(a) What is nuclear resonance absorption of gamma rays? Describe the working and applications of Mossbauer spectrometer. (15…

(a) Nuclear resonance absorption and the Mössbauer spectrometer

A nucleus in an excited state decays to the ground state by emitting a gamma ray; the same gamma ray, meeting a nucleus of the same kind in its ground state, can be absorbed to re-excite it. This is nuclear resonance absorption, the nuclear counterpart of the resonance absorption of light by atoms. For a free nucleus it is almost impossible to see: the recoil of the emitting nucleus takes away an energy R = E²/(2Mc²) (about 2 × 10⁻³ eV for the 14.4 keV line of ⁵⁷Fe), the absorbing nucleus needs the same again, and the natural line width Γ = ħ/τ (about 5 × 10⁻⁹ eV) is far smaller than 2R, so the emission and absorption lines do not overlap. Mössbauer's discovery (1958) is that when the nucleus is bound in a crystal a large fraction f of the events occur with the recoil momentum taken up by the crystal as a whole, with no phonon created; the recoil energy is then negligible, and the gamma ray is emitted and absorbed with the natural line width. Such recoilless emission and absorption make the resonance sharp enough to resolve energy shifts of one part in 10¹² or better.

The spectrometer consists of a source (⁵⁷Co diffusing in a rhodium or palladium matrix, decaying to the 14.4 keV level of ⁵⁷Fe), the absorber (the sample containing ⁵⁷Fe in its ground state), and a detector (proportional counter or scintillator) counting the gamma rays transmitted through the absorber. The source is mounted on an electromechanical drive that moves it towards and away from the absorber with a velocity v that is swept linearly over ±10 mm/s; the Doppler shift ΔE = (v/c)E changes the energy of the emitted line by about 5 × 10⁻⁸ eV per mm/s, comparable with the shifts and splittings in the absorber. Counts are stored in a multichannel analyser as a function of velocity; where the shifted source line coincides with an absorption line of the sample the transmitted intensity dips. The spectrum of dips against velocity is the Mössbauer spectrum, and it may be recorded in transmission or by counting the conversion electrons re-emitted from the surface.

Applications: the isomer shift gives the s-electron density at the nucleus and hence the oxidation state and bonding of iron (or tin, gold, europium); quadrupole splitting gives the electric-field gradient and hence site symmetry; magnetic hyperfine splitting into six lines gives the internal magnetic field and identifies magnetic ordering, used in the study of ferrites, steels, catalysts, minerals and haemoglobin; the recoil-free fraction measures lattice dynamics; and the sharpness of the line has been used for fundamental tests, the Pound–Rebka measurement of the gravitational redshift over 22 m and tests of time dilation.

(b)(i) Digital and analog systems

An analog system handles signals that vary continuously in amplitude and time, each value carrying information; a digital system handles signals that take only a finite set of discrete levels, in practice two, represented by voltage ranges for 0 and 1, and processes them as numbers. The principal reasons for the widespread use of digital electronics are: noise immunity and accuracy, since a signal only has to be recognised as 0 or 1 and can be regenerated exactly at each stage, whereas analog noise and drift accumulate; ease of storage and error detection, because numbers can be stored indefinitely and coded to detect and correct errors; programmability, the same hardware doing different jobs under software control; ease of design, using standard gates, flip-flops and integrated circuits whose behaviour is exact and repeatable and which can be integrated by the billion on one chip; economy and small size from that integration; and the ability to transmit, compress, encrypt and multiplex signals in ways analog systems cannot.

(b)(ii) Logic gates, and NAND and NOR as universal gates

AND gate: two inputs A, B; the logic diagram is the D-shaped symbol with the flat side receiving the inputs; Boolean equation Y = A·B. Truth table: 00→0, 01→0, 10→0, 11→1. Output is 1 only when all inputs are 1.

OR gate: the curved-back symbol; Y = A + B. Truth table: 00→0, 01→1, 10→1, 11→1. Output is 1 when any input is 1.

NOT gate: a triangle with a small circle at its output; Y = Ā. Truth table: 0→1, 1→0.

A NAND gate is an AND followed by NOT, Y = (A·B)‾, with table 00→1, 01→1, 10→1, 11→0; a NOR gate is an OR followed by NOT, Y = (A + B)‾, with table 00→1, 01→0, 10→0, 11→0. Either can build all three basic gates, and hence any logic function, which is what makes it universal.

From NAND: NOT by joining the two inputs, Y = (A·A)‾ = Ā. AND by a NAND followed by a NAND used as an inverter, Y = ((A·B)‾)‾ = A·B. OR by inverting each input with a NAND and feeding the results to a third NAND: Y = (Ā·B̄)‾ = A + B, by De Morgan's theorem.

From NOR: NOT by joining the inputs, Y = (A + A)‾ = Ā. OR by a NOR followed by a NOR inverter, Y = ((A + B)‾)‾ = A + B. AND by inverting each input with a NOR and feeding the results to a third NOR: Y = (Ā + B̄)‾ = A·B, again by De Morgan.

Since any Boolean function can be written as a sum of products (AND, OR, NOT), and each of these is realisable with NAND alone or NOR alone, NAND and NOR are universal gates.

(c) FET and MOSFET compared

Structure. The junction FET is a bar of n-type (or p-type) silicon, the channel, with source and drain contacts at its ends and a heavily doped p-type gate region diffused into its sides; the gate–channel p–n junction is always reverse-biased, and the width of its depletion layer sets the width of the conducting channel. The MOSFET has source and drain as two n⁺ regions diffused into a p-type substrate, with a metal (now polysilicon) gate separated from the substrate between them by a thin layer of silicon dioxide; the gate is insulated, and the channel is an inversion layer induced under the oxide by the gate voltage. The enhancement MOSFET has no channel until the gate voltage exceeds the threshold; the depletion MOSFET has a channel diffused in and can be operated either way.

Working and I–V (output) characteristics. In the JFET, drain current flows through the built-in channel with VGS = 0 and is reduced by making VGS more negative (n-channel), which widens the depletion layers; it is pinched off at VGS = VP. The drain characteristics, ID against VDS for fixed VGS, show an ohmic region at small VDS, saturation beyond VDS = VGS − VP, and breakdown at high VDS; the family lies between ID = IDSS (VGS = 0) and zero, all for VGS ≤ 0. In the enhancement MOSFET no current flows until VGS > VT (about 1–3 V); above threshold the inversion channel conducts and the drain characteristics have the same ohmic, saturation and breakdown regions, but the family is for positive VGS. The depletion MOSFET's family spans both negative and positive VGS.

Transfer characteristics. JFET: ID = IDSS(1 − VGS/VP)², a parabola from IDSS at VGS = 0 down to zero at VGS = VP, entirely in the negative-VGS half-plane; transconductance gm = gm0(1 − VGS/VP). Enhancement MOSFET: ID = k(VGS − VT)², a parabola starting at VGS = VT and rising in the positive-VGS half-plane; depletion MOSFET: the JFET-shaped curve extended through VGS = 0 into positive values, where ID exceeds IDSS.

Other differences. The MOSFET's insulated gate gives an input resistance of 10¹⁰–10¹⁵ Ω against 10⁸–10⁹ Ω for the JFET, and a gate current of picoamperes, but leaves it vulnerable to damage by static charge, so protective diodes are built in. The MOSFET is smaller and simpler to fabricate, which is why it dominates integrated circuits, both digital (CMOS) and analog; the JFET is more rugged, lower in noise, and is used in discrete low-noise input stages and as an analog switch. Both are voltage-controlled, majority-carrier devices with square-law transfer characteristics, in contrast with the current-controlled bipolar transistor.