The Impact of Special Relativity in Nuclear Physics: It’s not just E = Mc2

Of the 286,00 people living in Nagasaki at the time of the blast, 74,000 were killed and another 75,000 sustained severe injuries.

E = Mc2On August 9, 1945

San Onofre Nuclear Power Plant

E = Mc2

Nuclear Generation in California, 1960 through 2003Million Kilowatt Hours

http://www.eia.doe.gov/cneaf/nuclear/page/at_a_glance/states/statesca.html

About 13% of California’s electrical consumption came from nuclear power

E = Mc2

http://news.bbc.co.uk/1/shared/spl/hi/pop_ups/05/south_asia_pakistan_and_india_earthquake/html/6.stm

Radioactive decay supplies a significant fraction of the internal heat of the Earth’s mantle. Convection currents driven by this heat cause active plate tectonics.

E = Mc2

It would be difficult to find an area of physics which has not been profoundly influenced by Special Relativity.

Guiding Principles of Special Relativity

1) The speed of light c, is a constant for all observers in inertial reference frames.

2) The laws of physics must remain invariant in form in all inertial reference frames.

These two principles lead us to the Lorentz transformation, which gives us the translation table between two inertial reference frames O and O’.

z

y

x

ct

z

y

x

ct

1 0 0 0

0 1 0 0

0 0

0 0

'

'

'

'

O O’x x’

cv /

v

21/1

Both O and O’ see the event but they give different coordinates.

The Lorentz transformation shows that there are conserved quantities which have the same value measured in any inertial reference frame. These quantities are calculated from their respective 4-vectors.

22222222 ''')'()(

quantity conserved theand

),,,(

zyxctzyxct

zyxctx

Another extremely important 4-vector is the 4 momentum.

.)(')(

0, p' framerest sparticle' in the and

)'(')(

and

),(),,,(

22222

2222

mcEpcE

cpEpcE

pcEcpcpcpEp zyx

.)()( 2222 mcpcE

Since we want to describe microscopic systems we know we need to use quantum mechanics. The equation for E gives us two possible approaches to make a relativistic quantum mechanics. Call the wave function:

Emcpc

Emcpc2

2222 )()(

The first equation is the Klein-Gordon equation. The second is the Dirac equation.

K-G

equation

Dirac

equation

Particles Bosons Fermions

Negative energy states?

Yes antiparticles

Yes antiparticles

Basis states +E and –E if interactions are present

+E and –E if interactions are present

,1,0 ,2/3,2/

Under what circumstances should we expect relativity to be important in quantum systems?

An approach that focuses on the condition v/c <<1 is too limited. Q. Relativity gives us fermions and Fermi-Dirac statistics and the whole structure of matter relies on the nature of the fermions.

Q. Relativity explains low energy aspects of the microscopic structure of matter, such as atomic spectra.

http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/sodzee.html

Sodium D lines from the spin-orbit splitting of the 3p atomic state to the 3s1/2 state.

Relativity is essential in understanding atomic spectra, even when the energy of the state is a small fraction of the electron mass.

E(3p-splitting)/mec2 = 4 x 10-9 .

Relativistic Q.M. gives the right size of the spin-orbit splitting in atoms.

The Spin-Orbit Interaction

L

S

L

S

In the atom the S. O. interaction is generally attributed to the interaction of the electron’s magnetic moment and an induced magnetic field from the electron’s motion in the field of the nucleus.

However, it is a general property for any interacting fermion to show spin-orbit behavior. This is a consequence of Lorentz invariance (G. Breit, 1937).

How to make interacting fermions.

Emcpc 2 Dirac equation for a free particle.

)1

,( 0 Vc

VV

Introduce a 4-potential, Vand a scalar S.

Dirac equation for an interacting particle.

EVSmcV

cpc o)()

1( 2

For nuclei modern calculations generate a potential averaged over a scalar meson field and a vector meson field plus some smaller scalar and vector fields.

Relativity and Nuclear Structure

L = 1, p state in 11C,

ESO

Strong spin-orbit forces are seen in nuclei.

E(1p-splitting)/mpc2=2 x 10-3.

Velocity dependent forces are required in nuclear structure and are natural outcomes of a relativistic treatment using scalar and vector mesons

The magnitude of the nuclear spin-orbit potential is correctly given by a relativistic Q. Field theory using scalar and vector mesons.

e

e

eNZANZA

eNZANZA

)1,1(),(

)1,1(),(Radioactive decay and anti-particles

CSULA Proposal to search for other predicted relativistic effects in nuclei

1) Look for true nucleon-nucleon correlations as distinct from apparent correlations due to nonlocalities induced by relativistic effects.

3) Exploit the (e,e’p) asymmetry predicted by relativistic theories as a new observable for nuclear states.

2) Look for explicit evidence of the negative energy states in 208Pb.

Impulse Approximation limitations to the (e,e’p) reaction on 208Pb

- Identifying correlations and relativistic effects in the nuclear

medium

THOMAS JEFFERSON NATIONAL ACCELERATOR LABORATORY

K. Aniol , B. Reitz, A. Saha, J. M. Udias

Spokespersons

Hall A Collaboration MeetingJune 23, 2005

K.Aniol, Hall A Collaboration Mtg., June 23, 2005

THOMAS JEFFERSON NATIONAL ACCELERATOR LABORATORY

K.Aniol, Hall A Collaboration Mtg., June 23, 2005

(ii) Momentum distributions > 300 MeV/c

This was explained via long-range correlations in a nonrelativistic formalism [Bobeldijk,6], but also by relativistic effects in the mean field model [Udias,7].

I. Bobeldijk et al., PRL 73 (2684)1994

xB ≠ 1

E. Quint, thesis, 1988, NIKHEF

J. M. Udias et al. PRC 48(2731) 1994

J.M. Udias et al. PRC 51(3246) 1996

Excess strength at high pmiss

Negative Energy States- Complete Basis)ˆ)cos(ˆ)sin()sin(ˆ)cos()(sin()( 0 ztytxtRtR

The particle is in an orbit of radius R0 and constant angular velocity in 3 dimensions.

If we ignore the Z dimension and use a truncated basis of two dimensions in X and Y, we would interpret the particle’s projected motion in the XY plane as that of a harmonic oscillator.

rdt

rd 22

2

Asymmetry in the (e,e’p) reaction

peAeA '21

BF

BF

NN

NNA

q is the momentum transferred by the scattered electron.

qee

'

We detect protons knocked out forward and backward of q to determine the asymmetry A.

ATL in 3He, 4He and 16O

If relativistic dynamical effects are the main cause responsible for the extra strength, a strong effect on ATL would be seen.

There is a notable difference in ATL between 3He and 4He due to the density difference and in 16O. 16O: ATL

p1/2

p3/2

M. Rvachev et al. PRL 94:12320,2005

E04-107,2004J. Gao et al. PRL84:3265, 2000

THOMAS JEFFERSON NATIONAL ACCELERATOR LABORATORY

K.Aniol, Hall A Collaboration Mtg., June 23, 2005

ATL in 208Pb

THOMAS JEFFERSON NATIONAL ACCELERATOR LABORATORY

K.Aniol, Hall A Collaboration Mtg., June 23, 2005

ATL in 208Pb

Heavy Metal

Collaboration