Difference between revisions of "DPL94 math"

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__TOC__
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== Equation ==
 
== Equation ==
 
<math>
 
<math>
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</math>
 
</math>
  
Expressing in terms of $\textrm{w}_\textrm{1}, \textrm{w}_\textrm{eff}$ <br>
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Expressing in terms of {{:omega1}}, {{:omegaeff}} <br>
 
<math>
 
<math>
  \sin^2\theta \left( \frac{\Phi_\textrm{ex} \textrm{k}_\textrm{ex}}{\textrm{k}_\textrm{ex}^2 + \omega_\textrm{e}^2} \right)
+
  \sin^2\theta \left( \frac{\Phi_\textrm{ex} \textrm{k}_\textrm{ex}}{\textrm{k}_\textrm{ex}^2 + \omega_\textrm{e}^2} \right) =
 +
\frac{w_\textrm{1}^2}{w_\textrm{eff}^2} \cdot \left( \frac{\Phi_\textrm{ex} \textrm{k}_\textrm{ex}}{\textrm{k}_\textrm{ex}^2 + w_\textrm{1}^2 + \Omega^2} \right) =
 +
\frac{w_\textrm{1}^2}{w_\textrm{1}^2 + \Omega^2} \cdot \left( \frac{\Phi_\textrm{ex} \textrm{k}_\textrm{ex}}{\textrm{k}_\textrm{ex}^2 + w_\textrm{1}^2 + \Omega^2} \right)
 
</math>
 
</math>
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 +
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== Ramp code ==
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<source lang="python">
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import matplotlib.pyplot as plt
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import numpy as np
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from math import atan2
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phi = 2.0
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kex = 3.0
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def calc(w1_arr, Omega_arr, p, k):
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    val_arr = []
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    for w1 in w1_arr:
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        for Omega in Omega_arr:
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            theta = atan2(w1 , Omega)
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            val = w1**2/(w1**2+Omega**2) * (p*k/(k**2 + w1**2 + Omega**2))
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            #val = (p*k/(k**2 + w1**2 + Omega**2))
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            val_arr.append([w1, Omega, theta, val])
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    return np.array(val_arr)
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 +
w1 = np.arange(0., 20, 1)
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Omega = np.array([5])
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data = calc(w1, Omega, phi, kex)
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 +
plt.figure(1)
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plt.plot(data[:,2], data[:,3], '.')
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plt.ylabel(r'Imaginary Rex, ramping spin-lock field strength, nu1')
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plt.xlabel(r'Rotating frame tilt angle')
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Omega = np.arange(0., 20, 1)
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w1 = np.array([5])
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data = calc(w1, Omega, phi, kex)
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 +
plt.figure(2)
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plt.plot(data[:,2], data[:,3], '.')
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plt.ylabel(r'Imaginary Rex, ramping spin-lock offset')
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plt.xlabel(r'Rotating frame tilt angle')
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 +
plt.show()
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</source>
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=== Ramping ===
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[[File:ramp_w1_ramping_spin-lock_field_strength.png|thumb|center|upright=2|Ramping spin-lock field strength : w1]]
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[[File:ramp_Omega_spin-lock_offset.png|thumb|center|upright=2|Ramping spin-lock offset : Omega]]
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== Figure ==
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See Figure 1 and 10 in the reference:
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 +
* {{#lst:Citations|PalmerMassi06}}
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 +
[[File:Fig1 Palmer Massi 2006.png|thumb|center|upright=3|Try to reproduce Figure 1.]]
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Figure produced with script [[Relax_disp.spin_lock_offset%2Bfield_figure | found here. ]]
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 +
== See also ==
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[[Category:Relaxation_dispersion analysis]]

Latest revision as of 16:00, 6 November 2015

Equation

[math] \mathrm{R}_{1\rho}= \mathrm{R}_1\cos^2\theta + \left( \mathrm{R}_{1\rho}{´} + \frac{\Phi_\textrm{ex} \textrm{k}_\textrm{ex}}{\textrm{k}_\textrm{ex}^2 + \omega_\textrm{e}^2} \right) \sin^2\theta [/math]

Expressing in terms of ω1, ωe
[math] \sin^2\theta \left( \frac{\Phi_\textrm{ex} \textrm{k}_\textrm{ex}}{\textrm{k}_\textrm{ex}^2 + \omega_\textrm{e}^2} \right) = \frac{w_\textrm{1}^2}{w_\textrm{eff}^2} \cdot \left( \frac{\Phi_\textrm{ex} \textrm{k}_\textrm{ex}}{\textrm{k}_\textrm{ex}^2 + w_\textrm{1}^2 + \Omega^2} \right) = \frac{w_\textrm{1}^2}{w_\textrm{1}^2 + \Omega^2} \cdot \left( \frac{\Phi_\textrm{ex} \textrm{k}_\textrm{ex}}{\textrm{k}_\textrm{ex}^2 + w_\textrm{1}^2 + \Omega^2} \right) [/math]


Ramp code

import matplotlib.pyplot as plt
import numpy as np
from math import atan2

phi = 2.0
kex = 3.0

def calc(w1_arr, Omega_arr, p, k):
    val_arr = []
    for w1 in w1_arr:
        for Omega in Omega_arr:
            theta = atan2(w1 , Omega)
            val = w1**2/(w1**2+Omega**2) * (p*k/(k**2 + w1**2 + Omega**2))
            #val = (p*k/(k**2 + w1**2 + Omega**2))
            val_arr.append([w1, Omega, theta, val])
    return np.array(val_arr)

w1 = np.arange(0., 20, 1)
Omega = np.array([5])
data = calc(w1, Omega, phi, kex)

plt.figure(1)
plt.plot(data[:,2], data[:,3], '.')
plt.ylabel(r'Imaginary Rex, ramping spin-lock field strength, nu1')
plt.xlabel(r'Rotating frame tilt angle')

Omega = np.arange(0., 20, 1)
w1 = np.array([5])
data = calc(w1, Omega, phi, kex)

plt.figure(2)
plt.plot(data[:,2], data[:,3], '.')
plt.ylabel(r'Imaginary Rex, ramping spin-lock offset')
plt.xlabel(r'Rotating frame tilt angle')

plt.show()

Ramping

Ramping spin-lock field strength : w1
Ramping spin-lock offset : Omega

Figure

See Figure 1 and 10 in the reference:

  • Palmer, 3rd, A. G. and Massi, F. (2006). Characterization of the dynamics of biomacromolecules using rotating-frame spin relaxation NMR spectroscopy. Chem. Rev., 106(5), 1700-1719. (DOI: 10.1021/cr0404287)
Try to reproduce Figure 1.

Figure produced with script found here.

See also