- Proton-electron bremsstrahlung
- 1 Introduction
- 2 Kinematics of the process
- 3 Cross sections
- 4 Photon spectra from protons with power-law spectra
- References

A&A 406, 31-35 (2003)

DOI: 10.1051/0004-6361:20030782

**E. Haug**

Institut für Astronomie und Astrophysik, Universität Tübingen, 72076 Tübingen, Auf der Morgenstelle 10, Germany

Received 16 May 2003 / Accepted 22 May 2003

**Abstract**

The collision of energetic protons with free electrons is
accompanied by the emission of bremsstrahlung. If the target electrons are
approximately at rest, this process is designated electron-proton
bremsstrahlung or suprathermal proton bremsstrahlung. The kinematics and
the fully relativistic cross section of proton-electron bremsstrahlung
in Born approximation is given. The X-ray spectrum produced by protons
with a power-law spectrum is calculated for thin and thick targets.

**Key words: **radiation mechanisms: nonthermal - X-rays: general

The galactic and solar cosmic radiation consists largely of energetic
(suprathermal) protons. When a beam of these protons is incident on a
plasma, appreciable X- and gamma radiation is produced in collisions
with ambient electrons which are approximately at rest. This process is
much the same as the normal electron-proton bremsstrahlung except that
now the center of momentum of the proton-electron system is virtually
that of the energetic proton. Therefore it was designated suprathermal
proton bremsstrahlung (Brown 1970; Boldt & Serlemitsos 1969), inverse
bremsstrahlung^{}, or proton-electron
bremsstrahlung (PEB, Heristchi 1986). The PEB process was considered to
be a potential production mechanism for the diffuse -ray
background (Boldt & Serlemitsos 1969; Brown 1970; Pohl 1998) and for
solar flare hard X-rays (Boldt & Serlemitsos 1969; Emslie & Brown
1985; Heristchi 1986).

In the nonrelativistic case (proton velocity )
the
bremsstrahlung produced by a proton of kinetic energy *E* has the same
spectrum as that of an electron of kinetic energy
in
collisions with a stationary proton (
and
are the rest masses
of electron and proton, respectively). Since, however, the accuracy of
the nonrelativistic Bethe-Heitler cross section for bremsstrahlung falls
off rapidly at higher energies (Haug 1997) the corresponding PEB cross
section is of small value. At relativistic energies, the derivation of
the PEB cross section causes more trouble. The usual bremsstrahlung
cross section differential in both the energy and angles of the emitted
photon has to be transformed to the electron rest frame and to be
integrated over the emission solid angle (Brown 1970; Haug 1972). The
calculation by means of the Weizsäcker-Williams method (Jones 1971)
which is inherently simpler, yields poor results at the high-energy
tails of the PEB spectrum (Haug 1972).

In view of the renewed interest in the X- and gamma-ray production through PEB (Dogiel et al. 1998; Pohl 1998; Valinia & Marshall 1998; Baring et al. 2000) it is worthwile to provide a fully relativistic cross-section formula where nearly all of the angular integrations are performed analytically thus allowing to calculate the X-ray spectra for arbitrary energy distributions of the incident protons with substantially reduced computational expense.

The variables of the SPB process are displayed in Fig. 1. In the
following the proton energy
(including rest energy) is expressed
in units of
MeV and the momentum
in units of
,
whereas the electron energy
and the photon energy *k*
are given in units of
,
the electron momentum
and the
photon momentum k in units of
.
Energy and momentum of the
outgoing particles are designated by primed quantities. In order to
calculate the maximum photon energy,
,
the finite rest mass of
the proton has to be taken into account. Otherwise
could be
greater than the proton kinetic energy *E* for
,
in
contradiction to energy conservation (Heristchi 1986). In an arbitrary
frame of reference the conservation of energy and momentum is most
conveniently expressed in terms of the four-momenta which are denoted
by underlined quantities. Taking into account the different energy units
for protons, electrons, and photons the relation reads

Squaring this equation yields, utilizing and ,

or

where denotes the invariant product of the four-vectors and . Since

we have

yielding

where and is the photon emission angle relative to the incident proton and electron momentum, respectively, and is the angle between the incident proton and electron (Fig. 1).

For fixed values of the energies and it is easily seen that the absolute maximum of the photon energy is reached for and , resulting in

If the incident electron is at rest this reduces to

The last inequality implies that the photon energy never exceeds the kinetic energy of the proton, as it should be. Neglecting, however, in the denominator of (8), one obtains

which is only valid for , or GeV.

Solving Eq. (7) for
and *p* leads to the minimum energy
and momentum, respectively, of the proton required to produce photons
of energy *k*. In the following we will restrict to the case that the
target electron is at rest in the laboratory system, i.e.,
.
Then we have

where

Formula (10) is needed for the calculation of the photon spectra from protons with an energy distribution (see Sect. 4).

For
,
Eqs. (10) and (11) reduce to

We consider the collision of an energetic proton with a stationary
electron (*p*_{1}=0). Designating the variables in the proton rest system
by an asterisk, the invariance of the four-product (*pk*) yields

The invariance of the doubly differential bremsstrahlung cross section implies

where is the element of solid angle of the emitted photon.

Using the relativistic transformation formula for the photon emission
angle ,

one gets

The cross section is the common bremsstrahlung cross section (neglecting proton recoil) where the quantities , and are expressed by the corresponding quantities in the rest system of the electron by means of Eqs. (14) and (16). Using the fully relativistic bremsstrahlung cross section in Born approximation as given by Sauter (1934) it has the form

where is the fine-structure constant,

It is instructive to compare the differential cross section (18)
with that of the normal electron-proton bremsstrahlung where the electron
has the same velocity, i.e.,
.
One notes the following: at
low proton energies the behaviour of the two cross sections is similar.
If the energies become higher, this changes drastically. The
electron-proton cross section has a sharp maximum at small angles
originating from the denominator
.
Passing on to the rest frame of the proton, this expression transforms
to *k* so that the proton-electron cross section becomes more isotropic,
except for
.
In any case protons can emit photons of higher
energy than electrons. For instance, protons of kinetic energy *E*=1 GeV
produce photons of maximum energy
MeV, whereas electrons with the same velocity
keV) generate photons with .

The spectrum integrated over the photon angles is given by

The lower integration limit is obtained from (6), setting

Applying the cross section (18), nearly all of the integrations in (19) can be performed analytically. According to (20) one has to distinguish between two cases:

a) :

where

In deriving the expression (21) the approximations

were made. The exact value would be, e.g.,

which is a small quantity for . The non-vanishing magnitude of , etc. arises from the fact, that the proton recoil is neglected in the cross section (18), whereas it was allowed for in the kinematics, resulting in the additional term of

In the case
one has to take the limit

b) :

For small values of the photon energy, , it is convenient to expand the cross section (29) into powers of

The relative error of this formula is less than 0.3% for kinetic energies of the proton

In the cross section (18) the distortion of the electron wave
functions by the proton's Coulomb field is neglected. This can be taken
into account approximately if (18) is multiplied by the Elwert
factor (Elwert 1939; Elwert & Haug 1969)

where and . In particular, this results in a non-vanishing cross section at the short-wavelength limit of the bremsstrahlung process. Then, however, the integration in (19) has to be performed numerically. The cross sections given by Haug (1972) were calculated in this way. The effect of the Coulomb correction is appreciable only near the high-energy limit of the PEB spectrum.

4 Photon spectra from protons with power-law spectra

In a thin target the energy distribution of the incident protons is not
influenced by collisions with other particles. Denoting the spectral flux
of the protons by
,
the PEB photon spectrum (number of
photons emitted per second, cm^{3}, and MeV) in a plasma with the ambient
electron density
,
assumed to be uniform, is given
by

The lower integration limit is given by Eq. (10) and the factor originates from the different energy units of and

In many astrophysical applications
has the form of a
power law in the kinetic energy with spectral index ,

where

In nonrelativistic approximation the integration in (34) can be performed analytically. One gets, analogous to normal bremsstrahlung (Brown 1971),

where

The upper curve of Fig. 2 shows the photon energy distribution for the
spectral index
of the cosmic-ray protons. At low photon
energies *k*<0.07, corresponding to
keV, the spectrum has
also the form of a power law with spectral index
,
in agreement with Eq. (35). If *k* increases, the photon spectrum
flattens due to relativistic effects, and the spectral index becomes
for *k*>10 or MeV. Inclusion of the Coulomb
correction factor (31) would change the photon flux a little at
low energies *k*. However, the slope of the spectrum is virtually the
same.

In a thick target the impinging protons are slowed down by collisions
with ambient particles. Employing normal energy units, the number of
photons of energy
emitted by protons with the initial kinetic
energy *E*_{0} is

where

Switching to the energy units and

If we choose again a power law for the proton injection rate,

the second integral of (38) can be easily solved yielding

Considering protons impinging on a hydrogen target with number density , the energy loss rate is given by

where is the proton velocity. The Coulomb logarithm is different for plasmas consisting of neutral or ionized hydrogen (Emslie 1978).

Figure 2:
PEB photon spectra from protons with a power-law spectrum,
spectral index
,
for thin (upper curve) and thick targets.
The proton fluxes are arbitrary. |

The thick-target photon spectrum plotted in Fig. 2 was calculated for
a completely ionized hydrogen target with electron number density
using the energy loss rate (Lang 1980)

where

is the maximum kinetic energy transferred to free electrons. To facilitate the comparison with the thin-target curve the spectral index was again chosen (the magnitude of the photon fluxes is arbitrary). At very low energies the photon spectrum can be approximated by a power law with , again in agreement with the nonrelativistic result of Brown (1971). With increasing

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