Trapping of neutral molecules by the beam electromagnetic field

Neutral uncharged molecules are affected by the electromagnetic field of a charged particle beam if they carry either an electric or a magnetic dipole moment. The residual gas in an accelerator beam pipe consists of such molecules. In this paper we study their dynamics. Under a few approximations, whose validity we explore and justify, we derive the equations of motion of neutral molecules and their invariants, determine the conditions for these neutral molecules to become trapped in the field of the beams as a function of beam-pipe temperature, and compute the resulting enhancement of molecule density in the vicinity of the beam. We demonstrate that large agglomerates of molecules, “flakes,” are much more likely to be pulled into the beam than single molecules, and suggest that this phenomenon might help explain some beam observations at the Large Hadron Collider.

Figure 1

Magnetic field line pattern around a beam. Magnetic field strength decreases as $1/r$ with radial distance $r$ from the origin. The red spot at the center indicates the position of a proton beam leaving the picture plane towards the reader.

Figure 2

Electric and magnetic field of the beam as a function of radial distance as generated by a pointlike thin-wire beam of current 1 A.

Figure 3

Electric field lines around the round beam for a beam current of 1 A.

Figure 4

The rms transverse thermal velocity of a molecule, $v_{\mathrm{th}}$, as a function of temperature (top) and the average kinetic energy versus the rms thermal velocity $v_{\mathrm{th}}$ (bottom).

Figure 5

Motion of neutral molecules initially 1 mm from the beam, at rest, and subjected to the electromagnetic field of the beam, as seen in the horizontal direction. Different colors correspond to different molecules as indicated.

Figure 6

The motion of neutral oxygen, water and carbon monoxide molecules initially at rest, 1 mm from the beam, and subjected to the electromagnetic field of a beam, in the transverse plane.With zero initial velocity, all molecules possessing either a magnetic or electric dipole moment undergo an oscillatory motion.

Figure 7

Motion of neutral molecules initially at 1 mm from the beam with a thermal velocity corresponding to a temperature of 0.9 mK, and subjected to the electromagnetic field of the beam, in the horizontal (top) and the vertical direction (bottom). Different colors correspond to different molecules as indicated.

Figure 8

The motion of neutral oxygen, water and carbon monoxide molecules initially at 1 mm from the beam with a thermal velocity corresponding to a temperature of order 1 mK, and subjected to the electromagnetic field of a beam, in the transverse plane. The initial thermal velocities were for ${\mathrm{O}}_2$ $v_{\mathrm{th}}=0.2\mathrm{m}/\mathrm{s}$ (0.1 mK), for water $v_{\mathrm{th}}=3\mathrm{m}/\mathrm{s}$ (20 mK), and for CO $v_{\mathrm{th}}=0.3\mathrm{m}/\mathrm{s}$ (0.3 mK).

Figure 9

Motion of neutral molecules initially at 1 mm from the beam with a thermal velocity corresponding to a temperature of 15 K, and subjected to the electromagnetic field of the beam, in the vertical direction, with a hypothetical chamber wall at a radius of 6 mm. Different colors correspond to different molecules as indicated.

Figure 10

The motion of neutral oxygen, water and carbon monoxide molecules initially at 1 mm from the beam with a thermal velocity corresponding to a temperature of 15 K, and subjected to the electromagnetic field of a beam, in the transverse plane. The initial thermal velocities were $v_{\mathrm{th}}=62\mathrm{m}/\mathrm{s}$ for ${\mathrm{O}}_2$, $v_{\mathrm{th}}=82\mathrm{m}/\mathrm{s}$ for water, and $v_{\mathrm{th}}=67\mathrm{m}/\mathrm{s}$ for CO. Note the assumption of perfect elastic collisions with the chamber wall.

Figure 11

Motion of neutral flakes of molecules initially at 1 mm from the thin-wire beam with a thermal velocity corresponding to a temperature of about 15 K, and subjected to the electromagnetic field of the beam, in the horizontal (top) and the vertical direction (bottom). Different colors correspond to different types of constituent molecules as indicated.

Figure 12

The motion of neutral oxygen, water and carbon monoxide molecule flakes initially at rest, initially 1 mm from the thin-wire beam, and subjected to the electromagnetic beam field, in the transverse plane, for a pipe temperature of about 15 K.

Figure 13

The horizontal normalized force acting on an EDM (black) or MDM molecule (red), respectively, as a function of the transverse $(x,0)$ coordinate. The equilibrium position for the EDM case is indicated by the label $r_e$. Note that for both types of dipole moment the force is discontinuous at $x=0$.

Figure 14

Fraction of EDM molecules trapped in the beam as a function of $T/T_p^{*}$, for $T$ from 0 to $2T_p^{*}$ (left) and through $20T_p^{*}$ (right). For this calculation we considered trapping inside a radius of $R_b=3\sigma$, with $\sigma$ the rms beam size.

Figure 15

Fraction of MDM molecules trapped in the beam as a function of $T/T_{\mu }^{*}$, for $T$ from 0 to $2T_{\mu }^{*}$ (left) and through $1000T_{\mu }^{*}$ (right). For this calculation we assumed $R_b=3\sigma$.

Figure 16

Simulated pinch of the neutral EDM molecules in a cold gas, i.e., $T/T_p^{*}=0$. The particles are distributed either uniformly only in the $x$ plane (left) or randomly distributed (right) in a circular beam pipe of radius $R_p=10\sigma$.

Figure 17

EDM molecule density as a function of the radius and “time” (top) and the end time-averaged local molecule density versus radius (bottom). The molecule density is normalized with respect to the initial (or space-averaged) molecule density. The molecules are cold $T/T_p^{*}={10}^{-5}$. In this simulation, the particles were distributed throughout the transverse $x\text{−}y$ space. At the bottom, the large peak of density is $\sim 20$ times the initial uniform distribution. Note the smaller peak of density, which reflects the local oscillations of molecules around the equilibrium radius $r/\sigma \simeq \pi /2$; this phenomenon is clearly visible in the top panel as well.

Figure 18

Molecule density as a function of the radius and time (top), and the end time-averaged local molecule density versus radius (bottom). The center panel shows the $x\text{−}y$ structure of the molecule density at the point of maximum density. The molecule density is normalized with respect to the initial (or space-averaged) molecule density. In this simulation, the molecules are at a temperature $T/T_p^{*}=0.1$, and the particles initially distributed throughout the transverse $x\text{−}y$ space. According to the top panel (color scale), the peak density is $\sim 2.4$ times the initial uniform density.

Figure 19

Variation of the radial density distribution of EDM molecules with temperature $T/T_p^{*}$. Shown are the density enhancement due to the pinch in the beam field (black markers, left axis), the radial location of the maximum (blue markers, right axis), the average radial position of molecules (dash-dotted blue line), and the rms value of the radial position (dashed blue line).

Figure 20

Simulated pinch of the neutral MDM molecules in a cold gas, i.e., $T/T_{\mu }^{*}=0$. The particles are distributed either uniformly only in the $x$ plane (left) or randomly distributed (right) in a circular beam pipe of radius $R_p=10\sigma$.

Figure 21

MDM molecule density as a function of the radius and time (left) and the end time-averaged local molecule density versus radius (right). The molecule density is normalized with respect to the initial (or space-averaged) molecule density. In this simulation, the molecules are at a temperature $T/T_{\mu }^{*}=0.1$, and the particles initially distributed throughout the transverse $x\text{−}y$ space.

Figure 22

MDM molecule density as a function of the radius and time (left) and the end time-averaged local molecule density versus radius (right). The molecule density is normalized with respect to the initial (or space-averaged) molecule density. In this simulation, the molecules are at a low temperature of $T/T_{\mu }^{*}={10}^{-5}$, and the particles initially distributed throughout the transverse $x\text{−}y$ space.

Figure 23

Variation of the radial density distribution of MDM molecules with temperature $T/T_{\mu }^{*}$. Shown are the density enhancement due to the pinch in the beam field (black markers, left axis), the radial location of the maximum (blue markers, right axis), the average radial position of molecules (dash-dotted blue line), and the rms value of the radial position (dashed blue line).

Figure 24

Simulated angles of a molecule’s $\overrightarrow{p}$ (blue) and local electric field $\overrightarrow{E}$ (red), with respect to the horizontal direction, at the start (top left) and end (center left) of an adiabatic increase of the electric field of the beam, over a total time $T$ that would correspond to $\tau ={10}^5$ periods of phase advance ${\omega }_{E}T$. The panels on the right present the simulated angular direction of a molecule’s magnetic dipole moment $\overrightarrow{\mu }$ (blue) and of the local magnetic field $\overrightarrow{B}$ (red) during an analogous ramp-up of the beam’s magnetic field, over the same number of periods of phase advance ${\omega }_{B}T$. The top and center panels each present the behavior over a time interval $\mathrm{\Delta }\tau =200$. The bottom panels show the molecules’ trajectories over the full simulation. The red circle in the bottom left panel indicates the equilibrium position $r_e$—compare Fig. 13 and Eq. (35).

Figure 25

Electric dipole moment phase space orthogonal to the local field direction, $(p_{\perp }$, $d{p}_{\perp }/d\tau )$ (top left), and the adiabatic invariant $A_{\mathrm{inv}}$ as a function of the number of oscillations (top right). The right panel presents data for three different values of ${\tau }_{\max }$. The middle two panels present analogous simulation results for the case of the magnetic dipole moment. At the beginning of each simulation, large oscillations of $p_{\perp }$ or ${\mu }_{\perp }$ are seen, which later on converge onto an ellipse. The width of this ellipse in the direction of $p_{\perp }$, or ${\mu }_{\perp }$, becomes smaller the larger ${\tau }_{\max }$ is. Recalling Eq. (a15), ${\tau }_{\max }$ can be expressed as ${\tau }_{\max }=T_{\mathrm{ramp}}/(2\pi )(|p|E_0/I_i{)}^{1/2}$, and, therefore, the width of the ellipse in the direction of $p_{\perp }$ becomes smaller the more adiabatically the field changes (larger $T_{\mathrm{ramp}}$) and the higher the final field is (higher $E_0$). This is demonstrated in the bottom panel by the dependence of $A_{\mathrm{inv}}$ on ${\tau }_{\max }$, both for a linear (left) and for a Gaussian ramp (right).

Figure 26

Simulated density based on a rigid-molecule model, for a temperature $T/T_p^{*}=0.1$, at the normalized time $n_{\mathrm{osc}.}=1$, which is approximately the moment of maximum peak density. The picture presents a snapshot of the molecule distribution in the $x\text{−}y$ plane. The molecule density is normalized with respect to the initial (or space-averaged) molecule density. The particles are initially distributed uniformly throughout the transverse $x\text{−}y$ plane. For this simulation, we tracked $1.2\times {10}^7$ molecules with $L^2/{\sigma }^2={10}^{-5}$. The density is estimated on a grid of $400\times 400$ bins.