We derive an analytical expression of equivalent incoherent scatter spectral bandwidth and discuss its practical and interpretative usefulness at different altitude ranges. In the region above 120 km, where collision can be neglected, the equivalent bandwidth is a simple function of electron and ion temperature, ion mass, and ion composition. In the collision-dominated D region, equivalent bandwidth can be used to conveniently assess the effect of negative ions. As an example to illustrate the usefulness of the equivalent bandwidth, we also present nighttime incoherent scatter E region temperature measurement, which has not been reported previously for middle and low latitudes.
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 Incoherent scatter (IS) theory was first laid out in the early 1960s, and it matured very quickly in a short period of time [e.g., Farley, 1966, and references therein]. The complexity of the incoherent scatter spectra, although intimidating, has proved to be a rich source for the extraction of various ionospheric parameters including electron density, ion and electron temperatures, plasma drift, ion-neutral collision frequency, and ion composition [e.g., Evans, 1969]. The IS theory had remained largely the same for several decades until recently, when Sulzer and Gonzalez  included the effect of electron Coulomb collisions. Sulzer and Gonzalez have shown that electron Coulomb collision is important when the radar pointing is within a few degrees off perpendicular to the magnetic field line B and the radar wavelength is relatively long (say, several meters). It becomes largely negligible when the radar pointing is more than a few degrees away from the perpendicular-to-B direction. Using the incoherent scatter data taken by the Jicamarca radar, Aponte et al.  have shown that the inclusion of the Coulomb collisions results in the expected electron-to-ion temperature ratio for a large range of conditions while otherwise the ratio is often less than 1, which is physically unrealistic.
 The present study is concerned with the situation where the effect of electron Coulomb collisions can be neglected. Our purpose here is to reduce the complicated incoherent scatter spectra into one single parameter and discuss its practical usefulness. This single parameter is the equivalent spectral width, and its usefulness extends from quantitative determination of the effects of various variables to direct determination of ionospheric parameters. In section 2 we first give a general expression for incoherent scatter equivalent bandwidth and then discuss its approximations applicable to different regions. In section 3 we present the nighttime E region temperatures derived using the equivalent bandwidth method.
2.1. A General Expression for the Equivalent Incoherent Scatter Spectral Bandwidth
 An important characteristic of an incoherent scatter spectrum is its width. To quantitatively determine the IS spectral width, we define the equivalent bandwidth as the total backscatter power divided by the spectral power at the zero Doppler frequency. In order to derive the equivalent bandwidth we start with a general expression for the incoherent backscatter cross section of multiple-ion species having the same bulk motion,
where the summation is for all ion species, including negative ions [Swartz and Farley, 1979]. For our purpose here, we will assume that the ions do not feel the effect of the magnetic field while the electrons do. The admittance functions for the ions and electrons are [Dougherty and Farley, 1963]
respectively. In the above equations, and the subscripts are for the ith ion or the electron. The approximation in equation (3) is valid unless the radar pointing is very close to perpendicular to B. In the admittance functions, ψ = ην, θ = ηω, and φ = ηΩ = ηeB/m are the normalized angular frequency, collision frequency, and gyrofrequency, respectively, where η is the normalization factor defined by
In the above equations, we have largely used the same notations as used by Dougherty and Farley . Ne is the electron density, re = 2.82 × 10−15 m2 is the classical electron radius, m is the ion/electron mass, K =1.38 × 10−23 J K−1 is the Boltzmann's constant, T is the ion/electron temperature, k is the radar wave number, ω is the angular frequency variable, ωo is the center/transmitter frequency, ν is the ion/electron neutral collision frequency, μ is the electron-to-ion temperature ratio, and α is the angle between the radar pointing direction and the geomagnetic field B. In addition, pi is the ion partition function defined by
where Ni and qi are the number density and charge of ith species, respectively, and e is the amount of charge for an electron. Here h2 = Kεo/(Nee2) is the electron Debye length squared. G is similar to the plasma dispersion function and is defined as
When all the ion species have the same temperature, the total incoherent backscatter cross section for the expression shown in equation (1) is
This expression can be derived analytically using the method employed by Farley  for single-ion species. It agrees with Mathews'  continuum approach for multiply charged positive and negative ions. With the admittance functions as expressed in equations (2) and (3), the spectral power at the center frequency can be shown to be
where A is defined as
Equation (8) reduces to that given by Dougherty and Farley  for the case of a single-ion species, μ = 1, and no magnetic field. (The (1 + h2k2) term should be (1 + h2k2)2 in their equation (19).) For practical measurements the second term in equation (8) is negligible unless h2k2 is very large or α is very close to 90°. When α is 45°, as in the case of Arecibo, h2k2 needs to be as large as ∼10 for the second term to be comparable to the first term in the F region and has to be even bigger in the D region. For the Arecibo 430 MHz radar the measurable h2k2 is smaller than 3, corresponding to 100 el cm−3 in the D region. Thus for practical purposes, we can neglect the second term in equation (8) when the radar is not perpendicular to the field lines. When α is very close to 90°, one needs to include the effect of electron Coulomb collisions as well [Sulzer and Gonzalez, 1999].
 The equivalent bandwidth, designated as Wq, in units of radians, is then
We have neglected the contribution of the electron term in σb(ωo) as mentioned above. In addition, we have assumed that the plasma moves in one bulk velocity and that all the ion species are in thermal equilibrium. Wq is the equivalent bandwidth of a square spectrum with the spectral power as σb(ωo). This quantity can be easily measured from the observed spectrum. To appreciate the usefulness of the equivalent bandwidth, we discuss approximations applicable for different altitude ranges in section 2.2–2.4.
2.2. Approximation Above 120 km
 When the normalized ion collision frequency ψi is small (say, ψi smaller than 0.1), we have
Further, since negative ions and multiply charged ions are not known to exist above 120 km in a substantial amount compared to the total number of electrons, the sum of ion partition functions is unity. Thus we have
This approximation has a relative error less than 5% when ψi is less than 0.07 at 430 MHz. When h2k2 ∼ 0, which is typical of the daytime E and F regions, elevation of electron temperature above ion temperature results in a relative spectral broadening by a factor of (1 + Te/Ti)/2. In the case of single-ion species, Wq is linearly proportional to (Ti/mi)1/2.
 For the nighttime ionosphere, μ = 1, we have
If the ion composition is known, one can directly obtain the temperature using the equivalent bandwidth without going through any nonlinear least squares fittings. For small h2k2 and a single-ion species, ηi = 3.54/Wq, which leads to Ti = 4.6 × 10−7Wq2 for an ion mass of 31 amu and a probing frequency of 430 MHz.
 In addition to being a useful constraint on the different parameters the above equation can also be used to estimate errors caused by potential erroneous assumptions. For example, in deriving the F1 layer temperature, we need to assume a certain mixture of molecular (NO+, O2+) and atomic (O+) ions. The relative temperature difference between the assumption of 50% molecular ions and 50% atomic ions and the assumption of all atomic ions is about 44% in order to have the same equivalent bandwidth. This highlights the importance of having the correct ion composition for accurate determination of temperatures. This applies to the topside ionosphere as well, where one needs good estimation of the light ion composition in order to obtain accurate temperatures. Ion composition in the topside and, in fact, different ion temperatures as well are now regularly determined by fitting the IS spectra at Arecibo [Gonzalez and Sulzer, 1996; Sulzer and Gonzalez, 1996]. Compositional fitting in the F1 layer is more difficult because of the rapid altitude change of ion composition and the fact that the low altitude of the region prevents high spectral resolution measurements. Nevertheless, since molecular ions and atomic oxygen ions are sufficiently separated in mass, compositional fitting may still be realistic.
2.3. Approximation for the D Region
 In the altitude below 90 km, ψi is typically larger than 4, and Ti = Te. In such a case, we have
This approximation has a relative error smaller than 5% when ψi is larger than 4 for a radar frequency of 430 MHz.
 For singly (positively) charged ion species we have
When h2k2 is small, Dougherty and Farley  have shown that the spectrum is Lorentzian with the 3 dB full width as 2/(ηiψi). Equation (16) is more general since it is applicable even when h2k2 is reasonably large, which is often the case in the D region.
 A more interesting case to apply the equivalent bandwidth is the lower D region where negative ions exist. Let us consider only two types of ions, oppositely and singly charged but otherwise the same. If we denote the ion partition function as p− and p+ for negative ions and positive ions, respectively, charge conservation requires p+ = 1 + p−. Mathews  has shown that one effect of negative ions is to broaden the spectral width. This effect can be readily seen in the equivalent bandwidth. Assuming that the number of free electrons remains the same, the change in equivalent bandwidth due to the presence of negative ions is
When h2k2 is close to zero, the above equation reduces to (1 + p−); that is, the equivalent bandwidth increases linearly with the ratio of negative ion to electron concentration. In general, it can be shown that when h2k2 is smaller than 0.62, the above ratio is larger than 1 and increases with increasing p−. Otherwise, increasing p− actually decreases the spectral width. A physical interpretation of the spectral broadening due to negative ions in the case of small h2k2 is that the presence of negative ions reduces the hold of positive ions on the electrons [Mathews, 1978]. An intuitive interpretation of spectral narrowing due to negative ions when there are very few electrons is that negative ions are similar to electrons in reducing the Debye shielding distance, and hence the spectral width, although they do not contribute to the total scattering power. Generally speaking, however, the spectral width for an electron concentration Ne and a negative ion concentration N− is always wider than the spectral width of Ne + N− free electrons.
 Previous observations of negative ions were made by Ganguly et al.  using the powerful Arecibo incoherent scatter radar. Recently, Zhou  has shown that using a longer interpulse period, one can shorten the integration time by 2 orders of magnitude for incoherent scatter spectral measurements below ∼88 km. This improvement makes it much easier to study negative ions using IS radars and will be explored in the future.
2.4. Approximation for the Lower E Region
 In the lower E region between 95 and 120 km we have Ti = Te. For a single-ion species and daytime conditions, h2k2 ≪ 1, we have
In this altitude range (more precisely, when ψi is approximately between 0.1 and 4) the 1/A(ψi) − ψi term is a nonlinear function of ψi. Nevertheless, the following approximations can be used to simplify the above relation:
The second approximation corresponds to the altitude range of 91–104 km, and the first approximation applies roughly from 104 to 115 km. When one applies the above equations, the approximate bandwidth has an accuracy of 4%. If either νin or ηi is assumed to be known, one can solve the other parameter without fitting the spectra. If one needs to measure both parameters, one can use the above equations and fit for only one parameter.
 Current practice in IS fitting in this region is to assume that the ion-neutral collision frequency is known in order to derive the neutral temperature. In such an approach the ion-neutral collision frequency does not depend on time, and the temperature derived is assumed to represent tidal fluctuation. The problem is that the ion-neutral collision frequency has a time-varying component as well since it is proportional to the air density. Figure 1 plots the relative error of temperature and ion-neutral collision frequency for a probing frequency of 430 MHz. In both cases, the ion temperature is assumed to be 200 K, and the ion mass is assumed to be 31 amu. The relative error for temperature (collision frequency) is obtained by assuming that the ion-neutral collision frequency (temperature) is increased by 10% while the equivalent bandwidth is kept the same. If the density fluctuation is assumed to be about the same as the temperature fluctuation, the error in temperature fluctuation caused by assuming a non-time-varying ion-neutral collision frequency is more than half of the temperature fluctuation itself when the ion-neutral collision frequency is larger than 3000 Hz. This collision frequency corresponds to an altitude of approximately 105 km. Figure 1 also illustrates the difficulties of obtaining the ion-neutral collision frequency when it is smaller than 1000 Hz. In the collision-dominated regime (νin > 10 kHz) any error in temperature causes an equal amount of error in ion-neutral collision frequency, which can be readily seen from equation (16) as well.
2.5. An Example Incoherent Scatter Equivalent Bandwidth Profile
 In Figure 2c we plot the incoherent scatter bandwidth at a probing frequency of 430 MHz. The assumed ion-neutral collision frequency and electron concentration are plotted in Figure 2a, and the assumed ion temperature and mean ion mass are plotted Figure 2b. For the ion-neutral collision frequency we used the equation given by Mathews . The atmospheric density and temperature largely conform to those found in the U.S. Standard Atmosphere (1976). In addition, we have assumed the electron temperature to be the same as the ion temperature. The relative negative ions are assumed to change linearly with altitude from 60 to 75 km with a value of 4 at 65 km and 0 at 75 km. In Figure 2c we also plot the 3 dB full width, which is usually narrower than the equivalent bandwidth. The two dotted lines are the approximations for negligible ion-neutral collision frequency, applicable for higher altitudes, and for large ion-neutral collision frequency, applicable for lower altitudes, respectively. The dotted lines overlap with the solid line except in the 20 km altitude range around 100 km.
 Above 120 km the equivalent bandwidth is largely determined by the temperature and the ion composition, both in favor of widening the bandwidth with increasing altitude. In the altitude range of 90–120 km both temperature and ion-neutral collision frequency are important in controlling the spectral width. Between 75 and 90 km the bandwidth variation is essentially dominated by the change in ion-neutral collision frequency. Below 75 km, although ion-neutral collision frequency is still important, the low electron concentration and the presence of negative ions in our assumption make the bandwidth increase with decreasing altitude. Of interest is the fact that the equivalent bandwidth is much wider than the 3 dB width when the Debye length is large. The reason for this is that the 3 dB width is dominated by the ionic component, which is not very much affected by the Debye length, while the equivalent bandwidth is determined by both the electronic and ionic components. In the region where there is a sufficient amount of electrons the 3 dB width is about 10% narrower than the equivalent bandwidth. We further note that in the region where h2k2 ∼ 0 the equivalent bandwidth for unequal ion and electron temperatures is simply (1 + Te/Ti)/2 times that shown in Figure 2c.
3. Measurement of Nocturnal Ionospheric Temperature
 In this section we apply the equivalent bandwidth discussed in section 2 to derive the nocturnal ionospheric E region temperature, which has not been previously reported for an incoherent scatter radar at midlatitudes and low latitudes. The E region IS spectra at Arecibo were obtained using the coded-long-pulse (CLP) technique as described by Sulzer . The CLP program measured the IS spectra from 100 km up to ∼180 km with an 1.2 km height resolution for the data presented here although 150 m height resolution is now available at Arecibo. Further description of the CLP program is given by Zhou et al.  and Isham et al. .
 The theoretical basis for deriving the nighttime E and F region temperature is equation (13). Because the nighttime electron density is very low, one in general needs to consider h2k2, which is equal to 1.6 × 106Te/Ne for a probing frequency of 430 MHz where Ne is in m−3 and Te, assumed to be the same as Ti, is in kelvins. In addition, the near-field effect is considered in the same manner as it was by Zhou . Figure 3a is the electron density for the night of 8–9 March 1999. It was obtained by integrating the total spectra and calibrating to an ionosonde. Because of the fast recombination rate in the E region there is little ionization most of the time at night. However, ions/electrons often form layers and descend from the bottom of the F region to the E region shortly after sunset. This layer is variously known as the intermediate layer, the descending layer, or the descending tidal ion layer, and a detailed description of this phenomenon is given by Mathews et al. . Although the intermediate layers during the March 1999 observation were not very strong, we could derive the ion temperature fairly easily with an integration time of 40 s. The measured temperature corresponding to the electron density shown in Figure 3a is presented in Figure 3b. In deriving the ion temperature, which is the same as the ambient neutral temperature at night, we have assumed an ion mass of 31 amu. We also used the nonlinear least squares method to fit for Ti by assuming Te = Ti. Most of the nonlinear least squares fittings do not converge above 140 km.
 In Figure 4 we compare the temperature profiles measured on three consecutive nights with the Mass Spectrometer Incoherent Scatter (MSIS) model result. Because of the gradually descending nature of the intermediate layer the temperature profiles were measured during a time duration of several hours, with the top part measured around 2000 LT and the bottom part close to local midnight. The measurements for the three nights were largely the same for all the altitudes from 110 km to 170 km. The measured result on average is about 15% lower than the MSIS model from 125 km to 145 km. The lower temperature measured can be caused by incorrect assumption of ion mass. We have assumed that the intermediate layer is composed of molecular ions based on the observational results presented by Miller et al.  for the altitude range above 140 km at geomagnetically low latitudes. Nevertheless, it is possible that the layer composition below 140 km at Arecibo (dip angle ∼45°) is different. Comparison with independently measured temperature, such as that from a satellite, can resolve the ambiguity concerning ion compositions.
 We have given a general expression for incoherent scatter equivalent bandwidth, which is defined as the total power divided by the power at zero Doppler frequency. Although in its most general form the equivalent bandwidth is a complicated expression, its approximation forms can be made very simple for most regions of the ionosphere. The simplicity of the approximations makes it easier to quantitatively interpret the effect of electron/ion temperature, ion mass, and compositions. For example, the equivalent bandwidth can be used to estimate the temperature error caused by incorrect assumptions of ion composition and ion-neutral collision. The equivalent bandwidth can also be used to obtain ionospheric parameters directly under some circumstances or to reduce the number of free variables in nonlinear least squares fittings in general. We also present E region night temperature measurements from the intermediate layers using the equivalent bandwidth approach.
 I would like to thank Michael Sulzer for various discussions regarding the coded-long-pulse data processing. The Arecibo Observatory is part of the National Astronomy and Ionosphere Center, which is operated by Cornell University under a cooperative agreement with the National Science Foundation.