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Mass

1 unified mass unit: mass(12C)/12

Einstein: E=mc2 so: m=E/c2

1eV=1.60217733x10-19 J

1MeV=1.60217733x10-13 J

1u=931.494 MeV/c2

Binding energy

The total energy (mass) of a bound system is less than the combined energy (mass) of the separated nucleons

Example: deuteron 2H (1 proton + 1 neutron)

mp =1.007825 u

mn =1.008665 u

mp+n =2.016490 u m2H=2.014102 u

The deuteron is 0.002338 u lighter than the sum of

the proton and the neutron. This is the binding

energy and is the energy needed to break that

nucleus apart

222Rn: 222.017571 u

4He: 4.002602 u

Sum: 226.020173 u

difference: 0.005229 u

(or 4.87 MeV)

Now consider 226Radium226Ra: 226.025402 u

It is energetically more favorable for the 226Ra to

emit an alpha particle (4He).

4.87 MeV in energy is gained.

Where does this energy go to???

answer: kinetic energy of 222Rn and 4He

Radioactivity

Spontaneous emission of radiation by unstable nuclei

- 4He (alpha particles)
- particles (electrons or positrons)
- rays (energetic photons)

-decay

Because of ‘tunneling’ the

quantum mechanical wavefunction

is not zero outside the nucleus

and thus there is a small

probability it can escape.

-decay

--decay

electron emission

+-decay

positron emission

The atomic number is changed by 1, but the mass number

remains constant

…not the complete story

Besides the electron (positron) also an anti-neutrino

(neutrino) is produced.

if neutrinos would not

exist all electron had

this energy

Kinetic energy of the electron (MeV)

-decay

Just like in the case of electrons, the nucleus has

different energy levels and going from one to another

is associated with the release of a photon (MeV)

An example (226U)

In the lab:

137Cs (30.1 y)

- 0.512 MeV (94.6%)

- 1.174 MeV

(5.4%)

137mBa(2.6 min)

The milk source

0.662 MeV(85%)

137Ba (stable)

half-life

N0

N0/2

1 Curie (Ci) = 3.7x1010 decays/s

1 Bq = 1 decay/s

t1/2

For any given data point with N counts: error is N

14C dating

14C is produced from 14N by Cosmic

rays. While alive, organisms have

a fixed 12C/14C ratio (1/1.3x10-12)

(Carbon in CO2).

After dying, no more 14C is absorbed

and it decays away and the ratio of

12C/14C can be used for dating.

Shroud of Turin

Found to be 1320±60 years old

Radiation damage in matter

Radiation damage: ionization effects in cells when radiation

passes through it.

1 Roentgen (R): amount of radiation that will produce 2.08x109 ion pair in 1cm3 of air or the amount of radiation that deposits 8.76x10-3 J of energy into 1 kg of air.

1 rad (radiation absorbed dose): amount of radiation that deposits 10-2 J of energy into 1 kg of absorbing material

radiation safety

The damage done by radiation also depends on the type

of radiation:

RBE (relative biological effectiveness): number of rad of

gamma radiation that produces the same biological damage

as 1 rad of the radiation being used:

type RBE

gamma-rays 1.

beta-particles 1.0-1.7

Alpha particles 10-20

neutrons 4-10

dose in REM: dose in rad x RBE

Bush approval rating moves back up

Retired general and former CNN consultant Wesley Clark remains at the top of the list of Democrats vying to replace Bush, with five Democrats in double digits and North Carolina Sen. John Edwards possibly getting a delayed announcement bounce as he rose from 2 percent to 6 percent.

ExampleFor any given data point with N counts: error is N

Types of errors

- Statistical errors: Due to instrumental imprecision or statistical nature of observed phenomena (finite sample).
- Systematic errors: Uncertainties in the bias of the data
- Example: measurement of weight
- Statistical error: scale imprecision (every measurement is slightly different)
- Systematic error: The scale has an offset or is not properly calibrated

- Example:Opinion poll

Systematical error: poll group is biased

(e.g. only people in one town etc)

Statistical error: Sample error; if a

Similar group was polled the results

Would be different.

Average, Standard deviation and standard deviation of the mean

- If we measure a quantity X N times, the best estimate for the value of that quantity is the average:
- The spread in the measurements is given by the standard deviation:
- The error in the determination of the mean is the standard deviation of the mean:

i

Example: mass on a springWe measure by how much a spring

is stretched (d) if we hang a weight

from it. The measurement is repeated

10 times.

Measurement: 2.00±0.02 m

Standard deviation of the mean:

error

Average of 10 measurements

Known weight

Determination of a spring constantThe same measurement is repeated with

different weights (10 measurements per

weight). We want to determine the spring

constant k via the relation:

F=kd weight=kd

d(m)

Slope=1/k

w(N)

Fitting procedure

Fit the data with

the theoretical

curve: x=(1/k)F

[function type: y=ax]

Use Kaleidagraph to fit

(see details on LBS272L webpage)

Result from fitting

Results:

slope 1.0023±0.0067

so: (1/k)=1.002±0.007

Chisq=6.49 (2)

How can we tell how

good the fit was?

Use the 2-value

2-value and the goodness of fit

- We have N measurements that we want to compare with theory (the data points with the fitted curve).
- The data points are xi±dxi i=1,N
- The theoretical values are ti=(1/k)fittedFi i=1,N
- The the 2-value is:

If a data point is close to the theoretical value, relative

to the size of the error bar, it does not contribute strongly

to the 2-value. If it is far from the theoretical value, relative

to the error bar, the 2-value rises strongly

2-value and degrees of freedom

Degrees of freedom (D): Number of data points (N) minus the

number of fitted parameters (Z).

In spring example: N=6 Z=1 (1/k)

So: D=N-Z=6-1=5

Degrees of Freedom

2-value

D=5

2=6.49

g.o.f.=26.1%

Goodness of fit: probability that the data matches

the theory well (0-100%: use table or calculator on

webpage)

Goodness of fit

- From 2-value and D we find the goodness of fit (0-100%)
- if g.o.f. is very low (<5%) the data does not match the theory well:
- The theory could be wrong
- The error bars are estimated too small
- We were unlucky!

- if g.o.f. is very high (>95%) the data matches the theory “unlikely” well.
- This should only happen 1 in 20 measurements! Perhaps the error bars were estimated too large.
- We were lucky!

- A quick check of the goodness of the fit is the value 2/D.
- 2/D1 for random error (I.e. not too small/large)
- In the example: 2/D=6.49/5=1.3 okay!

Error propagation: addition/subtraction

We have measured 2 quantities and their errors. What is

the error in the sum/subtraction?

example: A=5±1 B=8±2

C=A+B=5+8=13

dC=(12+22)=2.2

so: C=13±2

D=A-B=5-8=-3

dD=(12+22)=2.2

so: D=-3±2

Error propagation: product/division

We have measured 2 quantities and their errors. What is

the error in the product/division?

example: A=5±1 B=-8±2

C=A*B=5*(-8)=-40

dC=|-40|[(1/5)2+(2/8)2]=12.8

so: C=-4·101±1·101

D=A/B=5/(-8)=-0.625

dD=|-0.625|[(1/5)2+(2/8)2]=0.2

so: D=-0.6±0.2

Error propagation: constants/polynomials

example: A=5±1 k=3

C=kA=3*5=15

dC=|3|1=3

so: C=15±3

n=-2

D=An=5-2=0.04

dD=|0.04||-2|(1/|5|)=0.016

so: D=0.04±0.02

More info on error analysis

- visit the webpage:
- http://www.nscl.msu.edu/~zegers/lbs272l/lbs272l.html
- link: help on error analysis
- this lecture
- extra explanations and examples
- how to fit in kaleidagraph and get the fit parameters
- 2-probability calculator

Enjoy the lab!

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