四极离子阱

四极离子阱是一种使用交变电场来束缚带电粒子的离子阱，也称无线射频 （RF）阱或者保罗离子阱，　为了纪念发明者沃尔夫冈·保罗。　沃尔夫冈·保罗发明了这种装置并分享此成果，因此而獲得了1989年的诺贝尔物理学奖。它一般用于质谱仪的一个组件或俘获离子量子计算机（英语：Trapped ion quantum computer）。 四极场中离子被施加了一个回复力使得它们回到阱的中心，场中的离子的运动由马丢函数（Mathieu equation）给出。对于离子阱中带电离子，可列出以下方程： d 2 u d ξ 2 + [ a u − 2 q u cos ⁡ ( 2 ξ ) ] u = 0 ( 1 ) {\displaystyle {\frac {d^{2}u}{d\xi ^{2}}}+[a_{u}-2q_{u}\cos(2\xi )]u=0\qquad \qquad (1)\!} 其中 u {\displaystyle u} 代表x, y, z 坐标， ξ {\displaystyle \xi } 是一个无量纲的参数，由 ξ = Ω t / 2   {\displaystyle \xi =\Omega t/2\ } 给出，并且 a u {\displaystyle a_{u}\,} 和 q u {\displaystyle q_{u}\,} 也是无量纲的限制参数。参数 Ω {\displaystyle \Omega \,} 是施加在环形电极上的电场频率。应用链式法则，我们可以得出： d 2 u d t 2 = Ω 2 4 d 2 u d ξ 2 ( 2 ) {\displaystyle {\frac {d^{2}u}{dt^{2}}}={\frac {\Omega ^{2}}{4}}{\frac {d^{2}u}{d\xi ^{2}}}\qquad \qquad (2)\!} 将（2）带入马修方程（1），可得： 4 Ω 2 d 2 u d t 2 + [ a u − 2 q u cos ⁡ ( Ω t ) ] u = 0 ( 3 ) {\displaystyle {\frac {4}{\Omega ^{2}}}{\frac {d^{2}u}{dt^{2}}}+\left[a_{u}-2q_{u}\cos(\Omega t)\right]u=0\qquad \qquad (3)\!} . 整理上式： m d 2 u d t 2 + m Ω 2 4 [ a u − 2 q u cos ⁡ ( Ω t ) ] u = 0 ( 3 ) {\displaystyle m{\frac {d^{2}u}{dt^{2}}}+m{\frac {\Omega ^{2}}{4}}\left[a_{u}-2q_{u}\cos(\Omega t)\right]u=0\qquad \qquad (3)\!} . 由牛顿运动学方程可知，以上的方程代表了施加在离子上的力。该方程可应用Floquet定理解得或用多尺度分析（multiple scale analysis）的标准计算方法得出。粒子动力学和保罗离子阱的带电粒子的时间平均密度也可以通过有质动力的概念得到。 每个维上的力没有耦合。例如，对于作用在离子上的力，在x轴上有： F x = m a = m d 2 x d t 2 = − e ∂ ϕ ∂ x ( 4 ) {\displaystyle F_{x}=ma=m{\frac {d^{2}x}{dt^{2}}}=-e{\frac {\partial \phi }{\partial x}}\qquad \qquad (4)\!} 其中， ϕ {\displaystyle \phi \,} 是四极势，由以下给出： ϕ = ϕ 0 r 0 2 ( λ x 2 + σ y 2 + γ z 2 ) ( 5 ) {\displaystyle \phi ={\frac {\phi _{0}}{r_{0}^{2}}}{\big (}\lambda x^{2}+\sigma y^{2}+\gamma z^{2}{\big )}\qquad \qquad (5)\!} 其中 ϕ 0 {\displaystyle \phi _{0}\,} 是外加电位， λ {\displaystyle \lambda \,} , σ {\displaystyle \sigma \,} , 和 γ {\displaystyle \gamma \,} 是权重，还有 r 0 {\displaystyle r_{0}\,} 是尺寸参数常数。为了满足拉普拉斯条件， ∇ 2 ϕ 0 = 0 {\displaystyle \nabla ^{2}\phi _{0}=0\,} 可由以下给出： λ + σ + γ = 0 {\displaystyle \lambda +\sigma +\gamma =0\,} . 对于一个离子阱， λ = σ = 1 {\displaystyle \lambda =\sigma =1\,} 和 γ = − 2 {\displaystyle \gamma =-2\,} 对于一个四极杆质量分析器, λ = − σ = 1 {\displaystyle \lambda =-\sigma =1\,} 并且有 γ = 0 {\displaystyle \gamma =0\,} 。 转换5式到圆柱坐标，即 x = r cos ⁡ θ {\displaystyle {x}={r}\,\cos \theta } , y = r sin ⁡ θ {\displaystyle {y}={r}\,\sin \theta } ,和 z = z {\displaystyle {z}={z}\,} 应用勾股定理 sin 2 ⁡ θ + cos 2 ⁡ θ = 1 {\displaystyle \sin ^{2}\theta +\cos ^{2}\theta =1\,} 给出以下： ϕ r , z = ϕ 0 r 0 2 ( r 2 − 2 z 2 ) . ( 6 ) {\displaystyle \phi _{r,z}={\frac {\phi _{0}}{r_{0}^{2}}}{\big (}r^{2}-2z^{2}{\big )}.\qquad \qquad (6)\!} 施加的电势是RF和DC的组合，由下式给出： ϕ 0 = U + V cos ⁡ Ω t . ( 7 ) {\displaystyle \phi _{0}=U+V\cos \Omega t.\qquad \qquad (7)\!} 其中 Ω = 2 π ν {\displaystyle \Omega =2\pi \nu } and ν {\displaystyle \nu } 是外加频率，单位是赫兹。 将7式带入5式， λ = 1 {\displaystyle \lambda =1} 得： ∂ ϕ ∂ x = 2 x r 0 2 ( U + V cos ⁡ Ω t ) . ( 8 ) {\displaystyle {\frac {\partial \phi }{\partial x}}={\frac {2x}{r_{0}^{2}}}{\big (}U+V\cos \Omega t{\big )}.\qquad \qquad (8)\!} 将8式带入4式，可得： m d 2 x d t 2 = − 2 e r 0 2 ( U + V cos ⁡ Ω t ) x . ( 9 ) {\displaystyle m{\frac {d^{2}x}{dt^{2}}}=-{\frac {2e}{r_{0}^{2}}}{\big (}U+V\cos \Omega t{\big )}x.\qquad \qquad (9)\!} 比较1式和9式的右手项，可得： a x = 8 e U m r 0 2 Ω 2 ( 10 ) {\displaystyle a_{x}={\frac {8eU}{mr_{0}^{2}\Omega ^{2}}}\qquad \qquad (10)\!} 和 q x = − 4 e V m r 0 2 Ω 2 . ( 11 ) {\displaystyle q_{x}=-{\frac {4eV}{mr_{0}^{2}\Omega ^{2}}}.\qquad \qquad (11)\!} 此外还有 q x = q y {\displaystyle q_{x}=q_{y}\,} ， a z = − 8 e U m r 0 2 Ω 2 ( 12 ) {\displaystyle a_{z}=-{\frac {8eU}{mr_{0}^{2}\Omega ^{2}}}\qquad \qquad (12)\!} 还有 q z = 4 e V m r 0 2 Ω 2 . ( 13 ) {\displaystyle q_{z}={\frac {4eV}{mr_{0}^{2}\Omega ^{2}}}.\qquad \qquad (13)\!} 离子的捕获可以从 q u {\displaystyle q_{u}} 和 a u {\displaystyle a_{u}} 空间稳定区域的角度来理解。图中阴影区域的边界是两个方向上的稳定边界（也称为带边界）。 两个区域的重叠域是陷阱域。 为了计算这些边界和类似的图表，请参阅Müller-Kirsten。 组合射频阱是四极离子阱和彭宁离子阱的组合。 四极离子阱的主要瓶颈之一是它只能限制单个带电物品或具有相似质量的多个物品。 但在某些应用中，如反氢生产，重要的是限制两种质量差异很大的带电粒子。 为了实现该目的，在四极离子阱的轴向上添加均匀的磁场。 Paul W., Steinwedel H. (1953). "Ein neues Massenspektrometer ohne Magnetfeld". RZeitschrift für Naturforschung A 8 (7): 448-450 DE 944900  "Verfahren zur Trennung bzw. zum getrennten Nachweis von Ionen verschiedener spezifischer Ladung", W. Paul and H. Steinwedel, filed on December 24, 1953, priority December 23, 1953 Wolfgang Paul. 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Paul Electromagnetic Traps for Charged and Neutral Particles Taken from Proceedings of the International School of Physics <<Enrico Fermi>> Course CXVIII “Laser Manipulation of Atoms and Ions”, (North Holland, New York, 1992) p. 497-517 R.I. Thompson, T.J. Harmon, and M.G. Ball, The rotating-saddle trap: a mechanical analogy to RF-electric-quadrupole ion trapping? (Canadian Journal of Physics, 2002: 80 12) p. 1433–1448 M. Welling, H.A. Schuessler, R.I. Thompson, H. Walther Ion/Molecule Reactions, Mass Spectrometry and Optical Spectroscopy in a Linear Ion Trap (International Journal of Mass Spectrometry and Ion Processes, 1998: 172) p. 95-114. G. Werth. Charged Particle Traps: Physics and Techniques of Charged Particle Field Confinement (Springer Series on Atomic, Optical, and Plasma Physics). Berlin: Springer. 2005. ISBN 3-540-22043-7. OCLC 231588573.  John Gillaspy. Trapping Highly Charged Ions: Fundamentals and Applications. Commack, N.Y: Nova Science Publishers. 2001. 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Plasmas 15, 062303 (2008), https://archive.today/20130223075038/http://link.aip.org/link/?PHPAEN/15/062303/1 Pradip K. Ghosh, Ion Traps, International Series of Monographs in Physics, Oxford University Press (1995), http://www.oup.com/us/catalog/general/subject/Physics/AtomicMolecularOpticalphysics/?view=usa&ci=9780198539957（页面存档备份，存于互联网档案馆） DE 944900  "Verfahren zur Trennung bzw. zum getrennten Nachweis von Ionen verschiedener spezifischer Ladung", W. Paul and H. Steinwedel, filed on December 24, 1953, priority December 23, 1953 GB 773689  "Improved arrangements for separating or separately detecting charged particles of different specific charges", W. Paul, claims priority of a German application filed on December 23, 1953 US 2939952  "Apparatus for separating charged particles of different specific charges", W. Paul and H. Steinwedel, claims priority of a German application filed on December 23, 1953 Nobel Prize in Physics 1989