Distorted Coulomb field of the scattered electron

Experimental results for the radiation emission from ultrarelativistic electrons in targets of 0.03%–5% radiation length is presented. For the thinnest targets, the radiation emission is in accordance with the Bethe-Heitler formulation of bremsstrahlung, the target acting as a single scatterer. In this regime, the radiation intensity is proportional to the thickness. As the thickness increases, the distorted Coulomb field of the electron that is the result of the first scattering events, leads to a suppressed radiation emission per interaction, upon subsequent scattering events. In that case, the radiation intensity becomes proportional to a logarithmic function of the thickness, due to the suppression. Eventually, once the target becomes sufficiently thick, the entire radiation process becomes influenced by multiple scattering and the radiation intensity is again proportional to the thickness, but with a different constant of proportionality. The observed logarithmic thickness dependence of radiation intensity at intermediate values of the thickness can be directly interpreted as a manifestation of the distortion of the electron Coulomb field resulting from a scattering event. The Landau-Pomeranchuk-Migdal effect is explored with high primary energy using materials with low nuclear charge ($Z$). Also, targets that should give rise to the claimed interference effect in high-energy radiation emission from a structured target of thin foils are investigated.

Figure 1

The electric field lines shortly after the electron has scattered once (a) and twice (b), respectively. For illustration purposes, the velocity of the particle has been set to $\beta =0.9$. The tangential component of the field lines can be interpreted as the radiation field, propagating outwards at the speed of light [6, 7]. As seen from one of the field lines (indicated by the thick red line), the rapid succession of scattering events may lead to closely spaced tangential component field lines pointing in opposite directions. To the observer, these field lines cancel each other for low frequencies.

Figure 2

A schematic drawing of the setup used in the experiment. The total length of the setup is about 20 m.

Figure 3

Exploded view of a $N=5$ target. The dotted line illustrates beam $z$ axis.

Figure 4

Relative area of the photo peak as a function the half-width of the selected region in units of $\sigma$ of the photo peak.

Figure 5

BGO calibrations: (a) The calibration at Aarhus. Data points are shown with error bars representing statistical errors only. (b) The calibration at CERN. The error bars of the data point obtained through a tagged electron beam is relatively large due to a momentum spread of the accepted beam. Nevertheless, the ASTRID calibration is well verified only with a slight energy shift.

Figure 6

Comparison between $e^{-}$ BGO calibration data obtained at ASTRID (full, red) and Geant4 simulation (dotted, black).

Figure 7

BGO backgrounds: blue lines are Geant4 simulations of synchrotron radiation generated in $\mathtt{B}\mathtt{16}$ at the two beam energies.

Figure 8

NHP Analysis: rejecting events following large pulses clearly changes the power spectrum. (a) NHP Logics: the first event (Trig) is a large pulse, which causes the analog PMT signal to enter an undefined state. PAT1 tags the possible, subsequent events that are considered flawed. (b) Results of applying the NHP condition.

Figure 9

BGO efficiency correction: a relatively flat polynomial can bring the data points in good agreement with simulations above 0.2 GeV.

Figure 10

The data points show the number of photons emitted $N_{\gamma }$ in a logarithmically binned interval of photon energies ($25\mathrm{bins}/\mathrm{decade}$) for various targets of the type $N\times \mathtt{A}\delta t\mathtt{A}\mathtt{i}\mathtt{r}1000$. The resulting power spectrum is corrected with the BGO efficiency and normalized to both the number of events $N_e$ and the combined target thickness in units of the material’s radiation length $\Delta t/X_0$. The lines show calculations of the power spectrum of each target using the Blankenbecler-Drell formalism [1]. As commented elsewhere, the calculations have been corrected by a factor 0.85 to take multiphoton emission and pair production into account.

Figure 11

The normalized power spectrum level as a function of the independent foil thickness ($⟨\delta t⟩$ in units of $X_0$ on lower scale, equivalent tantalum foil thickness on upper scale) in the different photon energy bins with the centroid value given. With squares and diamonds (with error bars denoting the statistical uncertainty only) are shown the measured values. The open, square data point at $\delta t=3\times {10}^{-4}$ is for the $80\times 25\mu \mathrm{m}$ Al foil, while the open, diamond data points are for the targets with $\Delta t/X_0\gtrsim 1.2\times 2.44\%$—i.e. not quite comparable to $100\mu \mathrm{m}$ Ta. The horizontal lines are the geant3 simulated values for $100\mu \mathrm{m}$ Ta according to Bethe-Heitler (long-dashed line) and LPM theory (full line). The falling curves are calculations of the logarithmic dependence of intensity on thickness (which decreases when divided by the thickness) following [33] (long-dashed), [12] (full), [1] (dashed), and the function in Eq. (7) (dotted). The vertical dashed line shows the thickness corresponding to the formation length, i.e. the critical thickness for the given photon energy.

Figure 12

Multiphoton correction factor for Ta foils of various total thickness $\Delta t$. These calculations presuppose a single-photon spectrum based on the LPM effect.

Figure 13

The $b$ parameter extracted from least-squares fits with the logarithmic function $a\ln (b\times ⟨\delta t⟩+1)/(b\times ⟨\delta t⟩)$, as a function of photon energy. The horizontal lines show the value of $b$ estimated from [33] and the leading order value found from [2], Eq. (30), shown in Eq. (8). Systematic uncertainties are estimated to be $\simeq 5\%$ (filled points) and $\simeq 10\%$ (open points).

Figure 14

Normalized power spectra for different materials at $E=207\mathrm{GeV}$. The experimental data points have been scaled by the BGO efficiency shown in Fig. 9. The three spectra carry similar statistics but are shown with different ordinate scales. The dotted and full lines are the geant3 simulations of the Bethe-Heitler and LPM theory, respectively.

Figure 15

Absolute and relative sandwich spectre measured at $E=207\mathrm{GeV}$. The reference and sandwich target spectra are close to identical. (a) Normalized power spectra for the different foil types. The LPM data for the $1\times 100\mu \mathrm{m}$ Ta target is shown again for reference. (b) Ratio of measured power spectra, $(10\times \mathtt{T}\mathtt{a}10\mathtt{A}\mathtt{i}\mathtt{r}90)/(10\times \mathtt{T}\mathtt{a}10\mathtt{A}\mathtt{i}\mathtt{r}1000)$, and corresponding ratio of BD calculations [1].

Figure 16

Characteristics of the logarithmic expression. (a) Terms of the Maclaurin series. (b) Relative deviation between the truncated Maclaurin series and Eq. (a2).

Figure 17

The normalized power spectrum level as a function of the independent foil thickness ($⟨\delta t⟩$ in units of $X_0$ on lower scale, equivalent tantalum foil thickness on upper scale) in the different photon energy bins with the centroid value given. With squares and diamonds (with error bars denoting the statistical uncertainty only) are shown the measured values. The open, square data point at $\delta t=3\times {10}^{-4}$ is for the $80\times 25\mu \mathrm{m}$ Al foil, while the open, diamond data points are for the targets with $\Delta t/X_0\gtrsim 1.2\times 2.44\%$—i.e. not quite comparable to $100\mu \mathrm{m}$ Ta. The horizontal lines are the geant3 simulated values for $100\mu \mathrm{m}$ Ta according to Bethe-Heitler (long-dashed line) and LPM theory (full line). The falling curves are calculations of the logarithmic dependence of intensity on thickness (which decreases when divided by the thickness) following [33] (long-dashed), [12] (full), [1] (dashed), and the function in Eq. (7) (dotted). The vertical dashed line shows the thickness corresponding to the formation length, i.e. the critical thickness for the given photon energy.

Figure 18

The normalized power spectrum level as a function of the independent foil thickness ($⟨\delta t⟩$ in units of $X_0$ on lower scale, equivalent tantalum foil thickness on upper scale) in the different photon energy bins with the centroid value given. With squares and diamonds (with error bars denoting the statistical uncertainty only) are shown the measured values. The open, square data point at $\delta t=3\times {10}^{-4}$ is for the $80\times 25\mu \mathrm{m}$ Al foil, while the open, diamond data points are for the targets with $\Delta t/X_0\gtrsim 1.2\times 2.44\%$—i.e. not quite comparable to $100\mu \mathrm{m}$ Ta. The horizontal lines are the geant3 simulated values for $100\mu \mathrm{m}$ Ta according to Bethe-Heitler (long-dashed line) and LPM theory (full line). The falling curves are calculations of the logarithmic dependence of intensity on thickness (which decreases when divided by the thickness) following [33] (long-dashed), [12] (full), [1] (dashed), and the function in Eq. (7) (dotted). The vertical dashed line shows the thickness corresponding to the formation length, i.e. the critical thickness for the given photon energy.