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The field emission effect was first discovered by Ralph H. Fowler and Lothar Wolfgang Nordheim in 19288_{.Field emission is the process whereby electrons tunnel through a barrier in} the presence of a high electric field (107 – 108 V/cm), due to quantum mechanical effects8,9. Since electron are emitted by the application of an electronic field instead of heating up the material, it is also referred to as “cold electron emission”. As illustrated in Figure 4.3, the external electric field lowers the effective work function, allowing the electrons near the Fermi surface with sufficient energy to tunnel through the potential barrier. The emission current density produced from the field emission is given by the Fowler-Nordeim equation:

𝐽 = 𝑎𝐹𝑒2 𝜑 𝑒

−𝑏𝜑3⁄2

𝐹𝑒 _{(4. 17 )}

Where J is the current density in A/cm2, a and b are constants with respective values of 1.54 x 10-6 A.eV.V-2 and 6.83 x 107 eV3/2 Vcm-1, φ is the work function of the material, and Fe is the calculated enhanced field10_{. From the equation it can be deduced that current is only dependent} on the applied electric field.

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Figure 4.19 Diagram of the field emission effect. The effective barrier is lowered by the electric field so that electrons near the Fermi level can tunnel through the barrier11.

Due to the high aspect ratios, and needle like tips of CNTs, the electric field at the tips far exceed the mean value of this parameter12. As a result, CNTs have high electric field enhancement factors, which enhance the electric field making it possible to achieve high emission currents12,11. The enhanced field, Fe, from Equation 4.1 is given by:

𝐹_𝑒 = 𝛽𝐹 (4. 18 ) where β is the electric field enhancement, and F is the electric field V/m. The field enhancement factor is most influential in the performance of CNT based cathodes. Compared to conventional field emission field emitters, CNTs function at relatively low voltages, allowing them to be used in miniature devices for a number of applications. Main applications of CNTs include, flat-panel field emission display, transparent conductive thin films, electron sources for electron

microscopes, and x-ray sources to name a few6. 4.2.2 CNT X-ray sources

The basic design of the x-ray tube has remained changed in the last century. Since CNT field emission is a better alternative to thermionic emission due to the emission of electrons at

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room temperature, compared to thermionic emission that requires heating of the cathode to high temperatures, CNTs are ideal candidates for cathodes in x-ray tubes11_{. However, there are} conditions that need to be satisfied for the stable operation of CNT cathodes, that determine its I- V characteristics12. Over a decade ago, our lab reported the optimization of the morphology of CNT films capable of producing high, and stable current emission. The cathodes were

constructed with purified SWNT bundles produced via laser ablation13. The SWNTs have an average diameter of 1.4 nm and an average bundle size of 50 nm. Presently, all CNT x-ray tubes cathodes are constructed using these SWNT in a triode type design13. Components of the CNT x- ray tube include one or an array of CNT cathodes, gate electrode, focusing electrodes, and anodes under a steady state vacuum around 10-10torr. A schematic diagram of a CNT based x-ray source is shown in Figure 4.4.

Figure 4.20 Diagram of typical CNT source design. The source consist of a CNT cathode, gate mesh, focusing electrodes and an anode. A small voltage is applied across the gate mesh and the cathode, while a large voltage is applied to the anode.

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The CNTs are electrophoretically deposited onto a molybdenum substrate by methods developed in our lab14_{. Areas of CNT deposition is dependent on the application of the x-ray} tube. Above the cathode is the gate mesh, constructed of tungsten bars. Voltages (Gate-cathode, Vgc) can be applied either to the cathode or gate while the other is grounded, on the order of 1000 – 2000 V. Focusing electrodes are placed above the gate, and are used to control the size of the desired focal spot by application a negative or positive voltage15. Anodes in CNT x-ray tubes are made of tungsten, and are angled at 16o15_{. Large DC voltages ranging from 30 kVp to 160 kVp} are applied to the anode depending on the imaging or therapy application. Because of the triode design in the CNT x-ray tubes, the anode current is less than current emitted from the cathode. The ratio of the tube current to the cathode current is known as the transmission rate. For CNT x- ray sources, usual transmission rates are about 60% to 70%15–17. Maximum tube current is limited to the heat capacity of the anode, and is calculated by the maximum power divided by the anode voltage. High tube currents allow for short x-ray pulses, which is beneficial in diagnostic imaging. High voltage feedthroughs and CNT source spacing is designed to prevent arcing in the tube. The focal spot size of CNT sources is dependent on electric field distribution between the gate through to the anode, and are of paramount importance in determining the system's spatial resolution15_.