DFT Studies of Temperature Effects in Nuclear …majewski/letters/dft-bayer.pdfJuly 2016 Allen...

DFT Studies of Temperature Effects in NuclearQuadrupole Resonance

Allen Majewski

Department of PhysicsUniversity of Florida

July 2016

Allen Majewski (University of Florida) DFT Studies of Temperature Effects in Nuclear Quadrupole ResonanceJuly 2016 1 / 36

Outline

1 NQR in the static latticeThe quadrupole interactionThe electric field gradient and the quadrupole copuling constantNQR transition frequencies

2 T-dependence of NQR frequenciesThe Bayer-Kushida (BK) model

3 Failures of the modelVolume dependence of NQRDensity functional theory for volume correction and moleculardynamics

Outline

NQR in the static latticeThe quadrupole interaction

Nuclear quadrupole interaction: splitting of the nuclear spin energylevels due to interaction of the nuclear electric quadrupole moment Qwith the external potential φ (~r).

φ (~r) is the (periodic) potential everywhere, provided by the chargedensity of the solid.

Q is the electric quadrupole moment of the nucleus under study. Allisotopes with spin ≥ 1 have one.

eQ =∫

(3z2 − r2)ρ(x)d3x

picture of lattice, φ (~r) ...

eQ =∫

(3z2 − r2)ρ(x)d3x

eQ =∫

(3z2 − r2)ρ(x)d3x

eQ =∫

(3z2 − r2)ρ(x)d3x

eQ =∫

(3z2 − r2)ρ(x)d3x

Outline

NQR in the static latticeThe electric field gradient

The nuclear quadrupole moment of a nucleus Qij = Q ineracts withthe gradient of the electric field gradient produced by the surroudingcharges

Uq = 16Qi j (∇E)ij

That’s the second spatial derivative of the potential due to thecharges surrouding the nucleus.

(∇E)ij = ∂2φ∂xi∂xj

= φij

We diagonalize the EFG tensor and are left with only φxx , φyy , φzznon-zero

X, Y, Z are the principal axes of the EFG at the nucleus.

= φij

NQR in the static latticeEFG at nuclear site

NQR in the static latticeThe quadrupole coupling constant Cq

Laplace’s equation: φxx + φyy + φzz = 0 → only two independentparameters describing the interaction.

letting eq = φzz = ∂2φ∂z2 , the coupling constant Cq and asymmetry

parameter η are defined as follows

Cq: quadrupole coupling constant

Cq = e2qQ~ = eφzz

η: asymmetry parameter

η =φyy − φxx

∂2φ∂y2 − ∂2φ

∂2φ∂z2

Cq = e2qQ~ = eφzz

η =φyy − φxx

∂2φ∂y2 − ∂2φ

∂2φ∂z2

Cq = e2qQ~ = eφzz

η =φyy − φxx

∂2φ∂y2 − ∂2φ

∂2φ∂z2

Cq = e2qQ~ = eφzz

η =φyy − φxx

∂2φ∂y2 − ∂2φ

∂2φ∂z2

Outline

NQR in the static latticePure quadrupole transition frequendcies

NQR frequencies are proportional to the principal EFG componentq = φzz

e times a factor involving η.

The NQR transition frequencies also depend on the isotope spin.Typical nuclei are 35Cl having spin 3

2 and 14N with spin 1

NQR frequencies are proportional to the principal EFG componentq = φzz

e times a factor involving η.

The NQR transition frequencies also depend on the isotope spin.Typical nuclei are 35Cl having spin 3

2 and 14N with spin 1

NQR transition frequency for nuclear spin I = 32

There is only one transition in this case

ν = 12e2qQ~

(1 + η2

= 12Cq

(1 + η2

NQR transition frequencies for nuclear spin I = 1

If η is not zero, there are three transitions.

νq =3

~(1± η

4Cq(1± η

3) (2)

recalling:

Cq =e2qQ

eφzz~

; η =φyy − φxx

ν = 12e2qQ~

(1 + η2

= 12Cq

(1 + η2

νq =3

~(1± η

4Cq(1± η

3) (2)

recalling:

Cq =e2qQ

eφzz~

; η =φyy − φxx

ν = 12e2qQ~

(1 + η2

= 12Cq

(1 + η2

νq =3

~(1± η

4Cq(1± η

3) (2)

recalling:

Cq =e2qQ

eφzz~

; η =φyy − φxx

Summary

NQR is like NMR except the transition frequency is controlled byelectric field gradient φij at the nuclear site.

Static Lattice: if we do not consider atomic motion (essentially, T =0K), NQR frequencies are determined entirely by the local electricfield gradient

NQR prediction <=> knowldge of a precise crystal structure andφ (~r) everywhere in it.

Description of NQR T-dependence requires consideration of internalmotions in the lattice.

Therefore the model described cannot address T-dependence.

Summary

Outline

T-dependence of NQR frequenciesaffects of internal motions

NQR frequencies have a sharp T-dependence

The static lattice theory discussed does not and cannot address theT-dependence

The T-dependence is a result of internal motions in the solid

T-dependence of NQR frequenciesHorst Bayer was the first to address NQR T-dependence in 1951

T-dependence of NQR frequenciesLattice vibrations: the Bayer-Kushida (BK) model

Bayer was first to adress NQR t-dependence. Kushida followed in1956.

He began by calculating the average EFG experienced by a nucleusundergoing harmonic motion about its principle axes.

Considering small angular displacements θx , θy , θz of a quadrupolarnucleus about the principal field gradient axes

Show image from Das, Hahn

Molecular motions are generally much faster than NQR frequencies →motions shouldn’t affect resultant EFG

However a quadrupole undergoing motion will experience an effectiveEFG resulting from the average EFG smeared over its oscillatorymotion

The effective EFG experienced by the nucleus in motion is smaller inmagnitude than that of the static lattice

Consider small rotations θx , θy , θz sendint the coordinates to theprimed system

Want to relate the efgs in the two coordinate systems, e.g. relate φijto φi ′j ′ = φ′ij

Consider small rotations θx , θy , θz sendint the coordinates to theprimed system

Want to relate the efgs in the two coordinate systems, e.g. relate φijto φi ′j ′ = φ′ij

Figure : Bayer considered small rotations θx , θy , θz about each principle EFG axis

Including terms up to second order in θx ,θy ,θz :

φ′xx =(1− θ2

y − θ2z

)φxx + θ2

zφyy + θ2yφzz

φ′yy = θ2zφxx +

(1− θ2

x − θ2z

)φyy + θ2

xφzzφ′zz = θ2

yφxx + θ2zφyy +

(1− θ2

x − θ2y

φ′xy = θzφxx + (θxθy − θz)φyy − θxθyφzzφ′yz = −θyθzφxx + θxφyy + (θyθz − θx)φzzφ′zz = −θyφxx − θxθzφyy +

(1− θ2

x − θ2y

Since 〈θx〉 = 〈θy 〉 = 〈θz〉 = 0 → off diagonal primed EFG componetsvanish

〈θ2x〉, 〈θ2

y 〉,〈θ2z 〉 are all positive however

φ′xx =(1− θ2

y − θ2z

)φxx + θ2

zφyy + θ2yφzz

(1− θ2

x − θ2z

)φyy + θ2

xφzzφ′zz = θ2

yφxx + θ2zφyy +

(1− θ2

x − θ2y

(1− θ2

x − θ2y

〈θ2x〉, 〈θ2

φ′xx =(1− θ2

y − θ2z

)φxx + θ2

zφyy + θ2yφzz

(1− θ2

x − θ2z

)φyy + θ2

xφzzφ′zz = θ2

yφxx + θ2zφyy +

(1− θ2

x − θ2y

(1− θ2

x − θ2y

〈θ2x〉, 〈θ2

Massaging, φ′zz and η′ are

φ′zz = φzz[1− 3

(〈θ2

x〉+ 〈θ2y 〉)− 1

2η(〈θ2

x〉 − 〈θ2y 〉)

2(3− η) 〈θ2

x〉〈θ2y 〉]

η′ =φzzφ′zz

[η− 3

(〈θ2

x〉 − 〈θ2y 〉)− 1

2η(〈θ2

x〉+ 〈θ2y 〉)

2(3− η) 〈θ2

x〉〈θ2y 〉]

Clearly φ and η are effectively reduced since 〈θx〉, θy 〉 are positive andthe fourth order term is small.

Image of NQR falling off with T

(〈θ2

x〉+ 〈θ2y 〉)− 1

2η(〈θ2

x〉 − 〈θ2y 〉)

2(3− η) 〈θ2

x〉〈θ2y 〉]

η′ =φzzφ′zz

[η− 3

(〈θ2

x〉 − 〈θ2y 〉)− 1

2η(〈θ2

x〉+ 〈θ2y 〉)

2(3− η) 〈θ2

x〉〈θ2y 〉]

(〈θ2

x〉+ 〈θ2y 〉)− 1

2η(〈θ2

x〉 − 〈θ2y 〉)

2(3− η) 〈θ2

x〉〈θ2y 〉]

η′ =φzzφ′zz

[η− 3

(〈θ2

x〉 − 〈θ2y 〉)− 1

2η(〈θ2

x〉+ 〈θ2y 〉)

2(3− η) 〈θ2

x〉〈θ2y 〉]

T-Dependence of NQRBK model: details

The model goes on to replace each 〈θ2i 〉 with harmonic oscillators

summed over all the modes of the lattice

1Aiω2i 〈θ2

i 〉 = ~ω(

12 + 1

e~ωi/kT−1

Assume further we can discard terms in η in the expression of φ′zz

(〈θ2

x〉+ 〈θ2y 〉) ]

We can express the NQR frequency νq preliminaryily as a function oftermpurature and a sum over all N lattice modes of vibration

νq (T ) = ν0

[1− 3

N∑i=1

ω2i~ω(

12 + 1

e~ωi/kT−1

1Aiω2i 〈θ2

i 〉 = ~ω(

12 + 1

e~ωi/kT−1

)Assume further we can discard terms in η in the expression of φ′zz

(〈θ2

x〉+ 〈θ2y 〉) ]

νq (T ) = ν0

[1− 3

N∑i=1

ω2i~ω(

12 + 1

e~ωi/kT−1

1Aiω2i 〈θ2

i 〉 = ~ω(

12 + 1

e~ωi/kT−1

(〈θ2

x〉+ 〈θ2y 〉) ]

νq (T ) = ν0

[1− 3

N∑i=1

ω2i~ω(

12 + 1

e~ωi/kT−1

1Aiω2i 〈θ2

i 〉 = ~ω(

12 + 1

e~ωi/kT−1

(〈θ2

x〉+ 〈θ2y 〉) ]

νq (T ) = ν0

[1− 3

N∑i=1

ω2i~ω(

12 + 1

e~ωi/kT−1

T-Dependence of NQRBK model: expansion of exponential

Suppose we are interested in some temperature T .

Consider only the first M < N modes which satisfy ~ωi ≤ kT

Now expand

12 + 1

e~ωi/kT−1= 1

x + x12 +O

The expansion is very accruate; only off by a few percent when x = 2

Now expand

12 + 1

e~ωi/kT−1= 1

x + x12 +O

Now expand

12 + 1

e~ωi/kT−1= 1

x + x12 +O

Now expand

12 + 1

e~ωi/kT−1= 1

x + x12 +O

T-Dependence of NQRBK model: obtaining fitting parameters

Substituting the expansion into the expression for ν (T ) we find

νq (T ) = ν0

[1− 3

N∑i=1

ω2i~ω(

12 + 1

e~ωi/kT−1

After hours and hours, we finally arrive at

ν (T ) = a + bT + c/T

a = ν0

b = −32kM〈

Aω2 〉

c = ~2

8kM〈A〉

νq (T ) = ν0

[1− 3

N∑i=1

ω2i~ω(

12 + 1

e~ωi/kT−1

)]After hours and hours, we finally arrive at

ν (T ) = a + bT + c/T

a = ν0

b = −32kM〈

Aω2 〉

c = ~2

8kM〈A〉

νq (T ) = ν0

[1− 3

N∑i=1

ω2i~ω(

12 + 1

e~ωi/kT−1

ν (T ) = a + bT + c/T

a = ν0

b = −32kM〈

Aω2 〉

c = ~2

8kM〈A〉

νq (T ) = ν0

[1− 3

N∑i=1

ω2i~ω(

12 + 1

e~ωi/kT−1

ν (T ) = a + bT + c/T

a = ν0

b = −32kM〈

Aω2 〉

c = ~2

8kM〈A〉

νq (T ) = ν0

[1− 3

N∑i=1

ω2i~ω(

12 + 1

e~ωi/kT−1

ν (T ) = a + bT + c/T

a = ν0

b = −32kM〈

Aω2 〉

c = ~2

8kM〈A〉

Failures of the modela constant volume theory

Model includes no volume dependence:

the EFG depends on volume

the phonon frequencies depend on volume

Outline

img/TNT-molecule.png

.png .pdf .jpg .mps .jpeg .jbig2 .jb2 .PNG .PDF .JPG .JPEG .JBIG2 .JB2

Outline

Summary

NQR spectroscopy is typically performed for various T at constantpressure.

The standard method of extrapolating T-dependence is the BK model.This is a constant volume theory → ignores thermal expansion.

Density functional theory calculatons of NQR frequencies canalleviate the problem by a direct calculation of the EFG in first fittingparameter in the BK model.

EFG calculation with DFT may bolster the model → better NQRprediction and molecular dynamics studies of suitable solids.

Outlook

Something you haven’t solved.Something else you haven’t solved.

Summary

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Summary

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For Further Reading I

A. Author.Handbook of Everything.Some Press, 1990.

S. Someone.On this and that.Journal of This and That, 2(1):50–100, 2000.

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