From 6b340faada4aee8ba6bbeca60d2f77a4be2516d6 Mon Sep 17 00:00:00 2001 From: Maddy Runyon Date: Thu, 19 Jun 2025 13:07:04 -0400 Subject: [PATCH 1/6] Consolidating and moving quadrupole info Shortening the description in "mass_spec_design" Expanding and outlining the section in "mass_analyzer_design" --- mass_analyzer_design.qmd | 30 +++++++++++++++++++++++------- mass_spec_design.qmd | 19 ++----------------- references.bib | 9 +++++++++ 3 files changed, 34 insertions(+), 24 deletions(-) diff --git a/mass_analyzer_design.qmd b/mass_analyzer_design.qmd index aa5eb82..4ad0707 100644 --- a/mass_analyzer_design.qmd +++ b/mass_analyzer_design.qmd @@ -22,19 +22,35 @@ Why use an ESA instead of a magnet? Before or after magnet? Why is it so often s Advantage is that you can change a voltage quickly (nanoseconds), for instance to deflect a beam. -## Quadrupole Mass Analyzers {#sec-analzer-quadrupoles} +## Quadrupole Mass Analyzers {#sec-analyzer-quadrupoles} -Quadrupoles are a dynamic mass analyzer. Quickly changing voltages. Radio frequency. +### Background +Using a combination of static and alternating voltages, quadrupole mass analyzers generate electric fields, rather than magnetic fields, that filter ions based on their mass to charge ratio (m/z). The possibility of using a quadrupole electric field for mass filtering was first proposed in the 1950's by Wolfgang Paul and Helmut Steinwedel, who drew from earlier use of quadrupole magnetic fields for high energy accelerators [@Dawson_1976]. In contrast to sector instruments, quadrupole mass analyzers are considered “dynamic” – to produce a mass spectrum, the strength and/or frequency of their electric fields must be incrementally adjusted throughout the duration of a scan. -Hexapoles and Octopoles. RF multipoles in general. +### Mass filtering - how does it work? {#sec-quadrupole-mass-filtering} -Rod shapes and resulting EM field lines. Compromises with number and shape of rods. +Quadrupole mass filters consist of four parallel rods, connected in crosswise pairs. A potential of (U+Vcosωt) is applied to one pair, while a potential of -(U+Vcosωt) is applied to the second pair. Here, Vcosωt represents an alternating RF voltage with an amplitude of V and frequency of ω; U represents a static voltage ("DC offset"). More simply, you could express the potentials as (DC+RF), and -(DC+RF). -Advantages: no moving parts, very fast, reliable, inexpensive (see also, vacuum systems and requirements). Disadvantages: One m/z out the back end at a time. Throughput can be low. +Though the voltage applied to each pair of rods oscillates continuously, the DC component can be thought of as the *average* voltage applied to each pair. In general, high m/z ions respond to the average voltage. If the DC component is positive, they are repelled by the rods, and thus stabilized. If the DC component is negative, they are attracted to the rods and destabilized. Lighter ions respond mainly to the RF component, though in a somewhat less intuitive way. -Stability triangle figure. +Because the forces acting on ions traveling through the quadrupole are uncoupled in the X, Y, and Z directions, we can think of mass filtering in terms of two separate planes: the XZ-plane, or X-direction, and the YZ-plane, or Y-direction. Note that in the Z-direction (along the length of the quadrupole), the net force acting on ions is zero. + +In the X-direction, ion stability is determined by the two rods with positive DC voltage. In this direction, low m/z ions generally follow the RF component of the field, gain energy, and oscillate with increasing amplitude until they collide with the rods. Meanwhile, high m/z ions are pushed toward the Z-axis and stabilized by the positive DC component. This creates a "high-pass" filter in the X-direction that allows only heavier ions to maintain their trajectory. + +In the Y-direction, ion stability is determined by the rods with negative DC voltage. While the negative DC component has a "de-focusing" or destabilizing effect on large ions, the RF component can keep lighter ions on course by periodically correcting their paths. In this direction, the applied electric fields create a "low-pass" filter through which only lighter ions can maintain their trajectory. + +By manipulating the RF and DC components of the field, the high and low-pass filters can be set to resolve individual atomic masses. The following section discusses the impact of RF and DC voltages on ion stability in more detail. + +### Equations of motion and stability diagrams + + +### Operation and typical geometry of instrument + + +### Advantages + +Quadrupoles can be used on their own, linked in series (e.g. triple quadrupole), or combined (either as filters or analyzers) with other mass analyzers. Compared to sector instruments, they are reliable, fast, and relatively inexpensive. However, they generally offer lower throughput and resolving power. -Lots of math with funny symbols. ## Time of flight mass analyzer (TOF) {#sec-analyzer-TOF} diff --git a/mass_spec_design.qmd b/mass_spec_design.qmd index 2e2542e..6427e59 100644 --- a/mass_spec_design.qmd +++ b/mass_spec_design.qmd @@ -91,26 +91,11 @@ RIMS ### Electrostatic analyzers ### Quadrupoles {#sec-design-quadrupoles} - -Quadrupole mass analyzers generate electric fields through which only ions of certain mass to charge ratios are able to pass. As shown in the figure -need fig-, ions enter the quadrupole along it's z-axis. Those with m/z within the selected range will maintain a stable trajectory along the z-axis and reach the spectrometer's detector. The electric fields will destabilize the paths of ions with m/z outside the selected range, knocking them "off course" and preventing them from passing through the analyzer. -Dynamic. +Quadrupole mass analyzers – also referred to as [quadrupole mass filters](#sec-quadrupole-mass-filtering) depending on the application and/or reference source – generate dynamic electric fields through which only ions of certain mass to charge ratios (m/z) can pass. By varying the strength and frequency of the field, quadrupoles can filter for a single m/z, or scan sequentially across a range of m/z to produce a mass spectrum. -Analyzer vs. filter? filtering vs. bending? +Variants on the quadrupole mass filter include monopoles, quadrupole ion traps, and other multipole geometries (e.g hexapoles, octopoles, dodecapoles), all of which can be used for focusing ion beams in addition to filtering/analyzing. -**How does quadrupole mass filtering work?** - -Quadrupole mass analyzers use four parallel rods that are connected in cross-wise pairs. A potential of (U+Vcosωt) is applied to one pair, while a potential of -(U+ωt) applied to the second pair. Here, Vcosωt represents an alternating RF voltage and U represents a DC voltage ("DC offset"). The applied potentials create orthogonal electrical fields--one in the x-plane, and one in the y-plane--that interact differently with ions of different m/z. - -In the X-direction, low m/z ions generally follow the RF component of the field, gain energy, and oscillate with increasing amplitude until they collide with the rods. This creates a "high-pass" filter in the X-direction that allows only heavier ions to maintain their trajectory. - -In the Y-direction, the DC component of the field has a "de-focusing" or destabilizing effect, but the RF component can keep lighter ions on course by periodically correcting their paths. In this direction, the applied electric fields create a "low-pass" filter through which only lighter ions can maintain their trajectory. By manipulating the RF and DC components of the field, the high and low-pass filters can be set to resolve individual atomic masses. - -**Some math!** or **More details!** or **How are potentials calculated?** -- follow an ex: stability of ion of a single mass -- mathieu equation and "stability triangle" - -**Advantages of quadrupoles** ### Collision/Reaction cells (CRCs){#sec-design-crc} diff --git a/references.bib b/references.bib index 9d5e653..b43a702 100644 --- a/references.bib +++ b/references.bib @@ -168,4 +168,13 @@ @article{DeLaeter_2006 pages = {847-854}, doi = { 10.1002/jms.1057} } +@book{Dawson_1976, + author={Dawson, Peter H.}, + title={Quadrupole Mass Spectrometry and Its Applications}, + pages={3}, + publisher={Elsevier Scientific Pub. Co.}, + place={Amsterdam, NY}, + year={1976} +} + From 4892c295602757d9760f59cec26ee1c960676421 Mon Sep 17 00:00:00 2001 From: Maddy Runyon Date: Thu, 19 Jun 2025 15:00:17 -0400 Subject: [PATCH 2/6] quad fig plus equations outline added a figure to quadrupole section of mass_analyzer_design and outlined the equations of motion section --- figs/quad-fig6.png | Bin 0 -> 124707 bytes mass_analyzer_design.qmd | 19 ++++++++++++++++++- 2 files changed, 18 insertions(+), 1 deletion(-) create mode 100644 figs/quad-fig6.png diff --git a/figs/quad-fig6.png b/figs/quad-fig6.png new file mode 100644 index 0000000000000000000000000000000000000000..dbb6d0457da600f04a8316ba5474ccfff4f6215a GIT binary patch literal 124707 zcmaI72{_d6_dae7Wy!9DnnbcU$xdY~MaYsRYZ#F=%h<-!h7_SNWGP#A*|)|>2!jx^ z3?utCc4PN{M(_9M^Zj1`-|zRjuCA_Ynx1){^PF>^``qWe!gaOP8BU%(Nk&G-a9abW zM@Dwyh>VQtIc*axJ*i+vX?uoE|Vn>ER zAjBM89(&kWyV;4kKCw?CD4ZoDJ5P2ScGJM;!y?W%;N8^M!__Wnx{w>cHoF}KS 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changed, 22 insertions(+), 3 deletions(-) diff --git a/mass_analyzer_design.qmd b/mass_analyzer_design.qmd index 99e26ea..62c78d8 100644 --- a/mass_analyzer_design.qmd +++ b/mass_analyzer_design.qmd @@ -46,8 +46,21 @@ By manipulating the RF and DC components of the field, the high and low-pass fil ### Equations of motion and stability diagrams -- Equations of motion depend both on the (a) geometry of the E-field and (b) on time-dependent potentials -- Beginning with a general geometric description of s quadrupole E-field, impose differential form of Gauss’ law (/Laplace eqn), integrate to get a general eqn for potential, move some things around and plug in the time dependent potentials ±(U+Vcosωt) +The equations of motion that describe ion paths within a quadrupole depend on: + (a) the hyperbolic geometry of the electric field + (b) the time-dependent potentials (voltages) applied to the rods + +Starting with the geometry, we can come up with a general equation for the electric potential throughout the quadrupole, which is just the equation for potential within a hyperbolic electric field: +$$\Phi=-\frac{\Phi_0}{2r^2}(x^2-y^2)$$ + +For a step-by-step derivation of this equation, check out [section II] in [@Dawson]. + +::: {.callout-note collapse=true} +## Or see this brief overview, for reference + Start with a general geometric description of a quadrupole electric field$$E=E_0(\lambda x+\sigma y+\gamma z)$$ Choose $\lambda$, $\sigma$, and $\delta$ such that they satisfy the differential form of Gauss’ Law (the Laplace equation, since the charge enclosed in the quadrupole is zero): $$\nabla \cdot E=0$$ There are multiple solutions, but we will use: $$\lambda=-\sigma , \gamma = 0$$ + Integrate general equation of electric field to get the potential and then plug in our values for $\lambda$, $\sigma$, and $\gamma$: $$\Phi = -\frac{1}{2}\lambdaE_0(x^2-y^2)$$ Now, assume the distance between the rods is $2r_o$ and the potential on each rod is $\frac{Phi_0}{2}$. Then, at the interior surface of the rods in the X-plane, $x=r_0$, and y=0, such that $$\Phi=-\frac{1}{2}E_0\lambda r_0^2$ and $\Phi = -\frac{\Phi_0}{2r^2} (x^2-y^2) +::: + - Plug potential into equations of motion (using F=ma=Eq), this is almost Mathieu’s eqn - SUMMARY → we can use the geometry of the electric field within the analyzer and the time dependent potentials on the rods (bc we already know the relationship between E and Φ) to express the force on an ion and thus determine its equation of motion…solving the equation of motion is trickier - Substitute in a and q → express the potential within the quadrupole as a second order diff eq. that resembles Mathieu eqn diff --git a/references.bib b/references.bib index b43a702..a8e61a9 100644 --- a/references.bib +++ b/references.bib @@ -176,5 +176,11 @@ @book{Dawson_1976 place={Amsterdam, NY}, year={1976} } - +@book{Dawson, + author={Dawson, Peter H.}, + title={Quadrupole Mass Spectrometry and Its Applications}, + pages={9-19}, + publisher={Elsevier Scientific Pub. Co.}, + place={Amsterdam, NY}, + year={1976} From b8f2f43113e49056849e477bddafd735fe47e53b Mon Sep 17 00:00:00 2001 From: Maddy Runyon Date: Thu, 26 Jun 2025 16:30:23 -0400 Subject: [PATCH 4/6] adding figs and expanding equations section added plots for RF +DC potentials, continued drafting equations of motion section --- mass_analyzer_design.qmd | 130 +++++++++++++++++++++++++++++++++++---- references.bib | 12 +--- 2 files changed, 121 insertions(+), 21 deletions(-) diff --git a/mass_analyzer_design.qmd b/mass_analyzer_design.qmd index 62c78d8..c363b72 100644 --- a/mass_analyzer_design.qmd +++ b/mass_analyzer_design.qmd @@ -29,10 +29,76 @@ Using a combination of static and alternating voltages, quadrupole mass analyzer ### Mass filtering - how does it work? {#sec-quadrupole-mass-filtering} -Quadrupole mass filters consist of four parallel rods, connected in crosswise pairs. A potential of (U+Vcosωt) is applied to one pair, while a potential of -(U+Vcosωt) is applied to the second pair. Here, Vcosωt represents an alternating RF voltage with an amplitude of V and frequency of ω; U represents a static voltage ("DC offset"). More simply, you could express the potentials as (DC+RF), and -(DC+RF). +Quadrupole mass filters consist of four parallel rods, connected in crosswise pairs. A potential of (U+Vcosωt) is applied to one pair, while a potential of -(U+Vcosωt) is applied to the second pair. Here, Vcosωt represents an alternating RF voltage with an amplitude of V and angular frequency of ω; U represents a static voltage ("DC offset"). More simply, you could express the potentials as (DC+RF), and -(DC+RF). Though the voltage applied to each pair of rods oscillates continuously, the DC component can be thought of as the *average* voltage applied to each pair. In general, high m/z ions respond to the average voltage – positive DC voltage repels and thus stabilizes them, negative DC voltage attracts and destabilizes them – while lighter ions are more sensitive to the RF component. +::: {.callout-note collapse=true} + +## RF Voltage with DC Offset? + +The plots below illustrate how a DC component, or U term, shifts the RF potential and changes its symmetry. A positive DC component shifts the signal upward, both increasing the average voltage and extending its time in the positive region; a negative DC component skews it downward, extending the length of time that the signal is negative. +```{python} +#| label: quad-potentials +#| fig-cap: "Quadrupole Potentials " + +import numpy as np +import matplotlib.pyplot as plt + +f = 100_000 # 100 kHz +omega = 2 * np.pi * f +period = 1 / f # = 10 μs + +V = 6000.0 +U = 1000.0 + +num_cycles = 2 +samples_per_cycle = 1000 +t_max = num_cycles * period +num_points = num_cycles * samples_per_cycle +t = np.linspace(0, t_max, num_points) + +phi1 = V * np.cos(omega * t) +phi2 = U + V * np.cos(omega * t) +phi3 = - (U + V * np.cos(omega * t)) + +fig, axs = plt.subplots(3, 1, figsize=(6, 10), sharex=True) # plot dimensions +colors = ['blue', 'green', 'red'] +functions = [ + (phi1, 'Φ = V·cos(ωt)', 0), + (phi2, 'Φ = U + V·cos(ωt)', 1000), + (phi3, 'Φ = –(U + V·cos(ωt))', -1000) +] + +ymin, ymax = -8000, 8000 + +for i, (ax, (y, label, true_mean), color) in enumerate(zip(axs, functions, colors)): + ax.plot(t * 1e6, y, label=label, color=color) # time in μs + ax.axhline(0, color='black', linewidth=2.0) + ax.axhline(true_mean, color='darkmagenta', linestyle='--', linewidth=2, + label=f'U = {true_mean} V') + ax.set_ylabel('Φ (V)') + ax.set_ylim(ymin, ymax) + ax.legend(loc='upper right') + ax.grid(True) + + if i in [1, 2]: + ax.annotate( + '', + xy=(t[50] * 1e6, true_mean), + xytext=(t[50] * 1e6, 0), + arrowprops=dict(arrowstyle='->', color='darkmagenta', lw=1) + ) + +axs[2].set_xlabel('Time (μs)') + +plt.suptitle('RF Potentials and DC Offsets (100 kHz, V = 6000 Volts, U = 1000 Volts)', fontsize=10.5, x = .49, fontweight='bold') +plt.subplots_adjust(hspace=10) +plt.tight_layout(rect=[0, 0, 1, 0.95]) +plt.show() +``` +::: + Because the forces acting on ions traveling through the quadrupole are uncoupled in the X, Y, and Z directions, we can think of mass filtering in terms of two separate planes: the XZ-plane, or X-direction, and the YZ-plane, or Y-direction. Note that in the Z-direction (along the length of the quadrupole), the net force acting on ions is zero. @@ -46,22 +112,64 @@ By manipulating the RF and DC components of the field, the high and low-pass fil ### Equations of motion and stability diagrams -The equations of motion that describe ion paths within a quadrupole depend on: - (a) the hyperbolic geometry of the electric field - (b) the time-dependent potentials (voltages) applied to the rods +The equations of motion that describe ion paths within a quadrupole depend on: -Starting with the geometry, we can come up with a general equation for the electric potential throughout the quadrupole, which is just the equation for potential within a hyperbolic electric field: -$$\Phi=-\frac{\Phi_0}{2r^2}(x^2-y^2)$$ +(a) the hyperbolic geometry of the electric field +(b) the time-dependent potentials (voltages) applied to the rods -For a step-by-step derivation of this equation, check out [section II] in [@Dawson]. +Starting with the geometry, we can come up with a general equation for the electric potential throughout the quadrupole, i.e. the equation for potential within a hyperbolic electric field: +$$ +\Phi = -\frac{\Phi_0}{2r^2}(x^2 - y^2) +$$ +For a step-by-step derivation of this equation, check out [Section II](https://books.google.com/books?id=A6o3BQAAQBAJ&lpg=PA9&dq=quadrupole%20mass%20analyzer&lr&pg=PA9&output=embed) in [@Dawson_1976]. ::: {.callout-note collapse=true} -## Or see this brief overview, for reference - Start with a general geometric description of a quadrupole electric field$$E=E_0(\lambda x+\sigma y+\gamma z)$$ Choose $\lambda$, $\sigma$, and $\delta$ such that they satisfy the differential form of Gauss’ Law (the Laplace equation, since the charge enclosed in the quadrupole is zero): $$\nabla \cdot E=0$$ There are multiple solutions, but we will use: $$\lambda=-\sigma , \gamma = 0$$ - Integrate general equation of electric field to get the potential and then plug in our values for $\lambda$, $\sigma$, and $\gamma$: $$\Phi = -\frac{1}{2}\lambdaE_0(x^2-y^2)$$ Now, assume the distance between the rods is $2r_o$ and the potential on each rod is $\frac{Phi_0}{2}$. Then, at the interior surface of the rods in the X-plane, $x=r_0$, and y=0, such that $$\Phi=-\frac{1}{2}E_0\lambda r_0^2$ and $\Phi = -\frac{\Phi_0}{2r^2} (x^2-y^2) +## Or see this brief overview, for reference: + +Start with a general geometric description of a quadrupole electric field: + +\begin{equation} +E = E_0(\lambda x + \sigma y + \gamma z) +\end{equation} + + +Choose weighting constants $\lambda$, $\sigma$, and $\gamma$ such that they satisfy the differential form of Gauss’ Law (the Laplace equation, since the charge enclosed in the quadrupole is zero): +$$ +\nabla \cdot E = 0 +$$ + +There are multiple solutions, but we will use: $\lambda = -\sigma$, $\gamma = 0$ + +Integrate the general equation of the electric field to get the potential and then plug in our values for $\lambda$, $\sigma$, and $\gamma$: + +\begin{equation} +\Phi = -\frac{1}{2}\lambda E_0(x^2 - y^2) +\end{equation} + +Now, assume the distance between the rods is $2r_0$ and the potential on each rod is $\frac{\Phi_0}{2}$. + +Then, at the interior surface of the two rods in the XZ-plane, $x = r_0$ and $y = 0$, such that: +$$ +\frac{\Phi_0}{2} = -\frac{1}{2}E_0 \lambda r_0^2 +$$ + +Solving for $E_0 \lambda$ and plugging back in to the first equation for potential gives: +$$ +\Phi = -\frac{\Phi_0}{2r^2} (x^2 - y^2) +$$ ::: -- Plug potential into equations of motion (using F=ma=Eq), this is almost Mathieu’s eqn +Now, to come up with equations of motion for ions in the X and Y directions, we will need to relate this potential to Newton's second law: $$F=ma$$ + +At any point in an electric field, the electrostatic force on a ion is the product of the electric field at that point and the charge of the ion: +$$F=Eq$$ + +So, splitting up Newton's second law into X and Y directions, and substituting Eq in for F, we re-write it as follows: +$$Eq=mẍ$$ $$Eq=mÿ$$ + +(Double dots represent the second derivative of a term, the second derivative of position being acceleration.) + +- finish setting up equations of motion - SUMMARY → we can use the geometry of the electric field within the analyzer and the time dependent potentials on the rods (bc we already know the relationship between E and Φ) to express the force on an ion and thus determine its equation of motion…solving the equation of motion is trickier - Substitute in a and q → express the potential within the quadrupole as a second order diff eq. that resembles Mathieu eqn - Not going to solve the mathieu equation here, bc Mathieu already did → link a paper diff --git a/references.bib b/references.bib index a8e61a9..0f6d9ea 100644 --- a/references.bib +++ b/references.bib @@ -171,16 +171,8 @@ @article{DeLaeter_2006 @book{Dawson_1976, author={Dawson, Peter H.}, title={Quadrupole Mass Spectrometry and Its Applications}, - pages={3}, + pages={3, 9-19}, publisher={Elsevier Scientific Pub. Co.}, place={Amsterdam, NY}, year={1976} -} -@book{Dawson, - author={Dawson, Peter H.}, - title={Quadrupole Mass Spectrometry and Its Applications}, - pages={9-19}, - publisher={Elsevier Scientific Pub. Co.}, - place={Amsterdam, NY}, - year={1976} - +} From 40e802642234f9b21ff05483b7519c5f9aa7178b Mon Sep 17 00:00:00 2001 From: Maddy Runyon Date: Mon, 4 Aug 2025 21:17:50 -0400 Subject: [PATCH 5/6] Update mass_analyzer_design.qmd introducing mathieu's eqn and making notation edits --- mass_analyzer_design.qmd | 120 +++++++++++++++++++++++++++++++++------ 1 file changed, 102 insertions(+), 18 deletions(-) diff --git a/mass_analyzer_design.qmd b/mass_analyzer_design.qmd index c363b72..4399918 100644 --- a/mass_analyzer_design.qmd +++ b/mass_analyzer_design.qmd @@ -112,76 +112,160 @@ By manipulating the RF and DC components of the field, the high and low-pass fil ### Equations of motion and stability diagrams -The equations of motion that describe ion paths within a quadrupole depend on: -(a) the hyperbolic geometry of the electric field +The equations of motion that describe ion paths within a quadrupole depend on: + + +(a) the hyperbolic geometry of the electric field (b) the time-dependent potentials (voltages) applied to the rods -Starting with the geometry, we can come up with a general equation for the electric potential throughout the quadrupole, i.e. the equation for potential within a hyperbolic electric field: + +Since, in the case of quadrupole mass analyzers, the time-dependent potentials contain a cosin function, the equations that describe ion motion within a quadrupole will be "Mathieu-type" equations. That is, linear second order differential equations with *periodic* (oscillating) coefficients. + +The canonical form of Mathieu's equation is: +$$\frac{d^2u}{dξ^2} + (a_u - 2q_ucos2ξ)u = 0$$ + +Since we are concerned with equations of *motion*, we can replace ξ with t, taking the second derivative of some position u with respect to time t: + +$$\frac{d^2u}{dt^2} + (a_u - 2q_ucos2t)u = 0$$ + +Importantly, there are two categories of solutions to this equation: bounded solutions, which correspond to stable ion trajectories through the analyzer, and unbounded solutions, which correspond to unstable ion trajectories. Here a and q are parameters which can be modified to control which m/z are stable in the analyzer and which are not, i.e. to modify *which solutions* to the equation are stable and which are unstable. Below is a loose derivation of the equations of ion motion which will relate the direct and alternating components of the applied potentials (U and V) to the parameters a and q. + +Starting with the geometry, we can come up with a general equation for the electric potential throughout the quadrupole, i.e. the equation for potential within a hyperbolic electric field: $$ \Phi = -\frac{\Phi_0}{2r^2}(x^2 - y^2) $$ For a step-by-step derivation of this equation, check out [Section II](https://books.google.com/books?id=A6o3BQAAQBAJ&lpg=PA9&dq=quadrupole%20mass%20analyzer&lr&pg=PA9&output=embed) in [@Dawson_1976]. + + + ::: {.callout-note collapse=true} ## Or see this brief overview, for reference: -Start with a general geometric description of a quadrupole electric field: + +Start with a general geometric description of a quadrupole electric field: + \begin{equation} E = E_0(\lambda x + \sigma y + \gamma z) \end{equation} -Choose weighting constants $\lambda$, $\sigma$, and $\gamma$ such that they satisfy the differential form of Gauss’ Law (the Laplace equation, since the charge enclosed in the quadrupole is zero): + + +Choose weighting constants $\lambda$, $\sigma$, and $\gamma$ such that they satisfy the differential form of Gauss’ Law (the Laplace equation, since the charge enclosed in the quadrupole is zero): $$ \nabla \cdot E = 0 $$ + There are multiple solutions, but we will use: $\lambda = -\sigma$, $\gamma = 0$ -Integrate the general equation of the electric field to get the potential and then plug in our values for $\lambda$, $\sigma$, and $\gamma$: + +Integrate the general equation of the electric field to get the potential and then plug in our values for $\lambda$, $\sigma$, and $\gamma$: + \begin{equation} -\Phi = -\frac{1}{2}\lambda E_0(x^2 - y^2) +\Phi = -\frac{1}{2}\lambda E_0(x^2 - y^2) \end{equation} -Now, assume the distance between the rods is $2r_0$ and the potential on each rod is $\frac{\Phi_0}{2}$. + +Now, assume the distance between the rods is $2r_0$ and the potential on each rod is $\frac{\Phi_0}{2}$. + Then, at the interior surface of the two rods in the XZ-plane, $x = r_0$ and $y = 0$, such that: $$ \frac{\Phi_0}{2} = -\frac{1}{2}E_0 \lambda r_0^2 $$ -Solving for $E_0 \lambda$ and plugging back in to the first equation for potential gives: + +Solving for $E_0 \lambda$ and plugging back in to the first equation for potential gives: $$ \Phi = -\frac{\Phi_0}{2r^2} (x^2 - y^2) $$ ::: -Now, to come up with equations of motion for ions in the X and Y directions, we will need to relate this potential to Newton's second law: $$F=ma$$ -At any point in an electric field, the electrostatic force on a ion is the product of the electric field at that point and the charge of the ion: +Now, to come up with equations of motion for ions in the X and Y directions, we will need to relate this potential to Newton's second law, $F=ma$. + + +This can be done by gradient of our potential function and making a series of substitutions, which result in the following two equations: + + +$$\frac{d^2x}{dt^2} + \frac{q}{mr_0^2}(\Phi_0)(x) = 0$$ + + +$$\frac{d^2y}{dt^2} + \frac{q}{mr_0^2}(\Phi_0)(y) = 0$$ + +::: {.callout-note collapse=true} +## For a walkthrough of this derivation, see below: +At any point in an electric field, the electrostatic force on a ion is the product of the electric field at that point and the charge (q) of the ion: $$F=Eq$$ + So, splitting up Newton's second law into X and Y directions, and substituting Eq in for F, we re-write it as follows: -$$Eq=mẍ$$ $$Eq=mÿ$$ +$$E_xq=m\frac{d^2x}{dt^2}$$ $$E_yq=m\frac{d^2y}{dt^2}$$ + + +(Double dots indicate the second derivative of a term, the second derivative of position being acceleration.) + + +Taking partial derivatives of the potential (above) with respect to x and y will give us equation for the Electric field as functions of x and y, which we can plug in for $E_x$ and $E_y$: + + +$$ \frac{x\Phi_0}{r_0^2}(q) = m\frac{d^2x}{dt^2}$$ + + +$$ \frac{y\Phi_0}{r_0^2}(q) = m\frac{d^2y}{dt^2}$$ + + +Moving all terms to the left side of the equation, we have: + + +$$\frac{d^2x}{dt^2} + \frac{q}{mr_0^2}(\Phi_0)x = 0$$ + + +$$\frac{d^2y}{dt^2} + \frac{q}{mr_0^2}(\Phi_0)y = 0$$ +::: + + +Now, we can replace $\Phi_0$ in each equation with the actual potentials applied to the rods. Note, again, that the applied potentials are sinusoidal functions of *time*: + + +$$\frac{d^2x}{dt^2} + \frac{q}{mr_0^2}(U+\cos(ωt))x = 0$$ + + +$$\frac{d^2y}{dt^2} - \frac{q}{mr_0^2}(U+\cos(ωt))y = 0$$ + + +These are the equations of motion! YAY! + + +In summary, we can use the (a) geometry of the electric field within the analyzer and (b) the time dependent potentials on the rods to express the force acting on an ion in the analyzer, and thus determine the equations of motion that will describe its path. + + +Solving the equations of motion is trickier, but fortunately, they are of the Mathieu "type" -- linear, second order differential equations with "harmonic" coefficients, i.e. a cosin function modifying the x and y terms -- and there are multiple well-documented methods for solving them. The canonical form of Mathieu's equation looks a bit different though: + + +INSERT THE EQUATION HERE + + +In order to get our equations in this form, some substitutions still have to be made. + + -(Double dots represent the second derivative of a term, the second derivative of position being acceleration.) -- finish setting up equations of motion -- SUMMARY → we can use the geometry of the electric field within the analyzer and the time dependent potentials on the rods (bc we already know the relationship between E and Φ) to express the force on an ion and thus determine its equation of motion…solving the equation of motion is trickier - Substitute in a and q → express the potential within the quadrupole as a second order diff eq. that resembles Mathieu eqn - Not going to solve the mathieu equation here, bc Mathieu already did → link a paper - And what does “solving” an equation of motion mean anyway? - Physical implications of bounded vs unbounded solutions to the equation: - - “Unbounded” = unstable ion trajectory - - “Bounded” = stable ion trajectory + - “Unbounded” = unstable ion trajectory + - “Bounded” = stable ion trajectory - Introduce stability diagram of U/V (specific to a particular m/z), and demonstrate how a and q can be used to generalize the stability diagram - Altering U and V but keeping a/q constant → ions of increasing or decreasing m/z traverse the mass filter, creating a mass spectrum - what happens if a/q is not kept constant? → alters resolution, ie pushes the a/q line either above the peak or too far down to provide unit mass resolution → question: how does this compare to a U/V plot? - ### Operation and typical geometry of instrument From b15b4d90c44f7cc07e60e9dd4791b38b4e922a1d Mon Sep 17 00:00:00 2001 From: Maddy Runyon Date: Wed, 13 Aug 2025 10:37:16 -0400 Subject: [PATCH 6/6] draft of equations section added/resized a fig, reorganized callouts, introduced parameters a and q --- figs/quad-fig7.png | Bin 0 -> 16980 bytes 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