Problem 4: Mass Spectrometry Introduction Problem 4: Mass Spectrometry The Quadrupole The Problem Initial Ideas Binary System Ion Trapping The Final Idea Simplifying the Problem August 14, 2014 Particle Trajectories Initial Parameters Numerical Analysis Analytics New Trajectories Jeremy Budd (Cambridge), Mike Lindstrom (UBC), Iain Moyles Conclusion (UBC), Mary Pugh (UofT), Kevin Ryczko (UOIT) and Future Work
Mass Spectrometry Problem 4: Mass Spectrometry Introduction The Quadrupole The Problem Initial Ideas Binary System Ion Trapping The Final Idea Simplifying the Problem Particle Trajectories Mass spectrometry is a technique used to determine the Initial Parameters chemical composition of an unknown substance. Numerical Analysis Analytics A typical device separates charged atoms and molecules New Trajectories based on their charge to mass ratio. Conclusion and Future Many different techniques and devices are used to do this; Work the one presented to us was the quadrupole method.
The Quadrupole Mass Spectrometer Problem 4: Mass Spectrometry Introduction A mass filter that uses a combination of AC and DC The Quadrupole The Problem voltages to create an electric field with a narrow range of Initial Ideas mass passing through to reach the detector. Binary System Ion Trapping By controlling both the AC and DC voltage, particles with The Final Idea Simplifying the a specific mass pass through the device. Problem Particle AC gets rid of particles with smaller mass, DC gets rid of Trajectories particles with larger mass. Initial Parameters Numerical Analysis Analytics New Trajectories Conclusion and Future Work
The Problem Problem 4: Mass Spectrometry Introduction The Quadrupole Want to measure multiple masses all at once with an area The Problem detector. Initial Ideas Binary System Don’t want to lose any ions. Ion Trapping The Final Idea Can we achieve higher mass resolution using only an Simplifying the Problem electric field? Particle Trajectories Don’t want to use a magnetic field. Initial Parameters Numerical Analysis Analytics New Trajectories Conclusion and Future Work
Initial Ideas Problem 4: Mass Spectrometry Introduction The Quadrupole The Problem Initial Ideas Binary separation system. Binary System Ion Trapping Ion trapping. The Final Idea Simplifying the Problem Particle Trajectories Initial Parameters Numerical Analysis Analytics New Trajectories Conclusion and Future Work
Binary System Problem 4: Mass Spectrometry Introduction The Quadrupole Use electric fields to constantly separate groups of The Problem particles until they can no longer be separated. Initial Ideas Binary System Solves the problem of finding all the masses all at once, Ion Trapping The Final Idea only uses an electric field, and we don’t lose any ions. Simplifying the Problem Downfall is that it would be impossible to model and Particle Trajectories manufacture. Initial Parameters Numerical Analysis Analytics New Trajectories Conclusion and Future Work
Ion Trapping Problem 4: Mass Spectrometry Send the particles into a quadrupole like device where Introduction there would be an electric field opposing the particles The Quadrupole motion. The Problem Initial Ideas Carefully place special curvature traps where the particles Binary System Ion Trapping would then be separated by mass. The Final Idea The opposing electric field acts like a potential barrier for Simplifying the Problem the particles, this allows the particles with not enough Particle Trajectories energy to get trapped. Initial Parameters Numerical Analysis Then can measure (possibly through the magnetic field) Analytics the charged particles in each trap. New Trajectories Conclusion and Future Work
The Final Idea Problem 4: Mass Spectrometry Introduction The Quadrupole Send particles into a positively charged solenoid. The Problem Initial Ideas The frequency of oscillation for the particles trajectory Binary System differ due to the particles mass. Ion Trapping The Final Idea Akin to how a prism can separate the different colours of Simplifying the Problem light, the solenoid will create a dispersion pattern of the Particle particles being studied ( � Trajectories E -prism). Initial Parameters Numerical Analysis Analytics New Trajectories Conclusion and Future Work
3D to 2D Problem 4: Mass PM Spectrometry Introduction The Quadrupole The Problem Initial Ideas Binary System Ion Trapping The Final Idea Simplifying the Problem Particle Trajectories Initial Parameters Numerical Analysis Analytics New Trajectories Conclusion and Future Work ge 1
Electrostatics Problem 4: Mass Using Coulombs law, Spectrometry N � Ze 1 1 Introduction V ( x , y ) = + � � 4 πǫ 0 ( x − l ) 2 + ( y − 2 nh ) 2 ( x − l ) 2 + ( y − (2 n − 1) h ) 2 The Quadrupole j =0 The Problem � �� � � �� � Initial Ideas rj ρ j Binary System Ion Trapping and the equations of motion come from The Final Idea Simplifying the Problem � ¨ x , ¨ y � = − β ∇ V ( x , y ) . Particle Trajectories where Initial Parameters Numerical Analysis Analytics Ze 2 New Trajectories β = , and U 0 is the initial speed. 4 πǫ 0 mWU 2 Conclusion 0 and Future Work
ODE’s Problem 4: Mass Equations of Motion: Spectrometry � � N + x − 1 x Introduction � x = β ¨ , The Quadrupole r 3 ρ 3 The Problem j j j =1 Initial Ideas Binary System � � N Ion Trapping y − 2 jh + y − (2 j − 1) h � ¨ y = β The Final Idea r 3 ρ 3 Simplifying the j j j =1 Problem Particle Initial Conditions: Trajectories Initial Parameters x (0) = x 0 , 0 < x 0 < 1 Numerical Analysis 2 , ˙ x (0) = 0 , Analytics New Trajectories Conclusion y (0) = y 0 , y < 0 , ˙ y (0) = u 0 , u 0 > 0 . and Future Work
Initial Velocity Problem 4: Mass Spectrometry Want to select an initial velocity so the charged particles can Introduction overcome the potential barrier it sees from the charges and still The Quadrupole have some velocity left over. The Problem Initial Ideas Binary System Ion Trapping The Final Idea After using conservation of energy: Simplifying the Problem � � Particle � � Z β N Z β N Trajectories � � − u 0 > . � � � � Initial Parameters 2 ) 2 + ( Nh ) 2 ] 2 ) 2 + ( y 0 − Nh ) 2 ] [( x − 1 [( x 0 − 1 Numerical Analysis Analytics New Trajectories Conclusion and Future Work
Numerical Trajectories Problem 4: Mass Spectrometry 40 35 Introduction The Quadrupole 30 The Problem Initial Ideas Binary System 25 Ion Trapping The Final Idea 20 Simplifying the Problem 15 Particle Trajectories Initial Parameters 10 Numerical Analysis Analytics 5 New Trajectories Conclusion 0 and Future 0 0.2 0.4 0.6 0.8 1 Work
Sums are gross! Problem 4: Mass Spectrometry Since N ≫ 1 approximate the potential sums as integrals but Introduction turning it into a Riemann sum: The Quadrupole The Problem Initial Ideas Binary System Ion Trapping � y − 2 M N x ≈ − β N x � The Final Idea β ( x 2 + s 2 ) 3 / 2 ds r 3 Simplifying the 2 M y Problem j j =1 Particle � y − 2 M N Trajectories y − 2 jh ≈ − β N t � β ( x 2 + t 2 ) 3 / 2 dt Initial Parameters r 3 2 M Numerical Analysis y j j =1 Analytics New Trajectories Conclusion and Future Work
New Trajectories Problem 4: Mass Spectrometry After integrating, Introduction The Quadrupole � � � � The Problem y − 2 M y − 2 M − β N y β N y x = ¨ ρ x , y − 2 M − − ρ x − 1 , y − 2 M − 2 Mx 2 M ( x − 1) Initial Ideas ρ x , y ρ x − 1 , y � � Binary System − β N 1 1 1 1 y = ¨ ρ x , y − ρ x , y − 2 M + ρ x − 1 , y − Ion Trapping 2 Mx ρ x − 1 , y − 2 m The Final Idea Simplifying the where Problem Particle � Trajectories ( x − a ) 2 + ( y − b ) 2 . ρ a , b = Initial Parameters Numerical Analysis Analytics New Trajectories Conclusion and Future Work
Sums vs. Integrals Problem 4: Mass Spectrometry 40 35 Introduction The Quadrupole 30 The Problem Initial Ideas Binary System 25 Ion Trapping The Final Idea 20 Simplifying the Problem 15 Particle Trajectories Initial Parameters 10 Numerical Analysis Analytics 5 New Trajectories Conclusion 0 and Future 0 0.2 0.4 0.6 0.8 1 Work
Vertial Acceleration Problem 4: Mass 0.208 Spectrometry 0.206 0.204 0.202 Introduction 0.2 The Quadrupole 0.198 The Problem 0.196 Initial Ideas 0.194 Binary System 0.192 Ion Trapping 0.19 0 100 200 300 400 500 600 700 800 900 1000 The Final Idea Simplifying the Problem We notice that the y acceleration essentially vanishes. Particle Trajectories Averaging over the entire domain for integral formulation, Initial Parameters ¨ y ≈ 0 like we see in numerics. Likewise, averaging ¨ x and Numerical Analysis letting M → ∞ , Analytics New Trajectories � 1 � 1 x = β Conclusion ¨ x + and Future x − 1 h Work
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