Computational project: Modelling a simple quadrupole mass spectrometer

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1 Computational project: Modelling a simple quadrupole mass spectrometer Martin Duy Tat a, Anders Hagen Jarmund a a Norges Teknisk-Naturvitenskapelige Universitet, Trondheim, Norway. Abstract In this project we have looked at numerical solution for a quadrupole mass spectrometer using the Forward Euler Method and the Runge-Kutta Method of Fourth order. We have investigated the parameters that determine which particles that reach the detector and which particles which do not. The two main parameters were the alternating and the direct voltage. Also, the mass of the particles and their initial position and velocity were varied to study how the detector was affected by these variations. For a N + particle it was determined that setting V AC = 6. V and V DC = 7.6 V where the optimal parameters, given that the particle starts within the detectors range and with a velocity directed no more than away from the z-axis. The calculations were mainly done in C++ while plotting was done in Matlab.. Introduction and theory.. Introduction A quadrupole mass spectrometer is a device that can detect particles with a particular mass relatively accurate if done correctly. It consists of four rod-shaped conductors placed in a square. When a combination of alternating, V AC, and direct, V DC, voltage is applied to the conductors, the resulting electric field in the middle of the square will accelerate charged particles. By adjusting the voltages, only particles with mass within a specific range will pass through the middle without being deflected. An approximation to the potential of a quadrupole is given by V = (V DC + V AC ) x y, () where it is assumed that the quadrupole lies in the xyplane such that the electrodes are parallel to the z-axis. The distance from the origin to each electrode is r. The positive electrodes are placed on the x-axis and the negative conductors are placed on the y-axis. The resulting differential equations of motion for the x, y and z direction are then d x dt d y dt r = q m (V DC + V AC ) x r, () = q m (V DC + V AC ) y r, (3) d z =, () dt where q is the charge of the particle and m is its mass... Numerical methods In general, the problems proposed in this report have no simple analytical solutions and thus have to be investigated by the means of numerical methods. Two such methods will be applied and compared; Euler s Method and Runge-Kutta Method.... Euler s Method Let an initial value problem be given as ẏ = f(t, y), y = a, t [ t, t ]. An approximation of y, w, may then be found by starting in a known point, eg. y, and follow the slope given by f a short distance h, or in general w = y, w i+ = w i + hf(t i, w i ), i t t = t + hi. t i h, (5) (6) Using Taylor s theorem, the global error of Forward Euler Method can be expressed as g = O(h). (7)... Runge-Kutta Method Another approach is using the Runge-Kutta Method of order four. The basic idea is similar to the one behind the Forward Euler Method, but we evaluate the slope at

2 several points and adding them with different weights; w i+ = w i + h 6 (s + s + s 3 + s ), where s = f(t i, w i ), s = f(t i + h, w i + h s ), (8) s 3 = f(t i + h, w i + h s ), s = f(t i + h, w i + hs 3 ), where the initial value problem is given as before. This time, the global error is g = O(h ). (9). Results and discussion.. Problem In figure and figure the potential and the electric field of four line charges with alternating polarity. Figure : A sketch of the field of four line charges, made by hand. In figure the lines are simply equipotential lines. For the special case of no alternating voltage the positive electrodes are the ones on the x-axis, and the negative electrodes are on the y-axis. We therefore see that the electric field in figure points towards the negative line charges on the y-axis. In figure 3 and figure similar sketches are made for the hyperbolic potential and with corresponding polarity. Therefore, in figure 3 the potential increases when moving away from the origin along the x-axis and decreases along the y-axis and the electric field in figure points away from the x-axis and towards the y-axis. Figure : A sketch of the potential of four line charges, made by hand. The gray colours was caused during scanning the sketch. It is not a part of the sketches. Figure 3: A sketch of the hyperbolic potential, made by hand.

3 y Figure 5: Plot of the potential from four line charges. Figure : A sketch of the hyperbolic field, made by hand. Quiver plot of E(x; y) If we study these figures closely, we see that the potential in figure 3 is very similar to the potential in the middle of the line charges in figure. It has the same curvature and also has the same asymptotic behaviour at the axes. Therefore we can safely approximate the potential of four line charges with a hyperbolic potential given by equation (), as long as we only consider the area close to the z-axis... Problem In order to plot the V (x, y) and E(x, y), dimensionless variables such as V (x,y) x V, r, etc has been used. r is, as before, the distance from the origin to each line charge, and V specifies the magnitude of the potential. Thus, if V AC =, V reduces to V = V DC. In figure 5 the potential of four line charges has been plotted. We see that near the charges the potential diverges to infinity since we have assumed that the charges are infinitely small. Also, V (x, y) diverges to near the charges on the x-axis, while it diverges to near the charges on the y-axis. r - Figure 6: charges x r Quiver plot of the electric field for the four line The corresponding electric field has been plotted in figure 6 as a quiver plot. It is clear the the electric field points away from the positive line charges and towards the negative line charges as they should, and the electric field lines are perpendicular to the equipotential lines. In figure 7 the hyperbolic potential has been plotted. The electric field has been plotted as a quiver plot in figure 8. Again, the electric field points away from the high potential as it should. 3

4 x (m) y y (m) 8, because the field in the x-direction points towards the origin and the magnitude increases as we move away from the origin. Moreover, in the y-direction in figure 9 the particle ends up really far away from the initial position. We also see from figure 8 that only a small deviance in the y-direction makes the particle fall quickly into the potential-void, or far away from our detector (note the axis dimensions in figure 9). Thus, the particle behaves exactly like expected in the y-direction too. # 3 5 Particle path r Figure 7: Plot of the hyperbolic potential Quiver plot of E(x; y) x r Figure 8: Quiver plot for the electric field for the hyperbolic potential. Fortunately, these plots are very similar to the sketches in Problem. The only difference is that the potential can now be plotted as a surface in three dimensions so we see more clearly where the potential is high or low. As a check, we set V DC = 5V, V AC = V and r = 3mm. Then we calculate the potential at some points: V (, ) =.V, V (, ) =.56V, V (, ) =.V, which obviously are the correct values x (m) # -3 Figure 9: Plot of projection of a particle s path, using the Forward Euler Method with timestep h =.µs. The oscillation in x-direction is also shown in figure with time step h =.µs. However, we also see that the amplitude is growing, which suggest that our numerical approximation has a non-negligible error. The energy of the particle is increasing, a trait common for the Forward Euler Method. This is certainly not in agreement with laws of physics. # -3 x-position over time.3. Problem 3 In this problem we want to begin modelling our spectrometer using V AC = V, V DC = 5 V and r = 3 mm. Then we send in a particle mimicking N +, so that m = 8 u and q = e, into the detector. The initial position of the particle is x = y = mm, z = mm and the initial velocity is v x = v y = m s, v z = 5 m s. To approximate the path given by equation (), (3), (), we implement the Forward Euler Method as described in section.. with h =. µs. A projection of this path into the xy-plane is found in figure 9. As we can see, the particle oscillates in the x-direction. This is as expected from figure 6, or more clearly, figure Time (s) # -5 Figure : Plot of the solution in the x-direction as a function of time, using the Forward Euler Method with timestep h =.µs. For the particular case when V AC = V an analytical

5 x (m) x (m) x (m) solution is achievable for the particle s path, namely r(t) = [x(t), y(t), z(t)] with ( ) x(t) = x cos qv DC mur t, ( ) y(t) = y cosh qv DC mur t, z(t) = z + v z t. () Here it is assumed that the initial conditions are the same as above. Any changes to the initial conditions will only give minor changes in these expressions. The x-component of such a movement is plotted in figure. The y- and z-components are trivial and therefore not shown. # -3 x(t) Numerical solution Analytical solution Time (s) # -5 Figure : Numerical and analytical solution in the x- direction. In the Forward Euler Method a time step of h =.µs has been used. # -3 Analytical solution.5 # -3.5 Numerical solution Analytical solution x(t) Time (s) # -5 Figure : Analytical solution for the x-position over time. This analytic solution supplies us with the opportunity to analyze how well our numerical results approximates the particles actual movement and how the error develops over time. We now change the initial position to x = mm, y = z = mm. Such a comparison is found in figure. As the numerical path deviates quite noticeable from the analytic, we may conclude that a smaller time step, h, is needed. If we decrease h by a factor, we get figure 3 where the path seems much more in accordance with the analytical solution. This relationship will be discussed in more detail in Problem Time (s) # -5 Figure 3: Numerical and analytical solution in the x- direction. In the Forward Euler Method a time step of h =.µs has been used... Problem We now set V AC = 5V and ω = 7 m s. The initial velocity of the particle is v z = 5 m s. The initial value of x(t) and y(t) are x = mm and y = mm. Now, the path of the particle has no simple analytical solution and all calculations will be done using Euler s method. For each iteration in the numerical calculations, the particle s distance from the z-axis is calculated. If the distance exceeds r, the particle is considered as lost, in other words it is not detected. The length of the electrodes are L = cm. At the end of the electrodes is a detector of radius r /. If the particle s distance from the z-axis is greater than this, it is not detected. In figure the particle s path has been plotted in three dimensions. 5

6 x (m) log(error) z (m) # D plot of the particle orbit -.5 y (m) Figure : 3D plot of the particle orbit. The solution was computed numerically using the Forward Euler Method with time step h =.µs. The particle was detected. It may not look like the particle s orbit is stable. However, upon a closer inspection the x- and y-components of the particle s position was found to be periodic, despite its chaotic motion. Therefore, it had to arrive at the detector after some time. The periodic motion i shown in figure 5. # -3 x(t) and y(t) x-position y-position x (m).5.5 # -3 some tolerance boundary. This is equivalent to determining the maximum value for the time step h and still have numerical errors that are smaller than some tolerance level. One way to analyze the error is to find the path for a particle over some chosen distance, equivalently, use a constant period. In this case a period of t = 6 µs was used, which corresponds to five oscillations. Then the difference between analytical and numerical, in other words the error, was found for each calculated point. For each value of h, the maximum error was found. Doing so, and then taking log of the maximum errors, a linear relation between h and the error reveals. This is shown in figure 6. Using linear regression, the slopes of error are found to be β Euler =.5 and β RK =., which fits remarkably well to the statements in section.. and.., that g Euler = O(h) log(g Euler ) log(h), () g RK = O(h ) log(g RK ) log(h). () Maximum error after 5 periods - Forward Euler Fourth order Runge Kutta Time (s) # -5 Figure 5: Plot of the x- and y-positions as a function of time using the Forward Euler method with timestep h =. µs log(h) Figure 6: loglog-plot of the absolute error as a function of the time interval h. The slope of the curves are.5 and.. Here we see clearly that the path is stable. Therefore it reached the detector and it was detected. Since it seems like the amplitude of the oscillations are not increasing significantly, there is probably no need to decrease the time step h..5. Problem 5 Until now we have been satisfied with the Forward Euler Method. Despite its simplicity, there exists methods which are much more precise, with only small increases in the number of calculations per iteration. One such method is Runge-Kutta Method. As we have got an analytical solution described by equation (), we are able to compare the two numerical methods and also investigate how few calculations we could have that are still sufficient to keep the error below 6 It should noted that as the global error increases over time, the greatest distances are found towards the end of the path in combination with maximum deflection. However, in our analysis we looked at all estimated points. Of course, one could have searched for the biggest relative error for each h instead of the absolute error, but since we are crossing zero several times, such an approach is vulnerable for dividing by zero and relating issues. To make sure that the global error is less than.%, each point in figure 6 was inspected. It was found that for Runge-Kutta for h =.5 µs the relative error was.5% and for h =. µs the relative error was.3%. Thus, for h =. µs the error is well below.%. In comparison, the lowest time step possible for an average computer was h =. µs and the maximum error after five periods using the Forward Euler Method was 3.%. These estimates only applies to the case with an analytical solution. To be sure that our results would not suffer

7 from significant numerical errors, h was set to h =. µs, and since the global error decreases in the order of h this will probably be good enough for all practical purposes. As the Runge-Kutta Method is quite supreme to the Forward Euler Method, the Runge-Kutta Method is always used if not otherwise specified. x/y (m) Unstable Stable 3 # -3 Unstable orbit x-position y-position. Figure 8: Plot of the x- and the y-position of a stable orbit. Here, VAC = V and VDC = 3V. Stability plot.5 z (m) x/y (m) VDC In figure 7 a stability plot is shown. A green point means that the particle s path reached the detector for the specified values of VAC and VDC, and a red point means that the particle s orbit was unstable. In this plot, the starting values where VAC = V and VDC = V. The increments for VAC is. V and the increments for VDC is.5 V. 8 Stable orbit x-position y-position.6. Problem 6 # -3 6 VAC - z (m) Figure 7: Overview of the stability of particle orbits as a function of VAC and VDC. 6 # -3 Figure 9: Plot of the x- and the y-position of an unstable orbit. Here, VAC = 3V and VDC = 6V. We see the that the stable area forms a triangle. The top of the triangle is where VAC = 6. V and VDC = 7.7 V. To distinguish between detected and undetected particles the same method as in section. was used. It may not be clear from figure 7, but there are some red dots appearing inside the green triangle, especially on the right hand side when VAC is high. This is probably because the particles are oscillating with a higher amplitude as VAC is increased. The particles will then sometimes arrive outside the detector, even though they follow a stable path. This could explain why the triangle boundary on the right side slowly fades away. On the other hand, on the left hand side, it seems like the triangle has a sharp boundary. This could be that the particles are falling off exponentially in the y-direction in the absence of VAC. Therefore, when VAC is increased beyond some point, the particle starts oscillating in the y-direction too and we get a stable path. Examples of stable and unstable paths are shown in figure 8 and 9, repectively. In the unstable orbit it is mainly because the particle fly away in the y-direction. This seems correct since the electrodes on the y-axis were set to be the negative electrodes, and we have implicitly assumed that our particle has a positive charge. Therefore it will be attracted towards the electrodes on the y-axis. To investigate this further, the particle mass was varied in integer steps in units of u. First, VAC and VDC were set to VAC = 6. V and VDC = 7.7 V respectively. For this combination of VAC and VDC it was only particles with mass m = 8 u which were detected. Thus we have created the perfect mass spectrometer. A plot of the proportion of detected particles against particle mass is plotted in figure. This is considered to be the ideal mass spectrometer where % of all particles of mass m = 8 u are detected and % of all other particles are detected. 7

8 Proportion detected (%) Proportion detected (%) Proportion detected (%) Figure : The proportion of particles detected as a function of mass in atomic masses. Here, V AC = 6V and V DC = 7.7V. However, we have also assumed that the particle entered the system at x = mm, y = mm and z = mm, and with initial velocity v z = 5 m s parallel to the z-axis. This is not applicable for a real mass spectrometer and therefore we must relax our conditions. We therefore set V AC = 5. V and V DC = 7. V. By using different masses from u to 8 u in integer steps, the proportion of particles detected as a function of mass is plotted in figure Figure : The proportion of particles detected as a function of mass in atomic masses. Here, V AC = 5V and V DC = 7V. Here we see that not only masses of m = 8 u are detected, but m = (8 ± ) u. Thus there will be some variations in the masses detected. However, the mass spectrometer described by figure requires completely idealistic conditions which will never be satisfied in reality. So V AC = 5. V and V DC = 7. V are probably the most optimal settings for detected particles with mass m = 8 u. On the other hand, one might end up with a completely useless mass spectrometer if the conditions are relaxed too much. One example of a mass spectrometer is shown in figure. Figure : The proportion of particles detected as a function of mass in atomic masses. Here, V AC = 37V and V DC = V. Here, V AC = 37. V and V DC =. V. We see now that the mass spectrometer detects particles of mass m = (8 ± 7) u. With such a high uncertainty it will not be capable of distinguishing different atoms from each other. Notice that in figure and figure, for each mass either % or % of the particles are detected. This is because all particles are sent in with the same initial position and velocity, so the outcome must necessarily be the same for each mass. A more realistic model is considered in section.8. After some trial and error, the relationship between V AC, V DC and the mass of the detected particles was determined. It was found that by decreasing V DC the interval of masses for the detected particles increased, mostly for heavier masses. By decreasing V AC instead the whole interval of detected masses was shifted towards lighter masses, but the interval did not necessarily increase. It mostly decreased when V AC was decreased..7. Problem 7 If a particle of mass m = u is to be detected using the same mass spectrometer, one must look back to equation () and (3). We already know that setting V AC = 5 V and V DC = 5 V will give a stable path. If the mass is only one half the original mass, then V AC and V DC must also be one half of the original value in order to reproduce the results in section.6. The stability diagram for m = u is shown in figure 3. 8

9 Proportion detected (%) Proportion detected (%) Proportion detected (%) 5 Unstable Stable Stability plot V DC V AC Figure 3: Overview of the stability of particle orbits as a function of V AC and V DC. We see that the main difference from figure 7 is that the axes are scaled by a factor of, which makes sense from the consideration of the equations of motion above. Also, it is more clear here that the green triangle fades away on the right side. In figure a plot of the proportion of detected particles as a function of particle mass has been plotted. This is an ideal mass spectrometer for m = u. Here, V AC = V and V DC = 3. V, which is not perfectly scaled by a factor of compared to figure, but it is very close. Figure 5: The proportion of particles detected as a function of mass in atomic masses. Here, V AC = V and V DC = 3.V. In figure 6 is a plot for a typically useless mass spectrometer. The particles that are detected have mass m = ( ± 7) u In conclusion, the parameters V AC and V DC scale approximately proportionally with the mass of the particles detected. This fits quite well with the differential equation of the particle s path Figure 6: The proportion of particles detected as a function of mass in atomic masses. Here, V AC = 3V and V DC =.7V. Figure : The proportion of particles detected as a function of mass in atomic masses. Here, V AC = V and V DC = 3.V. In figure 5 is a plot for a typically good mass spectrometer. We see that particles with mass m = ( ± ) u are detected. Thus the optimal parameters for detecting particles with mass m = u are V AC =. V and V DC = 3. V..8. Problem 8 In the so called real world it would be quite hard to make all our particle enter the spectrometer through the exact same point and with a velocity exclusively in z- direction. As we want our model to be somewhat applicable to this alleged real world, we supply our particles with randomized start position and velocity. But we still want the start positions to be within a distance of r / =.5 mm from the origin, and the velocity to be directed within from the z-axis, and we ve chosen to keep v z = 5 m s constant for all particles. Also the masses were set to random integers around m = 8u. In this way, we sent N = particles into our spectometer 9

10 received/sent received/sent received/sent and registered which one that reached the detector. The proportions received/sent is shown in figure 7 and 8 for a few, different configurations. We stress that small changes in the voltages do cause significant changes in distribution. Also, several of the bars has a height of zero dependently of start velocity and position. However, only half of these particles reach the detector. From figure 3 we know that only a small change in the voltages will change the particle s path and thus it make sense that even a minimal deviance in initial conditions is enough to cause the particle to leave its stable path. One may also note that V AC = 7 V, V DC = 7.7 V causes all particles with mass m = 9u to reach the detector, and no other particles. After some failing, similar results were achieved for m = 8 u, namely V AC = 6. V, V DC = 7.6 V (illustrated in figure 9) m(u) Figure 7: Proportion of detected particles by mass, m, for different voltages, with randomized start position (within r /) and velocity (max degrees deviance from ẑ). Voltage is given as V AC V DC. In figure 7 we see that setting V DC = 5V and V AC = 5V or V AC = 6V will let through particles with a mass range of atomic masses. A change in V AC will only shift this interval. In fiugre 8, however, we see that when V DC is cranked up to V DC = 7.7V, we manage to make the interval somewhat smaller m Figure 9: Proportion of detected particles by mass, m, for different voltages, with randomized start position (within r /) and velocity (max degrees deviance from ẑ). V AC = 6. V, V DC = 7.6 V. N = 5. Unfortunately we do not have any analytical solutions or any certain method of confirming these statistics. We also notice that the statistics are highly unpredictable because of its high sensitivity on V DC and V AC. However, in the end of Problem 6 some relations between the masses detected, V AC and V DC, were determined, and we see that those are confirmed in figure 7 and 8. The simple trends of how the particle s paths depended on V DC and V AC still agree! Therefore, we can most likely rely on these results. Since only one specific value of m is detected, our spectrometer is quite perfect, making mass resolution a somewhat imprecise concept (especially since there is no formal definition given to us) m(u) Figure 8: Proportion of detected particles by mass, m, for different voltages, with randomized start position (within r /) and velocity (max degrees deviance from ẑ). Voltage is given as V AC V DC. If you inspect figure 8 closely, you ll see that V AC = 6 V, V DC = 7.7 V only permits particles with m = 8u in-