the main reference plane of the TOF focusing of the immersion objective. Let us rewrite the Eq. (8) taking into account (25) as follows T = T0 −
1
0
(1)
(z − zT ),
(32)
where T0 =
1
0
(1)
(0)
(zT − zT )
(33)
is the time of flight of the central ion from the point of its emission (1) z = zu to the main reference plane of TOF focusing z = zT . This time is called the time interval of focusing. The dependence of the time interval of focusing on the ion mass determines the value of TOF dispersion by masses of the immersion objective Dtm = m
∂T0 1 = T0 . 2 ∂m
(34)
Fig. 1. Scheme of TOF mass spectrometer with a three-electrode immersion objective with rotational symmetry for direct injection of ions 1, 2, 3 are electrodes of the immersion objective, 4 are ion trajectories, d is the diameter of the cylinder.
The time interval of focusing also determines the effective drift distance L of the immersion objective L = 0 T0 .
(35)
If conditions (24) and (25) are fulfilled simultaneously, the limiting resolution R by masses of the TOF mass spectrometer with the immersion objective is determined by the equation R=
˚0 Dtm = R0 ε t
5/2 ,
(36)
where R0 =
L (5)
(37)
2Dtε
is a dimensionless coefficient. 6. Calculations of time-of-flight mass spectrometers Three-electrode immersion objectives with two types of symmetry: rotational and two-dimensional have been studied. In the immersion objective with rotational symmetry the accelerating non-uniform electrostatic field is created by two coaxial cylinders of equal diameter, whereas in the immersion objective with twodimensional symmetry it is created by two pairs of flat plates. In both cases, the role of the “cathode” is played by the exit window of the ionization region. The ratio between geometric and electrical parameters for the three-electrode immersion objective satisfying the condition (27) was obtained by numerical calculations. The values of electrical quantities are given in units of ˚0 (potential of drift space), and the values of the geometric quantities are given in units of d (the diameter of the cylinder in the rotational symmetry system or the distance between the plates in the two-dimensional system). The immersion objective with rotational symmetry (Fig. 1) has the following parameters: V1 = 0, V2 = 0.111˚0 , V3 = ˚0 , where V1 , V2 and V3 are potentials on the first, second and third electrodes, respectively, R0 = 1.24, zF = 8.06d, l = 0.743d is the width of the second electrode, d is the diameter of the cylinder. Here and further the plane z = 0 coincides with the plane of the exit window of the ionization region. The immersion objective with two-dimensional symmetry (Fig. 2) has the following parameters: V1 = 0, V2 = 0.121˚0 , V3 = ˚0 , R0 = 3.47, zF = 7.74d, l = 0.873d. Here d is the distance between the electrode plates. According to the method of formation of ion packets, the TOF mass spectrometer can be divided into two types: 1) TOF mass spectrometer with a conventional (direct) ion injection; 2) TOF mass spectrometer with orthogonal ion injection. In the first type, ions formed in the ionization region, under the action of the pushingout pulse generate pulsed ion packets in the form of flat discs, and enter the accelerating gap of the ion source. In the second type, the stationary ion beam in the form of a “cord” is transformed into pulsed packets in the form of short segments in the direction perpendicular to the movement of the stationary ion flow under
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are not worse than the same parameters of TOF mass reflectrons. Such high resolution is achieved due to possibility of eliminating of the first-order TOF chromatic aberration and high-quality TOF focusing of ion packets by energy in such a device. In addition, the non-uniform electrostatic field of the immersion objective provides spatial focusing of ion packets, which increases sensitivity of the device. Fig. 2. Scheme of TOF mass spectrometer with a three-electrode immersion objective with two-dimensional symmetry for orthogonal injection of ions. 1, 2, 3 are electrodes of the immersion objective, 4 are ion trajectories, d is the distance between electrode plates.
the action of periodically accelerating electric field, the so-called orthogonal accelerator. Figs. 1 and 2 show schematics of TOF mass spectrometer with direct and orthogonal ion injection, which use an immersion objective of rotational or two-dimensional symmetry, respectively. 7. Conclusions In conclusion it should be noted that the results of this research lay a physical basis for creation of a simple TOF mass spectrometer (without an ion reflector) whose resolution and sensitivity
Acknowledgments This work was supported in part by the Ministry of Education and Science of the Republic of Kazakhstan, grant no. 5 IPS GF3. References [1] W.C. Wiley, I.H. McLaren, Rev. Sci. Instrum. 26 (12) (1955) 1150. [2] E.K. Zavoisky, S.D. Fanchenko, Rep. USSR Acad. Sci. 108 (2) (1956) 218. [3] B.A. Mamyrin, V.I. Karataev, D.V. Shmikk, V.A. Zagulin, Sov. Phys. J. Exp. Theor. Phys. 64 (1) (1973) 82. [4] V.M. Kel’man, S.Ya. Yavor, Electronnaya Optika, AN SSSR, Moscow-Leningrad, 1968, 488 pp. [5] Z.Z. Ibraeva, A.A. Sapargaliev, E.M. Yakushev, Sov. Phys. Zhurnal Tekhnicheskoi Fiziki 55 (1985) 2170. [6] S.B. Bimurzaev, R.S. Bimurzaeva, Nucl. Instr. Methods A 645 (2011) 219. [7] S.B. Bimurzaev, R.S. Bimurzaeva, B.T. Sarkeev, Sov. Phys. Radiotekhn. Elektron. 36 (11) (1991) 2186.