It is well accepted that the gamma-ray burst (GRB) prompt emission is original from synchrotron radiation. Synchrotron radiation is the radiation of relativistic electrons in an ordered and large-scale magnetic field. If magnetic field is random and small-scale, synchrotron radiation is not valid. In this work, we propose that random and smallscale magnetic field can be generated by turbulence. The socalled jitter radiation is the radiation of relativistic electrons in random and small-scale magnetic field (Mao & Wang 2011). Jitter photons can be scattered by those relativistic electrons. We call this phenomenon as “jitter self-Compton (JSC)” process. We apply this physical process to the study of GRB. The mini-jets in a bulk jet structure is also introduced as well (Mao & Wang 2012). We present our model below.

The radiation by a single relativistic electron in the smallscale magnetic field was studied by Landau & Lifshitz (1971). The radiation intensity, which is the energy per unit frequency per unit time is

where

is the frequency in the radiative field, ω_pe is the background plasma frequency, γ is the electron Lorentz factor, and w_ω' is the Fourier transform of the electron acceleration. We simplify the radiation feature in one-dimensional case as

The dispersion relation q₀ = q₀(q) is in the fluid field, and the radiation field can be linked with the fluid field by the relation ω' = q₀ - q_v. We adopt the dispersion relation in the relativistic collisionless shocks presented by Milosavljevic et al. (2006). We find

. The relativistic electron frequency is ω_pe = (4πe²n/Г_shm_e)^1/2 = 9.8 × 10⁹Г_shs^-1. where n = 3 × 10¹⁰cm^-3 is the number density in the relativistic shock.

The stochastic magnetic field <δB(q)> generated by the turbulent cascade can be given by

where

is decided by the turbulent cascades (She & Leveque 1994). The famous Kolmogorov number is ξ_p = p/3.

In general, our JSC calculation is as same as Synchrotron Self-Compton calculation. The JSC emission flux density in the unit of erg s^-1 cm^-3 Hz^-1 is

where f(x) = x+2x²lnx+x²-2x³ for 0 ？ x ？ 1, f(x) = 0 for x ？ 1, and x≡v/4γ²v₀. Thomson scattering section is σ_T = 8πr₀²/3=6.65×10^-25cm². n_ph(v₀) is the number density of seed photons, and it can be easily calculated from the jitter radiation.

The electron energy distribution is given by Giannios & Spitkovsky (2009) as

for γ ≤ γ_nth and

for γ ？ γ_nth, where C is the normalization constant, γ_nth is the connection number between the Maxwellian and power law components, and Θ = kT/m_ec² is a characteristic temperature.

We further apply a “jet-in-jet” scenario, as shown in Fig. 1. Those microemitters radiating as minijets are within the bulk jet. The possibility of observing these minijets can be estimated by

. The microemitter has the length scale of l_s = γct_cool, where t_cool = 6πm_ec/σ_TγB² The total number of microemitters within the fireball shell is n = 4πR²δ_s/l_s³, where R~10¹³ cm is the fireball radius and δ_s = ct_cool is the thick of the shell. The length scale of the turbulent eddy is l_eddy ~R/Г. We can define a dimensionless scale as n_l = l_eddy/l_s. Therefore, we sum up the contributions of the microemitters within the turbulent

eddy and obtain the total observed duration of GRB emission as T = n_lnPГt_cool.

We apply our model and reproduce the multi-wavelength spectrum of GRB 100728A. The extremely powerful X-ray flares and GeV emission of GRB 100728A were observed by the Swift/X-ray telescope and the Fermi/LAT, respectively. In this work, as shown in Fig. 2, the emission of GRB 100728A can be well explained by the jitter radiation and JSC process.