Charge carriers have a tiny average drift inside the continuous wire. In a metal, negative electrons drift opposite the defined conventional-current arrow. They do not leap across the space between neighbouring windings; the wire, cryostat, and patient do not circulate.
Theme 02 · Fields and radio
Follow current into B₀, map and shim its variation, then use gradients and RF.
Manipulate 3D MRI magnet fields, perform a quantitative low-order B0 shim fit, then explore gradients, proton precession, RF transmit and receive hardware, I/Q detection, and a full-Bloch slice profile.
00 / WHERE THE MAIN FIELD COMES FROM
Cold wire carries current.
The current creates B₀.
A power supply first pushes electric current through many turns of superconducting wire. Those turns act together like one long solenoid: their magnetic fields add through the bore and return outside the magnet. After ramp-up, a closed superconducting path can keep that current circulating in persistent mode.
RAMP SUPPLY CONNECTED
PROBE · +2.29 T · +22,927 G
Yellow arrowheads follow the purple wire and connected ramp circuit through all 52 displayed turns. They show conventional-current direction, not electrons. Mint bore arrows show +Z B₀; nested closed curves show its exterior return direction.
Three-dimensional superconducting MRI magnet and magnetic-field model
Drag to orbit · pinch, wheel, or use + / ↺ / − to zoom · tap any object · field lines are a map, not physical tubes
Live superconducting-magnet model
IDEALIZED SOLENOID · NOT SERVICE CONTROLS
OPEN ANY WORD
- 01Ramp the current
A controlled DC power supply applies voltage so current rises in the cold winding. “Current” means net electric charge crossing a wire section each second, measured in amperes. Yellow arrows follow the conductor in the conventional positive-charge direction; the wire does not move, and electrons do not jump between turns.
- 02Every turn contributes
Each loop makes a magnetic field. Inside a long coil, neighbouring loop fields point mainly the same way and add; more amperes or more turns per metre gives a larger central field.
- 03Close the persistent path
Once the target current is reached, an internal superconducting connection completes a very-low-resistance loop. The ramp supply can be disconnected while the established current continues.
- 04B₀ stays on
The stable main field exists between scans. RF and gradient pulses switch during imaging; the superconducting main-magnet current normally does not pulse on and off for each patient.
CLICKABLE LONG-SOLENOID ESTIMATE · AT THE CENTER
Bcenter ≈ μ0nI = μ0(N/L)I
4π × 10⁻⁷ T·m/A × 4,000 turns/m × 600 A ≈ 3.02 T
0 A · no current600 A teaching value800 A
At 600 A and the selected winding density, the ideal long-solenoid estimate is 3.02 T. Raising current strengthens B₀ in direct proportion. Yellow arrowheads stay on the helical wire and mark conventional-current direction; they are not electrons and their display speed is not a measurement.
1,000 turns/m · 16 shownfixed 1.50 m winding6,000 turns/m · 76 shown
The winding length stays 1.50 m inside the 1.86 m package. The 3D winding shows 52 spaced turns to represent 6,000 effective turns. Move the slider: higher n visibly adds turns and closes their gaps; the drawing remains a compressed sample rather than literal manufacturer geometry.
4 · uncluttered10 · balanced view16 · denser drawing
This changes only how many direction-map curves the 3D lesson draws. It does not change current, B₀, tesla, gauss, fringe field, stored energy, or any clinical image.
−1.6 mcenter+1.6 m
axis0.64 m winding1.2 m outside
CURRENT × TURN DENSITY2.40 MA-turn/msource strength for this estimate
IDEAL CENTER B+3.02 T+30,159 G
NUMERIC PROBE B+2.29 T+22,927 G
FINITE-COIL CORRECTION×0.7611.50 m / 1.28 m coil is 1.17 diameters long
YOU CHANGED CURRENT / WINDING DESIGN
WHAT PHYSICALLY MOVES?
WHAT ARE FIELD LINES?
WHY TESLA AND GAUSS?
WHY SUPERCONDUCT?
More ampere-turns make a stronger main field.
At the center, increasing I or n increases the field almost linearly in this long-solenoid estimate. Clinically, a different B₀ changes the ¹H carrier frequency and many scanner/tissue behaviors; it is a magnet design and site platform, not a routine sequence knob.
Drawing lines connect the field direction at many points. They close into loops and crowd where the drawing represents stronger field. They are not strings, rays, current paths, or proton tracks.
Both measure magnetic flux density: 1 tesla = 10,000 gauss. Therefore 3 T = 30,000 G. Clinical MRI names B₀ in tesla; gauss is common in fringe-field discussions.
Ordinary resistance would turn sustained high current into continuous heat. Below its critical conditions, the magnet conductor can support persistent current with extremely small decay; cooling and protection remain essential.
3D FIELD BUILD-UP · MOVING CHARGE → CURRENT → LOOP → SOLENOID
One conductor makes a circular field.
Many loop fields add.
Blue moving markers now represent groups of negative charge crossing a counting plane; increasing current shows more marker crossings and a stronger calculated field. They are not a literal census of electrons in the wire. Because electrons are negative, their average drift is opposite the yellow conventional-current arrow. Build from a straight segment to one loop, then watch the field shape lengthen as aligned turns are added.
Three-dimensional conductor, current-loop, and solenoid field-summation model
01 · STRAIGHT CONDUCTOR
Five +X samples · B points +Y · r = 0.30 m gives |B| 2.67 µT
Yellow I points +Z and blue e⁻ drift is opposite. Uniform-thickness curves map circular direction; five tail-anchored tangent-arrow lengths share one 0–35 µT scale and fall exactly as 1/r.
Drag to orbit · pinch, wheel, or use + / ↺ / − · tap an object · animation pace and line count are teaching choices
Build the field one geometry at a time
MICROTESLA TEACHING SCALE · NOT MRI-MAGNET SERVICE DATA
OPEN THE EXACT MEANING
Count charge crossing a wire section—not electrons merely sitting in the wire.
The side view is kept two-dimensional because “crosses this plane during one second” is a measured rate. The rotatable model beside it shows the same conductor and circular field in 3D.
SIDE VIEW · CHARGE CROSSES THE GREEN GATE
conventional current I → +Z
COUNT HERE
negative-charge markers move toward −Z
END VIEW · B CIRCLES THE CURRENT
field glow = 50% of this teaching scale
CHARGE PER SECOND4.00 C/sthis equals 4.00 amperes
ELECTRON-CHARGE EQUIVALENTS2.50 × 10¹⁹ /scharge amount divided by |e|
VISIBLE MOVING PACKETS8 of 16compressed teaching markers, not eight electrons
ONE MARKER REPRESENTS3.12 × 10¹⁸ /sonly in this animation mapping
CLICKABLE RATE LAW · WHY “MORE ELECTRONS” NEEDS A TIME WINDOW
I = ΔQ / Δt = Ne|e| / Δt
Ampere measures charge crossing per second. Stationary charge gives no steady current-created B; current through the chosen geometry sets the calculated field.At 4.00 A, 4.00 coulombs of charge cross the chosen section each second—equivalent to about 2.50 × 10¹⁹ elementary electron charges per second. The conductor already contains vastly more carriers; this control changes net flow rate, not the mere existence of electrons.
0 A · carriers, no net flow4 A = 4 C/s8 A = 8 C/s
More charge crossing per second strengthens every B contribution in direct proportion. The number of bright blue packets is a compressed rate display; their visible speed is deliberately not a microscopic drift-speed measurement.
1 loop8 visible contributors12 loops
Moving this control opens the many-loop stage. More equally oriented loops add more same-direction axial field at the center; it does not mean the 3D drawing represents a real MRI winding count.
CONVENTIONAL CURRENTI points +Zdirection assigned to positive charge
AVERAGE ELECTRON DRIFTe⁻ drifts −Znegative carrier direction is opposite I
ACTIVE CONTRIBUTORSone straight conductorfield circles the wire
CALCULATED REFERENCE FIELD2.67 µT0.30 m from an ideal long straight wire
COIL / FIELD SHAPEstraight-wire circular fieldfield direction circles locally around the Z-directed conductor
NET CHARGE-FLOW RATE2.50 × 10¹⁹ e⁻ charges/s4.00 C/s; not total electrons stored in the wire
Vector addition, not field-line counting
QUANTITATIVE AXIAL CROSS-SECTION · SAME CURRENT AND WINDING CONTROLS
Color tells how much field.
Arrows tell which way it points.
The 3D guides above are useful direction guides, but their spacing is not a measurement. This map calculates a field value at a grid of positions through the coil. Drag the red probe directly: moving it changes only the observation coordinate, never the magnet.
CALCULATING FINITE-COIL FIELD GRID…
FINGER / POINTER: drag anywhere in the plot · KEYBOARD: focus the plot, then use arrow keys · the ordinary z and r sliders remain equivalent accessible controls
NEW QUANTITATIVE 3D LAB · MAPPED Δf → LOW-ORDER CORRECTION → RESIDUAL
Can broad shim fields flatten every B₀ error?
A field map describes where proton resonance is above or below a chosen reference. Fit the standard first- and second-order spatial shapes over one spherical region, then compare the original map, the added correction, and what the fit cannot remove.
Three-dimensional before, correction, and residual B₀ shimming comparison
SECOND-ORDER FIT · 22 CM SPHERE · 257 DISPLAY SAMPLES
19.31 Hz RMS → 4.77 Hz RMS
Broad variation falls 75.3%; a deliberately local unmodeled feature remains.
Drag to orbit · pinch, wheel, or use + / ↺ / − · tap a field volume · needles plot signed Δf along the B₀ axis; they are not tissue displacement
Low-order B₀ shim fit
SYNTHETIC MAP · UNCONSTRAINED LEAST SQUARES · NOT A CURRENT PRESCRIPTION
Why can broad shim fields reduce—but not erase—a localized field defect?
All three 3D volumes show the same coordinates and the same fixed hertz scale. Change only the fitted basis, region size, or local feature, then ask which spatial patterns the residual keeps.
OPEN THE EXACT MEANING
Increase model order without changing the measured map.
12 cm · local22 cm default28 cm · broader region
The fit uses 257 equally weighted Cartesian teaching samples inside a 22 cm sphere. Changing the region changes which physical positions contribute; it does not resize a patient or claim an optimal clinical shim volume.
0 Hz · low order only+45 Hz teaching feature+120 Hz
This synthetic local feature is not anatomy or a measured susceptibility field. Raising it adds spatial detail that eight broad harmonic shapes cannot reproduce exactly.
MAPPED SPREAD19.31 Hz RMS95.0 Hz peak-to-peak
RESIDUAL SPREAD4.77 Hz RMS40.0 Hz peak-to-peak
RMS REDUCTION75.3%0.112 µT · 0.037 ppm at 3 T
FIT CONTENT8 spatial shapes257 samples · ROI mean referenced
The residual is the map plus the fitted correction.
Δfres(r) = Δfmap(r) + Σ cj Φj(r)
Δf is proton off-resonance in hertz and Δf = γ̄ΔBz. Φj are explicit dimensionless X, Y, Z, Z², ZX, ZY, X²−Y², and 2XY shapes normalized at a 12 cm reference radius. cj are correction-field coefficients in hertz—not shim-coil current.Which broad shapes oppose the map?
X
Y
Z
Z²
ZX
ZY
X²−Y²
2XY
Signs describe the field added by this fit in the stated coordinate convention. A real scanner maps calibrated coil currents and constraints to fields; this lesson intentionally stops at field coefficients.
The low-order part shrinks; the local feature survives.
The least-squares correction lowers RMS off-resonance from 19.31 Hz to 4.77 Hz over the same samples. The residual is not a failure of arithmetic: the local Gaussian shape is deliberately absent from the allowed basis.
INTERACTIVE 3D · BODY → WATER → ¹H NUCLEUS → B₀ → RF → ENCODING → RECEIVE
Zoom from the patient to the nuclear physics.
Use the numbered journey in order. Every solid, sphere, arrow, field sheet, and ring can be tapped for its exact meaning. The jumps between body, tissue, atom, and nucleus are separate scale models—nothing literally swells inside the patient.
Three-dimensional body-to-hydrogen MRI physics journey
01 / 11 · BODY SCALE
LOCATE ONE TISSUE REGION
The yellow marker identifies the body region that later scale models magnify. It is not an MRI signal or a k-space address.
BODY · m
→TISSUE · mm
→MOLECULE · nm
→NUCLEUS · fm
→ENCODING · cm
→COIL SIGNAL · µV
Drag to orbit · pinch, wheel, or use + / ↺ / − · tap an object to identify it · model sizes are not one continuous scale
Live multiscale physics model
SLOWED + SCHEMATIC · VALUES PRINTED SEPARATELY
OPEN THE EXACT MEANING
First choose where inside the body the signal comes from.
The scanner surrounds the patient with B₀, gradient, and RF hardware. The yellow target marks one tiny tissue region inside the head. The next buttons magnify that region conceptually; the tissue does not move out of the body.
Patient, table, scanner bore, tissue water, fat, and many other materials.
A glowing target and connector make one future region easy to follow.
MRI never receives a ready-made picture from one location; it must excite and encode signals distributed through the body.
0.5 T3 T7 T
At 3.0 T, ¹H phase precesses at 127.73 million cycles per second. The 3D arrow is slowed enormously; its animation speed is only a trend cue.
0 · no slope20 mT/m40 mT/m
The selected stage assigns this field slope to physical Z for slice selection, Y for phase encoding, or X for readout. Gradient current changes precession rate with position; it does not push atoms along that axis.
0.5kHz6.0
−80mm+80
At 20.0 mT/m, a 2.0 kHz RF band selects an ideal 2.35 mm slab centered at z = 0 mm. Moving RF center frequency moves the slab; it does not move atoms.
¹H CARRIER AT ISOCENTER127.732 MHzcycles of transverse phase per second through time
SELECTED SLICE2.35 mm at z 0 mmRF center offset 0.00 kHz
PHASE LOBE · 0.80 msky +681.2 m⁻¹149.9 angle cycles across 220 mm
READOUT · Gx + ADCdkx/dt +0.852 × 10⁶ m⁻¹/sone dwell stores one whole-slice I + iQ sum
ANIMATION TRUTHMHz slowed to visible motionarrow direction and causal trends only; never timing calibration
CLICKABLE · FIELD → PRECESSION RATE
f0 = γ̄B0
CLICKABLE · RF BAND + GRADIENT → THICKNESS
Δz = BWRF / (γ̄|Gslice|)
CLICKABLE · READ GRADIENT → k-SPACE SPEED
dk/dt = γ̄G(t)
CLICKABLE · CHANGING FLUX → COIL VOLTAGE
vcoil(t) = −dΦ(t)/dt
01 / FROM FIELD TO POSITION
A controlled tilt in the magnetic field.
A gradient coil adds a small, nearly linear variation to the much larger main field. Proton frequency now carries an address: where a spin sits determines how fast its phase turns.
Three-dimensional gradient field, frequency, and phase model
01 · FIELD SLOPE
The gradient changes Bz magnitude—not its direction.
Every field arrow remains parallel to +Z. Height and color magnify only the millitesla-scale ΔB around the much larger B₀ carrier.
Live field model
¹H at B₀ = 3.0 T
LOCAL FIELD → FREQUENCY → ROTATING-FRAME PHASE
Bz(x) = B0 + Gxx
Δf = γ̄Gxx → Δφ = 2πΔfΔt
kx = γ̄GxΔt → Δφ(x) = 2πkxx
0.5 Tclinical 1.5 / 3 T7.0 T
−400+40 mT/m
−12 cmisocenter+12 cm
0exact short-time snapshot20 µs
PHASE ADDRESSAt x = +8.0 cm, Δf = +85.15 kHz accumulates +0.341 cycles (+122.6°) in 4.00 µs.
ΔB+2.00 mT
Δf = γ̄ΔB+85.15 kHz
local f127.818 MHz
Δφ / 2π+0.341 cycles
CARRIER LAW
f0 = γ̄B0
42.58 MHz/T × 3.0 T = 127.73 MHzB₀ CHANGESRF carrier127.73 MHz
B₀ DOES NOT SETGradient offset+85.15 kHz
B₀ DOES NOT SETOn-resonance flip92.0°
B₀ DOES NOT SETk-space speed1.064 × 10⁶ m⁻¹/s
At fixed G, position, B₁⁺, and pulse duration, changing B₀ retunes the MHz carrier. The gradient offset, ideal flip angle, and dk/dt stay unchanged.
Scope: this control retunes the field/RF frequency model. The TE/TR tissue constants remain the explicitly stated illustrative 3 T model; real coil fields, SAR, and relaxation are field-dependent.The gradients are tiny beside B₀, but they are switched quickly and precisely. The audible knocking in MRI is gradient hardware moving under Lorentz force.
BEGINNER GAME / THREE HIDDEN WATER BUCKETS
You are the operator.
Find water with fields.
Three water buckets are hidden inside the bore. From the next room, choose a gradient vector, tune a narrow RF frequency offset, send a short probe, and listen for the echo. Find each bucket’s X, Y, and Z coordinate—then see why a frequency under one gradient selects a plane rather than a unique point.
SCANNER ROOMTHE OPERATOR IS OUTSIDE THIS ROOM
3 WATER VOLUMES PRESENT · 0 / 3 FULLY LOCATED
START WITH X · MOVE THE TUNER + PROBE, OR RUN A GUIDED SWEEP
Three-dimensional hidden water bucket game
READY · 3 SIGNAL SOURCES · POSITIONS HIDDEN
Gx +12.0 mT/m · plane x = 0.0 cm
The translucent square is every position that has the tuned frequency under the current gradient. It is a plane, not a flashlight beam.
Drag to orbit · solid ring = bore · blue plane = same-frequency locations · buckets reveal only after detection
OPERATOR NOTEBOOK · MEASURED COORDINATES
Match peak height across X, Y, and Z.
water volume / signal tagX coordinateY coordinateZ coordinatestatus
0.6 L · small peak???0 / 3 axes
1.0 L · medium peak???0 / 3 axes
1.4 L · large peak???0 / 3 axes
0 OF 9 COORDINATES FOUND
One frequency measurement gives one projection—not a full 3D address.
With Gx on, frequency tells X but says nothing about Y or Z. Rotate the gradient job and repeat. Real MRI performs a much richer, systematic version with many phase and frequency encodings.
No gradient: no position clue
With G = 0, all three identical-water resonances sit near the same f₀. Frequency can say “water signal exists,” but not where it came from.
One gradient: one projection
Δf = γ̄G·r. A measured frequency gives position along G, while an entire perpendicular plane has that same frequency.
More encodings: reconstruct location
Real imaging collects many known phase/frequency patterns. Reconstruction solves the mixed whole-object voltages for a spatial image; an operator does not hunt voxel by voxel.
02 / RF TRANSMIT, RELAXATION & RECEIVE
Excite broadly.
Listen locally.
The transmit coil creates the rotating B₁⁺ field that tips magnetization. After excitation, the scanner stops transmitting and receive coils detect the tiny voltage induced by precessing transverse magnetization. A common 1.5 T / 3 T workflow is body-coil transmit with a close local array receiving.
RF WAVEFORM DECODER · THREE DIFFERENT SHAPES, THREE DIFFERENT JOBS
Frequency is horizontal spacing. B₁⁺ amplitude is height. Pulse duration is width.
The plots share the live controls below. They are separated because “a faster wave,” “a stronger pulse,” and “a larger received signal” do not mean the same thing.WHY THIS PLOT IS 2DLeft–right is elapsed time. Up–down is one field component or received-voltage magnitude. Neither axis is a patient direction, so a third spatial axis would be invented and misleading.
Change B₀ → retune the carrier in this comparison
T means tesla, the unit of magnetic-field strength.
The plotted field goes from a positive peak, through zero and a negative peak, back to the next positive peak: one complete 360° oscillation. It is a cycle of the RF magnetic field—not a proton traveling in a circle.
The slow outline connecting the peaks of thousands of fast carrier cycles. It is not a second radio wave and not a container; it summarizes how strongly the transmitter is driven over time.
The size of the complex voltage after the receiver mathematically removes the MHz carrier. Phase is stored too, even though this row shows only non-negative magnitude.
RECEIVER SIGNAL PATH · BEFORE AND AFTER DEMODULATION
How one fast RF coil voltage becomes I and Q.
Before demodulation there is one rapidly alternating voltage. The receiver compares it with two synchronized reference waves; low-pass filtering leaves two slower signed voltages that preserve the signal’s magnitude and phase.
WHY THIS EXPLANATION USES 2DThe wave rows plot voltage versus time; the round I/Q view plots two mathematical receiver-reference components. I and Q are not scanner X and Y. Use “See this split in the 3D receiver” to locate the calculation in the hardware chain.
Same signal · four viewsNORMALIZED PHASE +45°
Quantitative three-dimensional RF transmit and receive signal chain
TRANSMIT WINDOW · B₁⁺127.73 MHz
BODY COIL → B₁⁺ → SPINS
α = 92.0°
Local receive elements are detuned while the body coil transmits.
Drag to orbit · violet vectors are B₁⁺ · yellow arrow is net M · coral loops are local elements
dead time / switch
TRANSMIT FLIP ANGLE
α = γ ∫ B₁⁺(t) dt
RECEIVED VOLTAGE
vRx(t) ∝ −dΦM/dt
2 µTRF field strength18 µT
0.20 mspulse area3.00 ms
close · 1 cmreceive geometryfar · 12 cm
RF waveformpower amplifier
→
body coilB₁⁺ transmit
→
magnetizationprecessing spins
→
local arrayB₁⁻ receive
→
preampsADC / k-space
BODY TRANSMIT · LOCAL RECEIVE
The large built-in body coil drives a broad, comparatively uniform B₁⁺ field. During transmit the nearby receive array is actively detuned; during reception the transmitter is isolated and the local elements feed low-noise preamplifiers.
predicted rectangular-pulse flip92.0°
receive sensitivity proxy1.00× local
excitation coveragebroad / uniform
receive coveragelocal · 8 channels
transmit safety focuswhole-body SAR
switch stateRx array detuned
The local sensitivity number uses the exact on-axis field falloff of an ideal 50 mm-radius circular loop, C(d)/C(3 cm) = [(1 + (d/5 cm)²)/(1 + (3/5)²)]−3/2. It is a normalized geometry lesson, not a scanner specification or a full spatial array map. The 3D ruler declares 1 world unit = 100 mm. The shared B₀ control retunes the displayed ¹H carrier; it does not rescale geometry, SNR, relaxation, or SAR. Actual values require coil- and patient-specific electromagnetic measurements.
LIVE 3D LOCAL-COIL RECEIVE JOURNEY · USES THE MODEL ABOVE
What a local coil physically captures—and what happens after.
Drag the stage scrubber with a finger or press Play. The 3D camera follows the signal from the patient to the final combined image.
STAGE 00 OF 07 · HARDWARE HANDOFF
Stop transmitting before the tiny receiver listens.
The body coil finishes the high-power B₁⁺ pulse. The transmitter is isolated, the receive-only local loops come out of their protected detuned state, and their low-noise paths are connected.
WHAT CAUSES THIS?
The programmed pulse ends and the transmit/receive switching network changes electrical connections after a short recovery interval.
WHAT EXISTS NOW?
Excited transverse magnetization exists in the patient. The local array is now electrically able to respond to it.
MEASURED IN WHAT?
Switch timing is measured in seconds, commonly microseconds (µs). No image value has been measured yet.
WHY DO WE CARE?
The receive signal is tiny compared with transmit power. Isolation protects the preamplifier and prevents transmitter leakage from hiding early signal.
DO NOT CONFUSE IT WITH
The RF pulse did not travel into the local coil as an image. It prepared magnetization; reception is a later electromagnetic induction event.
transmit endsdrag / swipe through processingcombined image
0.25×display speed only2.00×
PREAMPLIFIER VOLTAGE GAIN · TAP FOR UNITS AND CLIPPING
GdB = 20 log10(Vout/Vin) ↔ Vout = Vin·10GdB/20
Gain scales signal and existing input noise; it does not create localization or improve input SNR.
20 dB · ×10voltage scaling60 dB · ×1000
32 kHzsample rate / noise tradeoff256 kHz
CURRENT SETTINGSAt 40 dB, voltage is multiplied by 100 before digitization. A 128 kHz sample rate gives 7.81 µs between samples in this one-sample-per-dwell model.
IN PATIENTMxy(r,t)
→
Moving transverse magnetization distributed over position r.
AT COIL cvc(t) ∝ −dΦc/dt
→
One continuous signed RF voltage per element.
AFTER DEMODULATIONSc(t) = Ic(t) + iQc(t)
→
Two slower signed components per element.
AFTER ADCSc[n] at k[n]
→
Digital complex samples assigned to gradient-created addresses.
SIMPLE COMBINATION EXAMPLEIRSS(r) = √Σc|Ic(r)|²
Aligned coil images become one relative magnitude image; tap for limits.
Transmit field
RF power at the Larmor frequency rotates magnetization. Amplitude and pulse duration set flip angle; spatial B₁⁺ variation makes flip angle nonuniform.
Receive sensitivity
Precessing transverse magnetization induces a tiny voltage. Close local elements couple strongly to nearby anatomy and admit less distant noise.
Isolation matters
Receive-only elements are detuned during the high-power transmit pulse. The transmit path is then isolated while low-noise preamplifiers listen for the echo.
ADVANCED 3D MODEL · FINITE-DURATION RF · FULL BLOCH ROTATION PRODUCT
A slice is not a plane.
It is a solved excitation profile.
A shaped RF pulse and a slice-select gradient act at the same time. Position becomes resonance offset; the pulse spectrum tips a band of those positions; a signed gradient moment then removes most of the phase ramp left by a symmetric pulse. Every vector, curve, and number below comes from the same finite-step Bloch calculation.
The gradient does not cut tissue. It makes resonance frequency vary continuously along the selected axis.
The sinc spectrum predicts the low-flip response; the full Bloch product remains valid when the tip is large.
Magnitude can look acceptable while residual phase still matters for coherent downstream sequences.
ONE-DIMENSIONAL POSITION ENSEMBLE · ROTATING FRAME · M / M₀
After rephasing · inspect the physical slice profile
The selected band is transverse while off-band magnetization remains near +Mz. Vector direction retains the complex phase that a magnitude-only slice drawing hides.
The quantitative RF waveform and slice-profile plots remain available if 3D rendering is unavailable.
VECTOR BASIS · ARROW DIRECTION
Mxworld x · parallel to the position rail
Myworld y
Mzworld z · camera up
Arrow base = physical position z. Arrow direction = magnetization M.
DRAG TO ORBIT · + / − TO ZOOM · TAP AN OBJECT TO IDENTIFY IT
RF amplitude + slice-gradient timeline
EXACT MIDPOINT SAMPLES · CURSOR FOLLOWS THE 3D STATE
B₁,x(t) · µT scale normalized into plot
Gslice during RF
signed post-RF gradient moment
displayed state
Complex excitation profile across physical z
FULL BLOCH SOLUTION + CENTER-SCALED SMALL-TIP SHAPE
full Bloch |Mxy|
full Bloch Mz
small-tip shape · center-scaled; values may cross +1
full Bloch phase · right axis
nominal BW / γ̄|G| band
equilibriumscrub exact stored statesslice profile
5°full Bloch regime120°
2bandwidth 1.25 kHz8
1.6 mslonger narrows bandwidth at fixed TBW6.0 ms
5nominal 1.47 mm40 mT/m
−1 kHzcenter z = 0.00 mm+1 kHz
60%center tip 90.0°140%
rectangular windowsidelobe ↔ transition tradeoffHamming
noneA −32.0 mT·ms/m · slope −30.5°/mm130%
FINITE-SAMPLE BLOCH PRODUCT + NOMINAL SLICE RELATION
Mn+1(z) = R[−γ|Beff,n(z)|Δt] Mn(z)
Δf(z) = γ̄Gz + Δf₀ − ΔfRF · BW = TBW / T · Δznom = BW / (γ̄|G|) · Arephase = −GT/2FULL-BLOCH CENTER TIP90.0°|Mxy| 1.000 · Mz 0.000
nominal slice thickness1.47 mmmeasured |Mxy| FWHM 1.60 mm
excitation bandwidth1.25 kHzpeak |B₁| 7.31 µT
through-slice phase slope−30.5°/mmbefore rephaser −521.0°/mm
raw complex RMS · |z−zc| ≤ 2Δznom0.2454large-tip nonlinearity is visible
profile quality6.97% center-zone variationlargest declared stopband |Mxy| 0.057
slice center0.00 mmRF 0 Hz · B₀ error held at 0 Hz
numerical closure< 2 × 10⁻¹⁴maximum ||M|| − 1 across the solved profile
Moving the RF carrier shifts the selected band; increasing |G| spreads the same RF bandwidth over a thinner physical slice.
Apodization reduces sidelobes but broadens the transition. The nominal thickness is not the exact FWHM of every pulse.
The dashed small-tip curve is center-scaled only to compare shape; the RMS metric uses its raw complex amplitude. Its error grows because Mz no longer remains approximately one.
A symmetric pulse accumulates approximately half the selection-gradient area after its effective excitation time; a negative half moment unwinds it.
EXPLICIT MODEL BOUNDARY
This is a one-dimensional, single-band, nonadiabatic windowed-sinc pulse. It uses 256 midpoint RF samples and 181 spatial positions with exact norm-preserving rotations. Relaxation, flow, diffusion, chemical shift, multiband excitation, Shinnar–Le Roux pulse design, gradient ramps, concomitant fields, eddy currents, RF amplifier limits, and measured B₀/B₁ maps are not simulated. The post-RF rephaser is represented by its signed moment, so its duration and off-resonance evolution are omitted.