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NE-CAT · Sector 24 · Advanced Photon Source

How NE-CAT works

NE-CAT runs two X-ray beamlines, 24-ID-C and 24-ID-E, at the Advanced Photon Source. Electrons circling the storage ring pass through rows of magnets and give off a narrow beam of X-rays. A silicon crystal picks out one energy, mirrors focus the beam to a few microns, and it lands on a protein crystal. This guide follows the X-rays from the ring to the detector.

Each section has a plain explanation and a figure you can adjust. The equations are folded away under the figures. Open any of them, or all of them with the button at the top, if you want the detail.

The path of the X-rays in five steps, not to scale. In the undulator, electrons wiggle between two rows of magnets and give off light, drawn as an orange wave. The monochromator reflects the light from two crystal faces and passes only one energy. The KB mirrors reflect it again and focus it. It hits the crystal at the sample, where most of it is caught by the beamstop, and the diffracted rays fan out to the detector. The sections below take these steps one at a time.

24-ID-C after the upgrade: 6 GeV ring, 3.0 cm undulator, Si(111) mono, 12.662 keV.

Beamline settings used in these guides

The Settings menu at the top of the page stays in view as you scroll. Choose 24-ID-C or 24-ID-E, before or after the APS Upgrade, and every figure, readout and worked number that depends on the ring or the beamline switches to those values. The highlighted column below is the one in use. The choice carries over when you follow a link to one of the other guides.

Sources. Ring values: APS Upgrade Final Design Report, chapter 1, table 1.1. The after-upgrade column is the brightness mode (324 bunches). Undulator periods and energy ranges: NE-CAT Communications, Winter 2022, the NE-CAT facilities page and the APS beamline directory. Monochromator crystals: APS beamline directory. Smallest undulator gaps (10.5 mm before the upgrade, 8.5 mm after): APS Upgrade Preliminary Design Report, chapter 5, table 5.3.

† Assumed, not taken from a published source. The undulator length (2.1 m, the APS-U canted design), and from it the number of periods. The plane in which each mono turns the beam, and that 24-ID-E uses a single reflection. A polarization of 0.98 arriving at the mono. Please correct any of these that differ on the real beamlines.

‡ Calculated in these guides from the other rows. These are not machine or beamline readings. The undulator gap comes from a general fit of field against gap (equation 2.5), not from the measured field of the NE-CAT undulators, so treat it as approximate.

§ As published by NE-CAT. Before the upgrade: NE-CAT Communications, Winter 2022 (1σ sizes). After: the NE-CAT facilities page. The two sources may not define beam size the same way.

It starts with electrons in a ring

The APS storage ring is a 1.1 km circle of magnets that holds a beam of electrons moving at almost the speed of light. The electrons travel in bunches, and each bunch goes around about 270,000 times a second. The guide How APS works follows the electrons from the gun to the ring and explains how the ring holds them.

With the 24-ID-C after the upgrade settings, the ring holds 200 mA of electrons at 6 GeV. Each electron carries 11,742 times its rest energy.

The electrons give off X-rays wherever their path is bent. NE-CAT is at sector 24 of the ring. In the straight part of that sector the electrons pass through two undulators, one after the other. Each undulator makes X-rays while the electrons are inside it, and each feeds one beamline: the first 24-ID-C, the second 24-ID-E.

The APS Upgrade replaced the ring's magnets with a new arrangement. The old ring shut down in April 2023. The energy came down from 7 to 6 GeV and the current doubled from 100 to 200 mA. The biggest change is to the electron beam itself: it went from 280 µm wide to about 15 µm, and its electrons now travel far more nearly parallel. A smaller, straighter electron beam makes a smaller, brighter X-ray beam.

Left: the storage ring from above, with bunches of electrons circulating. The ring is divided into 40 sectors, and NE-CAT is at sector 24. Right: the straight section of sector 24, enlarged, with one bunch passing through. The electrons give off X-rays only while they are inside an undulator. The first undulator sends its X-rays to 24-ID-C. A small steering magnet then turns the electrons slightly, so the second undulator points in a slightly different direction and sends its X-rays to 24-ID-E. After that the ring's magnets bend the electrons away, and the X-rays carry straight on down the beamlines. The angle between the two beams is drawn far larger than the real one, which is about a milliradian in the APS design. Only some of the bunches are drawn.

24-ID-C after the upgrade: 6 GeV, 200 mA, 324 bunches.

The math, if you want it

Equation 1.1How fast is the electron?

\gamma=\frac{E_e}{mc^{2}}\qquad\qquad \beta=\frac{v}{c}=\sqrt{1-\frac{1}{\gamma^{2}}}\;\approx\;1-\frac{1}{2\gamma^{2}}

$E_e$: The electron's total energy: 7 GeV in the APS before the upgrade, 6 GeV after it.
$mc^2$: The electron's rest energy, 0.511 MeV.
$\gamma$: The Lorentz factor: total energy in units of rest energy.
$\beta$: Speed as a fraction of the speed of light.

What it saysThe electron falls short of light speed by only 1/2γ².

In numbersFor 24-ID-C after the upgrade the ring energy is 6 GeV, so γ = 6000 / 0.511 = 11,742. Then 1 − β = 1/2γ² = 3.6×10⁻⁹. Over one 3.0 cm period, light gains just 1.09 Å on the electron.

The undulator makes the electrons wiggle

An undulator is two rows of permanent magnets, one above and one below the electron beam, with poles that alternate every half period. Between the rows the magnetic field points up, then down, then up again along the device.

A magnetic field pushes a moving charge sideways. So as an electron flies through, it is pushed left, then right, then left, and traces a gentle wiggle in the horizontal plane. Every time its path bends, it gives off light.

For 24-ID-C after the upgrade, each magnet period is 3.0 cm long. The wiggle is only 0.75 µm to either side.

The two rows can be moved toward or away from each other. The space between them is called the gap, and it sets how strong the field is on the beam. A smaller gap gives a stronger field and a bigger wiggle. In the figure, the gap is the first slider.

Gap between the magnet rows 12.00 mm

Magnet period λ_u 3.3 cm

Peak field B₀

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Deflection parameter K

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Wiggle amplitude

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Max steering angle K/γ

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N poleS poleB at the electronForce on the electron

Side view: the path looks straight, and B at the electron points up or down. Top view: the path is a sine wave in the horizontal plane, and the force points sideways. The wiggle is drawn hugely exaggerated. The real amplitude is about a micron against a period of centimeters. Close the gap and the field between the rows gets stronger (thicker purple arrows), so the wiggle grows. The gap is the one thing the beamline changes while it runs. The magnet period is fixed when the undulator is built.

Set for 24-ID-C after the upgrade: λu = 3.0 cm, K = 1.85 (harmonic 3 at 12.662 keV), γ = 11,742.

The math, if you want it

Equation 2.1The magnetic field

B_y(z)=B_0\cos\!\left(\frac{2\pi z}{\lambda_u}\right)

$B_y$: The vertical magnetic field on the beam axis.
$B_0$: Its peak value, set by the gap between the magnet rows. A smaller gap gives a larger B₀.
$\lambda_u$: The undulator period: the length of one N–S pair.

What it saysAlong the device the field swings smoothly between up and down, once per period.

Equation 2.2The sideways force

\mathbf{F}=-e\,\mathbf{v}\times\mathbf{B}\qquad\Longrightarrow\qquad \gamma m\,\ddot{x}=e\,c\,B_0\cos\!\left(\frac{2\pi z}{\lambda_u}\right)

$-e,\ m$: The electron's charge and rest mass.
$\mathbf{v}$: Its velocity, essentially c along the beam.
$\gamma$: The Lorentz factor, 11,742 at 6 GeV. It makes the electron γ times harder to push sideways.
$\ddot{x}$: Horizontal acceleration.

What it saysThe cross product of a forward velocity and a vertical field is horizontal. There is no vertical force at all, so the electron never leaves the horizontal plane. This one line is the origin of the beam's polarization.

Equation 2.3The path

\frac{dx}{dz}=\frac{K}{\gamma}\sin\!\left(\frac{2\pi z}{\lambda_u}\right)\qquad\qquad x(z)=-\frac{K\lambda_u}{2\pi\gamma}\cos\!\left(\frac{2\pi z}{\lambda_u}\right)

$dx/dz$: The slope of the path: the angle between the electron's direction and the axis.
$x(z)$: The sideways position along the device.
$K$: The deflection parameter, defined in equation 2.4.

What it saysIntegrating the force once gives the slope and twice gives the position. The path is a sine wave with the same period as the magnets. Its largest angle is K/γ and its amplitude is Kλ_u/2πγ.

In numbersFor 24-ID-C after the upgrade: K = 1.85 and λu = 3.0 cm give a largest angle K/γ = 157 µrad and an amplitude of 0.75 µm.

Equation 2.4The deflection parameter

K=\frac{e\,B_0\,\lambda_u}{2\pi\,m\,c}=0.934\;B_0[\text{T}]\;\lambda_u[\text{cm}]

What it saysK is a pure number that measures how hard the magnets steer. It is the largest steering angle in units of 1/γ, the natural width of the radiation cone (step 4 in the guide Why the beam is polarized). When K is about 1 or less, the electron's light always stays inside its own cone and the light from every period adds up in step. That is an undulator. When K is much larger, the cone sweeps side to side like a lighthouse. That is a wiggler.

In numbersK = 1.85 with λu = 3.0 cm needs B₀ = 1.85 / (0.934 × 3.0) = 0.66 T.

Equation 2.5Field against gap

B_0[\text{T}]\approx 3.88\,\exp\!\left[-5.07\,\frac{g}{\lambda_u}+1.52\left(\frac{g}{\lambda_u}\right)^{2}\right]

$g$: The gap between the upper and lower magnet rows.
$\lambda_u$: The undulator period.
$B_0$: The peak field on the beam axis, in tesla.

What it saysThe field on the axis falls steeply as the rows move apart. It roughly halves each time the gap grows by a fifth of a period. That makes the gap the tuning knob: motors move the rows, B₀ changes, and K follows from equation 2.4. This is a fit for undulators built from neodymium magnets with iron-alloy poles (Elleaume, Chavanne and Faatz, 2000), meant for g/λu between 0.1 and 1. The published fit begins with 3.69. These guides use 3.88, which reproduces the K values the APS quotes for its 3.3, 3.0 and 2.75 cm undulators at their smallest gap to within 2%. It is a general curve, not a measurement of the NE-CAT undulators.

In numbersFor 24-ID-C after the upgrade: K = 1.84 needs B₀ = 0.66 T, a gap of about 11.9 mm.

Why centimeter magnets make Ångström X-rays

A wiggle a few centimeters long should give off radio waves a few centimeters long. It gives off X-rays because the electron is traveling almost as fast as the light it emits.

Picture a race. Each time the electron starts a new magnet period it sends out a new wave crest. The light is only very slightly faster than the electron, so by the time the next crest is launched, the first one is only a tiny distance ahead. The crests end up packed close together, and closely packed crests are short-wavelength light.

For 24-ID-C after the upgrade, a 3.0 cm magnet period comes out as light of about 1.09 Å.

The beamline tunes the energy by opening and closing the gap between the magnet rows. A smaller gap means a stronger field and a bigger wiggle. A bigger wiggle slows the electron's forward progress a little, the crests spread out, and the photon energy drops.

Electron speed β = v/c 0.80

λ / λu, exact: 1/β − 1

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λ / λu, approx: 1/2γ²

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This ring: 1 − β = 3.6×10⁻⁹

3.0 cm → 1.09 Å

A race between the electron and its own light. Each time the electron starts a new magnet period it launches a new wave crest. Light is only slightly faster, so each crest is only slightly ahead of the next one. Push β toward 1 and the crests crowd together. The approximation 1/2γ² becomes accurate as β approaches 1.

The math, if you want it

Equation 3.1Slippage: the wavelength from a race

\lambda\;=\;\underbrace{\frac{\lambda_u}{\beta}}_{\text{light travels}}-\underbrace{\lambda_u}_{\text{electron travels}}\;=\;\lambda_u\!\left(\frac{1}{\beta}-1\right)\;\approx\;\frac{\lambda_u}{2\gamma^{2}}

What it saysThe electron needs a time λ_u/βc to cross one period. In that time light covers λ_u/β. The difference is how far one crest sits ahead of the next, and the spacing of crests is the wavelength. Putting in 1/β − 1 ≈ 1 − β ≈ 1/2γ² from equation 1.1 gives the result.

In numbers3.0 cm / (2 × 11,742²) = 1.09 Å, or 11.4 keV. This is the K = 0 limit. Equation 3.3 adds the corrections.

Equation 3.2The same answer by changing frames

\lambda'=\frac{\lambda_u}{\gamma}\qquad\text{then}\qquad \lambda=\lambda'\sqrt{\frac{1-\beta}{1+\beta}}\;\approx\;\frac{\lambda'}{2\gamma}\qquad\Longrightarrow\qquad \lambda\approx\frac{\lambda_u}{2\gamma^{2}}

$\lambda'$: The magnet period as the electron sees it. Riding with the electron, the undulator rushes past and is Lorentz-contracted by γ.
$\sqrt{\tfrac{1-\beta}{1+\beta}}$: The relativistic Doppler factor for a source coming straight at you. For β near 1 it equals 1/2γ.

What it saysIn its own frame the electron sits still while a field with a period of a few microns sweeps over it. It oscillates and radiates infrared light as a plain dipole. Seen from the lab, that dipole is approaching at nearly c, so its light is Doppler-compressed by another factor of 2γ. One γ comes from contraction and 2γ from Doppler.

In numbers3.0 cm / 11,742 = 2.55 µm in the electron's frame, then 2.55 µm / 23,484 = 1.09 Å in the lab.

Equation 3.3The undulator equation

\lambda_n=\frac{\lambda_u}{2n\gamma^{2}}\left(1+\frac{K^{2}}{2}+\gamma^{2}\theta^{2}\right)

$n$: Harmonic number, 1, 3, 5 … on axis. The crests also line up if light slips ahead by n wavelengths per period.
$K^2/2$: The wiggle correction. A wiggling path is longer than a straight one, so the electron's average forward speed drops to $\bar\beta_z=1-\frac{1+K^{2}/2}{2\gamma^{2}}$ . Light gains more per period and the wavelength grows.
$\gamma^2\theta^2$: The off-axis correction. Seen at an angle θ from the axis, the light's head start is measured along a slanted line, which lengthens it: $\lambda=\lambda_u(1/\bar\beta_z-\cos\theta)$ .

What it saysThe on-axis wavelength is set by three knobs: ring energy, magnet period and K. The beamline tunes by changing the gap. Closing it raises B₀ and K, which lengthens λ and lowers the photon energy. Light that leaves at an angle is redder than light on the axis.

Harmonics: one undulator, a ladder of energies

So far the undulator has made light of one wavelength. In fact it makes a ladder of energies at the same time. The lowest is called the fundamental, or first harmonic. Above it comes light at 3, 5, 7 … times that energy. These are the higher harmonics.

They come from the shape of the wave. When the field is weak the electron sways gently, and the electric field arriving on the axis is a smooth sine wave. A sine wave is a single energy. When the field is strong the electron swings through an angle wider than its own narrow cone of light. Someone on the axis is then lit only in a brief flash each time the electron points straight at them, which happens twice in every magnet period. The wave becomes a row of sharp spikes. A spiky wave is a smooth wave with faster ripples added at exact multiples of its rhythm, and those ripples are the harmonics.

Each flash points the opposite way to the one before, because the electron is swinging back. For that reason only the odd multiples survive on the axis. The even ones cancel there and show up only off to the side.

Closing the gap does two things. It slows the electron's forward progress, so every harmonic slides down in energy together. It also makes the wave spikier, which moves brightness out of the fundamental and into the higher harmonics. Opening the gap does the reverse. A beamline chooses the harmonic that is bright at the energy it wants.

For 24-ID-C after the upgrade, 12.662 keV is taken from harmonic 3 with K = 1.84.

All the harmonics travel down the beamline together. The monochromator of step 6 passes the one that is wanted. Light at a whole multiple of the chosen energy can also satisfy Bragg's law in the same crystal. The mirrors of step 7 reflect only below a cutoff energy, and that removes most of it.

Gap between the magnet rows 12.00 mm

Deflection parameter K

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Harmonic 1, the fundamental

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Harmonic 3

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Harmonic 5

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Brightest on the axis

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Harmonic 1Harmonic 3Harmonic 5Harmonic 7HigherEnergies 24-ID-C tunes over

Left: a snapshot of the electric field traveling down the axis. At a wide gap it is a weak, smooth sine wave. Close the gap and it grows, its crests move apart, and it sharpens into spikes that point up and down in turn. Right: the energies in that wave. Each bar is one harmonic at this gap, and its height is its brightness on the axis. The thin curve through a bar is that harmonic's tuning curve: where the bar goes as the gap changes. The small hollow dots mark the even harmonics, which are missing on the axis. The dashed line is the selenium edge. In a real beam each bar is a narrow peak, not a line.

Set for 24-ID-C after the upgrade: 6 GeV, λu = 3.0 cm.

The math, if you want it

Equation 4.1Why only odd harmonics on the axis

E\!\left(t+\tfrac{T}{2}\right)=-E(t)\qquad\Longrightarrow\qquad E(t)=\sum_{n=1,3,5,\dots}a_n\sin\!\left(\frac{2\pi n\,t}{T}\right)

$E(t)$: The electric field arriving at a point on the axis.
$T$: The time between crests of the fundamental, $\lambda_1/c$ .
$a_n$: How much of harmonic n the wave contains.

What it saysThe second half of every magnet period is the mirror image of the first. The electron swings back the other way, so it sends the same shape of field down the axis with the opposite sign. Any repeating wave can be built from sine waves at 1, 2, 3 … times its own rhythm. A wave that flips sign every half cycle needs only the odd ones. Away from the axis the two half periods are no longer seen equally, and the even harmonics appear there.

Equation 4.2How bright each harmonic is

F_n(K)=\frac{n^{2}K^{2}}{\left(1+K^{2}/2\right)^{2}}\left[J_{\frac{n-1}{2}}(\xi)-J_{\frac{n+1}{2}}(\xi)\right]^{2}\qquad\qquad \xi=\frac{nK^{2}}{4+2K^{2}}

$F_n$: The brightness of harmonic n on the axis, as a pure number. The number of photons sent along the axis is proportional to N² γ² I Fₙ, for N magnet periods and a ring current I.
$J$: Bessel functions. They appear whenever a wave is built from a motion that is itself a sine wave.
$n$: The harmonic number, odd.

What it saysThese are the thin curves in the figure. For a weak undulator only F₁ matters, and it grows as K². It reaches its largest value near K = 1.2. The third harmonic is brightest near K = 1.9, the fifth near 2.3, and so on: each higher harmonic takes over at a larger K. The same numbers come out if the spiky wave in the figure is split into its sine waves.

In numbersFor 24-ID-C after the upgrade at K = 1.84: F₁ = 0.31, F₃ = 0.43, F₅ = 0.38, F₇ = 0.30.

Undulator energy calculator

Ring energy (GeV)

Period λ_u (cm)

K 1.00

Harmonic n

Photon energy

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Wavelength

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K for Se edge, 12.66 keV

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The same sum as in the figure above, with every number open to change. Use it to try another ring energy, magnet period or harmonic. The last box gives the K that would put the chosen harmonic on the selenium edge.

Set for 24-ID-C after the upgrade: 6 GeV, λu = 3.0 cm, harmonic 3, K = 1.85. Change any value to explore.

The math, if you want it

Equation 4.3The working formula

E_n[\text{keV}]=\frac{0.950\;n\;E_e^{2}[\text{GeV}]}{\lambda_u[\text{cm}]\,\bigl(1+K^{2}/2\bigr)}

What it saysThis is equation 3.3 on axis, combined with equation 1.2 in the guide Why the beam is polarized and with the constants multiplied out. The calculator above uses it, starting from the settings chosen at the top of the page. The K it starts with is the one this formula needs for 12.662 keV, not a reading from the beamline.

In numbersFor 24-ID-C after the upgrade: harmonic 3 with K = 1.85 gives 0.950 × 3 × 36 / (3.0 × 2.70) = 12.66 keV. With K = 0 the fundamental would sit at 11.4 keV.

Relativity also makes the beam narrow

The electron gives off light in all directions, as it sees things. Seen from the lab, the electron's speed sweeps nearly all of that light forward into a narrow cone, like the headlight of a fast train. The faster the electron, the narrower the cone.

An undulator narrows it further. Light from every one of its periods adds up in step only very close to the axis, so the bright center of the beam is much narrower than the cone from a single wiggle.

For 24-ID-C after the upgrade, the cone from one wiggle is about 85 µrad wide and the bright center about 10 µrad.

The electrons in the bunch are not all pointing in exactly the same direction, and that spread blurs the X-ray beam. This is where the upgrade shows most. In the old ring the horizontal spread was as large as the bright center of the beam. In the new ring it is several times smaller.

Lorentz factor γ 1.0

Speed β

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Cone half-angle 1/γ

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This ring, γ = 11,742

1/γ = 85 µrad

In the wiggle plane (horizontal)In the vertical plane

Left: the radiation pattern of a charge accelerating sideways while moving to the right. At γ = 1 it is the doughnut from step 2 in the guide Why the beam is polarized. As γ grows, the pattern collapses forward. Right: the same pattern plotted against angle in units of 1/γ. Above γ of about 5 this view stops changing. In the horizontal plane the intensity falls to zero at γθ = 1. That ring is where the doughnut's hole ends up.

The math, if you want it

Equation 5.1The headlight effect

\tan\theta=\frac{\sin\theta'}{\gamma\,(\cos\theta'+\beta)}\qquad\qquad \theta'=90^{\circ}\;\Longrightarrow\;\theta\approx\frac{1}{\gamma}

$\theta'$: The angle of a light ray from the forward direction, measured in the electron's frame.
$\theta$: The angle of the same ray in the lab frame.

What it saysA ray that leaves sideways in the electron's frame (θ′ = 90°) comes out at only 1/γ in the lab. Half of everything the electron emits is squeezed into that forward cone.

In numbers1/γ = 85 µrad. That is 2.6 mm of beam radius at 30 m from the source.

Equation 5.2The pattern of a fast charge accelerated sideways

\frac{dP}{d\Omega}\;\propto\;\frac{(1-\beta\cos\theta)^{2}-(1-\beta^{2})\sin^{2}\theta\cos^{2}\phi}{(1-\beta\cos\theta)^{5}}\;\;\xrightarrow{\ \gamma\gg1\ }\;\;\frac{1}{(1+\gamma^{2}\theta^{2})^{3}}\left[1-\frac{4\gamma^{2}\theta^{2}\cos^{2}\phi}{(1+\gamma^{2}\theta^{2})^{2}}\right]

$\theta$: Angle from the direction of travel.
$\phi$: Angle around the direction of travel. φ = 0 is the wiggle plane (horizontal), φ = 90° is vertical.

What it saysSet β = 0 and the left side reduces to the sin²ψ doughnut of equation 2.2 in the guide Why the beam is polarized. For large γ everything depends on the product γθ, which is why the right-hand plot in the figure stops changing. The first factor is the forward cone. The bracket carries the dipole's hole: it reaches zero at γθ = 1 in the horizontal plane, and it never reaches zero in the vertical plane.

Equation 5.3The central cone and its bandwidth

\theta_{\text{cen}}\approx\frac{1}{\gamma}\sqrt{\frac{1+K^{2}/2}{n\,N}}\qquad\qquad \frac{\Delta\lambda}{\lambda}\approx\frac{1}{n\,N}

$N$: The number of magnet periods in the undulator.
$\theta_{\text{cen}}$: Half-angle of the cone that holds one harmonic's on-axis light.
$\Delta\lambda/\lambda$: The natural fractional bandwidth of that harmonic.

What it saysLight from N periods adds in step only for a narrow band of wavelengths and a narrow range of angles. More periods give a sharper line and a tighter cone, much tighter than 1/γ. These are rough single-electron values. The real beam is widened by the electron beam's own size and angular spread.

In numbersFor 24-ID-C after the upgrade, with about 70 periods, n = 3 and K = 1.85: θcen ≈ 85 µrad × √(2.70 / 210) ≈ 10 µrad, and Δλ/λ ≈ 0.5%.

Equation 5.4What the electron beam adds

\Sigma_{x'}\approx\sqrt{\theta_{\text{cen}}^{2}+\sigma_{x'}^{2}}\qquad\qquad \Sigma_{y'}\approx\sqrt{\theta_{\text{cen}}^{2}+\sigma_{y'}^{2}}

$\sigma_{x'},\ \sigma_{y'}$: The rms spread in the directions of the electrons, horizontally and vertically, in the undulator straight section.
$\theta_{\text{cen}}$: The single-electron central cone from equation 5.3.
$\Sigma_{x'},\ \Sigma_{y'}$: The rough angular size of the real photon beam.

What it saysEach electron lays down the pattern of equation 5.1 in the guide Why the beam is polarized around its own direction of travel. The beam is the sum of those patterns, so its angular size is the single-electron cone widened by the electrons' own spread. This is where the upgrade shows. The old ring's horizontal spread was as large as the cone itself. The new ring's spread is several times smaller than the cone in both directions, so the real beam is close to the single-electron picture.

In numbersFor 24-ID-C after the upgrade: electron spread 2.8 µrad horizontally and 1.3 µrad vertically, against a cone of about 10 µrad. Combined: about 10 µrad × 10 µrad.

A silicon crystal picks one energy

The light from the undulator is bright but not a single energy. The monochromator, or mono, selects one. It is a perfect crystal of silicon. X-rays bounce off the layers of atoms inside it, and the reflections from all the layers add up only when the wavelength and the angle match. Turn the crystal and a different energy gets through.

24-ID-C uses a channel-cut crystal: a single block of silicon with a slot cut in it. The beam bounces off one wall of the slot and then the other, and leaves traveling in its original direction. Because both walls are part of one crystal they are always exactly parallel, which makes the mono very stable. 24-ID-C can be tuned across a wide range of energies. 24-ID-E uses a different cut of silicon and stays at one energy, 12.662 keV, which suits selenium.

For 24-ID-C after the upgrade: Si(111), set to a Bragg angle of 8.98° for 12.662 keV.

Photon energy 12.662 keV

Crystal

Reflections

Bragg angle θ_B

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Beam walk on the second face

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Height step of the outgoing beam

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The mono seen from the side. With two reflections it is a channel cut: one piece of silicon with a slot, whose two inner faces do the reflecting. The pale plate behind the two blades is the part that joins them. Move the energy slider and the whole crystal turns about the pivot on the first face, marked with a small crosshair. The two faces stay locked together, the orange dot shows where the beam lands on the second face, and the dashed ticks mark where it lands at 6 and at 20 keV. The orange wiggles are wave packets, one for each photon.

Set for 24-ID-C after the upgrade: Si(111), 2 reflections, vertical deflection (assumed), 12.662 keV.

The math, if you want it

Equation 6.1Bragg's law

\lambda=2d\sin\theta_B\qquad\Longrightarrow\qquad \sin\theta_B=\frac{12.398}{2\;d[\text{\AA}]\;E[\text{keV}]}

$d$: Spacing of the reflecting atomic planes: 3.1356 Å for Si(111) and 1.9201 Å for Si(220).
$\theta_B$: The Bragg angle between the beam and those planes. The beam is turned through 2θ_B.

What it saysReflections from successive planes add in step only at one angle for each wavelength. Rotating the crystal selects the energy.

In numbersFor 24-ID-C after the upgrade: Si(111), d = 3.1356 Å. At 12.662 keV, sin θB = 12.398 / (2 × 3.1356 × 12.662) = 0.156, so θB = 8.98° and the beam turns through 18.0°.

Equation 6.2Channel-cut geometry

w=\frac{g}{\tan\theta_B}\qquad\qquad h=2g\cos\theta_B

$g$: The gap between the two reflecting faces of the channel.
$w$: How far along the second face the beam lands, measured from the point directly across from the first footprint.
$h$: The height step between the outgoing beam and the incoming beam.

What it saysBoth faces are cut into one crystal, so they are always exactly parallel. Whatever angle satisfies Bragg's law on the first face satisfies it on the second, and the outgoing beam is always parallel to the incoming one. The price is that the faces cannot move relative to each other. When the crystal turns to a new energy, the beam lands at a different place on the second face, and the outgoing beam shifts in height a little.

In numbersOn Si(111) at 12.662 keV, θ_B = 8.98°: w = 6.3 gap widths and h = 1.98 gap widths. Between 6 and 20 keV the footprint walks from 2.9 to 10.1 gap widths along the second face, and the height step changes from 1.89 to 1.99 gap widths.

Mirrors focus the beam onto the crystal

X-rays pass straight through ordinary lenses and mirrors. They do reflect from a very smooth surface if they strike it at a very shallow angle, a fraction of a degree. A gently curved mirror used this way can focus them.

24-ID-C uses a pair of such mirrors, one behind the other. One focuses the beam in height and the other in width. This arrangement is called a Kirkpatrick-Baez pair, or KB mirrors. Together the mirrors form a small image of the X-ray source on the sample.

Published focus for 24-ID-C after the upgrade: 23 × 5 µm (h × v).

Apertures just before the sample can then trim the beam to match the size of the crystal.

Mirror to adjust

Curve (bend) 1.00

Pitch 0.00°

Roll 0°

Yaw 0°

Up / down 0

Beam height at the sample

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Beam width at the sample

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Spot position

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Beam on the mirrors

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A KB mirror pair, drawn twice. Top: seen from the side. Mirror 1 lies under the beam and is bent into a shallow curve, so the rays that strike it come together in height at the sample. Mirror 2 stands on edge beside the beam and does nothing in this view. Bottom: seen from above. Now mirror 1 does nothing, and mirror 2 brings the rays together in width. The square shows the beam's cross-section at the sample. Choose a mirror and try each control. Curve moves the focus along the beam, so the spot grows in that mirror's direction only. Pitch changes the angle at which the beam strikes the mirror. It swings the reflected beam and is by far the most sensitive adjustment. Up / down (in / out for mirror 2) slides the mirror across the beam. The spot moves with it, and if the mirror goes too far part of the beam misses it or is blocked by its end. Roll twists the mirror about the beam and nudges the spot in the other mirror's direction. Yaw turns the mirror in its own plane and has almost no effect. The angles are drawn about forty times larger than life, and the effects of roll and yaw appear only in the square. Real mirrors meet the beam at a fraction of a degree and look flat to the eye. At such shallow angles even a magnified bend is too slight to see in the main drawing, so the curved bar under each mirror shows that mirror's bend on its own, magnified much further. The order drawn here, height first, is an assumption.

The math, if you want it

Equation 7.1How a mirror shrinks the source

s_{\text{focus}}\approx s_{\text{source}}\times\frac{q}{p}

$s_{\text{source}}$: Size of the X-ray source, which is the size of the electron beam inside the undulator.
$p$: Distance from the source to the mirror.
$q$: Distance from the mirror to the focus at the sample.

What it saysA focusing mirror makes an image of the source at the sample, shrunk by the ratio of the two distances. A mirror far from the source and close to the sample gives a small spot. For the same mirrors, a smaller source gives a smaller spot. Real mirrors add a little blur of their own from tiny errors in their shape.

In numbersThe upgrade shrank the horizontal source size from 280 µm to 14.7 µm, about 19 times, and the vertical size from 10 µm to 3.2 µm.

At the sample

The protein crystal is a fraction of a millimeter across. It sits in a small loop on the end of a pin, held at about 100 K in a stream of cold nitrogen gas. A goniometer holds the pin and rotates the crystal in the beam. 24-ID-C has an MD3-UP goniometer, which turns the crystal about a vertical axis. 24-ID-E has an MD2.

Most of the beam passes straight through the crystal and is caught by a small beamstop. A small part is diffracted by the ordered molecules in the crystal into thousands of separate rays. A large area detector behind the sample records where they land: an EIGER2 X 16M on 24-ID-C and an EIGER 16M on 24-ID-E. As the crystal rotates, the detector collects the pattern of spots from which the structure is worked out.

A second, much smaller detector sits beside the sample. It measures the energy of each X-ray it receives, and is used to find which elements are in the crystal and to scan across an absorption edge. The beam's polarization decides where that detector should go. The guide Why the beam is polarized explains where the polarization comes from, and the guide X-ray fluorescence at NE-CAT explains how to use it.

Photon energy 12.662 keV

Sample to detector distance 300 mm

Wavelength

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Resolution at the detector edge

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Resolution in the corner

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Lowest resolution past the beamstop

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Side view of the endstation. The beam comes from the left and hits the crystal. Most of it carries straight on and is caught by a small beamstop just behind the sample. A small part is diffracted into rays that fan out from the crystal. The dashed lines mark a few of them, each labeled with the resolution it carries: the wider the angle, the finer the detail. This fan is set by the crystal and the X-ray energy. Moving the detector does not change it. The detector simply catches whatever part of the fan lands on it, shown by the darker wedge and the solid red rays. Drag the detector along the beam, or use the slider. Closer in, it takes in more of the fan. Raise the energy and the whole fan narrows, so more of it fits on the detector. The detector is drawn to scale as an EIGER 16M, 311 mm across. The coarsest detail leaves the crystal at very small angles, close to the direct beam, and the beamstop must be small enough to let it past. The green rays carry 50 Å detail. The strip at the bottom is a close-up of the first 60 mm behind the sample with the height stretched several times. The gray wedge is the shadow of the beamstop, and the 50 Å rays pass outside it. The beamstop is drawn 0.3 mm across and 25 mm behind the sample. Those two numbers are illustrative.

The math, if you want it

Equation 8.1What resolution reaches the detector

\lambda=2d\sin\theta\qquad \tan 2\theta=\frac{r}{D}\qquad\Longrightarrow\qquad d_{\min}=\frac{\lambda}{2\sin\!\bigl(\tfrac12\arctan\frac{r}{D}\bigr)}

$d$: Spacing of a set of planes in the protein crystal. Small d means fine detail.
$2\theta$: Angle between a diffracted ray and the direct beam.
$r$: Distance from the beam center to a point on the detector.
$D$: Distance from the sample to the detector.
$d_{\min}$: The finest spacing whose reflection still lands on the detector: the resolution limit.

What it saysThe first part is Bragg's law again, this time for the sample. Finer detail diffracts to larger angles. A reflection is recorded only if its ray still hits the detector, so moving the detector closer or using a shorter wavelength brings finer detail onto it.

In numbersAt 12.662 keV (0.979 Å) with the detector at 300 mm, the edge at r = 155.6 mm sits at 2θ = 27.4°, which is 2.07 Å.

Where to go deeper

X-ray Data Booklet, section 2.1: Characteristics of synchrotron radiation (K.-J. Kim). Undulator equations and cones in four pages.
D. Attwood and A. Sakdinawat, X-Rays and Extreme Ultraviolet Radiation, chapter 5. Undulator radiation, derived by changing frames, and harmonics.
P. Elleaume, J. Chavanne and B. Faatz, Nucl. Instrum. Methods A 455, 503 (2000). The fit of undulator field against gap.
J. Als-Nielsen and D. McMorrow, Elements of Modern X-ray Physics. Sources, mirrors, perfect-crystal reflection.
APS Upgrade Final Design Report, chapter 1. Ring parameters before and after the upgrade.
NE-CAT Communications, Winter 2022 and the NE-CAT facilities page. Undulators, energy ranges, beam sizes, goniometers and detectors.
APS beamline directory entries for 24-ID-C and 24-ID-E.

Last updated November 11, 2026. Release 1.2. What changed.
The NE-CAT beamlines are funded by the National Institute of General Medical Sciences from the National Institutes of Health (P30 GM124165).
© 2026 NE-CAT, Cornell University. Released under the MIT License.