instrumentation primer for nmr - mit...3. explain why no nmr signal is usually detected when the...

Instrumentation Primer for NMR

P. J. Grandinetti

L’Ohio State Univ.

Jan. 10, 2018

1 Check out Terry Gullion’s ENC tutorial video link:***Basic Useful Circuits for NMR Spectroscopy***

2 Check out Kurt Zilm’s ENC tutorial link:Design, Care and Feeding of NMR Probes: A Tutorial

P. J. Grandinetti (L’Ohio State Univ.) Electronics Primer for NMR Jan. 10, 2018 1 / 48

https://www.youtube.com/watch?v=_Vhs-ZRN_i4

http://www.enc-conference.org/Portals/0/Probes_2011_Part_I.pps

Precessing tops

Precessing top

SpinAngular

Momentum

Precession Direction

Magnetic top in zero gravity precessingin a magnetic field

N

S

How do we measure precession frequency of a magnetic top?


SirMr. Michael Faraday (1791 - 1867)

“During his lifetime, he was offered a knighthood in recognition for his services toscience, which he turned down on religious grounds, believing that it was againstthe word of the Bible to accumulate riches and pursue worldly reward, and statingthat he preferred to remain ’plain Mr Faraday to the end’.” – Wikipedia, 2018


https://en.wikipedia.org/wiki/Michael_Faraday

How to measure precession frequency of a magnetic top?One approach is to exploit Faraday’s law of induction, discovered in 1831, whichtells us that a changing magnetic flux will induce a current in a surrounding loopof wire.

Faraday’s law of induction:

= −dΦdt

is the EMF and Φ is the magnetic flux.

Electromotive Force (EMF, i.e., voltage)induced in coil is related to change inmagnetic flux through the loop of wirewith time.

Φ(t) = ∫ B⃗(t) ⋅ da⃗

is surface attached to loop of wire.


http://farside.ph.utexas.edu/teaching/302l/lectures/node84.html

How to measure precession frequency of a magnetic top?Place a coil of wire of radius Rcoil around our spinning magnetic top to detect theprecession frequency.

N

S

Magnetic dipole vector of top changes with time according to

𝜇(t) = |𝜇| [sin𝜓 cos(𝜔t + 𝜉0) e⃗x + sin𝜓 sin(𝜔t + 𝜉0) e⃗y + cos𝜓 e⃗z]

|𝜇| is length of precessing vector, 𝜓 is angle between precessing vector and z-axis,𝜉0 is initial phase of precessing vector, and 𝜔 is angular precession frequency.


Advanced Exercise1. Show that the EMF induced in a loop of wire surrounding a precessing

magnetic dipole,

𝜇(t) = |𝜇| [sin𝜓 cos(𝜔t + 𝜉0) e⃗x + sin𝜓 sin(𝜔t + 𝜉0) e⃗y + cos𝜓 e⃗z]

is given by

x(t) = −dΦx(t)

dt= 𝜔

𝜇02Rcoil

|𝜇| sin𝜓 sin(𝜔t + 𝜉0)

Hint: Start with definition of magnetic flux and use Stoke’s Theorem,

Φ(t) = ∫ B⃗dip(t) ⋅ da⃗ = ∫(∇⃗ × A⃗dip(t)) ⋅ da⃗ = ∮ A⃗dip(t) ⋅ dl⃗

represents circumference of wire loop; A⃗dip(t) is magnetic vector potential forpoint dipole (see Griffiths 3rd Ed. E&M text, p. 244)

A⃗dip(r⃗) =𝜇04𝜋

𝜇 × e⃗r

r2 .


Faraday Detector of precessing magnetic dipole

x(t) = −dΦx(t)

dt= 𝜔

𝜇02Rcoil


From this signal we measure precession frequency.

Amplitude is directly proportional to magnetic dipole moment strength, |𝜇|.Signal scaled by sin𝜓 . The closer the magnetic dipole precesses to z-axis,the smaller the EMF signal.

Signal scaled by inverse of coil radius. EMF amplitude decreases withincreasing coil radius.

EMF amplitude increases with increasing precession frequency. Whenmagnetic top precesses in higher static magnetic flux densities we get higherprecessing frequencies and hence larger signal amplitudes.


Exercises2. Based on

x(t) = −dΦx(t)

dt= 𝜔

𝜇02Rcoil


calculate the increase in signal from doubling the external magnetic fieldstrength.

3. Explain why no NMR signal is usually detected when the long axis of theNMR receiver coil is parallel to the external magnetic field direction.


In 1820 Hans Christian Ørsted discovered that electric current produces amagnetic field that deflects compass needle from magnetic north, establishingfirst direct connection between fields of electricity and magnetism.


Biot-Savart Law

Jean-Baptiste Biot and Félix Savart worked out that magnetic flux densityproduced at distance r away from section of wire of length dl carrying steadycurrent is

dB⃗ =𝜇04𝜋

dl⃗ × r⃗r3 Biot-Savart law

Direction of magnetic field vector is given by “right-hand” rule: if you pointthumb of your right hand along direction of current then your fingers will curl indirection of magnetic field.

current


Calculate magnetic field produced by current in wire loop

Magnetic field along z axis away fromcurrent loop

B⃗(z) = e⃗z𝜇04𝜋

cos 𝜃R2

coil + z2 ∫ d𝓁

= e⃗z𝜇04𝜋

R2coil

(R2coil + z2)3∕2

Magnetic field at center of current loop

B⃗(0) = e⃗z𝜇04𝜋

1Rcoil

Magnetic field strength at center scaledby inverse of coil radius.


Let’s build an NMR Spectrometer!

RadioFrequency

Source

TransmitterSwitch

Output ofFreq. Source is

Continuous

N S

ReceiverSwitch

Sample Receiver

6 essential components in our primitive NMR spectrometer:

1 a radio frequency (rf) source tuned to resonance frequency of nuclei2 a switch (or gate) for turning rf irradiation on and off3 a magnet to polarize and split nuclear spin energy levels4 a transmitter and detector coil containing sample5 a switch (or gate) in front of receiver for protection6 receiver, which could simply be oscilloscope in this design


Let’s build an NMR Spectrometer!Radio

FrequencySource

TransmitterSwitch


Continuous

N S

ReceiverSwitch

Sample Receiver

Construction of such an instrument is straightforward.

Because time scale of NMR experiment is on order of microseconds,switching times for gates need to be on the order of nanoseconds for precisetime resolved measurements.

Computer controlled low power radio frequency gates having such switchingspeeds are readily available commercially. Check out link:www.minicircuits.com

In primitive spectrometer computer controls timing for opening and closingof transmitter and receiver gates. Check out link: Arduino boards


https://www.minicircuits.com/WebStore/Switches.html

https://www.arduino.cc

A Primitive NMR SpectrometerRadio

FrequencySource

TransmitterSwitch


Continuous

N S

ReceiverSwitch

Sample Receiver

Simplest experiment—pulse and detect signal—consist of 3 steps:

Step Transmitter switch state Receiver switch state Duration1 OFF OFF 30 seconds2 ON OFF 4 microseconds3 OFF ON 100 milliseconds

Elementary version of “Pulse Sequence” or “Pulse Program”Important part of pulse sequence is duration of each step, or event.NMR spectrometers have many more computer controlled switches thanreceiver and transmitter switches.NMR spectrometers have pulse sequence languages to write sequencescontaining loops, if statements, and other possibilities.


The NMR ProbeIn spectrometer coil wrapped around sample is called “transceiver coil”.

1 Used to produce oscillating B1 fieldthat rotates magnetization

2 Used as Faraday detector ofprecessing magnetization afterpulse

Let’s examine what is needed toenhance efficiency of this coil inproducing B1 fields.

These same changes, in turn, willalso enhance efficiency of this coilas a detector.

We start by reviewing the basics about voltages and currents in electroniccomponents like resistors, capacitors, and inductors.


The ResistorApply oscillating voltage (t) = 0 cos𝜔t

to a resistor in the circuit below

Oscillating current across resistor is given by

(t) = (t)R

=0R

cos𝜔t = 0 cos𝜔t

Maximum current through resistor, 0, related to 0 by Ohm’s law: 0 =0R


The InductorApply same oscillating voltage to inductor (like our transceiver coil)

Oscillating current across inductor is given by

(t) = ∫t

t0

(t)L

ds =0𝜔L

sin𝜔t = 0 sin𝜔t

Current oscillation is −90◦ out of phase with voltage oscillation.

(t) = 0 cos𝜔t and (t) = 0𝜔L

cos(𝜔t − 𝜋∕2) = 0 cos(𝜔t − 𝜋∕2)

Ignoring phase difference, 0 is related to 0 according to 0 =0𝜔L


The Inductor0 =

0𝜔L

Inductor behaves like frequency dependent resistance of 𝜔L, with currentoscillation −90◦ out of phase with voltage oscillation. Because it is 90◦ out ofphase it is called reactance (not resistance).Low frequency currents pass through inductors and high frequencies are blocked.

X L =

ωL

in out

ω

in out

Here you see problem of using inductor alone as transceiver coil.As receiver coil, oscillating current from EMF, (t), due to precessingmagnetization decreases rapidly as precession frequency increases.As a transmitter of oscillating B1 field, need to push higher currents throughcoil to get same B1 field at higher frequencies.For fixed oscillating voltage source (i.e., fixed power), current decreases withincreasing frequency, and so does B1 field strength.


The Inductor - Practical Note

Estimate inductance of coil from its dimensions. Inductance, L, in 𝜇H is

L = r2n2

9r + 10l

r is radius of coil in inches,

n is number of turns,

l is coil length in inches.

Quality factor or Q factor of inductor at operating frequency 𝜔 is defined as ratioof reactance of coil to its resistance

Q = 𝜔LR

Optimum Q is attained when the length of the coil (l) is equal to its diameter (2R)


The CapacitorApply oscillating voltage to capacitor

Oscillating current across inductor is given by

(t) = Cd(t)

dt= −C𝜔0 sin𝜔t = 0 sin𝜔t

1 Current oscillation is 90◦ out of phase with the Voltage oscillation

(t) = 0 cos𝜔t and (t) = C𝜔0 cos(𝜔t + 𝜋∕2) = 0 cos(𝜔t + 𝜋∕2)

2 Ignoring phase difference, 0 is related to 0 according to 0 =0

1∕(𝜔C)P. J. Grandinetti (L’Ohio State Univ.) Electronics Primer for NMR Jan. 10, 2018 20 / 48

The Capacitor

Capacitor behaves like a frequency dependent reactance of 1∕(𝜔C) with currentoscillation 90◦ out of phase with voltage oscillation.

Capacitor blocks (reacts against) low frequency currents, but allows highfrequencies to pass—the opposite behavior of inductor.

X c =

1/(ω

C)

ω

in out

in out


Complex Voltage and CurrentRelationships between maximum current and maximum voltage across a resistor,capacitor, and inductor

0 =0R

(resistor), 0 =0

1∕(𝜔C)(capacitor), and 0 =

0𝜔L

(inductor)

Need relationship for current and voltage at all times, not just for maximumvalues; and includes the phase information.Defining complex voltage c(t) = 0ei𝜔t, where actual voltage is real part,

(t) = ℜ{0ei𝜔t} = 0 cos𝜔t

Similarly, define complex current c(t) = 0ei𝜔t, where actual current is real part

(t) = ℜ{0ei𝜔t} = 0 cos𝜔t

To phase shift (t) by 90◦ multiply complex voltage c(t) by ei𝜋∕2,

(t) = ℜ{0ei𝜔tei𝜋∕2} = 0 cos(𝜔t + 𝜋∕2)


Capacitor Impedance

Applying this approach to current-voltage relationship for capacitors we write

(t) = 01∕(𝜔C)

cos(𝜔t + 𝜋∕2) = ℜ{c(t)ei𝜋∕2

1∕(𝜔C)

}Since ei𝜋∕2 = i we can simplify to

(t) = ℜ{ c(t)

−i∕(𝜔C)

}= ℜ

{c(t)ZC

}where ZC = −i∕(𝜔C) is the impedance of the capacitor.In terms of complex current across capacitor we write

c(t) =c(t)ZC

Obtain familiar Ohm’s law, describing relationship between current and voltage atall times, but now it includes phase information.


Inductor Impedance

Applying this approach to current-voltage relationship for inductors we find

(t) = ℜ{c(t)

i𝜔L

}

In terms of complex current across inductor we write

c(t) =c(t)ZL

where ZL = i𝜔L is the impedance of the inductor.


Impedance, Z

Z = R + iX

R is Resistance: impedes current flow from collisional processes and dissipatesenergy as heat. Analogous to friction.

X is Reactance: impedes current from changing electric and magnetic fieldsassociated with alternating currents. It is not associated with power dissipation.Analogous to inertia.

ZR = R for resistors, ZC = −i∕(𝜔C) for capacitors, and ZL = i𝜔L for inductors

Ohm’s law can be generalized to include inductors and capacitors.

For components in series we have

ZT = Z1 + Z2 + Z3 +⋯

For components in parallel we have

1∕ZT = 1∕Z1 + 1∕Z2 + 1∕Z3 +⋯


A Tuned CircuitReady to solve our problem with transceiver coil. To make problem more realisticinclude wire resistance, Rt, as in circuit below.

transceivercoil

Total impedance is

ZT = ZR + ZL = Rt + i𝜔L

Magnitude of impedance is

|ZT | = √ZTZ∗

T =√

R2t + 𝜔2L2

60

80

100

120

140

160

180

200

500 1500 2500 3500 4500ω/2π

|ΖΤ|

Only at 𝜔 = 0 do we have lowest impedance andhighest current.NMR signal oscillates at megaHertz frequenciesso this is not optimal.


A Tuned Circuit

We can solve this problem by adding a capacitor in series as shown below.

transceivercoil

Total impedance is

ZT = ZR+ZL+ZC = R+i(𝜔L − 1

𝜔C

)Set ZL = ZC, so 𝜔0 = 1∕

√LC

and then ZT = RHighest current amplitude at 𝜔0.

60

80

100

120

140

160

180

200

500 1500 2500 3500 4500ω/2π

|ΖΤ|


A Tuned and Matched Circuit

NMRProbe

rfxmitter

ZT = Rs + Rt + i(𝜔L − 1

𝜔C

)at resonant frequency 𝜔0 = 1∕

√LC

ZT = Rs + Rt

Current is

(𝜔0) =(𝜔0)Rs + Rt

Power in the coil is

P(𝜔0) = 2(𝜔0)Rt =2(𝜔0)Rt

(Rs + Rt)2

Want maximum power transfer fromtransmitter to coil.

dP(𝜔0)dRt

=(𝜔0)2

(Rm + Rt)2−

2(𝜔0)2Rm

(Rm + Rt)3= 0

Solving this expression gives thecondition Rt = Rs for the maximumpower transfer.

When Rt = Rs we say that theimpedance of the tuned circuit is“matched” to the transmitter’simpedance.


A Tuned and Matched Circuit

Transmitter’s impedance (resistance) isusually fixed at Rs = 50 Ω.

Rt is generally less than 1 Ω.

How do we match impedances? Addanother capacitor to circuit.

NMRProbe

rfxmitter

In this series tuned, parallel matched circuit probe impedance is

1ZT

= i𝜔Cm + 1

Rt + i(𝜔L − 1

𝜔Ct

)If Ct and Cm are adjusted so impedance is completely real (no imaginary part)and equal to ZT = 50 Ω at frequency 𝜔0 then we have maximum power transferbetween transmitter and sample, or also between sample and receiver.


Another Tuned and Matched Circuit

rfxmitter

NMRProbe

To learn more...1 Check out Terry Gullion’s ENC tutorial video link:

Basic Useful Circuits for NMR Spectroscopy





Jupiter: self-assembled


The DuplexorOur spectrometer needs to switch probe between transmitter and receiver toprotect receiver from high power rf pulse of transmitter called the duplexor.

RadioFrequency

Source

TransmitterSwitch

N S

ReceiverSwitch

Receiver

DuplexorSwitch

dB

High PowerAmplifier

Because duplexer is switching between high power rf from transmitter andlow power rf from probe there is an inexpensive and simple passive circuitthat can be used to rapidly perform this switching.

To understand how this works need to review 2 important devices:(1) the cross diodes, and (2) quarter-wave transmission lines.


Diode

Diode symbol:

Diodes is one-way valve for current(i.e. direction of arrow).

Current Flow

No Current Flow

Plot of current flow versus appliedvoltage for diode

AppliedVoltage

CurrentFlow

20 mA

10 mA

-1 μA

-2 μA

1 v 2 v

-100 v -50 v

Note axes ranges.For diode current flows forward aftervoltage of greater half a volt is applied.


Cross-Diodes

Two diodes connected antiparallel

=Symbol

for Cross diodes

Cross-diodes have property that currentflows in either direction as long asvoltage is greater than half a volt inmagnitude.

AppliedVoltage

CurrentFlow

20 mA

10 mA

-10 mA

-20 mA

1 v 2 v

-2 v -1 v


Cross-Diodes in NMRUse cross-diode to protectreceiver from high power rf pulse

Receiver50 Ω

fromprobe

High voltage pulse would turndiodes on and go to groundinstead of going into receiverwith its 50 Ω impedance.

Use cross-diode to block low power noise fromtransmitter from entering probe.

Transmitter50 Ω

toprobe

Broadband noise from transmitter caneasily saturate NMR signal and needs to beeliminated.

As long as noise voltage doesn’t exceedthreshold voltage of diode it will be blockedfrom going to probe.

When signal-to-noise ratio unexplainably drops it is often a blowncross-diode that is problem.

Solid-state NMR experiments which use long high power pulses such ascross-polarization are often responsible for blown out diodes.


Transmission linesImpedance of all devices (i.e. probes, transmitter, receiver) need to be matchedfor maximum power transfer.Connect all these devices together making sure impedance is 50 Ω everywhere.

Transmitter50 Ω

Load(Probe)

50 Ωtransmissionline

(usually a coaxial cable)If transmitter impedance

is 50 Ω then current and voltage oscillations will be in-phase

Will load (Probe) see 50 Ω impedance? Will the load see current and voltage oscillations

in phase?

To match impedance of source, Zs, and load, Zl, the characteristic impedance oftransmission line, Z0, must be

Z0 =√

ZsZl =√

50 ⋅ 50 = 50 Ω

If transmission line is terminated by load that doesn’t match its characteristicimpedance then voltage and current waves are partially reflected and standingwaves are set up in transmission line.


Quarter-Lambda lines - Shorted EndTerminate transmission line with short toground, that is, connect inner conductorto outer conductor

OuterConductor

InnerConductor

End shorted sothat Zl = 0 Ω

Voltage is zero and current is maximumat shorted-to-ground end.

All voltage and current oscillation willreflect and set up a standing wave.

Transmitter50 Ω

λ/4

xV(x)

xI(x)

Current is zero and voltage is maximum at 𝜆∕4 away from shorted-to-groundend—where source is connected.Cable impedance at point where source is connected looks like

Z = (0)(0) =

00 amps

= ∞ Ω

Transmitter can’t tell difference between nothing and 𝜆∕4 with shorted endconnected.


Quarter-Lambda lines - Open EndWhat if we didn’t short the end, but left the two conductors unconnected?

Transmitter50 Ω

λ/4

xV(x)

xI(x)

Voltage is maximum, and current is zero at open end.Current is maximum and voltage is zero at 𝜆∕4 away from open end—wheresource is connected.Cable impedance at point where source looks like

Z = (0)(0) = 0 volts0

= 0 Ω

All transmitter power is being sent to ground when 𝜆∕4 with an open end isattached.


Quarter-Lambda lines – Summary

𝜆∕4 length cables with shorted ends look like an infinite impedance at thesource.

𝜆∕4 length cables with open ends look like a zero impedance (short toground!) at the source.

Counter intuitive if you’ve only ever thought about DC circuits.


Duplexor Circuit

Duplexer switches probe between transmitter and receiver. Consider circuit below:

Transmitter50 Ohms

Probe50 Ohms

Receiver50 Ohms

λ/4

High voltage pulse from transmitter turns on all cross-diodes and transmitter sees

Transmitter50 Ohms

Probe50 Ohms

Receiver50 Ohms

λ/4PulseOn

At tee (marked with dot) transmitter sees 50 Ω load of probe, and infinite load infront of receiver. No rf pulse goes into receiver.


Duplexor CircuitWhen transmitter is off, then weak NMR signal from probe cannot turn on diodes:

Transmitter50 Ohms

Probe50 Ohms

Receiver50 Ohms

λ/4

All cross-diodes are off so probe signal goes only to receiver.

Cross diode also blocks noise from transmitter from reaching probe.

Since 𝜆∕4 length depends on frequency (i.e. 𝜆 = c∕𝜈) then any time youchange NMR frequency (i.e. when changing to different nucleus), duplexerhas to be changed (different 𝜆∕4 cable).

With wrong 𝜆∕4 length, then part of transmitter power will go to groundand not to probe.

If cross-diodes are blown (current flows in both directions with no resistancein blown diodes), then transmitter noise will saturate magnetization andsignal from probe will not go only to receiver and sensitivity will suffer.


Specifying Power Levels

Power is energy transfer per unit time.

P =2

rmsR

and rms =pp

2√

2

rms is rms voltage and pp is peak to peak voltage.

ExampleCalculate power from 50 Ω rf source with output of pp = 0.632 V

rms =pp

2√

2= 0.2236 V

2√

2= 0.2236 V

P = (0.2236 V)2

50Ω= 0.001 W or 1 mW.


Amplifier Gain

Amplifier gain given in decibels (dB). Logarithmic scale calculated according to

dB = 20 log10(pp)out(pp)in

If amplifier input is pp = 0.632 V then after 50 dB gain we get

(pp)out = (pp)in ⋅ 10dB∕20 = (0.632 volts) ⋅ 1050∕20 = 200 V

After 50 dB amplifier, 1 mW would be amplified to

P =(200 V∕(2

√2))2

50 Ω= 100 W.


Checking Power LevelsRF levels are also specified in units of dBm, given by

dBm = 10 ⋅ log P(mW)1 mW

dBm is gain in terms of dB’s with respect to 1 mW.

If rf source outputs 1 mW, then power in dBm is zero.

A 50 dB amplifier turns 0 dBm into 50 dBm—which is 100 W.

Amplifiers have a maximum input level.

Anything higher will overdrive amplifier and lead to distorted output (higherharmonics added).

Important to check rf power levels going into probe and make sure they arewithin specifications.

Never connect the output of a high power amplifier directly to theoscilloscope. Oscilloscope may not handle that much power and can bedamaged.


Checking Power Levels

Place high power attenuator (check power rating)between amplifier and oscilloscope.

50 dB 30 dB Attemuator(high power rating)

high powerrf attenuator

With setup above

1 measure voltage peak-to-peak on oscilloscope,

2 convert this to dBm,

3 add 30 dB to get output power level of amplifier.


Make Calibration Plot for Probe

Power/watts

Measure NMR signalas a function of

pulse length

Measure powergoing into probe


Let’s build an NMR Spectrometer!

N S

Transmitter ReceiverDuplexor

Probe

Others,e.g. gradients,temperaturecontrol, etc...

Computer


Further Reading

Transient Techniques in NMR of Solids: An Introduction to Theory andPractice, by Gerstein and Dybowski

Experimental Pulse NMR: A Nuts and Bolts Approach, by Fukushima andRoeder

The ARRL Handbook for Radio Communication

Radio-Frequency Electronics, Circuits and Applications, by Hagen

Also...

1 Check out Terry Gullion’s ENC tutorial video link:***Basic Useful Circuits for NMR Spectroscopy***





instrumentation primer for nmr - mit...3. explain why no nmr signal is usually detected when the...

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