## One-pulse NMR line intensity simulationfor a spin I = 3/2 in MAS powder. Contributor: R. Hajjar

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## One-pulse line intensity for MAS powder, Part 1

AIM: We show that, for simulating a powder sample (樣品), the number of summation steps that the Euler angle α is divided in [0, 2π[ rang must be a 4-fold number so that the results do not depend on the sign of the asymmetry parameter η.

Equipment: Mathematica-5 (or MathReader for reading the notebook if you do not have Mathematica-5).

Method: We simulate the central-line intensities of a spin I = 3/2 for pulse duration t increasing from 0 to 20 µs by steps of 1 µs in a powder rotating at the magic angle, using Mathematica-5 notebook.

The parameters for these simulations are:

• Observed line intensity: central transition
• Nucleus: 23Na
• Spin: 3/2
• 23Na Larmor frequency: 105.8731007 MHz
• Proton Larmor frequency: 400 MHz
• Strength of the radio-frequency pulse: 100 kHz
• Initial pulse duration: 0
• Final pulse duration: 20 µs
• Pulse duration increment: 1 µs
• Number of pulse duration increment: 20
• Rotor spinning speed: 15 kHz
• Quadrupole interaction: first- and second-orders
• Quadrupole coupling constant: 8 MHz
• Asymmetry parameter: -1 or 1
• Number maxα of summation steps of the Euler angle α of the rotor: variable
• Number maxβ of summation steps of the Euler angle β of the rotor: 3
• Number maxγ of summation steps of the Euler angle γ of the rotor: 3

### (A) Mathematica-5 notebook

1. Download the Mathematica-5 notebook called powder_MAS.nb or the notebook as PDF file powder_MAS.pdf (53 Kb)
2. Save this file into the software Mathematica 5 folder.
3. Open this file with Mathematica-5 and change the value of the asymmetry parameter.
4. Press "Ctrl-A" to select the notebook, then press "Shift-enter" to start the simulation.
5. A file called powderMAS.m is created in Mathematica-5 folder. MS Excel can open this file.

### (B) Result

The simulated line intensities are gathered in the following table. The two columns concerning maxα = 4 are identical.

t
(μs)
maxα = 2
maxβ = maxγ = 3
maxα = 3
maxβ = maxγ = 3
maxα = 4
maxβ = maxγ = 3
η = 1 η = -1 η = 1 η = -1 η = 1 η = -1
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
0
0.169246331
0.1247498304
-0.08616975267
-0.1710133737
-0.01113942073
0.1950035796
0.1548147068
-0.0489675948
-0.1595526638
-0.03961614394
0.1296392382
0.1491536129
-0.00958629567
-0.135343748
-0.08502827052
0.08936989872
0.130908683
0.01545694808
-0.07862456927
-0.04270830957
0
0.1643441541
0.09936157567
-0.100488862
-0.1545110357
0.04800325203
0.1782412882
0.06883221946
-0.134747615
-0.09579773525
0.08031402859
0.1456458828
0.00054177111
-0.1440298909
-0.08875314916
0.09126412118
0.1653669534
0.0272556453
-0.1459291249
-0.1133385316
0.1139546669
0
0.1724235685
0.1265569597
-0.08405763364
-0.1562376425
0.004517185206
0.165236943
0.1017903497
-0.08146709562
-0.1258229037
0.02332926274
0.1389571002
0.07368661782
-0.0615853822
-0.1012769246
0.007561202221
0.1189730994
0.0624802958
-0.07817301408
-0.08817115105
0.04068766838
0
0.1705219413
0.1230602849
-0.04796327787
-0.1171361231
0.002838116869
0.1058729311
0.06948357717
-0.06533060449
-0.07889613371
0.04585908482
0.1179377052
0.0276966918
-0.08996199743
-0.06170846661
0.04056470382
0.1109755805
0.01268431046
-0.08958616147
-0.0741772447
0.08267432174
0
0.1667952426
0.112055703
-0.09332930736
-0.1627622047
0.01843191565
0.1866224339
0.1118234631
-0.09185760491
-0.1276751995
0.02034894232
0.1376425605
0.07484769199
-0.07680809329
-0.1120484486
0.00311792533
0.1273684261
0.07908216416
-0.0652360884
-0.09598155045
0.03562317864
0
0.1667952426
0.112055703
-0.09332930736
-0.1627622047
0.01843191565
0.1866224339
0.1118234631
-0.09185760491
-0.1276751995
0.02034894232
0.1376425605
0.07484769199
-0.07680809329
-0.1120484486
0.00311792533
0.1273684261
0.07908216416
-0.0652360884
-0.09598155045
0.03562317864

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