File:Spin Echo diagram.svg

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Captions

Captions

Creation of a spin echo signal after 90° excitation pulse and 180° refocussing pulse

Summary

[edit]
Description
English: Prior to a 90° excitation pulse no transverse magnetization exists and no signal is recieved. After the 90° pulse, the created transverse magnetization starts its precession and a signal oscillating with the Larmor frequency can be oserved. This signal (also called the free induction decay (FID)) decays with the relaxation time T2* due to the spin-spin relaxation (T2) and a potentially inhomogeneous magnetic field in the sample. When at time TE/2 a 180° refocussing pulse is applied, the dephasing due to inhomogeneities is reversed and an spin echo forms at the echo time (TE). The echo again decays with T2* (dotted lines). The signal decay due to T2 cannot be reversed (dashed line). Tx: transmitted signal, Rx: recieved signal
Deutsch: Vor einem 90° Anregungspuls existiert keine tranverse Magnetisierung sodass kein Signal empfangen werden kann. Nach dem 90° Puls präzidiert die erzeugte transversale Magnetisierung mit der Larmor Frequenz und ein oszillierendes Signal kann gemessen werden. Das Signal (auch FID (engl. free induction decay) genannt) zerfällt mit der Relaxationszeit T2*, aufgrund der Spin-Spin Relaxation (T2) und einem möglicherweise inhomogenen Magnetfeld innerhalb der Probe. Wenn nach der Zeit TE/2 ein 180° Refokussierungspuls gesendet wird, wird die Dephasierung aufgrund des inhomogenen Magnetfelds umgekehrt und ein Spin Echo wird zur Echozeit (TE) erzeugt. Das Echo zerfällt erneut mit der Relaxationszeit T2* (gepunktete Linie). Der Signalzerfall aufgrund von T2 kann nicht rückgängig gemacht werden (gestrichelte Linie) . Tx: transmittiertes Signal, Rx: empfangenes Signal.
Date
Source Own work
Author FbrG
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Source code
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MATLAB code

MATLAB code 
% parameters for Larmor frequency, T2 and T2* are chosen for purposes of
% better visualization.

% define time axis in arbitrary units
tEnd = 20;
tStart = -1;
t = linspace(tStart, tEnd, 50000);

% Larmor frequency
w = 3;

%relaxation times
t2s = 0.9;
t2 = 15;

% echo time
TE = 2/3*tEnd;

margin = 20;
fPos = [50 50 1000 700];
figure('Position', fPos)
hold on
set(gca, 'units', 'pixels')
set(gca, 'Position', [margin margin fPos(3)-2*margin fPos(4)-2*margin])
xlim([tStart-0.5 tEnd])
axis off

%% signal

tFid = t(t>0 & t<=TE/2);
tEcho1 = t(t>=TE/2 & t<=TE);
tEcho2 = t(t>=TE & t<0.97*tEnd);
tT2 = t(t>0 & t<0.97*tEnd);

signal = 0*t;
signal(t>0 & t<=TE/2) = cos(2*pi*w*tFid).*exp(-tFid/t2s);
signal(t>=TE/2 & t<=TE) = cos(-2*pi*w*(tEcho1-TE)).*exp((tEcho1-TE)/t2s).*exp(-tEcho1/t2);
signal(t>=TE & t<0.97*tEnd) = cos(-2*pi*w*(tEcho2-TE)).*exp(-(tEcho2-TE)/t2s).*exp(-tEcho2/t2);



pT2 = plot(tT2, exp(-tT2/t2));
pT2s_1 = plot(tFid, exp(-tFid/t2s));
pT2s_2 = plot(tEcho2, exp(-TE/t2)*exp(-(tEcho2-TE)/t2s));

pSignal = plot(t, signal);

pRxAxis = plot([tStart tEnd], [0 0]);

pEnvelope = [pT2, pT2s_1, pT2s_2];

set(pSignal, 'LineWidth', 1.5)
set(pSignal, 'color', lines(1))

set(pEnvelope, 'LineWidth', 1.5)
set(pEnvelope, 'color', 0.3*[1 1 1])

set(pT2s_1, 'LineStyle', ':')
set(pT2s_2, 'LineStyle', ':')
set(pT2, 'LineStyle', '--')

set(pRxAxis, 'LineWidth', 1)
set(pRxAxis, 'color', 0*[1 1 1])

text(1.1*tStart, 0, 'Rx', 'HorizontalAlignment', 'right', ...
    'FontSize',     15)

text(1.5*t2s, 0.4, '$\sim \mathrm e^{-\frac{t}{T_2^*}}$', ...
    'Interpreter',  'latex', ...
    'FontSize',     20)
text(TE/2, 0.8, '$\sim \mathrm e^{-\frac{t}{T_2}}$', ...
    'Interpreter',  'latex', ...
    'FontSize',     20)


%% RF pulses

% pulse duration
dRF = 0.1;
% pulse amplitude
aRF = 0.25;

% vertical offset
vOff = 1.25;

p90 = plot(dRF*[-1 -1 0 0], vOff + aRF*[0 1 1 0]);
p180 = plot(TE/2+dRF/2 + dRF*[-1 -1 0 0], vOff + 2*aRF*[0 1 1 0]);

pTxAxis = plot([tStart tEnd], vOff+[0 0]);

pRF = [p90, p180];

set(pRF, 'LineWidth', 1.5)
set(pRF, 'color', 0*[1 1 1])

set(pTxAxis, 'LineWidth', 1)
set(pTxAxis, 'color', 0*[1 1 1])

text(1.1*tStart, vOff, 'Tx', 'HorizontalAlignment', 'right', ...   
    'FontSize',     15)

text(-dRF/2, vOff+aRF, '90°', 'HorizontalAlignment', 'center', ...
    'VerticalAlignment', 'bottom', ...
    'FontSize',     15)
text(TE/2, vOff+2*aRF, '180°', 'HorizontalAlignment', 'center', ...
    'VerticalAlignment', 'bottom', ...
    'FontSize',     15)

%% timings

% verticalPosition
vPos = 2.2;
% height
h = 0.1;

% vertical lines
vZero = plot([0 0], vPos+[-h h]/2);
vTE2  = plot([TE TE]/2, vPos+[-h h]/2);
vTE   = plot([TE TE], vPos+[-h h]/2);

% horizontal lines
hLine_1 = plot([0 TE/2], [vPos vPos]);
hLine_2 = plot([TE/2 TE], [vPos vPos]);


vLines = [vZero, vTE2 vTE];
hLines = [hLine_1, hLine_2];

set(vLines, 'LineWidth', 1)
set(vLines, 'color', 0*[1 1 1]);

set(hLines, 'LineWidth', 1)
set(hLines, 'color', 0*[1 1 1]);

text(0, vPos+h/2, '0', 'HorizontalAlignment', 'center', ...
    'VerticalAlignment', 'bottom', ...
    'FontSize',     15)
text(TE/2, vPos+h/2, 'TE/2', 'HorizontalAlignment', 'center', ...
    'VerticalAlignment', 'bottom', ...
    'FontSize',     15)
text(TE, vPos+h/2, 'TE', 'HorizontalAlignment', 'center', ...
    'VerticalAlignment', 'bottom', ...
    'FontSize',     15)

%% save to svg

saveas(gcf, 'SpinEcho.svg');

Licensing

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I, the copyright holder of this work, hereby publish it under the following license:
w:en:Creative Commons
attribution share alike
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Date/TimeThumbnailDimensionsUserComment
current16:37, 1 January 2021Thumbnail for version as of 16:37, 1 January 20211,250 × 875 (496 KB)FbrG (talk | contribs)Uploaded own work with UploadWizard

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