Space focusing of polar diatomic molecules using an electrostatic hexapole field is discussed, and it is shown that focused molecular trajectories in this type of field are properly described by a Jacobian elliptic function. Although a detailed mathematical treatment is given, emphasis is placed upon the physical nature of the trajectories. These results are applied to the design of a molecular beam resonance apparatus using hexapole fields, and it is shown that, in practice, several simplifying approximations (with little loss in accuracy) may be made which allow design calculations to be carried out using only a slide rule.

## REFERENCES

1.

H. G.

Bennewitz

, W.

Paul

, and C.

Schlier

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) and English transl. (unpublished) by T. C. English (a limited number of copies are available from the author on request).2.

P. Kusch and V. W. Hughes, “Atomic and Molecular Beam Spectroscopy,” in

*Encyclopedia of Physics*(Springer, Berlin, 1959), Vol. 37/1.3.

R. A.

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In practice, rods having circular cross section rather than hyperbolic‐like cross section are usually used because of the difficulty of machining hyperbolic‐like electrodes. This changes the potential to some extent, but Eq. (1) is still a fairly good approximation if the proper radius rod is used, as has been discussed for quadrupole fields by

I. E.

Dayton

, F. C.

Shoemaker

, and R. F.

Mozley

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25

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).5.

By the uniqueness theorem of electrostatics, this must be the potential since it is a solution of Laplace’s equation which satisfies all the boundary conditions.

6.

N. F. Ramsey,

*Molecular Beams*(Oxford U.P., London, 1956).7.

This equation is rigorously correct for molecules having zero angular momentum along the symmetry axis of the field; for a point source located on the symmetry axis, this will be the case for all molecules leaving the source. For an extended source, molecules leaving the source off axis will generally have some angular momentum along the axis, but for most practical sources, this angular momentum is small and can be ignored.

8.

See, for example: H. Jeffreys and B. S. Jeffreys,

*Methods of Mathematical Physics*(Cambridge U.P., London, 1956);E. D. Rainville,

*Special Functions*(Macmillan, New York, 1960);*Handbook of Mathematical Functions*, National Bureau of Standards AMS 55, edited by M. Abramowitz and I. A. Stegun (U.S. GPO, Washington, D.C., 1964).

For numerical calculations involving Jacobian elliptic functions, a very useful reference is L. M. Milne‐Thomson,

*Jacobian Elliptic Function Tables*(Dover, New York, 1950);although this book is now out of print, it is still available in many libraries. For tables, see also M. Schuler and H. Gebelein,

*Eight and Nine Place Tables of Elliptical Functions*(Springer, Berlin, 1955).Graphs (but no tables) of the Jacobian elliptic functions may be found in E. Jahnke and F. Emde,

*Tables of Functions*(Dover, New York, 1945).9.

T. C.

English

and T. F.

Gallagher

, Jr., Rev. Sci. Instrum.

40

, 1484

(1969

).10.

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