# 四极离子阱

## 理论

### 运动学方程

${\displaystyle {\frac {d^{2}u}{d\xi ^{2}}}+[a_{u}-2q_{u}\cos(2\xi )]u=0\qquad \qquad (1)\!}$

${\displaystyle {\frac {d^{2}u}{dt^{2}}}={\frac {\Omega ^{2}}{4}}{\frac {d^{2}u}{d\xi ^{2}}}\qquad \qquad (2)\!}$

${\displaystyle {\frac {4}{\Omega ^{2}}}{\frac {d^{2}u}{dt^{2}}}+\left[a_{u}-2q_{u}\cos(\Omega t)\right]u=0\qquad \qquad (3)\!}$.

${\displaystyle m{\frac {d^{2}u}{dt^{2}}}+m{\frac {\Omega ^{2}}{4}}\left[a_{u}-2q_{u}\cos(\Omega t)\right]u=0\qquad \qquad (3)\!}$.

${\displaystyle F_{x}=ma=m{\frac {d^{2}x}{dt^{2}}}=-e{\frac {\partial \phi }{\partial x}}\qquad \qquad (4)\!}$

${\displaystyle \phi ={\frac {\phi _{0}}{r_{0}^{2}}}{\big (}\lambda x^{2}+\sigma y^{2}+\gamma z^{2}{\big )}\qquad \qquad (5)\!}$

${\displaystyle \lambda +\sigma +\gamma =0\,}$.

${\displaystyle \phi _{r,z}={\frac {\phi _{0}}{r_{0}^{2}}}{\big (}r^{2}-2z^{2}{\big )}.\qquad \qquad (6)\!}$

${\displaystyle \phi _{0}=U+V\cos \Omega t.\qquad \qquad (7)\!}$

${\displaystyle {\frac {\partial \phi }{\partial x}}={\frac {2x}{r_{0}^{2}}}{\big (}U+V\cos \Omega t{\big )}.\qquad \qquad (8)\!}$

${\displaystyle m{\frac {d^{2}x}{dt^{2}}}=-{\frac {2e}{r_{0}^{2}}}{\big (}U+V\cos \Omega t{\big )}x.\qquad \qquad (9)\!}$

${\displaystyle a_{x}={\frac {8eU}{mr_{0}^{2}\Omega ^{2}}}\qquad \qquad (10)\!}$

${\displaystyle q_{x}=-{\frac {4eV}{mr_{0}^{2}\Omega ^{2}}}.\qquad \qquad (11)\!}$

${\displaystyle a_{z}=-{\frac {8eU}{mr_{0}^{2}\Omega ^{2}}}\qquad \qquad (12)\!}$

${\displaystyle q_{z}={\frac {4eV}{mr_{0}^{2}\Omega ^{2}}}.\qquad \qquad (13)\!}$

## 参考文献

1. ^ Paul W., Steinwedel H. (1953). "Ein neues Massenspektrometer ohne Magnetfeld". RZeitschrift für Naturforschung A 8 (7): 448-450
2. ^ 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
3. ^ Wolfgang Paul. Electromagnetic traps for charged and neutral particles. Rev. Mod. Phys.: 531–540. doi:10.1103/RevModPhys.62.531.
4. ^ March, Raymond E. An Introduction to Quadrupole Ion Trap Mass Spectrometry. Journal of Mass Spectrometry. 1997, 32 (4): 351–369. ISSN 1076-5174. doi:10.1002/(SICI)1096-9888(199704)32:4<351::AID-JMS512>3.0.CO;2-Y.
5. ^ N. W. McLachlan, Theory and Applications of Mathieu Functions (Oxford University Press, Oxford, 1947), p. 20
6. ^ 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.
7. ^ J. Walz; S. B. Ross; C. Zimmermann; L. Ricci; M. Prevedelli; T. W. Hansch. Confinement of electrons and ions in a combined trap with the potential for antihydrogen production. Hyperfine Interactions. 1996, 100: 133. Bibcode:1996HyInt.100..133W. doi:10.1007/BF02059938.

## 人物传记

### 相关专利

• 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