查看“︁四极离子阱”︁的源代码

{{校对翻译}}

[[Image:Paul-Trap.svg|thumb|right|300px|具有正电荷粒子（深红色）的经典装置的四极离子阱的方案，被类似带电粒子（浅红色）云包围。 电场“E”（蓝色）由四极端盖（a，正极）和环形电极（b）产生。 图1和图2显示了AC循环期间的两种状态。]]
'''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<ref>Paul W., Steinwedel H. (1953). "Ein neues Massenspektrometer ohne Magnetfeld". RZeitschrift für Naturforschung A '''8''' (7): 448-450</ref><ref>{{patent|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}}</ref>并分享此成果，因此而獲得了1989年的[[诺贝尔物理学奖]]。<ref>{{cite journal |author=Wolfgang Paul |title=Electromagnetic traps for charged and neutral particles |journal=[[Reviews of Modern Physics|Rev. Mod. Phys.]] |volume=62 |pages=531–540 |doi=10.1103/RevModPhys.62.531}}</ref>它一般用于[[质谱法|质谱仪]]的一个组件或{{le|俘获离子量子计算机|Trapped ion quantum computer}}。

==理论==
[[Image:lin quad thompson ucalgary.JPG|thumb|300px|right|卡尔加里大学的线性离子阱]]

===运动学方程===

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）<ref name='March 1997'>{{cite journal|last1=March|first1=Raymond E.|title=An Introduction to Quadrupole Ion Trap Mass Spectrometry|journal=Journal of Mass Spectrometry|volume=32|issue=4|year=1997|pages=351–369|issn=1076-5174|doi=10.1002/(SICI)1096-9888(199704)32:4<351::AID-JMS512>3.0.CO;2-Y}}</ref>给出。对于离子阱中带电离子，可列出以下方程：

<math> \frac{d^2u}{d\xi^2}+[a_u-2q_u\cos (2\xi) ]u=0 \qquad\qquad (1) \!</math>

其中<math>u</math> 代表x, y, z 坐标，<math>\xi</math> 是一个无量纲的参数，由<math>\xi=\Omega t/2\ </math>给出，并且<math>a_u\,</math> 和<math>q_u\,</math> 也是无量纲的限制参数。参数<math>\Omega\,</math> 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：

:<math> \frac{d^2u}{dt^2} = \frac{\Omega^2}{4}   \frac{d^2u}{d\xi^2}   \qquad\qquad (2) \!</math>

将（2）带入马修方程（1），可得：

:<math>  \frac{4}{\Omega^2}\frac{d^2u}{dt^2} + \left[a_u - 2q_u\cos (\Omega t) \right]u = 0 \qquad\qquad (3) \!</math>.

整理上式：

:<math> m \frac{d^2u}{dt^2} + m \frac{\Omega^2}{4}\left[a_u - 2q_u\cos (\Omega t) \right]u = 0 \qquad\qquad (3) \!</math>.

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。<ref>N. W. McLachlan, '''Theory and Applications of Mathieu Functions''' (Oxford University Press, Oxford, 1947), p. 20</ref>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：

:<math>F_x = ma = m\frac{d^2x}{dt^2} = -e \frac{\partial \phi}{\partial x} \qquad\qquad (4) \!</math>

其中， <math>\phi\,</math> 是四极势，由以下给出：

:<math>\phi=\frac{\phi_0}{r_0^2} \big(  \lambda x^2 + \sigma y^2 + \gamma z^2   \big) \qquad\qquad (5) \!</math>

其中<math>\phi _0\,</math>是外加电位， <math>\lambda\, </math>, <math>\sigma\,</math>, 和 <math>\gamma\,</math> 是权重，还有 <math>r_0\,</math>是尺寸参数常数。为了满足[[拉普拉斯条件]]，<math>\nabla^2\phi_0=0\,</math>可由以下给出：

<math>  \lambda  + \sigma  + \gamma   =0 \,</math>.

对于一个离子阱， <math>  \lambda  = \sigma  =1 \,</math> 和 <math>   \gamma   =-2 \,</math> 对于一个[[四极杆质量分析器]], <math>  \lambda  = -\sigma  =1 \,</math> 并且有 <math>   \gamma   =0 \,</math>。

转换5式到圆柱坐标，即 <math>{x}={r} \,\cos\theta</math>, <math>{y}={r} \, \sin\theta</math>,和 <math>{z}={z} \,</math> 应用[[勾股定理]] <math>\sin^2 \theta + \cos^2 \theta = 1 \,</math> 给出以下：

[[Image:Mass selective instability.gif|right|300 px|thumb|Diagram of the stability regions of a quadrupole ion trap according to the voltage and frequency applied to the ion trap elements.]]

:<math>\phi_{r,z} = \frac{\phi_0}{r_0^2} \big( r^2 - 2z^2 \big) . \qquad\qquad (6) \!</math>

施加的电势是RF和DC的组合，由下式给出：

:<math>\phi_0 = U + V\cos \Omega t .\qquad\qquad (7) \!</math>

其中<math>\Omega = 2\pi \nu</math> and <math>\nu</math> 是外加频率，单位是赫兹。

将7式带入5式， <math>\lambda = 1</math> 得：

:<math> \frac{\partial \phi}{\partial x} =  \frac {2x}{r_0^2} \big( U + V\cos \Omega t \big) . \qquad\qquad (8) \!</math>

将8式带入4式，可得：

<math> m\frac {d^2x}{dt^2} = - \frac {2e}{r_0^2} \big( U + V\cos \Omega t \big) x . \qquad\qquad (9) \!</math>

比较1式和9式的右手项，可得：

<math> a_x = \frac {8eU} {m r_0^2 \Omega^2} \qquad\qquad (10) \!</math>

和

:<math> q_x = - \frac {4eV} {m r_0^2 \Omega^2} . \qquad\qquad (11) \!</math>

此外还有 <math>q_x = q_y\,</math>，

:<math> a_z = -\frac {8eU} {m r_0^2 \Omega^2} \qquad\qquad (12) \!</math>

还有

:<math> q_z =  \frac {4eV} {m r_0^2 \Omega^2} . \qquad\qquad (13) \!</math>

离子的捕获可以从<math>q_u</math>和<math>a_u</math>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<ref>H.J.W. Müller-Kirsten, Introduction to Quantum Mechanics: Schrödinger Equation and Path Integral, 2nd ed., World Scientific (2012), Chapter 17 on Periodic Potentials, {{ISBN|978-981-4397-73-5}}.</ref>。

==组合射频阱==
组合射频阱是四极离子阱和[[彭宁离子阱]]的组合<ref>{{cite journal |author1=J. Walz |author2=S. B. Ross |author3=C. Zimmermann |author4=L. Ricci |author5=M. Prevedelli |author6=T. W. Hansch |year=1996 |title=Confinement of electrons and ions in a combined trap with the potential for antihydrogen production |journal=[[Hyperfine Interactions]] |volume=100 |page=133 |doi=10.1007/BF02059938 |bibcode = 1996HyInt.100..133W }}</ref>。 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。

==参考文献==
{{Reflist}}

==人物传记==
{{refbegin}}
* W. 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.&nbsp;497-517
* R.I. Thompson, T.J. Harmon, and M.G. Ball, ''The rotating-saddle trap<nowiki>:</nowiki> a mechanical analogy to RF-electric-quadrupole ion trapping?'' (Canadian Journal of Physics, 2002: '''80''' 12) p.&nbsp;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.&nbsp;95-114.
*{{cite book |author=G. Werth |title=Charged Particle Traps: Physics and Techniques of Charged Particle Field Confinement (Springer Series on Atomic, Optical, and Plasma Physics) |publisher=Springer |location=Berlin |year= 2005|pages= |isbn=3-540-22043-7 |oclc= 231588573|doi=}}
*{{cite book |author=John Gillaspy |title=Trapping Highly Charged Ions: Fundamentals and Applications |publisher=Nova Science Publishers |location=Commack, N.Y |year= 2001|pages= |isbn=1-56072-725-X |oclc= 42009394|doi=}}
*{{cite book |author=Todd, John F. J.; March, Raymond E. |title=Quadrupole Ion Trap Mass Spectrometry , 2nd Edition |url=https://archive.org/details/quadrupoleiontra0000marc |publisher=Wiley-Interscience |location=New York |year= 2005|pages= |isbn=0-471-48888-7 |oclc= 56413336|doi=}}
*{{cite book |author=Todd, John F. J.; March, Raymond E. |title=Practical aspects of ion trap mass spectrometry -  Volume I: Fundamentals of Ion Trap Mass Spectrometry |publisher=CRC Press |location=Boca Raton |year=1995 |pages= |isbn=0-8493-4452-2 |oclc= 32346425|doi=}}
*{{cite book |author=Todd, John F. J.; March, Raymond E. |title=Practical aspects of ion trap mass spectrometry: Ion Trap Instrumentation, Vol. 2 |url=https://archive.org/details/practicalaspects0002unse |publisher=CRC Press |location=Boca Raton |year=1995 |pages= |isbn=0-8493-8253-X |oclc= 32346425|doi=}}
*{{cite book |author=Todd, John F. J.; March, Raymond E. |title=Practical aspects of ion trap mass spectrometry, Vol. 3 |publisher=CRC Press |location=Boca Raton |year=1995 |pages= |isbn=0-8493-8251-3 |oclc= 32346425|doi=}}
*{{cite book |author=Hughes, Richard M.; March, Raymond E.; Todd, John F. J. |title=Quadrupole storage mass spectrometry |publisher=Wiley |location=New York |year=1989 |pages= |isbn=0-471-85794-7 |oclc= 18290778|doi=}}
* K. Shah and H. Ramachandran, ''Analytic, nonlinearly exact solutions for an rf confined plasma'', Phys. 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''{{Wayback|url=http://www.oup.com/us/catalog/general/subject/Physics/AtomicMolecularOpticalphysics/?view=usa&ci=9780198539957 |date=20121016170126 }}

{{refend}}

===相关专利===
* {{patent|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}}
*{{patent|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}}
* {{patent|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}}

==外部链接==
*[https://web.archive.org/web/20081013004002/http://nobelprize.org/physics/laureates/1989/illpres/ Nobel Prize in Physics 1989]

{{质谱法}}

[[Category:质谱]]
[[Category:测量仪器]]