Electron-Beam Lithography

Development of a Scintillating Reference Grid for

Spatial-Phase-Locked Electron-Beam Lithography

by

Mark Alan Finlayson

B.S.,University of Michigan(1998)

Submitted to the Department of Electrical Engineering and Computer Science

in partial ful?llment of the requirements for the degree of

Master of Science in Computer Science and Engineering

at the

MASSACHUSETTS INSTITUTE OF TECHNOLOGY

September2001

c Massachusetts Institute of Technology2001.All rights reserved. Author...................................................................................

Department of Electrical Engineering and Computer Science

August10,2001 Certi?ed by...............................................................................

Henry I.Smith

Keithley Professor of Electrical Engineering

Thesis Supervisor Accepted by..............................................................................

Arthur C.Smith

Chairman,Department Committee on Graduate Students

2

Development of a Scintillating Reference Grid for Spatial-Phase-Locked

Electron-Beam Lithography

by

Mark Alan Finlayson

Submitted to the Department of Electrical Engineering and Computer Science

on August10,2001,in partial ful?llment of the

requirements for the degree of

Master of Science in Computer Science and Engineering

Abstract

Spatial-phase-locked electron-beam lithography(SPLEBL)promises signi?cant improvement in the pattern ?delity and placement accuracy of electron-beam lithography by the introduction of feedbackinto the writing loop.In the planned formulation of SPLEBL,the feedbacksignal is produced by a reference grid which is bleached into a thin scintillating layer integrated into the electron-beam stack.Formulation and testing of a wide variety of material for use as the scintillating layer are described,in addition to a number of techniques to improve the contrast of the bleached grid.Production of?ducial gratings imaged in a SPLEBL-compatible electron beam system is described,and possible resist stacks that integrate the scintillator for use in SPLEBL are discussed.

Thesis Supervisor:Henry I.Smith

Title:Keithley Professor of Electrical Engineering

3

4

Acknowledgments

First,thanks to my advisor Professor Hank Smith.Without him I doubt I would have had the opportunity or desire to come to MIT and conduct graduate studies.He was gracious enough to extend to me an invitation to be a undergraduate researcher in his lab in the summer of1997,and since then has been extraordinarily generous in his postive regard and support.I am exceptionally grateful to him for the time and energy he has invested in me.

Second,thanks to Professors Roy Clarke,Donald Umstadter,and Duncan Steel at the University of Michigan,and Drs.Ted Bloomstein and Mordy Rothschild at MIT Lincoln Laboratories.These men were my mentors,and I count their encouragement and advice as crucial to my scienti?c development.

Third,thanks to my many colleages and collaborators at the Nanostructures Lab and MIT at large. Hindy Bronstein for guiding me in the kindest manner possible through my early forays into chemistry.Jim Daley for his clear and kind help,without which most of my e?orts would have been futile.Todd Hastings my close SPLEBL collaborator,for continually provided valuable insight and advice,as well as being one of the nicest guys I’ve ever met.Bernhard Vogeli for many incredibly instructive conversations and insights,as well as a lot of fun.James Goodberlet for being a source of many good ideas and suggestions.Euclid Moon for being indispensible font of information regarding anything computer,car,or electronic music related,and a willing ear for problems large and small.Mike Walsh for being an exceptionally entertaining(not to mention knowledgable and helpful)person.Jimmy Carter for helping out whenever needed.Maithilee Kunda,my RSI mentee,for being instrumental in the?nal round of data acquisition.Jaime Beers for being a great friend and a constant source of support.The many other members of the NSL for providing a rich,instructive,and fun environment in which to workand be productive:Solomon Assefa,Dave Carter,Maya F arhoud,Dario Gil,Joe Huang,Mike Lim,Mitch Meinhold,Rajesh Menon,Mark Mondol,Tom Murphy,Markus Peuker, Minghao Qi,Tim Savas,Oliver Stoltz,Shingo Uchiyama,and Feng Zhang.My many collaborators at MIT and elsewhere:Tim Long,Professor Tim Swager,Dr.Jason Hafner,Dr.Dave P?ug,and Professor Tayo Akinwande.

Fourth,thanks to my family and my many friends,girlfriends,and acquaintences here in Boston and in the world generally.They are too numerous to mention,but have made my life worth living.

Fifth,thanks to the greco-roman intellectual tradition,as well as western culture and civilization generally, for providing a setting in which neurotic obsession with analytical thinking would give rise to both modern society’s love a?air with science as well as a place like MIT.

Finally,thanks to all those that I have failed to mention through forgetfulness or willful neglect.

5

6

Contents

Abstract

3Acknowledgments

5

1Introduction &Motivation

131.1Planar Fabrication Techniques for Semiconductor Manufacturing ................131.2Electron-beam Lithography .....................................151.3Spatial-phase-locked Electron-beam Lithography .........................161.4The Scintillating Fiducial Grid ..........

.........

................172Overview of Scintillators

192.1Quantum Mechanics of Scintillating Organic Molecules ......................192.2Electron Interactions with Polymer ?lms ..............................222.3Energy Transfer in Plastic Scintilators ...............................232.4Photochemistry of Organic Fluors .................................242.5Solvation and Solvent Systems ...................................252.6Spin Casting and Baking Thin Polymer Films ...........................262.7Integrating the Scintillator into an Electron-beam Resist Stack

.................273Experimental Methods for Characterization of the Scintillators 313.1Preparation of Solutions .......................................313.2Spin Casting and Bak ing .......................................313.3Measurement of Scintillation ....................................323.4Area Bleaching with Ultraviolet Light ...............................363.5Bleaching &Imaging a Fiducial Grid ................................374E v

aluation of Scintillator Performance 394.1Standard Scintillating Systems ...................................394.2Novel Scintillating Systems .....................................424.3Survey Data on Performance of Systems ..............................434.4Detailed Data on Satisfactory Systems ...............................434.5Contrast Enhancement Schemes ...................................484.6Bleaching &Imaging a Fiducial Grid ..

.......

........

...............

49

5Summary 575.1F uture Work (57)

References 61

7

8

List of Figures

1-1The basic lithographic process (14)

1-2A basic MOSFET process (15)

1-3Semiconductor Industry Association Roadmap,MaskRequirements,1999[1] (15)

1-4Electron-beam lithography (16)

1-5Spatial-phase-locked electron-beam lithography (17)

1-6Interference lithography used to bleach the?ducial grid (18)

2-1Electron energy deposition per unit length versus path length inside a solid material (23)

2-2General energy pathway in a typical multi-component plastic scintillator (25)

2-3Possible SPLEBL stackcon?gurations (28)

3-1Schematic of a photomultiplier tube;adapted from[24,Figure9.1] (33)

3-2Response of the PMT installed on the LEO SEM;after data taken by J.G.Goodberlet[16].33 3-3Graph of Hamamatsu R6095photocathode response;after data from Hamamatsu datasheet for photomultiplier tubes R6094,R6095[21] (34)

3-4SEM mode used to probe for scintillation (35)

3-5Linear dependence of scintillation signal on beam current for di?erent accelerating voltages.35 3-6Spectrum of mercury lamp used for generic?ood exposures;data was taken with two di?erent gratings,and thus produced two spectrums which overlap from400to600nm (36)

4-1Standard polymers tested (40)

4-2Secondary scintillators tested (40)

4-3Primary scintillators tested (41)

4-4Polystyrene solutions excited byβ-particles;after[37,Figure1],with extraneous components omitted for clarity (41)

4-5p-Terphenyl grains formed during spin coating a Poly(vinyl toluene)?lm (42)

4-6Novel electronic polymers tested (43)

4-7Generic bleaching curve with?t equation (45)

4-8Various scintillator component emission spectra:polystyrene,poly(vinyl toluene),naphtha-lene,anthracene,p-terphenyl,and POPOP,along with PMT response(arb.units) (46)

4-9Bleaching of base polymers with mercury lamp (46)

4-10Bleaching of scintillating components with mercury lamp (47)

4-11Detailed data on scintillators#1,#5,#6and#20b,bleached with mercury lamp light (47)

4-12Detailed data on scintillators#1,#5,#6and#20b,bleached with351nm light (48)

4-13Rose Bengal singlet oxygen production and protonation reaction to deactivated form (49)

4-14Concept of the contrast enhancing layer as applied to bleaching scintillation (50)

4-15Bleachable diazonium salts and poly(N-vinyl pyrrolidone) (50)

4-16Diazonium salt absorbance bands at an arbitrary concentration,showing bleaching with in-creasing mercury lamp dose (51)

4-17Demonstrated CEL action (51)

4-18Bleaching of polymer bases with220nm light (52)

4-19Bleaching of naphthalene and anthracene with220nm light (52)

4-20Bleaching of p-terphenyl and POPOP with220nm light (53)

9

4-21Detailed data on scintillators#1,#5,#6and#20b,bleached at220nm (53)

4-22Detailed data on scintillators#1,#5,#6and#20b,bleached at351nm with an ARC (54)

4-235micron period gratings exposed with220nm light in scintillator#20b:(a)contact exposure,

(b)proximity exposure (55)

4-241micron period gratings exposed with351nm light:(a)scintillator#1,(b)scintillator#6,

(c)scintillator#20b (56)

10

List of Tables

2.1A selection of high-resolution electron-beam resists (28)

4.1Survey data of all scintillating systems tested (44)

4.2Contrasts of the best scintillators with mercury lamp and351nm bleaching (47)

4.3Contrasts of the best scintillators at220nm bleaching (49)

4.4Contrasts of the best scintillators at351nm bleaching,with and without an ARC (50)

4.5Final grating contrasts (54)

11

12

Chapter1

Introduction&Motivation

The computer is an extraordinarily powerful calculating tool.The speed and automation of its computa-tions have made possible the solution of problems that previously were too complex for numerical attack. This leverage the computer a?ords toward the processing of information has been applied across society to astounding e?ect,and as the device becomes smaller,cheaper,and faster,the scope of its application continuously increases.Indeed,so ubiquitous is the tool presently that its importance cannot overstated.

It is with visions of continuing the astounding increase of the computer’s scope and power that an extraordinary amount of time,money,and e?ort has been expended in improving the speed and reducing the size and cost of the technology.In the forefront of this e?ort is the thrust to improve the heart of the computer,the microchip.Therein lies the role of this thesis:it describes one facet of a technique which improves a core technology for microchip fabrication,and so?nds itself,from an applied point of view,to be a contribution to the e?ort to increase the computational power which our society has at its disposal. 1.1Planar Fabrication Techniques for Semiconductor Manufac-

turing

The general paradigm for the production of microchips was set dramatically in the late1950’s with the advent of planar semiconductor processing techniques.The advantage of the planar process is the speed with which it can produce high quality devices.Original electronic computers(before planar processes)made use of electronic gates called vacuum tubes:small enough to?t in the palm of one’s hand,they were painstakingly connected with wires in the complicated ways neccesary to fashion a computing device.The disadvantage with this technique was the amount of space required to house,and the amount of e?ort to construct the devices.Since computing power scales with the number of gates,early computers?lled large rooms.Planar production processes changed all this by making it possible to integrate,all at once,thousands of gates (now in the form of transistors)onto the surface of a thin silicon wafer.Before the planar revolution,each macroscopic gate was made inpidually and hooked together by human hands.After,microscopic gates were produced in parallel,thousands(and now millions)at a time.Thus the birth of the microchip.

The planar method is simple to understand.We will draw a quickanalogy to illustrate the principle. Suppose one has an acid which etches away a certain type of ceramic.Also suppose one has a?at dinner plate made out of this very sort of ceramic.If one wished to make a light relief pattern on the surface of this dinner plate(perhaps a pattern that would increase the aesthetic value of the plate),one might proceed as follows:take some wax resistant to the acid,and coat the plate with 1c0ec3e881c758f5f61f67dbing a stick or other appropriate scraping device,remove the wax in the desired pattern.Now dip the plate in the acid,wash o?the wax, and voila,the relief pattern is etched into the plate.Obviously the technique could be quite sophisticated, employing di?erent sorts of materials,acids,and waxes,di?erent methods for imprinting the wax with the pattern depending on the geometry of the piece,di?erent etching depths depending on how long you left the piece in the acid bath,etc.,etc.This artistic technique is called lithography.

Semiconductor lithography works much the same way.In this case the wax is replaced by a material called a photoresist(or just resist),which is generally a photosensitive polymer The dinner plate is replaced by a

13

Surface to be Patterned Spin

Expose Pattern

using Mask

Develop Photoresist

Etch Pattern

into Surface

Remove

Photoresist Figure1-1:The basic lithographic process

silicon wafer.The method for inscribing the pattern in the photoresist consists of projecting light through an optical mask(a stencil of the desired pattern),which then changes the chemical properites of the light-sensitive photoresist in such a way that the pattern can be developed in a solvent,much as a photographs are developed after exposure.Once the pattern has been developed in the photoresist,one proceeds with the acid dip and resist removal as usual.The pattern then has been transferred to the silicon.This process is illustrated by rough sketch in Figure1-1.

The particular transfer process described is termed a subtractive lithographic transfer process because it removes material from the surface of interest.Moreover,it is a wet process,because the etchant is a liquid. There are dry subtractive process which make use of,for example,a plasma to etch or sputter away the material of interest.There are also additive processes which add a layer of material to the surface of interest, both those that do and do not involve transferring a pattern.Coupled with a few other more specialized processing techniques(such as ion implantation),one has all the tools with which to make a transistor. An example process to make a metal-oxide-semiconductor?eld-e?ect transistor(MOSFET)is illustrated in Figure1-2.

Not only useful for making microchips,semiconductor processing techniques are applicable to the produc-tion of a wide range of interesting devices.Recent attention has focused on microelectromechanical systems (MEMS),integrated optoelectronics,and integrated optical waveguides and?lters.These technologies are found everywhere from presentation projectors to automobile airbag in?ation sensors to?beroptic telephone networks.

Once one has the ability to miniturize the transistor,one might ask:by how much do we miniturize it? The answer is“by as much as possible”[38].The smaller a transistor is,the faster they are able to switch, improving performance,and the more of them?t on a single wafer,lowering cost.

Currently the size of transistors is limited by the resolution of optical lithography.This resolution is set by the di?raction limit of light,in that features smaller than half the wavelength cannot be printed due to blurring caused by di?raction of the pattern.Consequently,to print smaller transistors industry is constantly moving toward shorter and shorter wavelengths.Eventually,it is likely that they will implement a mask-based next generation lithography(NGL)such as x-ray or deep-ultraviolet wavelengths,which will hand o?the size limitations to other factors[2].

14

p-Type

Single Crystal Silicon Wafer (for NMOS FETs)

Furnace

Oxidation

Etch Device

Window High-Quality

Furnace Oxidation of

Gate Oxide

Polysilicon

Deposition

Gate Etch

Ion

Implantation

of n-Type

Dopant

Oxide

Deposition

Etch

Contact

Windows

Deposition &

Etch of

Metal Contacts Figure1-2:A basic MOSFET process

1.2Electron-beam Lithography

One major hurdle in a mask-based next generation lithographic process would be the production of an error-free mask.Small feature sizes are irrelevant if feature placement is not well controlled–in other words, it hardly matters if you can print30nanometer features if you can’t put them where you want them to be within30nanometers.Indeed,for any particular critical feature size one wants to print,some small fraction of that size is usually required in placement accuracy to ensure proper device performance.Future requirements considered as a bare minimum for pattern placement are shown in the Semiconductor Industry Association roadmap(Figure1-3).

The current technique of choice for making lithographic masks is electron-beam lithography(EBL).Its operation is illustrated in Figure1-4.It is similar to optical lithography in that it uses a resist and deposits energy into that resist to change its development properties.For electron-beam lithography,however,the

Y EAR

T ECHNOLOGY N ODE

2008

70 n m

2011

50 n m

2014

35 n m

Lithography technology Optical NGL NGL NGL

I mage placement (nm, multi-point) 1515129

CD uniformity (nm, 3 sigma)

Isolated lines (MPU gates)10*753

Dense lines (DRAM half pitch)14*118 5.6

Contact/vias8129 6.4

L inearity (nm) 1075

C D mean to target (nm)654

Solutions Being Pursued No Kno w n Solutions

F igure1-3:Semiconductor Industry Association Roadmap,MaskRequirements,1999[1]

15

High Voltage (5 - 100 kV)

(less than

1 mm2)

Figure1-4:Electron-beam lithography

method of energy deposition is a computer-controlled electron beam,which writes features in a serial,rather than parallel,manner.Due to the nature of electron-beam optics the machine has only a small?eld of view,so a mechanical stage is usually required to move the substrate about(in a stop-and-go or continuous fashion)so as to access all areas on which patterns are to be written.After writing,the lithography is conducted normally,i.e.,by development and pattern transfer.

Electron-beam lithography su?ers from problems such as stage drift and electrical interference,inherent to its design.These can be problematic,as they a?ect the beam position on the sample after the computer calibration.Due to these inherent limitations,electron-beam lithography’s capabilities for pattern placement will be outstripped by the roadmap requirements by2008.

1.3Spatial-phase-locked Electron-beam Lithography

Spatial-phase-locked electron-beam lithography(SPLEBL)is today the only technique which promises to overcome these limitations[36].In its essence,the technique introduces feedbackinto the writing loop,as shown in Figure1-5),via the detection of the interaction between the electron beam and a reference grid

16

Figure1-5:Spatial-phase-locked electron-beam lithography

incorporated into the resist stack.As the electron-beam scans the wafer in the course of writing a pattern, the grid-beam interaction produces a periodic signal which is returned to the EBL computer.This allows for patterns to be directly registered to the grid in real-time.Ideally,production of such a highly accurate grid(the?ducial grid)is achieved via interferometric lithography(IL)[32].With SPLEBL,it is predicted that a pattern placement accuracy of less than one nanometer is achievable[15].

Key to the SPLEBL e?ort is the physical implementation of the?ducial grid.Several constraints govern the choice of a materials system and patterning process:First and foremost,the grid should be easily patterned,directly or indirectly,with interference lithography.This allows reference to the accurate standard of the wavelength of light.Second,electron-beam writing should proceed unmolested,and so the grid-beam interaction should not interfere with beam resolution or placement.Third,the grid signal must have a good signal-to-noise ratio and allow for fast transduction so as not to slow writing.Fourth,the grid must be compatible with subsequent processing(i.e.,pattern transfer).Fifth and?nally,as in any modi?cation of an already successful technique,a minimum of complexity is desirable.

1.4The Scintillating Fiducial Grid

It is likely there are many solutions which satisfy these grid design constraints.We will concern ourselves with a single one,detailed as follows.

The idea is based upon scintillation coutning,which employs ogranic molecules which quickly and e?-ciently emit light(usually in the visible)upon the absorption of high-energy radiations.The phenomenon is not the one familiar from everyday life in the form of glow-in-the-darktoys or faintly luminescent signs. That is called phosphorescence,which is the emission of light from a substance with a characteristic time that is generally very slow(many milliseconds at least,if not seconds).Rather,?uorescence,or scintillation,is the prompt emission(tens of nanoseconds at its fastest)of visible radiation directly following excitation[24, Chap.8].This latter phenomenon is the detection method in which we are interested.

In addition to being scintillators,many of these molecules degrade under ultraviolet illumination in the presence of oxygen,meaning they are susceptible to patterning with interference lithography.By spinning a solution of plastic and?ourophores in the same way resist is spun,the scintillating molecules can be held near the surface of the substrate in a plastic thin?lm.Proper choice of the?ourophores and plastic can insure a bright and robust signal with a good bleaching characteristic,while the low density of the layer

17

200-500 nm

e-beam resist

Figure1-6:Interference lithography used to bleach the?ducial grid

promises minimal interference with electron-beam lithography.Thus it can be seen that the scintillating ?ducial grid satis?es,at least at?rst glance,the?rst three requirements for the?ducial grid.

In regards to having the thin plastic scintillating layer(from now on,a scintillator)satisfy the forth requirement,there is potentially quite a bit of trouble in store.Ideally,it would be desired that the scintillator could be spun on and washed o?without a?ecting resist,no matter what the resist might be.In other words, no property of the scintillator would be tied to any property of the resist,so that allowing SPLEBL to be portable across a range of electron-beam lithography setups.In reality this may be di?cult to achieve, and certainly was not achieved under the auspices of this thesis.As will be covered in the next chapter, many of the common electron-beam resists dissolve in the same solvents that dissolve the scintillator,so in the absence of the identi?cation of an appropriate solvent,the stackmay require an interlayer.Moreover, the wavelengths of light used to achieve good bleaching contrast with the scintillators also expose certain electron-beam resists,requiring a scintillator-on-bottom and grid-exposure-?rst scheme,with an interlayer. With this sort of scheme,some sort of trilayer pattern-transfer technique would be required,thus clearly putting in danger satisfaction of the?fth requirement,that of simplicity.These issues will be discussed in more detail later in Section2.7.

Before workon this thesis was begun,one formulation of a scintillator had already been tested[18].This thesis will describe the search for scintillators that are brighter and have higher contrasts,and thereby make SPLEBL a feasible technique.More generally though,this thesis describes the development of an essential component of the SPLEBL technique,and so?nds itself as a contribution to the improvement of electron-beam written patterns,and thus can be thought of improving our ability to produce,via semiconductor lithography,large planar patterns of extraordinarly high resolution and?delity,an ability which is interesting for its application to a wide variety of important technologies.

18

Chapter2

Overview of Scintillators

This chapter gives an overview of the theoretical underpinnings of the experimental workin this thesis.

A familiarity with quantum mechanics at the graduate level and chemistry at the undergraduate level is assumed.The?rst three sections review the quantum mechanics of organic molecules and energy transitions, which gives an indication of the complicated connections between molecular structure and the observed properties of molecules,as well as allow us to understand how the molecules absorb energy from the incident electrons and transform it into light that is used in the SPLEBL feedbacksignal.Section2.4discusses brie?y the photochemistry of the bleaching of organic scintillators.Finally,solvents and their attendant properties,which are of key importance to laying down and removing the thin polymer layer in which the grid is bleached,are reviewed in the last three sections.

2.1Quantum Mechanics of Scintillating Organic Molecules Excellent references for the material in this section,drawn upon heavily here,are I.N.Levine,Quantum Chemistry[26],P.W.Atkins,Molecular Quantum Mechanics[3],and N.J.Turro,Modern Molecular Photo-chemistry[40].

Some organic molecules emit light and others do not.Why is this?What properties of molecules correlate with their?uorescence,and so might have an e?ect on the engineering of a plastic scintillating grid?Insights into the answers to these questions will be found in a quantum mechanical sketch of scintillating molecules.

Given some skeletal nuclear framework for a molecule,one possible starting point for any quantum mechanical analysis of that molecule would be the Schr¨o dinger Equation:

?Hψ(ˉx)=Eψ(ˉx)(2.1) which describes the wavefunction,ψ(ˉx),of that system.Theoretically an accurate wavefunction is a complete description of the system,and any information about that system it is possible to know can be had with the proper manipulations.

Key to an accurate wavefunction is an accurate Hamiltonian.A molecular Hamiltonian might be as follows:

?H

mol=?ˉh2

2

α

1

mα

?2α?ˉh

2

2m e

i

?2i+

α

β>α

ZαZβe2

rαβ

?

α

i

Zαe2

r iα

+

j

i>j

e2

r ij

(2.2)

where the?rst term describes the electronic kinetic energy,the second the nuclear kinetic energy,and the last three terms describe the various electrostatic forces,namely,the nuclei-nuclei,nuclei-electron,and electron-electron forces,respectively.This description is somewhat simlpistic in that it assumes the nuclei and electrons are point masses,and neglects spin-orbit and other relativitistic interactions.Again,some starting nuclear frameworkof a particular molecule under investigation is assumed.

As one might imagine,solving the Schr¨o dinger equation with this Hamiltonian would be quite onerous, and probably would yield results so inaccurate or convoluted as to be useless.A number of simplifying

19

approximations can be applied to the equation to facilitate the construction of more insightful wavefunctions. These will include:1)an approximation to separate nuclear and electronic portions of the wavefunction,and 2)an approximation to simplify the thus derived electronic portion of the wavefunction.

The approximation to separate the nuclear and electronic portions of the wavefunction is called the Born-Oppenheimer approximation,where the fact that nuclear masses are much larger than the electronic mass allows the molecular Hamiltonian to be written as a sum of nuclear and electronic Hamiltonians:

?H

elψel(q i;qα)=E elψel(q i;qα)(2.3)

?H

nucψnuc(qα)=E nucψnuc(qα)(2.4) To obtain a full solution to the equation,and thereby describe the molecule as accurately as allowed by the approximation,the electronic portion of the Born-Oppenheimer approximation to the Shr¨o dinger equation,Equation2.3,is solved for each set of nuclear coordinates,qα.The electronic energy,then being a function of the nuclear coordinates,serves as an e?ective potential,U(qα),in the nuclear part of the Born-Oppenheimer approximation,Equation2.4,allowing solutions for the wavefunctions of the nuclear masses and an understanding of their small departures from equilibrium.

The approximation is applied to the electronic portion of the wavefunction,ψel(q i;qα),is the Molecular Orbital approximation.This approach gives insight into the extended spatial shape of the molecular orbitals, and why conjugation of double bonds is important to the properties of scintillating molecules.

The molecular orbital approximation builds electronic wavefunctions from atomic wavefunctions(the basis set)by adding them linearly to one another,with weighting coe?cients determined by variational methods that minimize the energy.This method is also called the Linear Combination of Atomic Orbitals method,or LCAO method.The atomic wavefunctions in the basis set are generally drawn from the s-and p-orbitals in the valence shells of all atoms in the molecule,with symmetry conditions dictating which linear combinations are appropriate.Depending on whether these LCAO orbitals,or hybrid orbitals,change sign under a spatial inversion,they are called antibonding or bonding orbitals.

In the case of a diatomic homonuclear molecule the types of molecular orbitals(bonds)naturally fall into two classes,calledσandπ.Those orbitals that contain combinations of s-and p z-orbitals,and that have rotational symmetry about the internuclear axis,areσ-bonds.Those that are combinations of p x-and p y-orbitals,and are not rotationally symmetric about the internuclear axis,areπ-orbitals.The bene?ts of this classi?cation are numerous:σ-bonds are generally the core bonding orbitals,being held close to the internuclear axis,and form,in a sense,the‘skeletal’structure of the molecule,whereasπ-orbitals are on average further away from the nuclei and so are higher in energy(more weakly bound)and are thus more susceptible to chemical attack.The angular orientations of the bonds is found in the same manner as were coe?cients of the hybrid orbitals themselves,that is,by a variational calculation which minimizes the energy.Through this sort of calculation it is found that for minimum orbital energy(which results from a maximization of the orbital overlap integral)theσ-bonds impose no restriction on rotation about the internuclear axis,whileπ-bonds restrict its constituent p-orbitals to the same orientation.In other words,a p-orbital on one atom contributing to a particularπ-bond must point in the same direction as the p-orbital on its bonding partner which is associated in the same bond,and so aπ-bond is torsionally rigid.

For polyatomic molecules with more complex geometries,the s and p contributions to each molecular orbital become more complicated,but we can still bene?t from a roughσandπclassi?cation.Any particular σ-bond will still be roughly symmetric about whatever internuclear axis it is associated with,and will contain a large amount of s-orbital character,while aπ-bond will generally be found much further out from the same internuclear axis,and will have mostly p-orbital character.Once these polyatomicπ-orbitals are recognized as such,it is easily recognized that they have the same sort of torsional rigidity as do their diatomic counterparts.The molecular orbital technique shows,moreover,that under certain symmetry conditions, theseπ-orbitals can be conjugated,where p-orbitals on a set of neighboring atoms can join together to form one large distributedπ-system.It is theseπ-systems which give rise to the the most salient features of scintillating molecules,and which will most a?ect our engineering of a scintillating grid.

Aπ-system can be considered as merely one large orbital,the same as any other,except it is widely distributed across many atoms.Many important properties of theseπ-systems can be understood by analogy

20

with the particle-in-a-box problem from rudimentary quantum mechanics.In this conceptualization,the portion of the nuclear skeleton across which theπ-system is distributed can be thought of as the con?ning well,and the electrons the particles in that well.Since the size of the well increases(i.e.,as one moves to molecules with more conjugation)the energy of those levels(orbitals)decrease,and the spectrum shifts to the red.As the size of the molecule limits the possible number of conjugated bonds,we will?nd that the converse is also true for the most part:the smaller the molecule,the more toward the blue one will?nd its spectrum.

In addition to their spectroscopic properties,π-systems also lend torsional rigidity to the bonds they cover(relative rigidity,at least,in the face of normal thermal and collisional stresses encountered in the everyday life of the molecule).This rigidity,in the case of three or more atoms connected together in the system,gives rise to a planarity of that portion of the molecule.In the case of conjugated ring systems(so called aromatic systems),where aπ-orbital extends across a set of atoms connected head-to-tail,this rigidity can be especially robust.

Now that we have sketched a basic quantum mechanical picture of the orbitals of molecules,which are the font from which the emissive properties of molecules?ow,we can askthe following question:Are all molecules containingπ-systems equally good scintillators?The answer is,of course,‘No’.Some conjugated molecules emit very strongly and quickly when excited,others emit slowly,and still others emit not at all,or at best imperceptibly.Whence the di?erence?Simply,it arises out of the di?erent transition rates between orbitals,which are governed by Fermi’s Golden Rule:

rate=2π

h

ρ| ψi|?O c|ψf |2(2.5)

whereρis the density of?nal states capable of coupling with the intial state,and?O c is the operator corresponding to the perturbation that couples the initial and?nal states.It is important to note that this rule is a?rst order perturbation calculation,and thus an approximation.We are concerned with?ve types of transitions in the scintillating molecules,each with a di?erent form of operator and involving either the nuclear and electronic wavefunctions(or both):radiative absorption,radiative emission,nonradiative relaxation,vibrational relaxation,and intersystem crossing.Radiative processes involve a change of state from one electronic orbital to another,along with a change of parity.Emission or absorption of a photon accounts for a majority of the di?erence in momentum and energy.Nonradiative relaxation involves the conversion of electronic energy of a higher electronic state into nuclear vibrational energy of a lower electronic state.Vibrational relaxation involves a relaxation to lower nuclear vibrational sublevels inside of a single electronic state,sometimes through collisions with other molecules,other times radiatively.We do not include these under radiative transitions though,as the photons are of too low an energy to be of interest to us with regards to scintillation phenomenon.Finally,intersystem crossing involves a transition between states with a spin?ip,either with or without the subsequent emission of a photon.

Now we can begin to see how a molecule might contrive to be a scintillator,or not,depending on its transition rates.If the primary excited state(perhaps a singlet)couples strongly(perhaps through a nonradiative intersystem crossing)to a triplet state,and that triplet state is metastable because its primary means of deexcitation is through a very slow radiative intersystem crossing to the ground state(a singlet), the molecule will be a phosphor:that is,it will radiate,but only very slowly.If the excited state is strongly coupled to the ground state through a nonradiative transition,then the molecule will not scintillate at all. Moreover,if the method of initial excitation of the molecule is not absorption of a photon,but rather is a collision with a ionizing particle(such as a high-energy electron),there may be more than one initial excited state,perhaps with wildly di?erent energies,and perhaps also of both singlet and triplet character.This further complicates the problem,and can lead to a scintillator that is ine?cient under photon excitation but much more e?cient under electron excitation,or vice versa.With all these factors in mind,we might conclude that it is rare when the excited state of interest couples strongly in a radiative fashion to a state which is far removed from it in energy,thus making rare a good scintillator which is both bright and fast.

Now that we have sketched a picture of organic scintillators and how they work,it is easy to see that scintillation is a complex phenomenon.Even with an identity coupling operator in Equation2.5,we can see that the rates will depend in a complicated manner on the wavefunctions of the states,which themselves depend in a very complicated manner on molecular geometry and composition.A full treatment of the

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