Current and future applications of intense proton sources abound, including radiography, cancer therapy, warm dense matter generation, and inertial confinement fusion. With increasingly efficient acceleration and focusing mechanisms, proton current densities may soon approach and exceed $1010\u2009A/cm2$, e.g., via intense laser drivers. Simulations have previously shown that in this current density regime, beam-induced field generation plays a significant role in beam transport through dense plasmas. Here, we present a theoretical model for the generation of resistive magnetic fields by intense proton beam transport through solid density plasmas. The theoretical evolution of the magnetic field profile is calculated using an analytic model for aluminum resistivity, heat capacity, and stopping power, applicable from cold matter to hot plasma. The effects of various beam and material parameters on the field are investigated and explained for both monoenergetic and Maxwellian proton beams. For a proton beam with Maxwellian temperature 5 MeV and total energy 10 J, the model calculates resistive magnetic fields up to 150 T in aluminum. The calculated field profiles from several beam cases are compared with 2D hybrid particle-in-cell simulations, with good agreement found in magnitude and time scale.

## I. INTRODUCTION

Understanding the dynamics of intense particle beam propagation through plasma has numerous scientific applications, including for accelerators and colliders,^{1,2} neutron source generation,^{3} and inertial confinement fusion.^{4,5} In conventional linear accelerators, strong quadrupole magnets are used to steer and focus particle beams along their trajectory. Space-charge and current neutralization degree play key roles in ion beam transport through low-density plasma, which has been well investigated through experiments, theory, and simulations.^{6,7} In warm and hot dense plasma, however, the background electron density generally outnumbers the beam density, and beam transport largely depends on macroscopic characteristics such as self-generated fields, stopping power and conductivity. In this regard, the transport of intense ion beams through warm and hot dense plasma is wide open for scientific exploration.

The continuous advancement of short pulse lasers has opened this area of research, as laser-driven energetic proton beams with high current densities ($>109\u2009A/cm2$) and short bunch duration ($\u223cps$) are now routinely generated in experiments.^{8,9} Already, these laser-driven proton beams are used to produce warm dense matter samples for pump–probe experiments.^{10,11} As the laser to proton energy conversion efficiency, and subsequently proton beam current density, continues to increase, it will be necessary to account for collective effects as they propagate through dense plasma. For sufficiently intense proton beams incident on solids, self-generated resistive magnetic fields within materials may be capable of focusing or defocusing the proton beam itself. Simulations have shown that current densities $\u223c1010\u2009A/cm2$ can induce magnetic fields $\u2273100\u2009T$, enough to affect the trajectory of the protons within the material.^{12,13}

Here, we investigate and introduce a simple analytic model to estimate the magnetic field generation produced by intense proton beams. In Sec. II, we review the analytic model used to find the induced resistive magnetic fields, noting the mechanism differences between energetic protons and hot electrons. In Sec. III, the model is solved numerically and compared with hybrid particle-in-cell (PIC) simulations for both monoenergetic and Maxwellian beam sources. Various beam parameters are modified to observe their effects on magnetic field generation. In Sec. IV, these effects are discussed and explained in relation to the analytic model. This will ultimately aid in our understanding and estimation of field generation without the need to run computationally expensive simulations for several different cases.

## II. THEORY AND SEMI-ANALYTIC MODEL

In order to study the propagation and collective effects of intense proton beams, it helps consider that of hot electrons. When high-intensity lasers irradiate matter, the atoms are ionized and the freed electrons are accelerated (becoming “hot” electrons) through the material. If unimpeded, the energy stored within the magnetic field induced by their current is unreasonably large and unsustainable;^{14} therefore, the electron beam must inductively draw a return current composed of background electrons.^{15,16} This allows propagation of the hot electrons by neutralizing the total current within the material $Jc+Jb=0$, where $Jc$ and $Jb$ are background electron current density and beam current density, respectively.

However, the resistance encountered by the background electrons in the return current generates an electric field, which can be simply calculated by Ohm's Law $E=\eta Jc$ with material resistivity *η*. Furthermore, a spatial gradient in this resistive electric field drives a resistive magnetic field according to Faraday's Law

This resistive magnetic field is generated by two processes: (i) the neutralization of the beam current and (ii) any spatial gradients in either the current density or the material resistivity. This phenomenon has been investigated both computationally^{17,18} and experimentally.^{19,20} Importantly, this includes resistivity gradients due to temperature gradients since resistivity $\eta (Te)$ depends strongly on material temperature. In many practical cases where current density is strongly centralized and decays radially, e.g., Gaussian radial distribution, azimuthal resistive magnetic fields are generated.

Previously, assuming Ohmic heating via intense hot electron beam propagation, Davies found analytic solutions for temperature evolution and, subsequently, resistivity and field evolution from Eq. (1).^{21} These solutions assumed a simplified temperature dependence of resistivity and a constant heat capacity, yet proved remarkably useful in understanding the effects of different heating regimes. Two regimes were independently considered—cold $(\eta \u221dTe$ for $Te\u226aTF)$ and Spitzer $(\eta \u221dTe\u22123/2$ for $Te\u226bTF)$. The Fermi temperature *T _{F}* is typically within tens of eV for metals ($TF=12\u2009eV$ for aluminum and 7 eV for copper) and is a marker for the warm dense regime since electrons are partially degenerate at these temperatures. Nardi

*et al.*

^{22}used a more sophisticated model for resistivity and heat capacity to compare theoretical target heating with experimental results in femtosecond laser–matter interactions. This model takes into account collisional saturation in the warm dense regime, or $\eta \u223c\eta max$ for $Te\u223cTF$. Variations of this resistivity model were used also by Passoni

*et al.*

^{23}

In this work, we apply a variation of the above model to intense proton beams, which are capable of heating cold foils to temperatures over 1 eV, generating warm and hot dense plasmas on a time scale shorter than thermal expansion. Figure 1(a) is an apt illustration of the field generation from proton beams, wherein the beam current density induces a neutralizing current and resistive electric fields (yellow). Gradients in this field subsequently induce azimuthal magnetic fields (blue). Figure 1(b) illustrates the two primary heating mechanisms: Ohmic heating (yellow) brought on by the resistance encountered by the background electron return current and drag heating (red) as protons gradually deposit their energies during transport.

We neglect thermal conduction in this model since particle beam heating occurs on much shorter time scales. Ohmic heating is given by

where *C _{v}* is the temperature-dependent volumetric heat capacity. This is analogous to the macroscale power dissipation by resistors $P=RI2$. In this formulation, it is important to note that

*J*is technically the current density of the

*background electrons*, which was shown earlier to be equal and opposite to the

*beam*current density. As the beam particles themselves propagate through the material, they impart their energy to the bulk electrons via collisions. This collisional (drag) heating is expressed as

^{16}

where *e* is the elementary charge and $d\u03f5/dz$ is the temperature-dependent particle stopping power. Combining both heat sources, we obtain a first-order ordinary differential equation for temperature

which may be solved numerically. $Te(t)$ may, then, be used to calculate the resistivity and, in turn, the resistive fields from Eq. (1).

Before following this path, it is important to take a closer look at Eq. (4). In particular, let the ratio of the two source terms be given by

which can be interpreted as the ratio of averaged collisional (drag) force to the resistive electric force, both experienced by the beam particles. Since current densities and stopping power of hot electrons typically differ from those of protons, *S*_{0} will also differ. Considering resistivity of the form $\eta =\eta 0(T/T0)\alpha $, *α* = 1 approximates the cold regime and $\alpha =\u22121.5$ approximates the hot (Spitzer) regime. Between these extremes lies warm dense matter, which will be discussed later.

### A. Heating by hot electrons vs protons

Short-pulse laser-driven electron beams typically exhibit current densities $Je\u223c1013\u2009A/cm2$ and stopping power $d\u03f5e/dz\u223c5\u2009MeV/cm$ (for 5 MeV electrons at room temperature Al), yielding $S0\u223c0.06$ (resistivity is material-dependent). This indicates that Ohmic heating is dominant, accordant with Davies' neglect of drag heating.^{21} This is clearly seen in Fig. 2, which plots the numerical solutions to Eq. (4) for the cold (*α* = 1) and hot ($\alpha =\u22121.5$) regimes independently, assuming constant resistivity and heat capacity. In both regimes, $S0=0$ (blue) discounts drag heating completely. These curves match previous results.^{21} For hot electron beams which typically have $S0\u226a1$ (red), it is clear that drag heating does not significantly increase the temperature and resistivity. Because resistive fields arise from spatial gradients in resistivity, insignificant differences in resistivity beget insignificant differences in resistive fields.

Laser-driven proton beams, on the other hand, typically exhibit $Jp\u223c109\u2009A/cm2$ and $d\u03f5p/dz\u223c400\u2009MeV/cm$,^{8,9} yielding $S0>1000$. Contrary to hot electron beams, drag heating is now the dominant heating mechanism, shown clearly in Fig. 3. In the same vein as Fig. 2, $S0=0$ disregards drag heating, yielding the same blue curves. For $S0\u226b1$ (typical for intense proton beams), the numerical solutions to Eq. (4) are shown in red and yellow for both cold and hot regimes. When $S0>1000$, we clearly see orders of magnitude difference in heating. For *α* = 1, this can even be seen analytically by solving Eqs. (1) and (4) assuming constant stopping power (valid for particle energies $\u223cMeV$ and/or thin foils)

where $t0=CvT0/\eta 0J2$ is the characteristic Ohmic heating time scale. For $S0=0$, this reverts back to the single exponential solution shown in blue in Figs. 2 and 3 and in Ref. 21. For $S0\u226b1$, the temperature and resistivity increase by approximately a factor of *S*_{0}, which underlies the several orders of magnitude increase in temperature and resistivity for protons. Even though *S*_{0} has a spatial dependence through *J*, this factor of *S*_{0} will propagate through to augment the magnetic field as well.

### B. Resistivity and heat capacity model

So far, we have analyzed the cold and hot regimes independently, whereas in reality, resistivity follows a continuous function of temperature. In light of this, resistivity will be calculated from the Drude model

where *e* and *m _{e}* are, respectively, the electron charge and mass,

*n*is the free electron density, and

_{f}*ν*is the total electron collision frequency. Note that the free electron density implicitly depends on temperature via the mean ionization state of the material. Thus, calculating resistivity boils down to calculating the electron collision frequency. The Eidmann–Chimier model

_{e}^{24}smoothly interpolates the electron collision frequency among the cold, hot, and warm dense regimes. In the cold condensed matter regime $Te\u226aTF$ (Fermi temperature), electron–phonon collision frequency $\nu e\u2212ph$ and electron–electron collision frequency $\nu e\u2212e$ are dominant.

^{25,26}These collision frequencies describe the classical interaction of background electrons with lattice ions ($\nu e\u2212ph$) and with themselves ($\nu e\u2212e$). For the hot plasma regime $Te\u226bTF$, wherein atoms are approximately fully ionized, we use the classic

**Sp**itzer collision frequency

*ν*describing the Coulomb interaction among free electrons.

_{sp}^{27}For the intermediate regime $Te\u223cTF$, electron degeneracy and ion correlation effects come into play, invalidating the Spitzer formulation. An upper limit on the electron collision frequency is obtained by noting that the electron mean free path must exceed the inter-atomic distance

*r*

_{0}, or $\nu e<\nu max=ve/r0$ with electron thermal speed

*v*. This is equivalent to collisional or resistive saturation and has been shown both experimentally

_{e}^{28,29}and computationally.

^{30,31}

Thus, a harmonic average of the corresponding relaxation times is used to calculate the total electron collision frequency, and therefore resistivity, across a broad temperature range

Figure 4(a) displays the resultant $\eta (T)$ of solid-density aluminum from room temperature to 10 keV, along with experimental measurements.^{29,32–34} It should be noted that at low temperatures, the electron–phonon collision frequency naturally depends on the ion temperature. While electron–ion thermal equilibrium is assumed in the model above, this is invalid for ultra-short pulse laser–matter interactions, which formed the basis of resistivity measurements by Milchberg *et al.* and possibly the reason for their discrepancy. Nonetheless, resistive saturation is still apparent with a similar *η _{max}* as in the Eidmann–Chimier model.

The resistivity calculated from the model by Lee and More^{35} with Desjarlais ionization pressure correction^{36} (LMD model) is also plotted in Fig. 4(a) for comparison with the Eidmann–Chimier model. The hybrid-PIC code large scale plasma (LSP)^{37} was used here to calculate the electron collision frequency (LSP LMD module) for solid-density aluminum at the marked electron and ion temperatures. This was then plugged into Eq. (6) to calculate the corresponding resistivities, which show good agreement with the Eidmann–Chimier model for resistivity.

As in Eqs. (2)–(4), we must also consider the dependence of material heat capacity on bulk temperature. In the hot plasma regime, free electrons behave approximately as an ideal gas, yielding volumetric heat capacity $Cv=32nf$. For temperatures below this regime, the heat capacity calculation is deferred to tabulations made by Lin, Zhigilei, and Celli^{38} for various metals since the complexity of the calculation is outside the scope of this work. The two regimes are then smoothly interpolated to yield a piecewise formulation of $Cv(Te)$. Figure 4(b) displays the total heat capacity of Al at solid density from room temperature to 10 keV. Beyond 5 eV, the heat capacity is still temperature-dependent because the free electron density depends on the mean ionization state, which, in turn, is temperature dependent.

### C. Proton stopping power model

The final component to model before solving Eq. (4) is the particle stopping power. In this work, the proton stopping power is calculated by summing free- and bound-electron contributions independently to cover partially and fully ionized plasmas^{39}

with common stopping factor $\kappa 0=4\pi e4/mevp2$, bulk temperature-dependent mean ionization state $Z*$, and bound and free electron density $(Z\u2212Z*)ni$ and $Z*ni$, respectively. Estimation of the ionization degree $Z*$ of aluminum was interpolated from tabulated equation of state (EOS) properties generated by the software Prism PrOpacEOS.^{40} The bound electron stopping number *L _{b}* is calculated from the high-energy limit of the Bethe–Bloche expression

^{41–43}taking into account the excitation and ionization of target electrons. The free electron stopping number

*L*is calculated accounting for simple binary collisions and plasma oscillation excitations.

_{f}^{44}Ultimately, the total stopping power depends on the proton energy, the target material and its temperature, assuming constant density for time scales ∼tens of picoseconds.

## III. MAGNETIC FIELD GENERATION

Having modeled the temperature dependence of resistivity, heat capacity, and proton stopping power, and assuming a radially varying beam current density, we are finally in a position to solve Eqs. (4) and (1) numerically. After testing, the sufficient time resolution for the calculation was determined to be $1\u2009fs$ to start converging to the real solution. Aluminum resistivity and heat capacity have been well investigated and modeled as in Fig. 4 so will be used as the surrogate bulk material in this work with initial temperature *T*_{0}. In all cases, we consider a rigid beam model where the beam current density is unidirectional and axisymmetric with a Gaussian radial profile $J\u221d\u2009exp\u2009(\u2212r2/r02)z\u0302$ with characteristic radius *r*_{0}. Note that the full width at half-maximum is FWHM $\u22481.67r0$.

### A. Monoenergetic proton beams

Let us first consider a monoenergetic beam with on-axis current density *J*_{0}. Because thermal conductivity is ignored, the spatial dependence ultimately derives from time-independent *J*(*r*). Equation (1) then simplifies to

$Te(r,t)$ can be solved numerically from Eq. (4) and used to find $\eta (r,t)$ and, subsequently, the resistive fields as above. Note that since there is yet no longitudinal (*z*) dependence, this model applies to a thin sliver *δz* of bulk material, across which stopping power does not vary significantly (valid for $\u223cMeV$ protons). The free parameters in this model are, therefore, initial Al temperature *T*_{0} at solid density $2.7\u2009g/cm3$, beam parameters *J*_{0}, *r*_{0}, and proton energy *ϵ _{p}*.

To start, we investigate the impact of initial temperature *T*_{0} on magnetic field generation. Figure 5 displays $\eta (r,t)$ and $B\varphi (r,t)$ due to a monoenergetic $2.25\u2009MeV$ proton beam with $J0=1010\u2009A/cm2$ and $r0=17\u2009\mu m$, for initial temperatures [(a) and (b)] 0.03 eV, [(c) and (d)] 10 eV (warm dense Al, middle), and [(e) and (f)] 200 eV (hot dense Al). The first peculiar feature is—even though the initial temperatures 0.03 and 10 eV differ by three orders of magnitude, both conditions yield the same magnetic field evolution capping at $25\u2009T$. This can be explained by the characteristic rate of proton heating. The inset of Fig. 5(a) displays the temperature of Al along the beam axis in the first $500\u2009fs$. The protons instantly heat the Al to about 5 eV, reaching 10 eV after only 100 fs. One can imagine that this temperature evolution scales locally with the smoothly varying Gaussian current density. Since 10 eV is reached so quickly on the time scale of the field generation and since the resistivity explicitly depends on temperature, both scenarios follow an almost identical evolution.

However, the maximum magnetic field reached with initial temperature 200 eV is less than half that of the above, as shown in Fig. 5(f). This is because the initial resistivity is much lower at 200 eV than at 10 eV. Equation (10) shows that the field generation is proportional to the resistivity, so the sharp decrease in Spitzer resistivity from resistive saturation largely inhibits magnetic field generation. The significance of the initial heating period on field generation will be explained further in Sec. IV.

Before proceeding, it is important to benchmark these calculations with simulations. The hybrid-PIC code LSP^{37} is used here to benchmark the fields calculated numerically by Eq. (10). As a PIC code, LSP advances electromagnetic fields via Maxwell's equations (Faraday's and Ampère's Laws). To model the collisions of background particles in the cold and warm dense regimes, the LMD and Spitzer modules within LSP were used. Prism PrOpacEOS^{40} tables were also imposed to calculate the average ionization degree of aluminum as a function of density and temperature. The comparison of total electron collision frequency between the Eidmann–Chimier model and LSP LMD module was shown in Fig. 4(a). Proton energy deposition in LSP was calculated the same way as outlined in Sec. II C, where at each time step, stopping power was calculated dynamically based on temperature and density.^{39}

Comparisons of simulated theoretical calculations of magnetic field are shown in Fig. 6. Figures 6(a), 6(b), 6(d), and 6(e) display the simulation results of monoenergetic (5 MeV) proton beam propagation ($J0=1010\u2009A/cm2$) through Al with initial temperature 10 eV, where the beam was injected at *z* = 0 in the $+z$ direction. Although a smaller viewing window is displayed, the simulation box spanned $\u221250$–$400\u2009\mu m$ longitudinally and $70\u2009\mu m$ radially to avoid boundary effects. Note that the appropriate magnetic field profile is within the dotted regions in Figs. 6(b) and 6(e). Figures 6(c) and 6(f) show good agreement between the simulated (dotted curves) and calculated (solid curves) magnetic field profiles. In both cases, the simulated and theoretical fields converge with time in both shape and magnitude, with only a minor discrepancy in radial spread. The radial discrepancy is possibly due to the rigid nature of the theoretical model, which assumes perfect beam neutralization at all radii. In simulations, beam neutralization takes time and background electrons may even be pulled in from the beam periphery. Despite this discrepancy, it is important to note that simulations respond almost identically to theoretical calculations when beam radius is doubled, i.e., field magnitude halves and radial spread doubles.

Looking closer at the simulated fields in Figs. 6(b) and 6(e), it is interesting to note that there are actually two significant sets of azimuthal magnetic field—one as described above and another which aligns closely with the temperature contours near the beam front. The current model does not take into account the magnetic field development longitudinally, but one may speculate that these fields affect beam focusing or defocusing. This will be further discussed in Sec. IV.

### B. Proton beams with Maxwellian energy distribution

Laser-driven proton beams have proven a useful means of generating warm and hot dense plasmas isochorically and uniformly.^{45} In a typical proton-heating configuration, a high-intensity short-pulse laser irradiates a primary (source) target and couples primarily to bulk electrons. A cloud of hot electrons forms outside the source target, which generate an electric field capable of accelerating protons to $\u223cMeV$ energies from the rear surface.^{46} Protons and electrons co-propagate as a quasi-neutral beam across a vacuum gap before encountering a secondary (sample) target. There, co-propagating electrons generally have $\u223ckeV$ energies and stop within a thin ($\u223c\mu m$) front layer, leaving protons to propagate deeper into the sample. Still, protons would dominate over electrons in heating the front surface. Our theoretical model for proton beam-driven magnetic field development may also be applied to the front *δz* depth of the sample, simply by incorporating a time dependence in the current density and stopping power.

Laser-driven proton beams are accelerated from a source foil into vacuum and often exhibit Maxwellian energy spectra

with beam temperature *T _{p}* and total beam energy

*ϵ*. This distribution of protons would then disperse across a vacuum gap before encountering a sample foil. Due to this dispersion, incident proton energy and current density are time-varying at the sample front. Assuming the proton beam originates from an instantaneous “burst” source (valid for laser pulses $\u226aps$) and exhibits characteristic radius

_{tot}*r*

_{0}, the current density is expressed as

where $\tau =mpd2/2Tp$ is the characteristic impact time; *e* and *m _{p}* are the proton charge and mass, respectively; and

*d*is the vacuum gap distance. Similar expressions were given in Ref. 47 for the time-varying beam power.

This configuration is depicted in Fig. 7(a). Note that because the energy spectrum exponentially decays, beam density also decreases with distance from the source foil. Figure 7(b) shows the characteristic current density and beam power felt by the sample foil from a Maxwellian beam. Current density is near zero as very few high-energy protons reach the sample first, followed by protons with energy $\u223cTp$ forming the peak around $t\u223c\tau $, and ending with a $t\u22123$ decay ($t\u22125$ for beam power) of low-energy protons. Here, *t* = 0 represents the burst source time. Previously, current density incident on a sample target was held constant. To compare to these cases, it helps define the maximum current density incident on a sample

Importantly, $\u03f5(t)=mpd2/2t2$ must be used in calculating the time-varying stopping power in Eq. (4). Beyond this, the same strategy for calculating the resistive magnetic fields holds—solve for $Te(r,t)$, obtain $\eta (r,t)$, and calculate $B\varphi (r,t)$.

The magnetic fields resulting from varying Maxwellian beams ($Tp=5\u2009MeV$) are shown in Fig. 8. For all cases, the field develops steadily at early times before converging to a maximum profile. Convergence to this maximum field profile occurs when the current density decays, approximately at the inflection point past its peak, $t\u223c\tau $. This is reasonable since current density decays rapidly after this time, and total magnetic flux along with it. This point will be further discussed in Sec. IV.

Distribution . | $\u03f5tot\u2009(J)$ . | $\u27e8J\u27e9\u2009(\xd71010\u2009A/cm2)$ . | $r0\u2009(\mu m)$ . | $d\u2009(\mu m)$ . | $\Delta t\u2009(ps)$ . | $Bmax\u2009(T)$ . |
---|---|---|---|---|---|---|

Maxwellian (a) | 0.33 | 0.56 | 8.4 | 50 | 4.8 | 30 |

Maxwellian (b) | 10 | 17 | 8.4 | 50 | 4.8 | 135 |

Maxwellian (c) | 10 | 4.3 | 8.4 | 200 | 19.4 | 145 |

Maxwellian (d) | 10 | 0.12 | 50 | 200 | 19.4 | 5 |

Monoenergetic | 10 | 19 | 8.4 | ⋯ | 4.8 | 100 |

Maxwellian | 100 | 0.5 | 50 | 200 | 19.4 | 16 |

Distribution . | $\u03f5tot\u2009(J)$ . | $\u27e8J\u27e9\u2009(\xd71010\u2009A/cm2)$ . | $r0\u2009(\mu m)$ . | $d\u2009(\mu m)$ . | $\Delta t\u2009(ps)$ . | $Bmax\u2009(T)$ . |
---|---|---|---|---|---|---|

Maxwellian (a) | 0.33 | 0.56 | 8.4 | 50 | 4.8 | 30 |

Maxwellian (b) | 10 | 17 | 8.4 | 50 | 4.8 | 135 |

Maxwellian (c) | 10 | 4.3 | 8.4 | 200 | 19.4 | 145 |

Maxwellian (d) | 10 | 0.12 | 50 | 200 | 19.4 | 5 |

Monoenergetic | 10 | 19 | 8.4 | ⋯ | 4.8 | 100 |

Maxwellian | 100 | 0.5 | 50 | 200 | 19.4 | 16 |

As an example, Fig. 8(a) displays the magnetic field evolution from a Maxwellian proton beam with $\u03f5tot=0.33\u2009J,\u2009d=50\u2009\mu m$ ($\tau \u22481.6\u2009ps$), and $r0=8.4\u2009\mu m$. The calculated fields (solid curves) rapidly increase until $t=2\u2009ps$, within which time the current density peaks and slightly decays. For $t>2\u2009ps$, the field develops at a slower pace before converging to a profile with $Bmax\u224830\u2009T$. The parameters of this beam were chosen to approximately match the overall conditions of the monoenergetic beam shown in Fig. 6(a), i.e., the average proton energy *T _{p}* matches that of the monoenergetic beam, and the vacuum gap

*d*was chosen such that the time-averaged current density of the Maxwellian beam over $3\u2009ps$ is $8\xd7109\u2009A/cm$

^{2}, close to that of the monoenergetic beam. Interestingly, the magnetic field profile after $3\u2009ps$ of the both beams approximately match, even though the field development at earlier times does not. Simulations were also conducted for the beam conditions in Fig. 8(a), with results shown as dashed curves. The magnetic field evolution from simulations agrees quite well in magnitude to theoretical results, but again, varies slightly with radial spread.

Figure 8(b) shows the field evolution from the same proton beam configuration as in (a), except that the beam energy is increased to $10\u2009J$. This energy increase results in a total *B _{max}* increase by a factor of $\u223c4.5$. All else the same, amplifying the beam total energy amounts to multiplying the overall particle count and, therefore, current density. Conservation of energy may be employed to estimate the magnetic field amplification — since the field energy density is proportional to

*B*

^{2}, one may roughly estimate $Bmax\u221d\u03f5tot$. Amplifying

*ϵ*by a factor of 30 in the above case, this would place $Bmax\u223c160\u2009T$, slightly higher than the calculated $135\u2009T$ shown in Fig. 8(b). Obviously, conservation of energy here is more intricate than these relations, but this provides a rough estimate of the field amplification.

_{tot}Figure 8(c) shows the field profile evolution for the same beam parameters as in (b), except the vacuum distance has now been increased to $d=200\u2009\mu m$. To compare the field evolutions, both (b) and (c) contain profiles at intervals with respect to *τ*. To this end, the plots are remarkably similar in magnitude, suggesting self-similarity with the corresponding time scales. Compared to (b), the maximum current density in (c) decreases by a factor of 4 [Eq. (13)], yet the magnetic field achieved is slightly greater. This is because the decrease in current density $J\u221d1/d$ is compensated by the increase in time scale $\tau \u221dd$. The link between current density and time for magnetic field generation will be further discussed in Sec. IV.

Finally, the profiles shown in Fig. 8(d) are a result of beam parameters identical to (c), except that beam radius is increased to $50\u2009\mu m$. This change has by far the greatest impact on field generation since it alters not only the current density but also its radial gradient. Because both *J*(*r*, *t*) and $Jmax\u221d1/r02$, widening the beam by a factor of 6 would presumably decrease the field generation by a factor of 36. This is exactly what is observed, with $Bmax\u223c4.5\u2009T$ down from $150\u2009T$.

## IV. DISCUSSION

In all of the scenarios introduced here, it is important to point out that the field generation is most significant at early times. In other words, the gain in magnetic flux during equal time intervals generally decreases with time. This can be seen in Figs. 5(d)–5(f) and Fig. 8 and can be shown by integrating Eq. (10) with radius

assuming $J\u21920$ at large radii and where $\phi $ represents a quasi-2D azimuthal magnetic flux, i.e., $\phi =\u2202\Phi \u2202z$ with true azimuthal magnetic flux $\Phi =\u222cB\varphi drdz$. This is equivalent to invoking Stokes' Theorem on Eq. (1), or Faraday's Law in integral form. The magnetic fields apply up to a certain depth below which stopping power remains approximately constant, on the order of several *μm* [see Figs. 6(b) and 6(e)]. In this case, the true magnetic flux may be approximated by $\phi \u223c\Phi \Delta z$.

Equation (14) shows that the rate of change of the azimuthal magnetic flux is principally dependent on the *axial* field, made of both the axial resistivity and current density. At first, this may sound counterintuitive because the resistive magnetic field predominantly depends on the *gradients* of resistive electric field, i.e., resistivity and current density. Yet/However while the magnetic flux *density* ($B\varphi $) may take on large values at steep gradients, the total magnetic flux ($\phi $) looks only at the “area under the curve,” e.g., in Fig. 8.

Assuming a constant current density (as in Figs. 5 and 6), the change in magnetic flux with time is primarily determined by the resistivity, which depends on material temperature. Resistivity is maximized at the early times, while temperature increases toward resistive saturation and only decreases with temperature (Spitzer $\eta \u221dTe\u22123/2$) beyond $\u223c100\u2009eV$, decreasing the gain in magnetic flux along with it.

Current density is likewise a contributor to magnetic flux gain. This is more clearly observed with Maxwellian proton beams, where the current density exhibits a rapid rise followed by a steady decay. Near the current density peak, the magnetic flux gain is maximized, as shown by the first two plots in Figs. 8(b) and 8(c), corresponding to $t=\tau /2$ and $t=\tau $. In this phase, the increase in magnetic flux may be approximated by

The combination of $Jmax\u221d1/d$ and $\tau \u221dd$ cancels the dependence on *d* entirely, equalizing the total magnetic flux gain. For the same spatial scale (*r*_{0}), and self-similar spatial profile with time, *B _{max}* is bound to match as well. This explains the stark resemblance between the field evolution in Figs. 8(b) and 8(c). The minor increase when $d=200\u2009\mu m$ is likely due to the nonlinear heating and slightly prolonged phase in the warm dense regime, where magnetic flux gain is maximized. For $t>\tau $, the current density decays as $t\u22123$, and the magnetic flux gain significantly slows down. The field profile quickly converges accordingly.

It is important to note that within this framework, electron–ion thermalization is not taken into account, which may result in inaccurate electron temperature evolution. Equation (4) implies that all the energy deposited by protons goes into heating the background electrons. In reality, this energy would pass from the background electrons into background ions as they thermally equilibrate. Over several tens of picoseconds, the lattice ions are heated enough to hydrodynamically expand. Material properties including average ionization state, electron heat capacities, and stopping power calculated herein assume time-independent solid-state density, which breaks down upon material expansion.

In the extreme case where ions remain at room temperature, the resistivity curve would resemble that in Fig. 4(a) only in the Spitzer regime ($Te>10\u2009eV$). From room temperature up to $Te\u22481\u2009eV$, the resistivity would be constant (*T _{i}*-dependent), followed by a steep rise with plateau at $Te\u224810\u2009eV$. Comparing the two resistivity models, significant differences in resistive magnetic fields are only seen for current densities at or below $J\u223c109\u2009A/cm2$. For these current densities, the material spends a significant duration within the warm dense regime, where the resistivity models differ. Higher current densities heat the material quickly to $10\u2009eV$, after which the field generation does not differ between models.

In theory, if electron temperature is overestimated, resistivity in the Spitzer regime would be underestimated, leading to underestimated magnetic fields. Simulations conducted for the beam conditions shown in Fig. 8(b), for example, show maximum B-fields reach $450\u2009T$, over three times what theoretical calculations show. One possible reason that this is not seen where simulations agree with calculations is that the magnetic field maximizes within the warm dense matter regime, where resistivity varies less around saturation. The model's exclusion of electron–ion energy transfer is, thus, a major source of discrepancy when compared to self-consistent PIC simulations.

Maxwellian proton beams typically emerge from high-intensity ($>1018\u2009W/cm2$) laser interactions with solid targets. As previously mentioned, intense lasers primarily couple with and accelerate electrons beyond the target rear. What results from this are two electron populations—hot electrons ($\u223cMeV$) which reach the sample foil before the protons, and co-propagating electrons ($\u223ckeV$) which travel with and neutralize the protons.^{48} The co-propagating electrons are stopped within a skin depth of the sample foil, but the hot electrons may have an effect on the generated fields. The analysis thus far has assumed only protons incident on the sample foil, in which case background electrons have more than enough density to form a return current, but including a forward-propagating hot electron beam may alter the physics.

By analysis of a three-species system, one may approximately find the electric field necessary to reach a steady-state balance of proton, hot electron, and background electron currents, utilizing the appropriate collision frequencies. In a steady state, the equations of motion for hot and background electrons are

with collision frequencies *ν* and velocities *v* among hot electrons, background electrons, and protons. Note that we neglect the hot electron collisions with protons in the first equation and cold electron collisions with hot electrons in the second—both collision frequencies *ν _{hp}* and $\nu eh\u226a\nu he$ and

*ν*. Assuming all currents are neutralized $Je+Jh+Jp=0$ and $nh\u223cnp\u226ane$, we obtain for the electric field

_{ep}This may be simplified further by assuming $\nu ei\u226b\nu he,\nu ep$

where $eE0m=\u2212npne\nu eivp$ is the electric field if hot electrons were not taken into account, that is, if protons were neutralized solely by background electrons. Here, $\alpha =nhne\nu ei\nu he+\nu hi$ is a factor representing the hot electron contribution. Upon deeper analysis, we find that $\alpha \u226a1$, yielding electric fields that are minimally affected by the co-propagating electrons.

The fields observed in this work are relatively mild in terms of proton beam focusing ability. Simulations have shown that azimuthal magnetic fields of $\u223c30\u2009T$ do not significantly focus the protons, but those $\u227380\u2009T$ are sufficient to affect the beam radius.^{12} As previously mentioned, the fields responsible for focusing may be those that contour around the beam front, as seen in Figs. 6(b) and 6(e). Similarly, stronger fields may be seen in Fig. 3 of Ref. 39 and Fig. 5 of Ref. 12. One explanation for this is the following. Expanding Eq. (10) yields two terms

If the beam front always “sees” a cold solid ahead and around, the rapid proton-heating induces an instantly steep resistivity gradient

since $d\eta /dTe\u22650$ up to the warm dense regime and $\u2202Te/\u2202r<0$ because $dJ/dr<0$. Therefore, both terms add constructively to generate the magnetic field most efficiently.

As mentioned previously, the fields investigated in this work apply to the first several micrometers within a material on which the proton beam is incident. Field generation deeper within a dense plasma requires further development of the model, in which proton stopping power (especially near the Bragg peak) will play a more significant role. In addition to collisional stopping power, the model would have to account for beam collective effects in order to accurately reproduce beam transport.^{13,49} A more advanced model would include the $J\xd7B$ force at depth to self-consistently predict the beam evolution. When the beam starts to focus, longitudinal gradients come into play in magnetic field development. The model currently assumes *J _{r}* = 0, but for nonzero

*J*

_{r}As the beam front begins to focus, a positive feedback loop emerges since $\u2202\eta /\u2202z<0,\u2009Jr<0$, and $\u2202Jr/\u2202z>0$, further strengthening the magnetic field.

## V. SUMMARY AND CONCLUSIONS

We have investigated the development of resistive magnetic fields driven by intense proton beam propagation through matter. Unlike electron beams, proton beams significantly heat the target through direct collisional heating, as opposed to Ohmic heating brought about by the background electron return current. This necessitates a dynamic treatment of resistivity and heat capacity for the duration of the beam pulse since both depend strongly on temperature. Using the Eidmann–Chimier resistivity model for aluminum and the spatial and temporal evolution of temperature and resistivity, the resulting magnetic fields are solved for varying proton beam conditions. It was found that proton beams with Gaussian radial profiles induced azimuthal, annular magnetic fields.

For monoenergetic, constant current density proton beams, initial material temperature plays a significant role in total field generation. For Maxwellian proton beams, the development of resistive magnetic field is found to be self-similar with the characteristic impact time *τ*, as evidenced by Figs. 8(b) and 8(c) and explained by Eq. (14). It was found that beam radius has the most significant impact on the maximum field produced, assuming constant total beam energy. For a Maxwellian beam with characteristics of a typical laser experiment with two foils, the model calculated fields as strong as $145\u2009T$. The agreement of the calculated field profiles with hybrid-PIC simulations (notably limited by electron–ion thermalization) shows that the model can provide a good estimate of the magnetic field. The advantage of this model is to provide the essential physics involved in resistive field generation from proton beams. Since the model results are benchmarked with hybrid-PIC, resistive fields can be well estimated without the need for large processing power and time.

Aluminum was used as a sample metal in this work because the resistivity in several temperature regimes is well known, but the model can similarly be applied to other materials with known resistivity and heat capacity. This would be particularly important for proton fast ignition studies since understanding the self-generation of fields and resulting proton beam transport between the cone tip and the dense core is critical. Since aluminum's electric resistivity is lower than most, other transport media may produce stronger fields that are capable of self-focusing the beam.

## ACKNOWLEDGMENTS

This work was partially supported under the auspices of the U.S. DOE/NNSA HEDLP program by the University of California under Contract No. DE-NA0003876.

M.S.'s work was performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under Contract No. DE-AC52–07NA27344.

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

### Author Contributions

**Krish Bhutwala:** Conceptualization (equal); Formal analysis (equal); Investigation (equal); Software (equal); Visualization (equal); Writing – original draft (lead); Writing – review & editing (equal). **Joohwan Kim:** Conceptualization (equal); Formal analysis (supporting); Project administration (lead); Software (supporting); Supervision (equal); Visualization (supporting); Writing – original draft (equal). **Christopher McGuffey:** Funding acquisition (equal); Supervision (supporting); Writing – review & editing (equal). **Mark Sherlock:** Formal analysis (supporting); Investigation (supporting); Supervision (supporting); Validation (lead); Writing – review & editing (equal). **Mathieu Bailly-Grandvaux:** Formal analysis (supporting); Supervision (supporting); Writing – review & editing (equal). **Farhat N Beg:** Funding acquisition (equal); Supervision (equal); Writing – review & editing (equal).

## DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding author upon reasonable request.

## References

_{2}O-ice layered targets

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