Reshaping Data & Pivot Tables in Grid Measurements
Mathematical Problem Formulation
Physical Law: Spatial Magnetic Field Uniformity in a Helmholtz Coil:
A Helmholtz coil pair consists of two identical circular coils of radius $R$, separated by distance equal to their radius ($d = R$). On the central transverse plane ($z = 0$), the axial magnetic field $B_z(x, y)$ exhibits maximal uniformity due to cancellation of second-order spatial derivatives:
$$\left. \frac{\partial B_z}{\partial z} \right|_{0} = 0, \quad \left. \frac{\partial^2 B_z}{\partial z^2} \right|_{0} = 0$$
Off-axis in the transverse plane, the field diminishes quadratically with radial distance $r = \sqrt{x^2 + y^2}$ according to the multipole expansion:
$$B_z(r) \approx B_0 \left( 1 - \alpha \frac{r^2}{R^2} \right)$$
Spatial Matrix Transformation (Pivot Table):
Laboratory data collected as sequential serial tuples $(x_i, y_i, B_i)$ in "long format" is mapped via a pivot transformation into a 2D Cartesian spatial grid matrix $\mathbf{B} \in \mathbb{R}^{M \times N}$:
$$\mathbf{B}_{j, k} = B_z(x_k, y_j)$$
Theoretical Background & Explanation
1. Long vs Wide Formats in Experimental Physics:
Laboratory automated scanning rigs (such as 2D Hall probe positioners or automated spectrometer goniometers) output continuous streaming logs where each line records a point coordinate and a sensor reading ("long format"). However, physical analysis and 2D contour visualization require a 2D matrix structure ("wide format").
2. pd.pivot_table() for Spatial Grid Construction:
The pivot_table() function maps coordinates to index rows and columns, reorganizing 1D lists of records into coordinate matrices. If multiple measurements exist at the same coordinate $(x, y)$, it automatically aggregates them via mean or median.
3. Stacking & Unstacking:
.unstack(): Pivots an index level into column headers..stack(): Condenses column headers back into a compact multi-index Series.
4. Field Homogeneity Evaluation:
By subtracting the central field $B(0, 0)$ across the matrix, researchers quantify spatial field deviations $\frac{\Delta B}{B_0}\%$, establishing the effective uniform working volume of the magnet apparatus.