========
Examples
========
.. raw:: html
.. role:: orange
.. |13C| replace:: :sup:`13`\ C
.. |1H| replace:: :sup:`1`\ H
.. |CH| replace:: :orange:`C`\ H
.. |CH2| replace:: :orange:`C`\ H\ :sub:`2`
.. |CH3| replace:: :orange:`C`\ H\ :sub:`3`
.. |html_br| raw:: html
.. |latex_br| raw:: latex
\\
Example 1: Excitation and Precession of a Single Spin
-----------------------------------------------------
Let us start with a simple example of an uncoupled spin, for which the
DROPS display is equivalent to the well-known vector representation.
Selecting :menuselection:`Spin System --> 1-Spin` selects a
single-spin system. By default, the initial state is :pton:`I1z`
(representing “z magnetization”). The red vector in the transparent
red sphere should point along the z axis:
.. drops:: I1z
:prefs: Current Layout=1
:sequence: Standard/PulDel100msec
:class: border
:height: 350px
By default, initially, the single spin is seen approximately from the
positive z axis, dragging upward results in, for example, the view in
Figure :numref:`examples_drugup`. You can manipulate the 3D display
directly. Use one finger or the mouse to rotate the display like a
trackball, or two fingers to pinch-zoom and rotate. For more
information, see :ref:`Manipulating the DROPS Display`.
.. drops:: I1z
:prefs: Current Layout=1
:sequence: Standard/PulDel100msec
:class: border
:height: 350px
:view: C
:name: examples_drugup
:caption: Rotated view
:link:
E1 Display Operator
===================
In order to see the currently displayed product operator, select
:menuselection:`View --> List Prod. Ops`.
As shown below, the list of product operators representing the current
state contains only the term `z`, corresponding to :pton:`I1z`. Its
coefficient has no imaginary part and the real part is 1.
(If a different initial state is shown, select :menuselection:`Initial
State --> I1z`.)
.. drops:: I1z
:nspin: 1
:class: border
:link:
:sequence: Standard/PulDel100msec
:window: List Basis Elems
E1 Edit Sequence
================
Also by default, the pulse sequence should be a 90°y pulse followed by
a delay of 100 milliseconds. The pulse sequence is displayed
schematically at the bottom of the screen. The 90°y pulse is
represented by a yellow rectangle.
In order to see the detailed parameters of the currently selected
pulse sequence, select :menuselection:`Pulse Sequence --> Edit
Sequence...`. This displays the editable list shown in Figure
:numref:`examples_listedit`. As expected, it consists of two elements:
A 90°y pulse followed by a delay of 100 milliseconds. If a different
pulse sequence is listed, select :menuselection:`Pulse Sequence -->
90°-T --> Pulse Delay T=100 ms`.
.. drops:: I1z
:name: examples_listedit
:sequence: Standard/PulDel100msec
:window: SequenceEditTable
:caption: The sequence editor
:class: border
:link:
E1 Change Spin Params
=====================
By default, the offset frequency of the spin **I1** is `ν1 = 1
Hz`. This can be checked (or modified) by selecting
:menuselection:`Spin System --> Parameters...`.
.. drops:: I1z
:nspin: 1; v1=1
:sequence: Standard/PulDel100msec
:window: SysParamWindow
:class: border
:link:
The displayed parameter and its current value is indicated above the
slider: “v1 = 1.00 Hz”.
E1 View Simulation
==================
Pressing the play button at the bottom of the screen starts the
simulation of the pulse sequence, and immediately shows the effects on
the state. The pulse flips the spin by 90° around the y axis from z
to x. During the delay, the spin rotates (precesses) around the z
axis. The current time point is indicated by a circle and a vertical
line.
.. drops:: I1z
:nspin: 1; v1=1
:sequence: Standard/PulDel100msec
:window: SysParamWindow
:time: 0.08
:class: border
:link:
At the end of the sequence, the spin points along the x axis. As
described in more detail in the :doc:`tutorial`, the buttons at the
bottom of the screen allow you to interactively :ref:`control the
simulation `. For example, you can slow
down or speed up the simulation or run the simulation in a loop.
After stopping or pausing the simulation, you can change the current
time point by moving the time cursor.
E1 View Param Changes
=====================
.. drops:: I1z
:nspin: 1; v1=0.2
:sequence: Standard/PulDel100msec
:window: SysParamWindow
:time: 0.08
:class: border
:link:
You can also change the offset frequency (i.e. the rotation frequency
during the delay) interactively while the simulation is running and
immediately see the effect of the parameter change. Here, for
:math:`v1=0.2 Hz`, the magnetization vector at the end of the delay
points closer to the x axis than was the case with :math:`v1=1 Hz`.
.. note::
In this simple example, you learned how to define a spin
system, the spin system parameters, the initial state, the pulse
sequence and how to run and control the simulation. As we
considered a single, uncoupled spin, the spin dynamics can be
completely described by the well-known vector representation. In
addition to the “magnetization vector”, the corresponding
single-spin droplet (consisting of a red and green sphere) of the
DROPS representation was displayed. Although this did not provide
any additional information in this case, it is interesting to note
that at all times, the magnetization vector is parallel to the
corresponding droplet, i.e. to the axis formed by the centers of
the green and red spheres. Hence, for the simple case of an
uncoupled spin 1/2, the vector picture can be viewed as a special
case of the DROPS representation. However, in contrast to the
vector picture, the DROPS representation is not limited to
uncoupled spins.
E1 Further Exploration
======================
Suggestions for further exploration:
Predict the effect of the pulse sequence for different initial states
and check your prediction by running the simulation.
Examples:
* Observe the effect of the pulse sequence for the initial state
:pton:`I1x` rather than :pton:`I1z`.
* Observe the effect of the pulse sequence for the initial state
:pton:`I1y`.
What is the effect of the phase and/or flip angle of the excitation
first pulse is modified?
Selecting :menuselection:`Pulse Sequence --> Edit Sequences...`
provides the Sequence List, which in this case consists of two
sequence elements: (1) The pulse 90°y and (2) the Delay 100 ms. If
you double tap on 90°y, a sub menu opens in which you can change
the flip angle and the phase of the pulse. In most cases, the
pulses of interest will be 90° or 180° pulses, which can be chosen
directly. If you would like to change the 90°y pulse e.g. to a
45°y pulse, select `Other Pulse` and enter the desired flip angle
in the number pad.
Example 2: Rotation of :math:`I^+` Around the z Axis
----------------------------------------------------
In this example, we use the SpinDrops app to visualize the droplet of
the +1-quantum coherence term :pton:`I1p = I1x + i I1y` and to see its
evolution under z rotations. Choose :menuselection:`Spin System -->
1-Spin` and select :menuselection:`Initial State --> I1(+)`. For the
sequence, select :menuselection:`Pulse Sequence --> z Rotation -->
360°(z)` To see the corresponding Cartesian product operators, select
:menuselection:`View --> List Prod. Ops.`
.. drops:: I1p
:sequence: Standard/Pulse360degz
:window: List Basis Elems
:link:
The left column of the list indicates the operator terms using the
short-hand three-letter code (:ref:`Cartesian Basis`). On the right, the
real and imaginary parts of the corresponding coefficients are
displayed, which in general can be complex.
E2 The Droplet
==============
In the vector picture, the state :pton:`I1p = I1x + i I1y` is
represented by two vectors: A real vector (red) pointing along the x
axis and an imaginary vector (yellow) pointing along the y axis.
.. list-table::
:class: no-cap-num nocol w100
* - :drop:`I1p`
- `=`
- :drop:`I1x`
- `+`
- :drop:`i*I1y`
The corresponding droplet is a combination of the droplet for the
operator :pton:`I1x` (consisting of a red and a green sphere) and the
droplet for the operator :pton:`i I1y` (consisting of a yellow and
blue sphere).
E2 Progression
==============
As the sequence progresses over time, the :pton:`I1p` droplet will
rotate around the z-Axis.
.. list-table::
:class: no-cap-num nocol w100
* - :drop:`I1p`
- :drop:`exp(-i*pi/2)*I1p`
- :drop:`exp(-i*pi)*I1p`
- :drop:`exp(-i*3*pi/2)*I1p`
- :drop:`exp(-i*2*pi)*I1p`
* - :math:`0°_z`
- :math:`90°_z`
- :math:`180°_z`
- :math:`270°_z`
- :math:`360°_z`
E2 Further Exploration
======================
Suggestions for further exploration:
Observe the effect of the z rotation for different initial states,
such as :pton:`I1m` or :pton:`I1x`.
Apply different rotations (e.g. around the x or y axis) using pulses.
Example 3: Weak Coupling Evolution in a Two-Spin System
-------------------------------------------------------
Select :menuselection:`Spin System --> 2-Spin` and choose :pton:`I1x`
as the initial state by selecting :menuselection:`Initial State -->
I1x`. Select :menuselection:`Pulse Sequence --> Delay T --> T = 1
s`. In :menuselection:`Pulse Sequence --> Edit Sequence` double tap
the delay and set it to :math:`2 s`. Change the spin system
parameters to `v1=0 Hz, v2=0Hz, J12=1 Hz` (if the parameter window is
not visible, select :menuselection:`Spin System --> Parameters...`).
.. drops:: I1x
:nspin: 2; v1=0; v2=0; J12=1
:height: 350px
:sequence: Standard/FreeEvo1sec
:window: SysParamWindow
:class: border
:link:
Push the play button to run the simulation.
Initially, the droplet representing the linear spin operators of
**I1** shrinks and a new droplet emerges between **I1** and **I2**,
corresponding to the antiphase operator :pton:`2I1yI2z` (confirm using
:menuselection:`View --> List Prod. Ops.` or using the right-hand
rule). After 0.5 seconds (corresponding to :math:`1/(2 J12)`, the
**I1** spin droplet has completely vanished and the :ref:`bilinear
` :math:`\{I1,I2\}` droplet has reached its maximum size.
.. drops:: I1x
:nspin: 2; v1=0; v2=0; J12=1
:height: 350px
:sequence: Standard/FreeEvo1sec
:time: 0.5
:window: List Basis Elems
:class: border
:link:
E3 Evolution
============
.. drops:: I1x
:nspin: 2; v1=0; v2=0; J12=1
:height: 350px
:sequence: Standard/FreeEvo1sec
:time: 1.0
:window: List Basis Elems
:class: border
:link:
|html_br|
|latex_br|
.. image:: _images/page111_jevo_in_2spin.png
:width: 45%
:align: right
After 1 second (corresponding to :math:`1/J12`), the :math:`\{I1,I2\}`
droplet vanishes and the **I1** droplet reaches its maximum size
again. However, notice that the sign of the **I1** droplet (and of the
magnetization vector) is inverted compared to the initial state at
:math:`t = 0 s`. After 2 seconds (corresponding to :math:`2/J12`),
the initial state :pton:`I1x` is reproduced. If the simulation is set
to repeat mode, this will result in a continuous oscillation between
the two droplets.
This simulation visualizes the well-known evolution of :pton:`I1x` in
the presence of a J coupling (in the weak coupling limit), which is
given by :pton:`I1x cos(π J12 t ) + 2I1yI2z sin(π J12 t)`. For
:pton:`J12 = 1 Hz` and :math:`t = 0.5 s`, the argument :pton:`(π J12 t
) = π/2`, i.e. the state is given by :pton:`I1x cos (π/2) + 2I1yI2z
sin (π/2) = 2I1yI2z`. For :math:`t = 1 s, 1.5 s`, and :math:`2 s`, the
state is :pton:`-I1x`, :pton:`-2I1yI2z`, and :pton:`I1x`,
respectively.
E3 Further Exploration
======================
Suggestions for further exploration:
#. What happens if the offset frequency **v1** is set to non-zero
values?
#. What is the effect if the offset frequency **v2** is set to
non-zero values?
#. What happens if the initial state is :pton:`I1z`?
Example 4: Refocusing of Offset and Coupling Effects
----------------------------------------------------
In this example, we illustrate the effect of 180° pulses on the
evolution of a two-spin system. Select :menuselection:`Spin System -->
2-Spin` and set :pton:`J12 = 1 Hz` using the parameter slider
(in :menuselection:`Spin System --> Parameters...`).
.. list-table:: Evolution under coupling
:class: no-cap-num nocol w100
:name: examples_e4t
* - .. drops:: I1x
:nspin: 2; v1=0; v2=0; J12=1
:time: 0
:width: 40%
:sequence: Standard/Delay1over2J12
:aspect: 120%
:caption: t=0
:event: id:MainFrame,e:KDOWN,k:109
:link:
- .. drops:: I1x
:nspin: 2; v1=0; v2=0; J12=1
:time: 0.25
:width: 40%
:sequence: Standard/Delay1over2J12
:aspect: 120%
:caption: t=0.25
:event: id:MainFrame,e:KDOWN,k:109
:link:
- .. drops:: I1x
:nspin: 2; v1=0; v2=0; J12=1
:time: 0.5
:width: 40%
:sequence: Standard/Delay1over2J12
:aspect: 120%
:caption: t=0.5
:event: id:MainFrame,e:KDOWN,k:109
:link:
|html_clear|
|latex_clear|
We consider the evolution of the initial state :pton:`I1x` (select
:menuselection:`Initial State --> I1x`) during a delay :pton:`T = 1/(2
J12) = 0.5 s` (select :menuselection:`Pulse Sequence --> Delay T -->
T=1/(2·J12)`.
The panels above show the DROPS visualization of the system at
:math:`t = 0 s`, :math:`t = 1/(4 J12) = 0.25 s`, and :math:`t = 1/(4
J12) = 0.5 s`, for the case where both offset frequencies are zero
(**ν1** = **ν2** = :math:`0 Hz`).
In this case, initial x magnetization of the first spin (:pton:`I1x`)
evolves under the weak coupling Hamiltonian during the delay :math:`T
= 1/(2 J12)` completely to the anti-phase operator :pton:`2I1yI2z`.
In the following, we simulate the resulting final state without and
with additional 180° pulses in the center of the delay T, assuming a
non-zero offset frequency of the first spin (:math:`ν1 = 0.5 Hz`).
E4 Refocusing of Offset Effects
===============================
.. image:: _images/page114_refocusing.png
For :math:`ν1 = 0.5 Hz` and :math:`J12 = 1 Hz`, simulations are shown
for :math:`t1 = 0 s`, :math:`t2 = T/2` = :math:`1/(4 J12)` =
:math:`0.25 s`, :math:`t3 = t2` (assuming a negligible duration of the
180° pulses) and :math:`t4 = T = 1/(2 J12) = 0.5 s`.
:pton:`I1x` evolves
a. to :pton:`-2I1xI2z` in the absence of pulses during the delay T,
b. to :pton:`I1x` if a spin **I1**-selective 180°x pulse is irradiated at T/2,
c. to :pton:`I1y` if a spin **I2**-selective 180°x pulse is irradiated at T/2,
d. to :pton:`2I1yI2z` if **I1**- and **I2**-selective 180°x pulses are
irradiated at T/2.
.. drops:: I1z
:nspin: 2; v1=0.5; v2=0; J12=1
:sequence: Standard/Echo1over4J12
:aspect: 210%
:time: 0.25
:class: border
:caption: [click to try (d) interactively in another window]
:event: id:MainFrame,e:KDOWN,k:109
:link:
These results reflect the well-known fact that in case
a. offset and coupling terms of the Hamiltonian are active.
b. the effect of the coupling and of the offset ν1 are refocused.
c. the effect of the coupling and of the offset ν2 are refocused but ν1
is active.
d. the simultaneously irradiated **I1**- and **I2**-selective 180°x
pulses at T/2 (corresponding to a non-selective 180°x pulse for the
two-spin system) refocus frequency-offset effects but the coupling
evolution is active. For the initial state :pton:`I1x`, this is
confirmed by the resulting final state of the system at t4 = T =
1/(2 J12) = 0.5 s for different offsets ν1 of 0 Hz, 0.5 Hz, 1 Hz
and 1.5 Hz.
.. image:: _images/page115_refocus2.png
E4 Further Exploration
======================
Suggestions for further exploration:
- Does the offset ν2 of the second spin have any effect in the
experiment from the example? What if we start with the initial state
:pton:`I2x` instead of :pton:`I1x`?
- How would the results change if 180°y pulses rather than 180°x pulses
were used in the example?
- Explore the effects of selective and non-selective 180° pulses in
three-spin systems.
- Explain the results of the refocusing experiments using the product
operator formalism.
Example 5: Spectral Editing
---------------------------
The :term:`DEPT` experiment (Distortionless Enhancement of
Polarization Transfer) makes it possible to distinguish :orange:`C`,
|CH|, |CH2|, and |CH3| groups. The |1H| spins are excited and the
amplitude and sign of the detected |13C| spin depends on the number of
attached protons.
The basic pulse sequence of the DEPT experiment has the form
:code:`90°x(H) - T - 180°x(H),90°x(C) - T - θ y(H),180°x(C) - T`
where the flip angle :math:`θ` of the editing |1H| pulse is 45°, 90°
or 135° and the delay T is :math:`1/(2 J_{CH})`. The experiment
relies on polarization transfer so that only signal originating from
|1H| polarization is detected on the |13C| frequency. Hence, the
spins of |13C| atoms without attached |1H| atoms do not yield any
detectable signal. Predicting the |13C| signals of |CH|, |CH2|, and
|CH3| is less simple.
As up to three spins 1/2 can be considered in the current version of
SpinDrops, it is only possible to simulate the relative size and sign
of of the final |13C| magnetization for |CH| and |CH2| groups (and
hence of the relative size and sign of the corresponding |13C| NMR
signal), but not yet |CH3|. In the following, we will assume that
spin **I2** represents a |13C| spin and spins **I1** and **I3**
represent |1H| spins:
.. list-table::
:class: nocol
* - |CH| system
- .. graph:: foo
node[shape=circle]
rankdir=LR
"I1 (H)" -- "I2 (C)";
"I2 (C)"[color=orangered]
- with J12 ≠ 0
* - |CH2| system
- .. graph:: foo2
node[shape=circle]
rankdir=LR
"I1 (H)" -- "I2 (C)";
"I2 (C)" -- "I3 (H)";
"I2 (C)"[color=orangered]
- with J12 = J23 ≠ 0, and J13 = 0.
E5 DEPT-45
==========
Select :menuselection:`Spin System --> 3-Spin Chain`. To simulate the
|CH| system, set J12 = 1 Hz, J13 = 0 Hz, and J23 = 0 Hz
(:menuselection:`Spin System --> Parameters`) and define the initial
state as :pton:`I1z` (Initial Selecting :menuselection:`Initial State
--> I1z`). For the |CH2| system, set J12 = 1 Hz and J23 = 1 Hz and
define the initial state as :pton:`I1z+I3z` (:menuselection:`Initial
State --> Edit Operator`). Select :menuselection:`Pulse Sequence -->
Sequence List --> Heteronuclear --> DEPT-45` and run the simulation.
Notice that in both cases the final magnetization vector of spin
**I2** is pointing in the positive x direction.
.. drops:: I1z
:nspin: 3; v1=0; v2=0; J12=1; J13=0; J23=0
:sequence: Standard/DEPT45
:aspect: 180%
:time: 1.6
:view: Bp130
:height: 350px
:prefs: Current Layout=3
:caption: DEPT-45 :math:`CH` : J12 = 1 Hz
:class: border
:event: id:MainFrame,e:KDOWN,k:109
:link:
.. drops:: I1z + I3z
:nspin: 3; v1=0; v2=0; J12=1; J13=0; J23=1
:sequence: Standard/DEPT45
:aspect: 180%
:time: 1.6
:view: Bp130
:height: 350px
:prefs: Current Layout=3
:caption: DEPT-45 :math:`CH_2` : J12 = 1 Hz, J23 = 1 Hz
:class: border
:event: id:MainFrame,e:KDOWN,k:109
:link:
E5 DEPT-90
==========
Select :menuselection:`Pulse Sequence --> Sequence List -->
Heteronuclear --> DEPT-90` and run the simulation. Note that compared
to the DEPT-45 sequence shown on the previous page, the flip angle
:math:`θ` of the editing pulse has changed from 45° to 90° (the yellow
pulses).
In DEPT-90, the final magnetization vector of spin **I2** is pointing
in the positive x direction for |CH| (orange ellipse). However, for
|CH2| the final magnetization vector of spin **I2** is zero (orange
ellipse). This results in positive |13C|-NMR signals for |CH| groups
but no |13C|-NMR signals for |CH2| groups.
.. drops:: I1z
:nspin: 3; v1=0; v2=0; J12=1; J13=0; J23=0
:sequence: Standard/DEPT90
:aspect: 180%
:time: 1.6
:view: Bp130
:height: 350px
:prefs: Current Layout=3
:caption: DEPT-90 :math:`CH` : J12 = 1 Hz
:class: border
:event: id:MainFrame,e:KDOWN,k:109
:link:
.. drops:: I1z + I3z
:nspin: 3; v1=0; v2=0; J12=1; J13=0; J23=1
:sequence: Standard/DEPT90
:aspect: 180%
:time: 1.6
:view: Bp130
:height: 350px
:prefs: Current Layout=3
:caption: DEPT-90 :math:`CH_2` : J12 = 1 Hz, J23 = 1 Hz
:class: border
:event: id:MainFrame,e:KDOWN,k:109
:link:
E5 DEPT-135
===========
Selecting :menuselection:`Pulse Sequence --> Sequence List -->
Heteronuclear --> DEPT-135` changes the flip angle θ of the editing
pulses to 135°.
In this case of DEPT-135, the final magnetization vector of spin
**I2** is pointing in the positive x direction for the |CH| system,
whereas it is pointing in the negative x direction for the |CH2|
system. This results in positive and negative |13C|-NMR signals for
|CH| and |CH2| groups, respectively.
.. drops:: I1z
:nspin: 3; v1=0; v2=0; J12=1; J13=0; J23=0
:sequence: Standard/DEPT135
:aspect: 180%
:time: 1.6
:view: Bp130
:height: 350px
:prefs: Current Layout=3
:caption: DEPT-135 :math:`CH` : J12 = 1 Hz
:class: border
:event: id:MainFrame,e:KDOWN,k:109
:link:
.. drops:: I1z + I3z
:nspin: 3; v1=0; v2=0; J12=1; J13=0; J23=1
:sequence: Standard/DEPT135
:aspect: 180%
:time: 1.6
:view: Bp130
:height: 350px
:prefs: Current Layout=3
:caption: DEPT-135 :math:`CH_2` : J12 = 1 Hz, J23 = 1 Hz
:class: border
:event: id:MainFrame,e:KDOWN,k:109
:link:
E5 Further Exploration
======================
Suggestions for further exploration:
- What are the expected relative signal amplitudes for |CH| and |CH2|
groups in DEPT-45, DEPT-90 and DEPT-135? (Tip: Remember that you can
always display the coefficients of the product operator terms by
selecting :menuselection:`View --> List Prod. Ops.`)
- Does the final DROPS display (and hence the final state of the spin
system) depend on the offset frequencies ν1, ν2, ν3?
- Which product operator terms are created at the end of DEPT-45,
DEPT-90, and DEPT-135 in addition to the desired magnetization of
spin **I2**?
- Calculate the effect of the DEPT experiments analytically using the
standard product operator formalism and compare the results with the
DROPS simulations. What are the expected relative signal amplitudes
for |CH3| groups in DEPT-45, DEPT-90 and DEPT-135?
Example 6: TOCSY Transfer in a Two-Spin System
----------------------------------------------
In this example, we explore the transfer of x magnetization in the
isotropic mixing period of TOCSY experiments, where isotropic mixing
conditions are created by a multiple-pulse sequence. Select
:menuselection:`Spin System --> 2 Spins` and choose :pton:`I1x` as the
initial state by selecting :menuselection:`Initial State -->
I1x`. Choose the isotropic mixing sequence for two spins by selecting
:menuselection:`Pulse Sequence --> Sequence List --> Homonuclear -->
Isotropic Mixing`.
.. drops:: I1x
:nspin: 2; v1=0; v2=0; J12=1
:sequence: Standard/IM12
:time: 0
:window: SysParamWindow
:aspect: 180%
:view: Br131
:class: border
:caption: Before isotropic mixing
:link:
The isotropic mixing period is indicated by a grey rectangle. Set the
coupling J12 to 1 Hz. (Here, the offset frequencies v1 and v2 are
irrelevant as they are effectively suppressed by the isotropic mixing
sequence.) Interestingly, after 0.5 seconds, corresponding to
:math:`t = 1/(2 J12)`, the initial state :pton:`I1x` has turned
completely into :pton:`I2x`, i.e. x magnetization has been fully
transferred from the first to the second spin).
.. drops:: I1x
:nspin: 2
:sequence: Standard/IM12
:time: 0.5
:window: List Basis Elems
:aspect: 180%
:view: Br131
:class: border
:caption: After isotropic mixing
:link:
E6 TOCSY Transfer Return
========================
Under the isotropic mixing Hamiltonian, the initial state
.. math::
\rho(0)\ =&\ I_{1x} \\
\\
\text{evolves to} \\
\\
\rho(t)\ =&\ I_{1x} \cos^2 (\pi J_{12} t) \\
&+ (2I_{1y}I_{2z} - 2I_{1z}I_{2y}) \cos(\pi J_{12} t) \sin(\pi J_{12} t) \\
&+ I_{2x} \sin^2 (\pi J_{12} t)
Creating a circulation like:
.. digraph:: foo3
layout="circo";
"I1x" -> "I1yI2z - I1zI2y" -> "I2x" -> "I1zI2y - I1yI2z" -> "I1x";
This can be verified by using Figure :numref:`example_6a` as a
starting point, and extending the duration of the isotropic mixing
period to :math:`1 s` (or more, by selecting :menuselection:`Pulse
Sequence --> Edit Sequences...`, double tapping Hiso12, and changing
the pulse length from :code:`0.5/J12` to :code:`1/J12`.)
.. drops:: I1x
:name: example_6a
:nspin: 2
:sequence: Standard/IM12
:time: 0.5
:window: SequenceEditTable
:aspect: 180%
:view: Br131
:class: border
:caption: Change the pulse length here
:link:
For :math:`t = 1/(4J12)` (corresponding to :math:`t = 0.25 s` for
:math:`J12 = 1 Hz`), the argument :pton:`(π J12 t) = π J12/(4J12) =
π/4` and :math:`\cos(π J12 t ) = \sin(π J12 t ) = \frac{1}{\sqrt 2}`.
Hence the state is :pton:`0.5 I1x + (I1yI2z - I1zI2y) + 0.5 I2x` and
the bilinear droplet located between **I1** and **I2** reaches its
maximum value.
For :math:`t = 1/(2J12) = 0.5 s`, :math:`(π J12 t ) = π J12/(2J12) =
π/2` and :math:`cos(π J12 t ) = 0`, :math:`sin(π J12 t ) = 1`. Hence,
the state is :pton:`I2x`, i.e. the initial x magnetization has been fully
transferred from spin :math:`I_1` to spin :math:`I_2`. For :math:`t =
1/J12 = 1 s`, the state is again :pton:`I1x` and the cycle starts anew.
Example 7: Inversion of Multiple-Quantum Coherence
--------------------------------------------------
In the standard SpinDrops representation, a non-selective 180°y pulse
simply rotates the droplets by 180° around the y axis. This is
illustrated here for the initial operator :pton:`I1pI2pI3p`
(select :menuselection:`Initial State --> MQ (+/- ops) -->
3Q(I1,I2,I3) --> 2*I1(+)*I2(+)*I3(+)` ).
.. drops:: I1pI2pI3p
:sequence: Standard/Pulse180degy
:window: List Basis Elems
:class: border
In addition to the initial orientation of the droplet, the snapshots
show its orientations after rotations of 45°, 135°, and 180° around
the y axis.
.. list-table:: Inverting :pton:`I1pI2pI3p`
:class: nocol no-cap-num w100
:name: examples_e7
* - .. drops:: 3I1pI2pI3p
:sequence: Standard/Pulse180degy
:time: 0
:width: 40%
:view: B180
:caption: 0°
:link:
:event: id:MainFrame,e:KDOWN,k:109
:nspin: 3; Homonuclear=1; v1=0; v2=0; v3=0; J12=0; J13=0; J23=0
- .. drops:: 3I1pI2pI3p
:sequence: Standard/Pulse180degy
:time: 0.0125
:width: 40%
:view: B180
:caption: 45°
:link:
:event: id:MainFrame,e:KDOWN,k:109
:nspin: 3; Homonuclear=1; v1=0; v2=0; v3=0; J12=0; J13=0; J23=0
- .. drops:: 3I1pI2pI3p
:sequence: Standard/Pulse180degy
:time: 0.0375
:width: 40%
:view: B180
:caption: 135°
:link:
:event: id:MainFrame,e:KDOWN,k:109
:nspin: 3; Homonuclear=1; v1=0; v2=0; v3=0; J12=0; J13=0; J23=0
- .. drops:: 3I1pI2pI3p
:sequence: Standard/Pulse180degy
:time: 0.05
:width: 40%
:view: B180
:caption: 180°
:link:
:event: id:MainFrame,e:KDOWN,k:109
:nspin: 3; Homonuclear=1; v1=0; v2=0; v3=0; J12=0; J13=0; J23=0
The fact that this operation also inverts the coherence order from
:math:`p=+3` to :math:`p=-3` can be inferred from the direction of the
rainbow colors of the initial and final droplet orientations. However,
this can be seen more directly by separating the droplet according to
coherence order :math:`p`:
In the previous examples, the :ref:`Standard Plane` display mode was
selected. To separate the droplets according to coherence order
:math:`p`, choose :menuselection:`View --> Separation --> Coh. Order
p`. In :ref:`this display mode `, the change of coherence
order from :math:`p=+3` to :math:`p=-3` (via all the intermediate
coherence orders) can be clearly followed. In :ref:`Coh. Order p
` display mode, the droplets are arranged in planes with
coherence order :math:`p=3` at the top and :math:`p=-3` at the bottom,
as explained in :doc:`drops_props`. The black triangle in the center
corresponds to :math:`p=0`.
.. drops:: I1pI2pI3p
:sequence: Standard/Pulse180degx
:prefs: Current Separation=3
:class: border
:view: C80
:link:
E7 Further Exploration
======================
Suggestions for further exploration:
- Study the effect of pulses and delays on various initial states of
pure or mixed coherence orders.
- Test the different display modes with droplet separations based on
coherence order :math:`p`, the absolute value of coherence order
:math:`|p|` and/or tensor rank :math:`j`.