Transformations of Harmonics Under Reflection and Rotations

Hank Glass

3/5/98

Consider the case of a dipole (or gradient) magnet where the dipole
field may be either pointing up or down. The harmonics will depend on
the coordinate system, for which one has the following choices:

-   observe magnet looking at lead (or label) end or other end (LE vs
    OE)

-   the frame is chosen so that *y* (the normal direction) points in the
    direction of the field

-   the "lab" frame is chosen, so that *y* points up.

Transformations among these frames are made by appropriate rotations and
reflections as follows \[1\]:

1.  **Rotation:**

Under a rotation of angle *φ*, the normal and skew harmonics transform
according to

$\left( B_{n^{\prime}} + \text{iA}_{n^{\prime}} \right) = \left( B_{n} + \text{iA}_{n} \right)e^{\text{in}\varphi}$

(We are using the "European" convention where *n* = 1 corresponds to the
dipole.) Flipping coordinate system from **B↑ **(field points up) to
**B↓ **(field points down) is simply *φ = π*, and so

$\left\{ \begin{matrix}
B_{n^{\prime}} = ( - 1)^{n}B_{n} \\
A_{n^{\prime}} = ( - 1)^{n}A_{n}
\end{matrix} \right.\ $ **(** **)**

Odd harmonics (dipole, sextupole,...) change sign while even harmonics
(quadrupole, octupole,...) do not.

2.  **Reflection:**

In changing the view from Lead end to Other end (or vice-versa), the
harmonics change according to

$$\left\{ \begin{matrix}
B_{n^{\prime}} = ( - 1)^{n + 1}B_{n} \\
A_{n^{\prime}} = ( - 1)^{n}A_{n}
\end{matrix} \right.\ $$

The even normals and the odd skews (normal quadrupole, skew sextupole,
...) change sign while the other harmonics do not.

The situation can be summarized in the following table. Assume one
measures the harmonics in the Lead End **B↑** reference frame, and one
finds that all of the normals and skews are positive in this frame. The
table shows whether the harmonics change sign or keep the same sign in
the other frames.

+-------------+-------------+-------------+-------------+-------------+
|             | LE          | LE          | OE          | OE          |
|             |             |             |             |             |
|             | **B↑ **     | **B↓**      | **B↑ **     | **B↓**      |
+-------------+-------------+-------------+-------------+-------------+
| *b1*        | +1          | −1          | +1          | −1          |
+-------------+-------------+-------------+-------------+-------------+
| *b2*        | *+*         | *+*         | −           | −           |
|             |             |             |             |             |
| *(even      |             |             |             |             |
| normals)*   |             |             |             |             |
+-------------+-------------+-------------+-------------+-------------+
| *b3*        | *+*         | −           | *+*         | −           |
|             |             |             |             |             |
| *(odd       |             |             |             |             |
| normals)*   |             |             |             |             |
+-------------+-------------+-------------+-------------+-------------+
| *a2*        | *+*         | \+          | *+*         | \+          |
|             |             |             |             |             |
| *(even      |             |             |             |             |
| skews)*     |             |             |             |             |
+-------------+-------------+-------------+-------------+-------------+
| *a3*        | *+*         | −           | −           | \+          |
|             |             |             |             |             |
| *(odd       |             |             |             |             |
| skews)*     |             |             |             |             |
+-------------+-------------+-------------+-------------+-------------+

Note that for the Recycler project, the preferred coordinate system
\[2\] is OE **B↓**. **Nominal harmonics for the gradient magnets are
listed (in the LE B↑ frame) in Ref. \[3\], and the measurement system
also reports them in that frame.**

**References:**

\[1\] A.K Jain, "Basic Theory of Magnets," in proceedings of the *CERN
Accelerator School on Measurement and Alignment of Accelerator and
Detector Magnets*, April 11-17, 1997, Anacapri, Italy.

\[2\] C.S. Mishra et al., discussions at Fermilab MIMP meetings.

\[3\] <http://www-ap.fnal.gov/~gpj/FNAL/Recycler/Magnet_list.html>