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Energy resolution specifies the characteristic energy separation or broadening that an instrument can distinguish, with a stated width convention and measurement geometry. Resolving a phonon in inelastic scattering requires measuring a small energy transfer as well as momentum transfer. It is distinct from the incident beam's central energy.

Robert Hofstadter's Stanford installation selected Electrons up to $190\mathrm{MeV}$. Magnetic spectrometer scans counted shielded Lucite Cherenkov radiation pulses per monitor charge. Carbon-12 at $187\mathrm{MeV}$ and $80^{\circ }$ resolved $185.1\mathrm{MeV}$ elastic scattering and $180.7\mathrm{MeV}$ inelastic scattering ($4.43\mathrm{MeV}$ excitation). Calibration used forward-beam energy, hydrogen nuclear recoil and carbon-12 energy levels. Closed entrance slits checked background.^[ref]

The key distinction is between an Electron counter and an energy measurement. Here the counter registers particles admitted by the magnetic spectrometer; changing its magnetic field selects another outgoing momentum. A peak is therefore reconstructed from a scan, rather than read as one pulse whose height equals the scattered Electron energy. Dividing counts by monitored incident electric charge compensates for changes in beam intensity. A peak's height alone is insufficient: its area and the instrument's accepted momentum interval determine the selected yield.

An inelastic scattering peak corresponds to a different final nuclear quantum state, so including it in the elastic scattering yield would answer a different structural question. Conversely, finite energy resolution can hide sufficiently nearby excitations beneath an apparent elastic scattering peak. Background subtraction, target-loss corrections, angular acceptance and counter efficiency all enter before counts become a differential scattering cross-section. A dip in uncorrected counts is therefore not automatically a zero of a nuclear charge form factor.

In inelastic scattering, the participants exchange energy with internal degrees of freedom, so their total translational kinetic energy need not be conserved.

Consider an original illustrative model with exactly two outgoing scattering channels: a central beam $0$ and one diffracted beam $g$. Let $t\ge 0$ be specimen thickness and $\kappa >0$ a constant coupling with units of inverse length. Normalize the probability amplitudes $A_0,A_g$ to incident probability current, assume equal longitudinal speeds, and specifyThese ordinary differential equations define a lossless toy model, not a fit to the silicon example or a complete theory of crystal scattering. Differentiating once more gives $A_0^{\prime \prime }=-{\kappa }^2{A}_0$, soThe identity $p_0+p_g=1$ expresses conservation within these two scattering channels. Alternatively, substituting the ordinary differential equations into $d(|A_0{|}^2+|A_g{|}^2)/dt$ makes the two cross terms cancel. Here ideal bright-field transmission electron microscopy accepts fraction $p_0$ and rejects fraction $p_g$; ideal dark-field transmission electron microscopy accepts precisely the rejected channel. This complementarity requires the declared two-channel completeness and absence of loss.

Even with known $\kappa$, thickness is ambiguous: $\kappa t=\pi /4$ and $3\pi /4$ both give $(p_0,p_g)=(1/2,1/2)$. Near $t=0$, $p_g\simeq (\kappa t{)}^2$; extending that approximation through a full oscillation would erase the ambiguity incorrectly. The model isolates reversible transfer between probability amplitudes; it does not say that every thicker specimen must look darker.

Real dark-field transmission electron microscopy usually collects only a subset of the directions rejected in bright-field transmission electron microscopy. Other elastic scattering directions, inelastic scattering and detector losses prevent a general two-image conservation rule. To keep loss and background separate, extend the counting model explicitly. Let $\ell$ be the fraction absent from both recorded beam channels, $p_0+p_g=1$ describe the remaining fraction, $N$ be incident count per exposure, ${\eta }_j$ the detector efficiency and $b_j$ its mean additive background. Then the expected value of recorded count $n_j$ isFor separately acquired equal-illumination exposures with known $b_j,{\eta }_j$, the two corrected mean counts sum to $N(1-\ell )$, not necessarily $N$. Removal from these channels is not synonymous with physical absorption: an Electron can leave the chosen acceptance or change kinetic energy without being stopped in the specimen. Background subtraction removes a mean offset, but its counting variance remains.

Raman scattering is inelastic scattering of light in which the outgoing photon has a different frequency because matter gains or loses internal energy. Molecular vibration is a particularly useful recipient: a visible photon can reveal an internal energy spacing normally associated with infrared frequencies. Rayleigh scattering supplies the much stronger, approximately unchanged-frequency reference. The distinction concerns the transferred energy, rather than merely a change of direction or the colour of the illuminating beam.

The elementary bookkeeping is conservation of energy:Here $h$ is the Planck constant, ${\nu }_0$ and ${\nu }_s$ are the incident and scattered photon frequencies, and $E_i,E_f$ are initial and final internal molecular energies. This expression neglects translational recoil and Doppler broadening; those can matter in sufficiently precise gas measurements. A positive internal energy change gives Stokes Raman scattering, and a negative one gives Anti-Stokes Raman scattering. The energy difference determines a possible line position, but it does not by itself establish that a transition is optically allowed or predict its strength.

The spontaneous molecular process developed here has a connection to the Raman transition used in atomic measurement methods: two photon interactions transfer an internal energy difference. Coherently driving that process with two applied light fields is a different experimental regime from collecting weak spontaneous Raman scattering beside an illuminating beam.