EISCAT RadarSchool 2005 : Basic ScatteringTheory A generic radar system The radar equation Basic scattering theory Thomson scattering Electron radar cross section Scattering from many electrons Scattering from plasma Ion-acoustic waves Gudmund Wannberg, EISCAT HQ

RADAR acronym for RAdio Detection And Ranging: Generic name for measuring systems deriving information about distant objects (radar targets) by illuminating them with RF energy and recording the reflected and/or back- scattered energy, In the early days, only the presence of, and range to, the radar targets could be inferred, (that is how the acronym became RADAR...) Today, many radar systems also allow spectral analysis and (sometimes) target imaging.

A generic radar system Transmitting antenna: G T Receiving antenna: A r A/DRX Power: P TX Timing & Control To computer Signal generator Radar target:  Physics Engineering

Basic Radar Equation: P r =  P G T A r /[(4  R 2 ) (4  R 2 )] Most radars are monostatic, i.e. transmitter and receiver are co- located and share a common antenna with gain G. From antenna theory: G = 4  A/ 2 Monostatic Radar Equation: P r =  P G 2 2 /[ (4  ) 3 R 4 ] The physics is contained in ... We first consider a single electron:

Scattering from a single electron Let us assume that the periodic electric field (generated e.g. by a radar transmitter) at the location of the electron is E(t): E(t) = Re (E 0. exp (–i  0 t)), (1) where E 0 is a complex electric field. E(t) exerts a time-varying force on the electron. If  0 >>  0, its equation of motion becomes: -e E 0 exp (–i  0 t) = m e. dv e /dt(2) where e = Coulomb, m e = kg, and  0 is the electron gyrofrequency. The velocity of the electron is v real = Re (v). Solving for v(t): v 0 = -i e E 0 / (m e  0 )(3)

Scattering from a single electron The current density associated with the motion of this electron becomes: j (r,t) = -e v(t)  (r – r e (t)) (4) where r e (t) is the position of the electron and  (r) is a spatial delta function. The vector potential at the receiver due to this current becomes: A (r,t) = (  0 / 4  )  (j’/  r – r’  ) d (r’)(5) where the integral is evaluated at the retarded time t’ = t -  r – r’  /c Using the current density defined by (4), we can integrate (5): A (r,t) = (  0 e / 4  ) v (t’) / (  r – r e (t’)  )(6)

Scattering from a single electron Assuming non-relativistic conditions and choosing the origin at the location of the electron, we can rewrite (6) as: A (r,t) = - (  0 e / 4  ) v (t’) /  r  = - [  0 e 2 / (4  m e  0 )] E 0 (1/  r  ) exp –i (  0 (t -  r  /c)) (7) We now introduce the wave vector k 1, defined through k 1 R =  0  r  /c where  k 1  =  0 /c and the direction of k 1 is along r : A (r,t) =- [  0 e 2 / (4  m e  0 )] E 0 (1/ R 1 ) exp –i(  0 t - k 1. r ) (8) where R 1 =  r 

Scattering from a single electron Now compute the magnetic induction B = (1/  0 ) H at r (the location of the receiver) in the far field approximation: B (r, t) =  x A   - [  0 e 2 / (4  m e  0 )] [(k 1 x E 0 ) /R 1 ] exp –i(  0 t - k 1 r ) (10) The Poynting vector at the receiver due to this induction becomes S r = ½   H  2 = ½  [ e 2 / (4  m e  0 )] 2 [  k 1 x E 0  / 4  ] 2 (11) where  = (  0 /  0 ) ½ = ohms is the impedance of free space.

Scattering from a single electron Introducing the power density incident on the electron, S in, S in = ½  E 0  2 /  and the polarisation angle, χ sin χ =  k 1 x E 0  / (  k 1 .  E 0  ) we can finally compute S r, the power density at the receiver: S r = ¼ [e 2 / (  0 m e c 2 )] 2 sin 2 χ S in /(4  R 1 2 ) = (4  r 0 2 ) sin 2 χ S in /(4  R 1 2 ) where r 0 is the classical electron radius, defined as r 0 = e 2 / (4   0 m e c 2 ) = m (12)

Scattering from a single electron The total power scattered by the electron is found by integrating (12) over the whole sphere : P T =  S r. R 2 d  = =  T S in 3/2 1/4  sin 2 χ d  =  T S in (13) where  T = 8  /3 r 0 2 = m 2, the electron Thomson cross section, is the ratio of total scattered power to incident power density.  T is a constant, independent of  0 The scattered power density along the r direction is S r = (4  R 1 2 ) -1 [3/2  T S in sin 2 χ ] where the factor in brackets is the electron radar cross section,  0 When sin 2 χ = 1 (backscatter),  0 = 3/2  T  m 2

Scattering from many electrons Consider an ensemble of electrons contained in a small box of volume V, sufficiently far away from the radar that a plane wave approximation is OK. The transmitter-generated field in the volume is: E 1 (r,t) = Є 0 p exp –i (  0 t - k 0. r )(27) where r is some arbitrary position within the volume, relative to an origin somewhere in the volume. The scattered field from an electron at position r p (t) is: E r (R 1,t) = - (r 0 /R 1 )  p Є 0 exp –i (  0 t’ - k 0. r p (t’))(28) where the retarded time t’ is = t – (1/c)  R 1 – r p (t’) 

Scattering from many electrons Since V is far away from both transmitter and receiver, we can approximate: t’ ~ t + (1/c) [R 1 + n 1. r p (t’)](29) Here, n 1 is a unit vector from the origin within the volume towards the receiver. We now introduce a time-dependent electron density N(r,t). The total number of electrons in a volume element d(r) is then N(r,t). d(r) and the field at the receiver due to these electrons is dE r (R 1,t) = - (r 0 /R 1 )  p Є 0 N(r,t’) exp –i (  0 t’ - k 0. r) d(r) (30)

In the ionosphere, the electron density is typically some m- 3 – a very large swarm of mosquitos indeed ! As with mosquitos, the electron density will fluctuate with time due to random thermal motions and/or density waves of one kind or another. Rather than treating so many electrons individually, we expand the density time variation in a Fourier integral:

Scattering from many electrons We expand the density variation vs. time in a Fourier integral: N (r,t) = (1/2  )  N (r,  ) exp -i  t d  T where N (r,  ) =  N(r,t) exp i  t dt 0 After some substitutions, we can compute the Fourier component of the elementary contribution from the volume element d(r) to the received field E r : dE r (R 1,  ) = - (r 0 /R 1 )  p Є 0 d(r). (1/2  )  d  ’ N(r,  ’) exp i [(  ’ +  0 ) R 1 /c - (k 1 - k 0 ) r N ]  dt exp it (  -  ’ -  0 ) where r N is a unit vector along r and the integral over t is taken from 0 to T.

After a further bit of algebra (omitted due to lack of time), we arrive at the following expression for the scattering cross section per unit volume and frequency interval:  V (  1 ) = 4  r 0 2 sin 2 χ. (1/T) (1/V) where  k 0  =  0 /c and  k 1  =  1 /c Scattering from many electrons The scattering is a three-wave interaction, which satisfies a Bragg condition: k = k 1 – k 0  =  1 -  0

Only the spatial Fourier component of the density distribution with wave- vector k = k 1 - k 0 contributes to the scattering at (k 1,  1 ) ; i.e. exists. We can illustrate this using a stationary model: If a small, randomly distributed, stationary ensemble of electrons is illuminated, they will scatter in different relative phases and the effective cross section will depend on the look direction...

Scattering from a plasma Plasma is electrically quasi-neutral, i.e. in any volume there are, on average, equal numbers of electrons and positive ions, and seen from the outside the volume has zero net electric charge. Both electrons and ions can scatter EM radiation, BUT Ionospheric ions (H +, O +, NO +, O 2 + ) are – times heavier than electrons and therefore scatter that much more weakly, as we can see from the equation of motion: e E 0 exp (–i  0 t) = m i dv i /dt = - m e dv e /dt ; dv i /dt = - (m e / m i ) dv e /dt The current density and induction due to ions are thus negligible BUT: In plasma, electrons no longer move freely; they are bound to the ions by electric polarisation fields. The plasma dispersion function has a low-frequency branch, the ion-acoustic branch, which is largely controlled by the ions...

Scattering from ion-acoustic waves The ion-acoustic dispersion relation in the one-dimensional case is: (  /  ) 2 = m i -1 (kT e + k T i ) = c s 2 Here  designates the wave vector and k is Boltzmann’s constant. The large ion mass determines the period of the associated ion acoustic waves. Note that in the ionosphere, the electron and ion temperatures can be (and probably are) unequal !

Scattering from ion-acoustic waves in plasma We can re-write the dispersion relation as f = (1/2  )  [m i -1 k (T e + T i )] ½ = Λ -1 [m i -1 k (T e + T i )] ½ which shows that the ion-acoustic frequency is - inversely proportional to wavelength, - directly proportional to mean plasma thermal velocity In a mono-static radar situation, ion-acoustic waves with Λ = radar /2 will scatter a signal back towards the transmitter !

Scattering from ion-acoustic waves in plasma Since the power spectral density of an ion-acoustic radar echo is controlled by - electron density n e - ion temperature T i - electron temperature T e - ion mass m i we can derive physical parameters from the echo spectrum !!! NOTE: The total ion-acoustic scattering cross section  ion  n e (1+ T e /T i ) –1 is typically only one-half of that expected from an electron gas of equal charge density; even less when the T e /T i ratio is >1

Some model IS ion line spectra from different ionospheric regions LEGEND: Red – F region (300 km) n e = T e = 2000 K O + T i = 1000 K Green - F region (300 km) n e = T e = 3000 K O + T i = 500 K Blue – E region (120 km) n e = T e = 300 K NO + / O 2 + T i = 300 K Black – topside (1000 km) n e = T e = 4000 K 90%O + 10% H + T i = 3000 K Spectra computed for the EISCAT UHF radar wavelength of 0.33 m (930 MHz). Power spectral density (y-axis) plotted to linear scale

IS and bulk plasma drift In addition to the internal wave motion, the whole plasma may be drifting in some particular direction under the influence of external forces (e.g. convection electric fields). In an IS measurement, a bulk drift manifests itself as a Doppler shift of the whole received scatter spectrum. The line-of-sight component of the drift velocity can be determined from: v drift = ½ c (  f/f o ) where  f = Doppler shift f o = ISR operating frequency fofo ff Note:  f > 0 drift towards radar ! f

Bulk plasma drift and convection E fields At altitudes above 200 km, the plasma is essentially collisionless (the mean free path is in the order of tens of meters) and thermal effects tend to drive both electrons and ions in spiral trajectories along the magnetic field lines. However, if there are electric fields present, these can drive the plasma across the magnetic field lines with a velocity v  : v  = E x B / B 2 Since B is known to fair accuracy and/or can be extracted from magnetic field models, once we have determined v  (which we do by measuring three components of it), we can compute the magnitude and direction of the convection electric field E